Exam Revision Flashcards
There are constraints of getting valuable info for hypothesis from study’s design such as - (2)
duration of the study
how many people you can recruit
What is a sample?
A sample is the specific group that you will collect data from.
What is a population?
A population is the entire group that you want to draw conclusions about.
Example of population vs sample (2)
Population : Advertisements for IT jobs in the UK
Sample: The top 50 search results for advertisements for IT jobs in the UK on 1 May 2020
What is inferential statistics?
Inferential statistics allow you to test a hypothesis or assess whether your data is generalisable to the broader population.
Why is there a focus to do parametric tests than others in research? - (3)
- they are more rigorous, powerful and sensitive than non-parametric tests to answer your question
- This means that they have a higher chance of detecting a true effect or difference if it exists.
- They also allow you to make generalizations and predictions about the population based on the sample data.
We can obtain multiple outcomes from the
same people
We can obtain outcomes under
different conditions, groups or both
What are the 4 types of outcomes we measure? (4)
- Ratio
- Interval
- Ordinal
- Nominal
What is a continous variables? - (2)
: there is an infinite number of possible values these variables can take on-
entities get a distinct score
2 examples of continous variables (2)
- Interval
- Ratio
What is an interval variable?
: Equal intervals on the variable represent equal differences in the property being measured
Examples of interval variables - (2)
e.g. the difference between 600ms and 800ms is equivalent to the difference between 1300ms and 1500ms. (reaction time)
temperature (Farenheit), temperature (Celcius), pH, SAT score (200-800), credit score (300-850)
What is ratio variable?
The same as an interval variable and also has a clear definition of 0.0.
Examples of ratio variable - (3)
E.g. Participant height or weight
(can have 0 height or weight)
temp in Kelvin (0.0 Kelvin really does mean “no heat”)
dose amount, reaction rate, flow rate, concentration,
What is a categorical variable? (2)
A variable that cannot take on all values within the limits of the variable
- entities are divided into distinct categories
What are 2 examples of categorical variables? (2)
- Nominal
- Ordinal
What is nominal variable? - (2)
a variable with categories that do not have a natural order or ranking
Has two or more categories
Examples of nominal variable - (2)
genotype, blood type, zip code, gender, race, eye color, political party
e.g. whether someone is an omnivore, vegetarian, vegan, or fruitarian.
What is ordinal variables?
categories have a logical, incremental order
Examples of ordinal variables - (3)
e.g. whether people got a fail, a pass, a merit or a distinction in their exam
socio economic status (“low income”,”middle income”,”high income”),
satisfaction rating [Likert Scale] (“extremely dislike”, “dislike”, “neutral”, “like”, “extremely like”).
Using the term ‘variables’ for continous and categorical variables as - (2)
both outcome and predictor are variables
We will see later on that not only the type of outcome but also type of predictor influences our choice of stats test
Likert scale is ordinal variable but sometimes outcomes measured on likert scale are treated as - (3)
continuous after inspection of the distribution of the data and may argue the divisons on scale are equal
(i.e., treated as interval if distribution is normal)
gives greater sensitivity in parametric tests
What is measurement error?
The discrepancy between the actual value we’re trying to measure, and the number we use to represent that value.
In reducing measurement error in outcomes, the
values have to have the same meaning over time and across situations
Validity means that the (2)
instrument measures what it set out to measure
refers to the accuracy of a measure (whether the results really do represent what they are supposed to measure
Reliability means the
ability of the measure to produce the same results under the same conditions
Test-retest reliability is the ability of a
measure to produce consistent results when the same entities are tested at two different points in time
3 types of variation (3)
- Systematic variation
- Unsystematic variation
- Rnadomisation
What is systematic variation - (2)
Differences in performance created by a specific experimental manipulation.
This is what we want
What is Unsystematic variation (3)
Differences in performance created by unknown factors
.
Age, Gender,** IQ, **Time of day, Measurement error etc.
These differences can be controleld of course (e.g., inclusion/exclusion of pps setting age range of 18-25)
Randomisation (other approaches) minimises - (2)
effects of unsystematic variation
does not remove unsystematic variation
What is the independent variable (Factors)? ( 3)
- The hypothesised cause
- A predictor variable
- A manipulated variable (in experiments)
What is depenedent variable? (measures)- (3)
- The proposed effect , change in DV
- An outcome variable
- Measured not manipulated (in experiments)
In all experiments we have two hypotheses which is (2)
- Null hypothesis
- Alternative hypothesis
What is null hypothesis?
that there is no effect of the predictor variable on the outcome variable
What is alternative hypothesis?
is that this is an effect of the predictor variable on the outcome variable
Null Hypothesis Signifiance Testing computes the probability the (2)
the probability of the null hypothesis being true (referred as p-value) by computing a statistic and how likely it is that the statistic has that value by chance alone
Referred as p-value
The NHST does not compute the probability of the
null hypothesis
There can be directional and non-directional hypothesis of
an alternate hypothesis
non-directional alternate hypothesis is..
The alternative hypothesis is that this is an effect of the group on the outcome variable
Directional alternate hypothesis is…
The alternative hypothesis is that this the mean of the outcome variable for group 1 is larger than the mean of group 2
Example of directional alternate hypothesis
There would be far greater engagment in stats lecture if they were held at 4 PM and not 9AM
For a non-directional hypothesis you will need to divide your alpha value at
two ends of the tail of normal distirbution
The 3 misconceptions of Null Hypothesis Signifiance Testing (NHST) - (3)
- A significant result means the effect is important
- A non-significant result means the null hypothesis is true
- A significant result means the null hypothesis is false (just give probability that data occured given null hypothesis, doesn’t say huge evidence that null hypothesis is categorically false)
P-Hacking and HARKING is another issue with
NHST
p-Hacking and HARKINGS are the - (2)
researchers degrees of freedom
cchange after results are in and some analysis has been done
P-hacking refers to a
selective reporting of significant results
Harking is
Hypothesising After the Results are Known
P-hacking and HARKING are often used in
combination
What does EMBERS stand for? (5)
- Effect Sizes
- Meta-analysis
- Bayesian Estimation
- Registration
- Sense
EMBERS can reduce issues of
NHST
Uses of Effect sizes and Types of Effect Size (3)
- There a quite a few measures of effect size
- Get used to using them and understanding how studies can be compared on the basis of effect size
- A brief example: Cohen’s d
Meaning of Effect Size (2)
Effect size is a quantitative measure of the magnitude of the experimental effect.
The larger the effect size the stronger the relationship between two variables.
Formula of Cohen’s d
What is meta-analysis?
Meta-analysis is a study design used to systematically assess previous research studies to derive conclusions about that body of research
Meta-analysis brings together.. and assesses (2)
- Bringing together multiple studies to get a more realistic idea of the effect
- Can assess effect siz that are averaged across studies
Funnel plots in meta-analysis can be made to….. values stuides…. (2)
investigating publication bias and other bias in meta-analysis
values studies by their sample size and observe bias
Bayesian approaches capture
probabilities of the data given the hypothesis and null hypothesis
Bayes factor is now often computed and stated alongside
conventional NHST analysis (and effect sizes)
Registration is where (5)
- Telling people what you are doing before you do it
- Tell people how you intend to analyze the data
- Largely limits researcher degrees of freedom (HARKING p-hacking)
- A peer reviewed registered study can be published whatever the outcome
- The scientific record is therefore less biased to positive findings
Sense is where (4)
- Knowing what you have done in the context of NHST
- Knowing misconceptions of NHST
- Understanding the outcomes
- Adopting measures to reduce researcher degrees of freedom (like preregistration etc..)
most of the statistical tests in this book rely on
having data measured
at interval level
To say that data are interval, we must be certain that
equal intervals on the scale represent
equal differences in the property being measured.
The distinction between continous and discrete variables can often be blurred - 2 examples- (2)
continuous variables can be measured in discrete terms; we measure age we rarely use nanoseconds but use years (or possibly years and months).
In doing so we turn a continuous variable into a discrete
one
treat discrete variables as if they were continuous, e.g., the number of boyfriends/girlfriends
that you have had is a discrete variable. However, you might read a magazine that says ‘the average number of boyfriends that women in their 20s have has increased from 4.6 to 8.9’
a device for measuring sperm motility that actually measures sperm count is not
valid
Criterion validity is whether the
instrument is measuring what it claims to measure (does
your lecturers’ helpfulness rating scale actually measure lecturers’ helpfulness?).
The two sources of variation that is always present in independent and repeated measures design is
unsystematic variation and systematic variation
effect of our experimental manipulation
is likely to be more apparent in a repeated-measures design than in a
between-group design,
effect of experimental manipulation is more apparent in repeated-design than independent since in independent design,
differences between the characteristics of the people allocated to each of the groups is likely to create considerable random variation both within
each condition and between them
This means that, other things being equal, repeated-measures designs have
more power to
, repeated-measures designs have
more power to d
We can use randomization in two different ways depending on
whether we have an
independent and repeated design measure
Two sources of systematic variation in repeated design measure - (2)
- Practice effects
- Boredom effects
What is practice effects?
Participants may perform differently in the second condition because
of familiarity with the experimental situation and/or the measures being used.
What is boredom effects?
: Participants may perform differently in the second condition because
they are tired or bored from having completed the first condition.
We can ensure no systematic variation between conditions in repeated measure is produced by practice and boredom effects by
counterbalancing the order in which a person participates in a condition
Example of counterbalancing
we randomly determine whether a participant
completes condition 1 before condition 2, or condition 2 before condition 1
*
What distribution is needed for parametric tests?
A normal distribution
The normal distribution curve is also referred as the
bell curve
Normal distribution is symmetrical meaning
This means that the distribution curve can be divided in the middle to produce two equal halves
The bell curve can be described using two parameters called (2)
- Mean (central tendency)
- Standard deviation (dispersion)
μ is
mean
σ is
standard deviation
Diagram shows:
e.g., If we move 1σ to the right then it contains 34.1% of the valeues
Many statistical tests (parametric) cannot be used if the data are not
normally distributed
The mean is the sum of
scores divided by the number of scores
Mean is a good measure of
central tendency for roughly symmetric distributions
The mean can be a misleading measure of central tendency in skewed distributions as
it can be greatly influenced by scores in tail e.g., extreme values
Aside from the mean, what are the 2 other measured of central tendency? - (2)
- Median
- Mode
The median is where (2)
the middle score when scores are ordered.
the middle of a distribution: half the scores are above the median and half are below the median.
The median is relatively unaffected by … and can be used with… (2)
- extreme scores or skewed distribution
- can be used with ordinal, interval and ratio data.
The mode is the most
frequently occurring score in a distribution, a score that actually occurred
The mode is the only measure of central tendency that can be used with
with nominal data
The mode is greatly subject to
sample fluctuations and is therefore not recommended to be used as the only measure of central tendency
Many distributions have more than one
mode
The mean, median and mode are identical in
symmetric distribtions
For positive skewed distribution, the
mean is greater than the median, which is greater than the mode
For negative skewed distribution
usually the mode is greater than the median, which is greater than the mean
Kurtosis in greek means
bulge or bend in greek
What is central tendency?
the tendency for the values of a random variable to cluster round its mean, mode, or median.
Diagram of normal kurotsis, positive excess kurotsis (leptokurtic) and negative excess kurotsis (platykurtic)
What does lepto mean?
prefix meaning thin
What is platy
a prefix meaning flat or wide (think Plateau)
Tests of normality (2)
Kolmogorov-Smirnov test
Shapiro-Wilks test
Tests of normality is dependent on
sample size
If you got a massive sample size then you will find these normality tests often come out as …. even when your data visually can look - (2)
significant
normally disttibuted
If you got a small sample size, then the normality tests may look non-siginificant, even when data is normally distributed, due to
lack of power in the test to detect a significant effect
There is no hard or fast rule for
determining whether data is normally distributed or not
Plot your data because this helps inform on what decisions you want to make with respect to
normality
Even if normality test is sig and data looks visually normally distributed then still do
parametric tests
A frequency distribution or a histogram is a plot of how many times
each score occurs
2 main ways a distribution can deviate from the normal - (2)
- Lack of symmetry (called skew)
- Pointyness (called kurotsis)
In a normal distribution the values of skew and kurtosis are 0 meaning…
tails of the distribution are as they should be
Is age nominal or continous?
Continous
Is gender continous or nominal?
Nominal
Is height continous or nominal?
Continous
Which of the following best describes a confounding variable?
A. A variable that affects the outcome beingmeasured as well as, or instead of, theindependent variable
B. A variable that is manipulated by theexperimenter
C. A variable that has been measured using an unreliable scale
D.A variable that is made up only of categories
A
If a test is valid , what does it mean?
A.The test measures what it claims to measure.
B. The test will give consistent results. (Reliability)
C.The test has internal consistency (measure for correlations between different items on same test = see if it measures same construct)
D. Test measures a useful construct or variable = test can measure something useful but not valid
A
A variable that measures the effect that manipulating another variable has is known as:
A. DV
B. A confounding variable
C. Predictor variable
D. IV
A
The discrepancy between the numbers used to represent something that we are trying tomeasure and the actual value of what we are measuring is called:
A. Measurement error
B. Reliability
C. The ‘fit’ of the model
D. Variance
A
A frequency distribution in which low scores are most frequent (i.e. bars on the graph arehighest on the left hand side) is said to be:
A. Positively skewed
B. Leptokurtic = distribution with positive kurotsis
C. Platykurtic = negative kruotsis
D. Negatively skewed = frequent scores
A
Which of the following is designed to compensate for practice effects?
A. Counterbalancing
B. Repeated measures design = practice effects issue in repeated measures
C. Giving a participants a break between tasks = this compenstates for bordeom effects
D. A control condition = provides reference point
A
Variation due to variables that has not been measured is
A. Unsystematic variation
B. Homogenous variance = assumption variance each population is equal
C. Systematic variation = due to exp manpulation
D. Residual variance = confirms how well regression line constructed fit to actual data
A
Purpose of control condition is to
A. Allow inferences about cause
B. Control for participants’ characteristics = randomisation
C. Show up relationship between predictor variables
D. Rule out tertium quid
A Allow inferences of cause
If the scores on a test have a mean of 26 and a standard deviation of 4, what is the z-score for a score of 18?
A. -2
B. 11
C. 2
D. -1.41
A (18-26) = -8/4 = -2
The standard deviation is the square root of the
A. Variance
B. Coefficient of determination = r squared
C. Sum of squares = sum of squared deviances
D. Range = largest = smallest
A
Complete the following sentence:A large standard deviation (relative to the value of the mean itself
A. Indicate data points are distant from man (i.e., poor fit of data)
B. Indicate the data points are close to mean
C. Indicate that mean is good fit of data
D. Indicate that you should analyse data with parameteric
A
The probability is p = 0.80 that a patient with a certain disease will besuccessfully treated with a new medical treatment. Suppose that thetreatment is used on 40 patients. What is the “expected value” of thenumber of patients who are successfully treated?
A. 32
B. 20
C. 8
D. 40
A = 80% of 40 is 32 (0.80 * 40)
Imagine a test for a certain disease.
Suppose the probability of a positive test result is .95if someone has the disease, but the probability is only .08 that someone has the disease if his or her test result was positive.
A patient receives a positive test, and the doctor tellshim that he is very likely to have the disease. The doctor’s response is:
A. confusion of intervse
B. Law of small numbers
C. Gambler’s fallacy
D. Correct, because test is 95% accurate when someone has the disease = incorrect as doctor based assumption on incorrect inverse proability
A
Which of these variables would be considered not to have met the assumptions ofparametric tests based on the normal distribution?
(Hint many statistical tests rely on data measured on interval level)
A. gender
B. Reaction time
C. Temp
D. Heart rate
A
The test statistics we use to assess a linear model are usually _______ based on thenormal distribution
(Hint: These tests are used when all of the assumptions of a normal distribution havebeen met
A. Parametric
B. Non-parametric
C. Robust
D. Not
A
Which of the following is not an assumption of the general linear model?
A. Dependence
B. Addictivity
C. Linearity
D. Normally distributed residuals
A = independence is an assumption of parametric and not dependence
Looking at the table below, which of the following statements is the most accurate?
Hint: The further the values of skewness and kurtosis are from zero, the more likely it is that thedata are not normally distributed
A. For the number of hours spent practicsing , there is not an issue of kruotsis
B. For level of msucial skill, data are heavily negatively skewed
C. For number of hours spent practicsing there is an issue of kruotsis
D. For the number of hours spent practicsing, the data is fairly positively skewed
A - correct
B. Incorrect as value of skewnessis –0.079, which suggests that the dataare only very slightly negatively skewedbecause the value is close to zero
C. Incorrect as value of kurtosis is0.098, which is fairly close to zero,suggesting that kurtosis was not aproblem for these data
D. Incorrect as value of skewnessfor the number of hours spent practisingis –0.322, suggesting that the data areonly slightly negatively skewed
Diagram of skewness
In SPSS, output if value of skewness is between -1 and 1 then
all good
In SPSS, output if value is below -1 or above 1 then
data is skewed
In SPSS, output if value of skewness is below -1 then
negatively skewed
In SPSS, output if value of skewness is above 1 then
positively skewed
Diagram of lepto kurotic, platykurtic and mesokurtic( normal)
What does kurotsis tell you?
how much our data lies around the ends/tails of our histogram which helps us to identify when outliers may be present in the data.
A distribution with positive kurtosis, so much of the data is in the tails, will be
pointy or leptokurtic
A distribution with negative kurtosis, so the data lies more in the middle, will be more
be more sloped or platykurtic
Kurtosis is the sharpness of the
peak of a frequency-distribution curve
If our Kurtosis value is 0, then the result is a
normal distribution
If kurotsis value in SPSS between -2 and 2 then
all good! = normal distribution
If kurotsis value in SPSS less than -2 then
platykurtic
If kurotsis value is greater than 2 in SPSS then
leptokurtic
Are we good for skewness and kurotsis in this output SPSS?
Good because both the skewness is between -1 and 1 and kurtosis values are between -2 and 2.
Are we good for skewness and kurotsis in this output SPSS?A
Bad because although the skewness is between 1 and -1, we have a problem with kurtosis with a value of 2.68 which is larger than 2 and -2
Correlational research doesn’t allow to rule out the presence of a
third variable = confounding variable
e.g, we find that drownings and ice cream sales are correlated, we conclude that ice cream sales cause drowning. Are we correct? Maybe due to the weather
The tertium quid is a variable that you may not have considered that could be
influencing your results e.g., ice cream and drowning session
How to rule out tertium quid? - (2)
Use of RCTs.
Randomized Controlled Trials allow to even out the confounding variables between the groups
Correlation does not mean
causation
To infer causation,
we need to actively manipulate the variable we are interested in, and control against a group (condition) where this variable was not manipulated.
Correlation does not mean causation as according to Andy
causality between two variables cannot be assumed because there may be other measured or unmeasured variables affecting the results”
Aside from checking of kurotsis and skewness assumptions in data also check if it has
linearity or less commonly additivity
Additivity refrs to the combined
effect of many predictors
What does this diagram show in terms of additivty /linearity? - (5)
There is a a linear effect when the data increases at a steady rate like the graph on the left.
Your cost increases steadily as the number of chocolate bars increases.
The graph on the right shows a non-linear effect when there is not this steady increase rather there is a sharp change in your data.
So you might feel ok if you eat a few chocolate bars but after that the risk of you having a stomach ache increases quite rapidly the more chocolates you eat.
This effect is super important to check or your statistical analysis will be wrong even if your other assumptions are correct because a lot of statistical tests are based on linear models.
Discrepnacy between measurement and actual value in population is .. and not..
measurement error and NOT variance
Measurement error can happen across all psychological experiments from.. to ..
recording instrument failure to human error
What are the 2 types of measurement errors? - (2)
- Systematic
- Random
What is systematic measurement error?
: predictable, typically constant or proportional to the true value and always affect the results of an experiment in a predictable direction
Example of systematic measurement error
for example if I know I am 5ft2 and when I go to get measured I’m told I’m 6ft this is a systematic error and pretty identifiable - these usually happen when there is a problem with your experiment
What is random measurement error?
measurable values being inconsistent when repeated measures of a constant attribute or quantity are taken.
Example of random measurement error
for example my height is 5ft2 when I measure it in the morning but its 5ft when I measure myself in the evening. This is because my measurements were taken at different times so there would be some variability – for those of you who believe you shrink throughout the day.
What is variance?
Average squared deviation of each number from its mean.
Variability is an inherent part of
things being measured and of the measurement process
Diagram of variance formula
In central limit theorem - (2)
states that the sampling distribution of the mean approaches a normal distribution, as the sample size increases. This fact holds especially true for sample sizes over 30.
Therefore, as a sample size increases, the sample mean and standard deviation will be closer in value to the population mean μ and standard deviation σ .
What does histogram look at? - (2)
Frequency of scores
Look at distribution of data, skewness, kurotsis
What does boxplot look at? - (2)
To identify outliers
Shows median rather than mean (good for non-normally distributed data)
What do line graphs are?
simply bar charts with lines instead of bars
Bar charts are a good way to display
display means (and
standard errors)
What do scatterplot illustrates? - (2)
a relationship between two variables, e.g. correlation or regression
Only use regression lines for regressions!
What are matrix scatterplots? - (2)
Particular kind of scatterplot that can be used instead of the 3-D scatterplot
clearer to read
Using data provided how would you summarise skew?
A. The data has an issue with positive skew
B.The data has an issue with negative skew
C.The data is normally distributed
B
What is the median number of bullets shot at a partner by females?
67.00
What descriptive statistics does the red arrow represents?
A. Inter quartile range
B. Median
C. Mean
D. Range
A
What is the mean of males and females SD? - (2)
Males M = 27.29
Females SD = 12.20
What is the respective standard error of mean for femals and males?
3.26 & 3.42
Answering the question: Meets assumption of parametric tests will determine whether our continous data can be tested with
with parametric or non-parametric tests
A normal distribution is a distribution with the same general shape which is a
bell shape
A normal distribution curve is symmetric around
the mean μ
A normal distribution is defined by two parameters - (2)
the mean (μ) and the standard deviation (σ).
Many statistical tests (parametric) cannot be used if the data is not
normally distributed
What does this diagram show? - (2)
μ = 0 is peak of distribution
Block areas under the curve and gives us insight to way data is distributed and certain scores occuring if they belong to normally distribution e.g., 34.1% of values lie one SD below mean
A z score in standard normal distribution will reflect the number of
SD above or below the mean of a particular score is
How to calculate a z score?
Take a value of participant (e.g., 56 years old) and take away mean of distribution (e.g., mean age of class is 23) divided by SD (class like 2)
If a person scored a 70 on a test with a mean of 50 and a standard deviation of 10
Converting the test scores to z scores, an X of 70 would be…
What the result means…. - (2)
a z score of 2 means the original score was 2 standard deviations above the mean
We can convert our z scores to
pecentiles
Example: What is the percentile rank of a person receving a score of 90 on the test? - (3)
Mean - 80
SD = 5
First calculating z score: graph shows that most people scored below 90. Since 90 is 2 standard deviations above the mean z = (90 - 80)/5 = 2
Z score to pecentile can be looked at table that z score of 2 is equivalent to the 97.7th percentle:
The proportion of people scoring below 90 is thus .977 and proportion of people scoring above 90 is 2.3% (1-0.977)
What is the sample mean?
an unbiased estimate of the population mean.
How can we know how that our sample mean estimate is representative of the population mean?
Via computing standard error of mean - smaller SEM the better
Standard deviation is used as a measure of how
representative the mean was of the observed data.
Small standard deviations represented a scenario in which most data points were
most data points were close to the mean
Large standard deviation represented a situation in which data points were
widely spread
from the mean.
How to calculate the standard error of mean?
computed by dividing the standard deviation of the sample by the the square root of the number in the sample
The larger the sample the smaller the - (2)
standard error of the mean
more confident we can be that the sample mean is representative of the population.
The central limit therom proposes that
as samples get large (usually defined as greater than 30), the sampling distribution has a normal distribution with a mean equal to the population mean, SD = SEM
The standard deviation of sample means is known as the
SEM (standard error of the mean)
A different approach to assess accuracy of sample mean as estimate of - population mean, aside from SE, is to - (2)
calculate boundaries and range of values within which we believe the true value of the population mean value will fall.
Such boundaries are called confidence intervals.
Confidence intervals are created by
samples
A 95% confidence intervals is consructed such that
these intervals (created by samples) will contain the population mean
95% Confidence interval for 100 samples (CI constructed for each) would mean
95 of these samples, the confidence intervals we constructed would contain the true value of the mean in the population.
Diagram shows- (4)
- Dots show the means for each sample
- Lines sticking out representing Ci for the sample means
- If there was a vertical line down it represents population mean
- If confidence intervals don’t overlap then it shows significant difference between the sample means
In fact, for a specific confidence interval, the probability that it contains the population value is either - (2)
0 (it does not contain it) or 1 (it does contain it).
You have no way of knowing which it is.
if our sample means were normally distributed with a mean of 0 and a
standard error of 1, then the limits of our confidence interval
would be –1.96 and +1.96 -
95% of z scores fall between
-1.96 and 1.96
Confidence intervals can be constructed for any estimated parameter, not just
μ - mean
. If the mean represents the true mean well, then the confidence interval of that mean should be
small
if the confidence interval is very
wide then the sample mean could be
very different from the true mean, indicating that it
is a bad representation of the population
Remember that the standard error of the mean gets smaller with the number of observations and thus our confidence interval also gets
smaller - make sense as more we measure more certain sample mean close to population mean
Calculating Confidence Intervals for sample means - rearranging in z formula
LB = Mean - (1.96 * SEM)
UB = Mean + (1.96 * SEM)
The standard deviation of SAT verbal scores in a school system is known to be 100. A researcher wishes to estimate the mean SAT score and compute a 95% confidence interval from a random sample of 10 scores.
The 10 scores are: 320, 380, 400, 420, 500, 520, 600, 660, 720, and 780.
Calculate CI
* M - 530
* N = 10
* SEM = 100/ square root of 10 = 31.62
* Value of z for 95% CI is number of SD one must go from mean (in both directions) to contain 0.95 of the scores
* Value of 1.96 was found in z-table
* Since each tail is to contain 0.025 of the scores, you find the values of z for which is 1-0.025 = 0.975 of the socres below
* 95% of z scores lie between -1.96 and +1.96
* Lower limit = 530 - (1.96) (31.62) = 468.02
* Upper limit = 530 + (1.96)(31.62) = 591.98
Think of test statistic capturing
signal/noise
Hypo
A testStatistic for which the frequency of particular values is known (t, F, chi-square) and thus we can calculate the
probability of obtaining a certain value or p value.
To test whether the model fits the data or whether our hypothesis is a good explanation of the data, we compare
systematic variation against unsystematic
If the probability (p-value) less than or equal to the significance level, then
the null hypothesis is rejected; When the null hypothesis is rejected, the outcome is said to be “statistically significant”
If the proabilibty (p-value) is greater than the signifiance leve, the
null hypothesis is not rejected.
What is a type 1 error in terms of variance? - (2)
think the variance accounted for by the model is larger than the one unaccounted for by the model (i.e. there is a statistically significant effect but in reality there isn’t)
Type 1 is a false
positive
What is type II error in temrs of variance?
think there was too much variance unaccounted for by the model (i.e. there is no statistically significant effect but in reality there is)
Type II error is false
negative
Example of Type I and Type II error
Type I and Type II errors are mistakes we can make when testing the
fit of the model
Type 1 errors when we believe there is a geniue effect in
population, when in fact there isn’t.
Acceptable level of type I error is usually
a-level of usually 0.05
Type II error occurs when we believe there is no effect in the
population when, in reality, there is.
Acceptable level of Type II error is probability/-p-value is
β-level (often 0.2)
An effect size is a standardised measure of
the size of the an effect
Properities of effect size (3)
Standardized = comparable across studies
Not (as) reliant on the sample size
Allows people to objectively evaluate the size of observed effect.
Effect Size Measures
r = 0.1, d = 0.2 (small effect):
the effect explains 1% of the total variance.
Effect size measures
r = 0.3, d = 0.5 (medium effect) means
the effect accounts for 9% of the total variance.
Effect size measures
r = 0.5, d = 0.8 (large effect)
effect accounts for 25% of the variance
Beware of the ‘canned’ effect sizes (e.g., r = 0.5, d = 0.8 and rest) since the size of
effect should be placed within the research context.
We should aim to achieve a power of
.8, or an 80% chance of detecting
an effect if one genuinely exists.
When we fail to reject the null hypothesis, it is either that there truly are no difference to be found,
OR
it may be because we do not have enough statistical power
Power is the probability of
correctly rejecting a false H0 OR the ability of the test to find an effect assuming there is one in the population,
Power is calculated by
1 - β OR probability of making Type II error
To increase statistical power of study you can increase
your sample sizee
Factors affecting the power of the test: (4):
- Probability of a type 1 error or a-level [level at which we decide effect is sig - p-value) –> bigger [more lenient] alpha then more power)
- True alternate hypothesis H1 [effect size] (degree of overlap, less means more power) - if you find large effect in lit then better chance of detecting something
- The sampel size [N]) –> bigger the sample, less the noise and more power
- The particular tests to be employed - parametric tests greater power to detect sig effect since more sensitive
How to calculate the number of pps they need for reasonable chance of correctly rejecting null hypothesis?
Sample size calculation at a desired level of power (usually power set to 0.8 in formula)
With power, we can do 2 things - (2)
- Calculate power of test
- Calculate sample size necessary to detect an decent effect size and achieve a certain level of power based on past research
Diagram of Type I error, Type II error, power - (4) and making correct decisions
Type 1 error p = alpha
Type II error p = beta
Accepting null hypothesis which is correct - p = 1- alpha
Accepting alternate hypo which is correct - p = 1 - beta
If there is a less degree of overlap in h0 and h1 then
bigger difference means higher power and and correctly reject the null hypothesis than distributions that overlap more
If distribution between h0 and h1 are narrower then
This means that the overlap in distributions is smaller and the power is therefore greater, but this time because of a smaller standard error of our estimate of the means.
Most people want to assess how many participants they need to test to have a reasonable chance of correctly rejecting the null hypothesis (the Power). This formula shows - (2)
us how.
We usually set the power to 0.8.
What is z scores? - (2)
A measure of variability:
The number of standard deviations from the population mean or a particular data point is
Z-scores are a standardised measure, hence they ignore measurement units
Why should we care about z scores? - (2)
Z-scores allow researchers to calculate the probability of a score occurring within a standard normal distribution
Enables us to compare two scores that are from different samples (which may have different means and standard deviations)
Diagram of finding percentile of Trish
Trish takes a test and gets 25
Mean of the class is 20
SD = 4
25-20/4 = 1.25
Z-score = 1.25
Let’s say Trish takes a test and scores 25 and the mean is 20 You may calculate the z-score to be 1.25 you would use a z-score table to see what percentile they would be in (marked in red) so to read the table you would go down to the value 1.2 and you would go across to 0.05 which totals to 1.25 and you can see about 89.4% of other students performed worse.
Diagram of z score and percentile
Josh takes a different test and gets 1150
Mean of the class is 1000
SD = 150
1150 – 1000/150 = 1.0
Z score = 1.0
Who performed better Trish or Josh?
Trish had z score of 1.25
We would use our table and look down the column to a z-score of 1 and across to the 0.00 column (in purple) and we can see 84.1% of students performed worse than Josh so Trish performed better than Josh.
Diagram of z scores and normal distribution - (3)
68% of scores are within 1 SD of the mean,
95% are within 2 SDs and
99.7% are within 3 SDs.
Whats standard error?
: by taking into account the variability and size of our sample we can estimate how far away from the real population mean our mean is!
If we took infinite samples from the population, 95% of the time the population mean will lie within the
the 95% confidence interval range
What does narrow CI represent?
high statistical power
Wide CIs represent?
low statistical power
Power bring the probability of catching a real effect (as opposed to
missing a real effect – Type II error)
We can never say the null hypothesis is
FALSE (or TRUE).
TheP valueor calculated probability is the estimated probability of us
us finding an effect when the null hypothesis (H0) is true.
p = probability of observing a test statistic at least as a big as the one we have if the
H0 is true
Hence, a significant p value (p <.05) tells us that there is a less than 5% chance of getting a test statistic that is
larger than the one we have found if there were no effect in the population (e.g. the null hypothesis were true)
Statistical signifiance does not equal importance - (2)
p = .049, p = .050 are essentially the same thing- the former is ‘statistically significant’.
Importance is dependent upon the experimental design/aims: e.g., A statistically significant weight increase of 0.1Kg between two adults experimental groups may be less important than the same increase between two groups of babies.
Children can learn a second language faster before the age of 7’. Is this statement:
A. One-tailed
B. A non scientific
C. Two-tailed
D. Null hypothesos
A as one-tailed is directional and two tailed is non-direcitonal
Which of the following is true about a 95% confidence interval of the mean:
A. 95 out of 100 CIs wll contain population mean
B. 95 out of 100 sample means will fall within the limits of the confidence interval.
C. 95% of population means will fall within the limits of the confidence interval.
D. There is a 0.05 probability that the population mean falls within the limits of the confidence interval.
A as If we’d collected 100 samples, calculated the mean and then calculated a confidence interval for that mean, then for 95 of these samples the confidence intervals we constructed would contain the true value of the mean in the population
What does a significant test statistic tell us?
A. That the test statistic is larger than we would expect if there were no effect in the population.
B. There is an important effect.
C. The null hypothesis is false.
D. All of the above.
A and just because test statistic is sig does not mean its important effect
Of what is p the probability?
(Hint: NHST relies on fitting a ‘model’ to the data and then evaluating the probability of this ‘model’ given the assumption that no effect exists.)
A.p is the probability of observing a test statistic at least as big as the one we have if there were no effect in the population (i.e., the null hypothesis were true).
B. p is the probability that the results are due to chance, the probability that the null hypothesis (H0) is true.
C. p is the probability that the results are not due to chance, the probability that the null hypothesis (H0) is false
D. p is the probability that the results would be replicated if the experiment was conducted a second time.
A
A Type I error occurs when:
(Hint: When we use test statistics to tell us about the true state of the world, we’re trying to see whether there is an effect in our population.)
A. We conclude that there is an effect in the population when in fact there is not.
B. We conclude that there is not an effect in the population when in fact there is.
C. We conclude that the test statistic is significant when in fact it is not.
D. The data we have typed into SPSS is different from the data collected.
A as If we use the conventional criterion then the probability of this error is .05 (or 5%) when there is no effect in the population
True or false?
a. Power is the ability of a test to detect an effect given that an effect of a certain size exists in a population.
TRUE
True or False?
We can use power to determine how large a sample is required to detect an effect of a certain size.
TRUE
True or False?
c. Power is linked to the probability of making a Type II error.
TRUE
True or False?
d. The power of a test is the probability that a given test is reliable and valid.
FALSE
What is the relationship between sample size and the standard error of the mean?
(Hint: The law of large numbers applies here: the larger the sample is, the better it will reflect that particular population.)
A. The standard error decreases as the sample size increases.
B. The standard error decreases as the sample size decreases.
C. The standard error is unaffected by the sample size.
D. The standard error increases as the sample size increases.
A The standard error (which is the standard deviation of the distribution of sample means), defined as σ_Χ ̅ =σ/√N, decreases as the sample size (N) increases and vice versa
What is the null hypothesis for the following question: Is there a relationship between heart rate and the number of cups of coffee drunk within the last 4 hours?
A. There will be no relationship between heart rate and the number of cups of coffee drunk within the last 4 hours.
B. People who drink more coffee will have significantly higher heart rates.
C. People who drink more cups of coffee will have significantly lower heart rates.
D. There will be a significant relationship between the number of cups of coffee drunk within the last 4 hours and heart rate
A The null hypothesis is the opposite of the alternative hypothesis and so usually states that an effect is absent
A Type II error occurs when :
(Hint: This would occur when we obtain a small test statistic (perhaps because there is a lot of natural variation between our samples.)
A. We conclude that there is not an effect in the population when in fact there is.
B. We conclude that there is an effect in the population when in fact there is not.
C. We conclude that the test statistic is significant when in fact it is not.
D. The data we have typed into SPSS is different from the data collected.
A A Type II error would occur when we obtain a small test statistic (perhaps because there is a lot of natural variation between our samples)
In general, as the sample size (N) increases:
A. The confidence interval gets narrower.
B. The confidence interval gets wider.
C. The confidence interval is unaffected.
D. The confidence interval becomes less accurate
A
Which of the following best describes the relationship between sample size and significance testing?
(Hint: Remember that test statistics are basically a signal-to-noise ratio, so given that large samples have less ‘noise’ they make it easier to find the ‘signal’.)
A. In large samples even small effects can be deemed ‘significant’.
B. In small samples only small effects will be deemed ‘significant’.
C. Large effects tend to be significant only in small samples.
D. Large effects tend to be significant only in large samples.
A
The assumption of homogeneity of variance is met when:
A. The variances in different groups are approximately equal.
B. The variances in different groups are significantly different.
C. The variance across groups is proportional to the means of those groups.
D. The variance is the same as the interquartile range.
A - To make sure our estimates of the parameters that define our model and significance tests are accurate we have to assume homoscedasticity (also known as homogeneity of variance)
Next, the lecturer was interested in seeing whether males and females reacted differently to the different teaching methods.
Produce a clustered bar graph showing the mean scores of teaching method for males and females.
(HINT: place TeachingMethod on the X axis, Exam Score on the Y axis, and Gender in the ‘Cluster on X’ box. Include 95% confidence intervals in the graph).
Which of the following is the most accurate interpretation of the data?
A.Females performed better than males both the reward and indifferent conditions. Regarding the confidence intervals, there was a large degree of overlap between males and females in all conditions of the teaching method.
B.Males performed better than females in the reward condition, and females performed better than males in the indifferent condition. Regarding the confidence intervals, there was no overlap between males and females across any of the conditions of teaching method.
C.Males performed better than females in all conditions. Regarding the confidence intervals, there was a small degree of overlap between males and females for the reward and indifferent conditions, and a large degree of overlap between males and females for the punish condition.
D.Males performed better than females in the reward condition, and females performed better than males in the indifferent condition. Regarding the confidence intervals, there was a small degree of overlap between males and females for the reward and indifferent conditions, and a large degree of overlap between males and females for the punish condition.
D
Produce a line graph showing the change in mean anxiety scores over the three time points.
NOTE: this is a repeated measures (or within subjects) design, ALL participants took part in the same condition.
Which of the following is the correct interpretation of the data?
A.Mean anxiety increased across the three time points.
BMean anxiety scores were reduced across the three time points, and there was a slight acceleration in this reduction between the middle and end of the course.
CMean anxiety scores were reduced across the three time points, though this reduction slowed down between the middle and end of the course.
DMean anxiety scores did not change across the three time points.
B
A general approach in regression is that our outcomes can be predicted by a model and what remains
is the error
The i in the general model in regression shows
e.g., outcome 1 is equal to model plus error 1 and outcome 2 is equal to model plus error 2 and so on…
For correlation, the outcome is modelled by
scaling (multiplying by a constant) another variable
Equation of correlation model
If you have a 1 continous variable which meets assumtpion of parametric test then you can conduct a
pearson correlation or regression
Variance is a feature of outcome measurements we have obtained and we want to predict with a model in correlation/regression that…
captures the effect of the predictor variables we have manipulated or measured
Variance of a single variable represents the
average amount that the data cary from the mean
Variance is the standard deviation
squared (s squared)
Variance formula - (2)
xi minus average of all scores of pp which is squared and divided by total number of participants minus 1
done for each participant (sigma)
Variance is SD squared meaning that it captures the
average of the squared difference the outcome values from the mean of all outcomes (explaining what the formula of variance does)
Covariance gathers information on whether
one variable covarys with another
In covariance if we are interested whether 2 variables are related then interested whether changes in one variable are met with changes in other
therefore.. - (2)
when one variable deviates from its mean we
would expect the other variable to deviate from its mean in a similar way.
So, if one variable increases then the other, related variable, should also increase or even decrease at the same level.
If one variable covaries with another variable then it means these 2 variables are
related
To get SD from variance then you would
square root variance
What would you do in covariance formula in proper words? - (5)
- Calculate the error between the mean and each subject’s score for the first variable (x).
- Calculate the error between the mean and their score for the second variable (y).
- Multiply these error values.
- Add these values and you get the product deviations.
- The covariance is the average product deviations
Example of calculaitng covariance and what does answer tell you?
The answer ispositive: that tells us the x and y values tend to risetogether.
What does each element of covariance formula stand for? - (5)
X = the value of ‘x’ variable
Y = the value of ‘y’ variable
X(line) = mean of ‘x’ - e.g., green
Y(line) = mean of ‘y’ - e.g., blue
n = the number of items in the data set
covariance will be large when values below
the mean for one variable
What does a positive covariance indicate?
as one variable deviates from the mean, the other
variable deviates in the same direction.
What does negative covariance indicate?
a negative covariance indicates that as one variable deviates from the mean (e.g. increases), the other deviates from the mean in the opposite direction (e.g. decreases).
What is the problem of covariance as a measure of the relationship between 2 variables? - (5)
dependent upon the units /scales of measurement used
So covariance is not a standardised measure
e.g., if 2 variables measured in miles and covariance is 4.25 then if we convert data to kilometres then we have to calculate covariance again and see it increases to 11.
Dependence of scale measurement is a problem as can not compare covariances in an objective way –> can not say whether covariance is large or small to another data unless both data sets measured in same units
So we need to STANDARDISE it.
What is the process of standardisaiton?
To overcome the problem of dependence on the measurement scale, we need to convert
the covariance into a standard set of units
How to standardise the covariance?
dividing by product of the standard deviations of both variables.
Formula of standardising covariance
Same formula of covariance but multipled of SD of x and SD of y
Formula of Pearson’s correlation coefficient, r
Example of calculating Pearson’s correlation coefficient, r - (5)
standard deviation for the number of adverts watched (sx)
was 1.67,
SD of number of packets of crisps bought (sy) was 2.92.
If we multiply these together we get 1.67 × 2.92 =
4.88.
.Now, all we need to do is take the covariance, which we calculated a few pages ago as being 4.25, and divide by these multiplied standard deviations.
This gives us r = 4.25/
4.88 = .87.
The standardised version of covariance is the
correlational coefficient or Pearson’s r
Pearson’s R is … version of covariance meaning independent of units of measurement
standardised
What does correlation describe? - (2)
Describes a relationship between variables
If one variable increases, what happens to the other variable?
Pearson’s correlation coefficient r was also called the
product-moment correlation
Linear relationship and normally disturbed data and interval/ratio and continous data is assumed in
Pearson’s r correlation coefficient
Pearson Correlation Coefficient varies between
-1 and +1 (direction of relationship)
The larger the R Pearson’s correlation coefficient value, the closer the values will
be with each other and the mean
The smaller R Pearson’s correlation coefficient values indicate
there is unexplained variance in the data and results in the data points being more spread out.
What does these two graphs show? - (2)
- example of high negative correlation. The data points are close together and are close to the mean.
- On the other hand, the graph on the right shows a low positive correlation. The data points are more spread out and deviate more from the mean.
The Pearson Correlation Coefficient measures the strength of a relationhip
between one variable and another hence its use in calculating effect size
A Pearson’s correlation coefficient of +1 indicates
two variablesare perfectly positively correlated, so as one variable increases, the other increases by a
proportionate amount.
A Pearson’s correlation coefficient of -1 indicates
a perfect negative relationship: if one variable increases, the other decreases by a proportionate amount.
Pearson’s r
+/- 0.1 means
small effect
Pearson’s r
+/- 0.3 means
medium effect
Pearson’s r
+/- 0.5 means
large effect
In Pearson’s correlation, we can test the hypothesis that - (2)
correlation coefficient is different from zero
(i.e., different from ‘no relationship’)
In Pearson’s correlation coefficient, we can test the hypothesis that the correlation is different from 0
If we find our observed coefficient was very unlikely to happen if there was no effect in population then gain confidence that
relationship that
we have observed is statistically meaningful.
. In the case of a correlation
coefficient we can test the hypothesis that the correlation is different from zero (i.e. different
from ‘no relationship’).
There are 2 ways to test this hypothesis
- Z scores
- T-statistic
Confidence intervals tells us something
likely correlation in the population
Can calculate confidence intervals of Pearson’s correlation coefficient by transforming formula of CI
As sample size increases, so the value of r at which a significant result occurs
decreases e.g 20 n p is not < 0.05 but at 200 pps it is p < 0.05
Pearson’s r = 0 means - (2)
indicates no linear relationship at all
so if one variable changes, the other stays the same.
Correlation coefficients give no indication of direction of… + example - (2)
causality
e.g., although we conclude no of adverts increase nmber of toffees bought we can’t say watching adverts caused us to buy toffees
We have to be caution of causality in terms of Pearson’s correlation r as - (2)
- Third variable problem - causality between variables can not be assumed in any correlation
- Direction of causality: Correlation coefficients give nothing about which variables causes other to change.
If you got weak correlation between 2 variables = weak effect then take a lot of measurements for that relationship to be
significant
R correlation coefficient gives the ratio of
covariance to a measure of variance
Example of correlations getting stronger
R squared is known as the
coefficient of determination
of cor
R^2 can be used to explain the
proportion of the variance for a dependent variable )outcome) that’s explained by an independent variable . (predictor)
Example of R^2 coefficient of determination - (2)
X = exam anxiety
Y = exam performance
If R^2 = 0.194
19.4% of variability in exam performance can be explained by exam anxiety
the variance in y accounted for by x’,
R^2 calculate the amount of shared
variance
Example of r and R^2
Multiply 0.1 * 0.1 for example
R^2 gives you the true strength of.. but without
the correlation but without an indication of its direction.
What are the three types of correlations? - (3)
- Bivarate correlations
- Partial correlations
- Semi-partial or part correlations
Whats bivarate correlation?
relation between 2 variables
What is a partial correlation?
looks at the relationship between two variables while ‘controlling’ the effect of one or more additional variables.
The partial correlation partials out the
the effect of one or more variables on either X or Y
A partial correlation controls for third variable which is made from - (3)
- A correlation calculates each data points distance from line (residuals)
- This is the error relative to the model (unexplained variance)
- A third variable might predict some of that variation in residuals
The partial correlation compares the unique variation of one variable with the
unfiltiered variation of the other
The partial correlation holds the
third variable constant (but we don’t manipulate these)
Example of partial correlation- (2)
For example, when studying the effect of a diet, the level of exercise might also influence weight loss
We want to know the unique effect of diet, so we need to partial out the effect of exercise
Example of Venn Diagram of Partial Correlation - (2)
Partial Correlation between IV1 and DV = D / D+C
Unique variance accounted for by the predictor (IV1) in the DV, after accounting for variance shared with other variables.
Example of Partial Correlation - (2)
Partial correlation: Purple / Red + Purple
If we were doing just a partial correlation, we would see how much exam anxiety is influencing both exam performance and revision time.
Example of partial correlation and semi-partial correlation - (2)
The partial correlation that we calculated took
account not only of the effect of revision on exam performance, but also of the effect of revision on anxiety.
If we were to calculate the semi-partial correlation for the same data, then this would control for only the effect of revision on exam performance (the effect of revision
on exam anxiety is ignored).
In partial correlation, the third variable is typically not considered as the primary independent or dependent variable. Instead, it functions as a
control variable—a variable whose influence is statistically removed or controlled for when examining the relationship between the two primary variables (IV and DV).
The partial correlation is
The amount of variance the variable explains
relative to the amount of variance in the outcome that is left to explain after the contribution of other predictors have been removed from both the predictor and outcome.
These partial correlations can be done when variables are dichotomous (including third variable) e.g., - (2)
we could look at the relationship between bladder relaxation (did the person wet themselves or not?) and the number of large tarantulas crawling up your leg controlling for fear of spiders
(the first variable is dichotomous, but the second variable and ‘controlled for’ variables are continuous).
What does this partial correlation output show?
Revision time = partial, controlling for its effect
Exam performance = DV
Exam anxiety = X - (5)
- . First, notice that the partial correlation between exam performance and exam anxiety is −.247, which is considerably less than the correlation when the effect of
revision time is not controlled for (r = −.441). - . Although this correlation is still statistically significant (its p-value is still below .05), the relationship is diminished.
- value of R2 for the partial correlation is .06, which means that exam anxiety can now account for only 6% of the variance in exam performance.
- When the effects of revision time were not controlled for, exam anxiety shared 19.4% of the variation in exam scores and so the inclusion of revision time has severely diminished the amount of variation in exam scores shared by anxiety.
- As such, a truer measure of the role of exam anxiety has been obtained.
Partial correlations are most useful for looking at the unique
relationship between two variables when
other variables are ruled out
In a semi-partial correlation we control for the
effect that
the third variable has on only one of the variables in the correlation
The semi partial (part) correlation partials out the - (2)
Partials out the effect of one or more variables on either X or Y.
e.g. The amount revision explains exam performance after the contribution of anxiety has been removed from the one variable (usually the predictor- e.g. revision).
The semi-partial correlation compares the
unique variation of one variable with the unfiltered variation of the other.
Diagram of venn diagram of semi-partial correlation - (2)
- Semi-Partial Correlation between IV1 and DV = D / D+C+F+G
Unique variance accounted for by the predictor (IV1) in the DV, after accounting for variance shared with other variables.
Diagram of revision and exam performance and revision time on semi-partial correlation - (2)
- purple/red + purple + white+ orange
- When we use semi-partial correlation to look at this relationship, we partial out the variance accounted for by exam anxiety (the orange bit) and look for the variance explained by revision time (the purple bit).
Summary of partial correlation and semi-partial correlation - (2)
A partial correlation quantifies the relationship between two variables while accounting for the effects of a third variable on both variables in the original correlation.
A semi-partial correlation quantifies the relationship between two variables while accounting for the effects of a third variable on only one of the variables in the original correlation.
Pearson’s product-moment correlation coefficient (described earlier) and Spearman’s rho (see section 6.5.3) are examples of
of bivariate correlation coefficients.
Non-parametric tests of correlations are… (2)
- Spearman’s roh
- Kendall’s tau test
In spearman’s rho the variables are not normally distributed and measures are on a
ordinal scale (e.g., grades)
Spearman’s rho works on by
first ranking the data n(numbers converted into ranks), and then running Pearson’s r on the ranked data
Spearman’s correlation coefficient, rs, is a non-parametric statistic and so can be used when the data have
data have violated parametric assumptions such as nonnormally distributed data
In spearman correlation coefficient is sometimes called
Spearman’s rho
For spearman’s r we can get R squared but it is interpreted slightly different as
proportion of
variance in the ranks that two variables share.
Kendall’s tau used rather than Spearman’s coefficient when - (2)
when you have a small data set with a large number of
tied ranks.
This means that if you rank all of the scores and many scores have the same rank, then Kendall’s tau should be used
Kendall’s tau test - (2)
For small datasets, many tied ranks
Better estimate of correlation in population than Spearman’s ρ
Kendall’s tau is not numerically similar to r or rs (spearman) and so tau squared does not tell us about
proportion of
variance shared by two variables (or the ranks of those two variables).
The Kendall’s tau is 66-75% smaller than both Spearman’s r and Pearson’s r so
tau is not comparable to r and r s
There is a benefit using Kendall’s statistic than Spearman as it shows - (2)
Kendall’s statistic is actually a better estimate of the correlation in the population
we can draw more accurate generalizations from Kendall’s statistic than from Spearman’s.
Whats the decision tree for Spearman’s correlation? - (4)
- What type of measurement = continous
- How many predictor variables = one
- What type of continous variable = continous
- Meets assumption of parametric tests - No
The output of Kendall and Spearman can be interpreted the same way as
Pearson’s correlation coefficient r output box
The biserial and point-biserial correlation coefficients used when
one of the two variables is dichotomous (e.g., example of dichotomous variable is women being pregnant or not)
What is the difference between biserial and point-biserial correlations?
depends on whether the dichotomous variable is discrete or continuous
The point–biserial correlation coefficient (rpb) is used when
one variable is a
discrete dichotomy (e.g. pregnancy),
biserial correlation coefficient (rb) is used
when - (2)
one variable is a continuous dichotomy (e.g. passing or failing an exam).
e.g. An example is passing or failing a statistics test: some people will only just fail while others will fail by
a large margin; likewise some people will scrape a pass while others will clearly excel.
Example of when point=biserial correlation used - (3)
- Imagine interested in relationship between gender of a cat and how much time it spent away from home
- Time spent away is measured in interval level –> mets assumptions of parametric data
- Gender is discrete dichotomous variable coded with 0 for male and 1 for female
Can convert point-biserial correlation coefficient into
biseral correlation coefficient
Point biserial and biserial correlation differ in size as
biserial correlation bigger than point biserial
Example of queston conducting Pearson’s r (4) -
The researchers was interested in whether the amount someone gets paid and amount of holidays they take from work, whether these two variables would be related to their productivity at work
- Pay: Annual salary
- Holiday: Number of holiday days taken
- Productivity: Productivity rating out of 10
Example of Pearson’s r scatterplot :
relationship between pay and productivity
If we have r = 0.313 what effect size is it?
medium effect size
±.1 = small effect
±.3 = medium effect
±.5 = large effect
What does this scatterplot show?
o This indicates very little correlation between the 2 variables
What will a matrix scatterplot show?
the relationship between all possible combinations of your variables
What does this scatterplot matrix show? - (2)
- For Pay and Holiday, we can see the line is very flat and indicates the correlation between the two variables is quite low
- For pay and productivity, the line is steeper suggesting the correlation is fairly substantial between these 2 variables and same for holidays and pay and productivity and holidays here
What is degrees of freedom for correlational analysis?
N-2
What does this Pearson’s correlation r output show? - (4)
- The relationship between pay and holidays is very low correlation is -0.04
- Between pay and productivity, there is a medium size correlation of r = 0.313
- Between holidays and productivity there is medium going on large effect size of 0.435
- Relationship between pay and productivity and also holidays and productivity is sig but correlation with pay and holidays was not sig
Another examp;e of Pearson’s correlation r question - (3)
A student was interested in the relationship between the time spent preparing an essay, the interestingness of the essay topic and the essay mark received.
He got 45 of his friends and asked them to rate, using a scale from 1 to 7, how interesting they thought the essay topic was (1 - I’ll kill myself of boredom, 4 - it’s not too bad!, 7 - it’s the most interesting thing in the world!) (interesting).
He then timed how long they spent writing the essay (hours), and got their percentage score on the essay (essay).
Example of interval/ratio continous data needed for Pearson’s r for IV and DV - (2)
- Interval scale: difference between 10 degrees C and 20 degrees is same as 80 F and 90 F, 0 degrees does not mean absence of temp
- Ratio: Height as 0 cm means no weight and weight, time
Pearson’s correlation r , spearman and kendall equires
one IV and one DV
What does this SPSS output show?
A. There was a non-significant positive correlation between interestingness of topic and the amount of time spent writing. There was a non-significant positive correlation between time spent writing an essay and essay mark
There was a significant positive correlation between interestingness of topic and essay mark, with a medium effect size
B. There was a significant positive correlation between interestingness of topic and the amount of time spent writing, with a small effect size.There was a significant positive correlation between time spent writing an essay and essay mark, with a large effect size. .There was a non-significant positive correlation between interestingness of topic and essay mark
C. There was a significant negative correlation between interestingness of topic and the amount of time spent writing, with a medium effect size.. There was a non-significant positive correlation between time spent writing an essay and essay mark. There was a non-significant positive correlation between interestingness of topic and essay mark
D. There was a significant positive correlation between interestingness of topic and the amount of time spent writing, with a large effect size. There was a non-significant positive correlation between time spent writing an essay and essay mark There was a non-significant positive correlation between interestingness of topic and essay mark
D. There was a significant positive correlation between interestingness of topic and the amount of time spent writing, with a large effect size. There was a non-significant positive correlation between time spent writing an essay and essay mark There was a non-significant positive correlation between interestingness of topic and essay mark
r = 0.21 effect size is..
in between small and medium effect
Effect size is only meaningful if you evaluatte it witth regards to
your own research area
Biserial correlaion is when
one variable is dichotomous, but there is an underlying continuum (e.g. pass/fail on an exam)
Pointt biserial correlation is when
When one variable is dichotomous, and it is a true dichotomy (e.g. pregnancy)
Example of dichotomous relationship - (3)
- example of a true dichotomous relationship.
- We can compare the differences in height between males and females.
- Use dichotomous predictor of gender
What is the decision tree for multiple regression? - (4)
- Continous
- Two or more predictors that are continous
- Multiple regression
- Meets assumptions of parametric tests
Multiple regression is the same as simple linear regression expect for - (2)
every extra predictor you include, you have to add a coefficient;
so, each predictor variable has its own coefficient, and the outcome variable is predicted from a combination of all the variables multiplied by their respective coefficients plus a residual term
Multiple regression equation
In multiple regression equation, list all the terms - (5)
- Y is the outcome variable,
- b1 is the coefficient of the first predictor (X1),
- b2 is the coefficient of the second predictor (X2),
- bn is the coefficient of the nth predictor (Xn),
- εi is the difference between the predicted and the observed value of Y for the ith participant.
Multiple regression uses the same principle as linear regression in a way that
we seek to find the linear combination of predictors that correlate maximally with the outcome variable.
Regression is a way of predicting things that you have not measured by predicting
an outcome variable from one or more predictor variables
Can’t plot a 3D plot of MR as shown here
for more than 2 predictor (X) variables
If you got two prediictors thart overlap and correlate a lot then it is a .. model
bad model can’t uniquely explain the outcome
In Hierarchical regression, we are seeing whether
one model explains significantly more variance than the other
In hierarchical regression predictors are selected based on
past work and the experimenter
decides in which order to enter the predictors into the model
As a general rule for hierarchical regression, - (3)
known predictors (from other research) should be entered into the model first in order of their importance in predicting the outcome.
After known predictors have been entered, the
experimenter can add any new predictors into the model.
New predictors can be entered either all in one go, in a stepwise manner, or hierarchically (such that the new predictor
suspected to be the most important is entered first).
Example of hierarchical regression in terms of album sales - (2)
The first model allows all the shared variance between Ad budget and Album sales to be accounted for.
The second model then only has the option to explain more variance by the unique contribution from the added predictor Plays on the radio.
What is forced entry MR?
method in which all predictors are forced
into the model simultaneously.
Like HR, forced entry MR relies on
good theoretical reasons for including the chosen predictors,
Different from HR, forced entry MR
makes no decision about the order in which variables are entered.
Some researchers believe that about forced entry MR that
this method is the only appropriate method for theory testing because stepwise techniques are influenced by random variation in the data and so rarely give replicable results if the model is retested.
Why select colinearity diagnostics in statistics box for multiple regression? - (2)
This option is for obtaining collinearity statistics such as the
VIF, tolerance,
Checking assumption of no multicolinearity
Multicollinearity poses a problem only for multiple regression because
simple regression requires only one predictor.
Perfect collinearity exists in multiple regression when at least
e.g., two predictors are perfectly correlated , have a correlation coefficient of 1
If there is perfect collinearity in multiple regression between predictors it
becomes impossible
to obtain unique estimates of the regression coefficients because there are an infinite number of combinations of coefficients that would work equally well.
Good news is perfect colinearity in multiple regression is rare in
real-life data
If two predictors are perfectly correlated in multiple regression then the values of b for each variable are
interchangable
As colinearity increases in multiple regression, there are 3 problems that arise - (3)
- Untrustory bs
- Limit size of R
- Importance of predictors
One way of identifying multicollinearity in multiple regression is to scan a
a correlation matrix of all of the predictor
variables and see if any correlate very highly (by very highly I mean correlations of above .80
or .90)
The VIF indicates in multiple regression whether a
predictor has a strong linear relationship with the other predictor(s).
If VIF statistic above 10 or approaching 10 in multiple regression then what you would want to do is have a - (2)
look at your variables to see if you need to include all variables whether all need to go in model
if high correlation between 2 predictors (measuring same thing) then decide whether its important to include both vars or take one out and simplify regression model
Related to the VIF in multiple regression is the tolerance
statistic, which is its
reciporal (1/VIF) = inverse of VIF
In Plots in SPSS, you put in multiple regression - (2)
ZRESID on Y and ZPRED on X
Plot of residuals against predicted to asses homoscedasticity
What is ZPRED in MR? - (2)
(the standardized predicted values of the dependent variable based on the model).
These values are standardized forms of the values predicted by the model.
What is ZRESID in MR? - (2)
(the standardized residuals, or errors).
These values are the standardized differences between the observed data and the values that the model predicts).
SPSS in multiple linear regression gives descriptive outcoems which is - (2)
- basics means and also a table of correlations between variables.
- This is a first opportunity to determine whether there is high correlation between predictors, otherwise known as multi-collinearity
In model summary of SPSS, it captures how the model or models explain in MR
variance in terms of R squared, and more importantly how R squared changes between models and whether those changes are significant.
Diagram of model summary
What is the measure of R^2 in multiple regression
measure of how much of the variability in the outcome is accounted for
by the predictors
The adjusted R^2 gives us an estimate of in multiple regression
fit in the general population
The Durbin-Watson statistic if specificed in multiple regresion tells us whether the - (2)
assumption of independent errors is tenable (value less than 1 or greater than 3 raise alarm bells)
value closer to 2 the better = assumption met
SPSS output for MR = ANOVA table which performs
F-tests for each model
SPSS output for MR contains ANOVA that tests whether the model is
significantly beter at predicting the outcome than using the mean as a ‘best guess’
The F-ratio represents the ratio of
improvement in prediction that results from fitting the model, relative to the inaccuracy that still exists in the model
We are told the sum of squares for model (SSM) - MR regression line in output which represents
improvement in prediction resulting from fitting a regression line to the data rather than using the mean as an estimate of the outcome
We are told residual sum of squares (Residual line) in this MR output which represents
total difference between
the model and the observed data
DF for Sum of squares Model for MR regression line is equal to
number of predictors (e.g., 1 for first model, 3 for second)
DF for Sum of Squares Residual for MR is - (2)
Number of observations (N) minus number of coefficients in regression model
(e.g., M1 has 2 coefficents - one for predictor and one for constant, M2 has 4 - one for each 3 predictor and one for constant)
The average sum of squares in ANOVA table is calculated by
calculated for each term (SSM, SSR) by dividing the SS by the df. T
How is the F ratio calculated in this ANOVA table?
F-ratio is calculated by dividing the average improvement in prediction by the model (MSM) by the average
difference between the model and the observed data (MSR)
If the improvement due to fitting the regression model is much greater than the inaccuracy within the model then value of F will be
greater than 1 and SPSS calculates exact prob (p-value) of obtaining value of F by change
What happens if b values are positive in multiple regression?
there is a positive relationship between the predictor and the outcome,
What happens if the b value is negative in multiple regression?
represents a negative relationship between predictor and outcome variable?
What do the b values in this table tell us what relationships between predictor and outcome variable in multiple regression? (3)
Indicating positive relationships so as advertising budget increases, record sales increases (outcome)
plays on ratio increase as do record sales
attractiveness of band increases record sales
The b-values also tell us, in addition to direction of relationship (pos/neg) , to what degree each in multiple regression
predictor affects the outcome if the effects of all other predictors are held constant:
B-values tell us to what degree each predictor affects the outcome if the effects of all other predictors held constant in multiple regression
e.g., advertising budget - (3)
(b = 0.085):
This value indicates that as advertising budget (x)
increases by one unit, record sales (outcome, y) increase by 0.085 units.
This interpretation is true only if the
effects of attractiveness of the band and airplay are held constant.
Standardised versions of b-values are much more easier to interpret as in muliple regression
not dependent on the units of measurements of variables
The standardised beta values tell us that in multiple regression
the number of standard deviations that the outcome will change as a result of one standard deviation change
in the predictor.
The standardized beta values are all measured in standard deviation
units and so are directly comparable: therefore, they provide a in MR
a better insight into the
‘importance’ of a predictor in the mode
If two predictor variables (e.g., advertising budget and airplay) have virtually identical standardised beta values (0.512, and 0.511) it shows that in MR
both variables have a comparable degree of importance in the model
A good regression model will have a narrow and small CI interval indicating in MR
value of b in this sample is close to the true value of b in the populatio
A bad regression model have CI that cross zero indicating that in MR
in some samples the predictor has a negative
relationship to the outcome whereas in others it has a positive relationship
In image below, which are the two best predictors based on CIs and one that isn’t as (2) in MR
two best predictors (advertising and airplay) have very tight confidence intervals indicating that the estimates for the current model are likely to be representative of the true population values
interval for attractiveness is wider (but still does not cross zero) indicating that the parameter for this variable is less representative, but nevertheless significant.
If you do part and partial correlations in descriptive box, there will be another coefficients table which looks this in MR like:
The zero-order correlations are the simple in MR
Pearson’s correlation coefficients
The partial correlations represent the in MR
represent the relationships between each predictor and the outcome variable, controlling for the effects of the other two predictors.
The part correlations in MR - (2)
represent the relationship between each predictor and the outcome, controlling for the effect that the other two variables have on the outcome.
representing the unique relationship each predictor has with otucome
Partial correlations in example is calculated by in MR- (2)
unique variance in outcome (ignore all other predictors) explained by predictor divided by variance in outcome not explained by all other predictors
A/A+E
Part correlations are calculated by - (2) in MR
unique variance in outcome explained by predictor divided by total variance in outcome
A/A+B+C+E
How to interpret this image in terms of colinearity - VIF and tolerance in MR
For our current model the VIF values are all well below 10 and the tolerance statistics all well above 0.2;
therefore, we can safely conclude that there is no collinearity within our data.
We can produce casewise diagnostics to see a in MR to see (2)
summary of residuals statistics to be examined of extreme cases
To see whether individual scores (cases) influence the modelling of data too much
SPSS casewise diagnostics shows cases that have a standardised residuals that are in MR (2)
less than -2 or greater than 2
(We expect about 5% of our cases to do tha and 95% to have standardised residuals within about +/- 2.)
What does this casewise diagnostic show? - (2) MR
- 99% of cases should lie within ±2.5 so expect 1% of cases lie outside limits
- From cases listed, clear two cases (1%) lie outside of limits (case, 164 [investigate further has residual 3] and 179) - 1% which isconform to accurate model
If there are many more cases we likely have (more than 5% of sample size) in case wise then in MR
broken the assumptions of the regression
If cases are a large number of standard deviations from the mean, we may want to in casewise diagnostics in MR
investigate and potentially remove them because they are ‘outliers’
Diagram of assumption of homoscedasticity and linearity of ZRESID againsr ZPRED in MR
What does this partial plot show? - (2) in MR
the partial plot shows the strong positive relationship to album sales.
There are no obvious outliers and the cloud of dots is evenly spaced out around the line, indicating homoscedasticity.
What does this plot show in MR(2)
the plot again shows a positive relationship to album sales, but the dots show funnelling,
There are no obvious outliers on this plot, but the funnel-shaped cloud indicates a violation of the assumption of homoscedasticity.
P plot and histogram of normally distributed in MR
P plot for skewed distirbution histogram for MR
If R^2 is 0.374 (outcome var in productivity and 3 predictors) then it shows that in MR
37.4% of the variance in productivity scores was accounted for by 3 predictor variables
This table tells us in terms of standardised beta values that (outcome is productivity in MR)
holidays had standardized beta coefficient of 0.031 whereas cake had a much higher standardized beta coefficient of 0.499 which tells us that amount of cake given out much better predictor of productivity than the amount of holidays taken
For pay we have a beta coefficient of 0.323 which tells us that pay was also a pretty good predictor in the model of productivity but slightly less than cake
What does this table tells us in terms of signifiance? - (3) in MR
- P value for holidays is 0.891 which is not significant
- P value for cake is 0.032 is significant
- P value for pay is 0.012 is significant
In ANOVA it is comparing M2 with all its predictor variables with in MR
baseline not M1
To see if M2 is an improvement of M1 in HR we need to look at … in model summary in MR
change statistics
What does this change statistic show in terms of M2 and M1 in MR
M2 explains an extra 7.5% which is sig
For this output interpret whether predicotrs are sig predictors of record scales and magnitude t statistic on impact of record sales in MR - (2)
For this model, the advertising budget (t(196) = 12.26, p < .001), the amount of radio play prior to release (t(196) = 12.12, p < .001) and attractiveness of the band (t(196) =4.55, p < .001) are all significant predictors of record sales.
From the magnitude of the t-statistics we can see that the advertising budget and radio play had a similar impact,
whereas the attractiveness of the band had less impact.
All residuals should lie within ….. SDs for no outliers /normal amount of outliers in MR
-3 and 3 SD
Which variables (if any) are highly correlated in MR?
Weight, Activity, and the interaction between them are statistically significant
What does homoscedasticity and hetrodasticity mean in MR? - (2)
Homoscedasticity: similar variance of residuals (errors) across the variable continuum, e.g. equally accurate.
Heteroscedasticity: variance of residuals (errors) differs across the variable continuum, e.g. not equally accurate
Diagram of normal, skewed to left (pos) and skewed to right (neg) of p-plots in MR
Diagram of excluded variables table in SPSS - (3) in MR
- Tells that OCD interpretiotn of intrustrions would have not have a significant impact on model’s ability to predict social anxiety
Beta value of Interpretation of Intrusions is very small, indicating small influence on outcome variable
Beta is the degree of change in the outcome variable for every 1 unit of change in the predictor variable.
When checking assumption fo regression, what does this graph tell you in MR
Normality of residuals
A consumer researcher was interested in what factors influence people’s fear responses to horror films. She measured gender and how much a person is prone to believe in things that are not real (fantasy proneness). Fear responses were measured too. In this table, what does the value 847.685 represent in MR
The residual error in the prediction of fear scores when both gender and fantasy proneness are included as predictors in the model.
A psychologist was interested in whether the amount of news people watch predicts how depressed they are. In this table, what does the value 3.030 represent in MR
The improvement in the prediction of depression by fitting the model
A consumer researcher was interested in what factors influence people’s fear responses to horror films. She measured gender (0 = female, 1 = male) and how much a person is prone to believe in things that are not real (fantasy proneness) on a scale from 0 to 4 (0 = not at all fantasy prone, 4 = very fantasy prone). Fear responses were measured on a scale from 0 (not at all scared) to 15 (the most scared I have ever felt).
Based on the information from model 2 in the table, what is the likely population value of the parameter describing the relationship between gender and fear in MR
Somewhere between −3.369 and −0.517
A consumer researcher was interested in what factors influence people’s fear responses to horror films. She measured gender (0 = female, 1 = male) and how much a person is prone to believe in things that are not real (fantasy proneness) on a scale from 0 to 4 (0 = not at all fantasy prone, 4 = very fantasy prone). Fear responses were measured on a scale from 0 (not at all scared) to 15 (the most scared I have ever felt).
How much variance (as a percentage) in fear is shared by gender and fantasy proneness in the population in MR
13.5%
Recent research has shown that lecturers are among the most stressed workers. A researcher wanted to know exactly what it was about being a lecturer that created this stress and subsequent burnout. She recruited 75 lecturers and administered several questionnaires that measured: Burnout (high score = burnt out), Perceived Control (high score = low perceived control), Coping Ability (high score = low ability to cope with stress), Stress from Teaching (high score = teaching creates a lot of stress for the person), Stress from Research (high score = research creates a lot of stress for the person), and Stress from Providing Pastoral Care (high score = providing pastoral care creates a lot of stress for the person). The outcome of interest was burnout, and Cooper’s (1988) model of stress indicates that perceived control and coping style are important predictors of this variable. The remaining predictors were measured to see the unique contribution of different aspects of a lecturer’s work to their burnout.
Which of the predictor variables does not predict burnout in MR
Stress from research
Using the information from model 3, how would you interpret the beta value for ‘stress from teaching’ in MR
As stress from teaching increases by one unit, burnout decreases by 0.36 of a unit.
How much variance in burnout does the final model explain for the sample in MR
80.3%
A psychologist was interested in predicting how depressed people are from the amount of news they watch. Based on the output, do you think the psychologist will end up with a model that can be generalized beyond the sample?
No, because the errors show heteroscedasticity.
Diagram of no outliers for one assumption of MR
Note that you expect 1% of cases to lie outside this area so in a large sample, if you have one or two, that could be ok
Output diagram what does output show in MR? - (2)
Difference between no predictors and model 1 (a).
Difference between model 1 (a) and model 2 (b).
Our model 2 is significantly better at predicting the value of the outcome variable than the null model and model 1 (F (2, 197) = 167.2, p<.001) and explains 66% of the variance in our data (R2=.66
What does this output show in terms of regression model in MR? - (3)
y = 0.09x1 + 3.59x2 + 41.12
For every £1,000 increase in advertising budget there is an increase of 87 record sales (B = 0.09, t = 11.99, p<.001).
For every number of plays on Radio 1 per week there is an increase of 3,589 record sales (B = 3.59, t = 12.51, p<.001).
Report R^2, F statistic and p-value to 2DP for overall model - (3)
o R squared = 0.09
o F statistic = 22.54
o P value = p < 0.001
Report beta and b values for video games, resitrctions and parental aggression to 2DP and p-value in MR
Which of the following assumptions of homosecdasticity and linearity is correct?
AThere is non-linearity in the data
BThere is heteroscedasticity in the data
CThere is both heteroscedasticity and non-linearity in the data
DThere are no problems with either heteroscedasticity or non-linearity
D - data poiints show random pattern
Determine the proportion of variance in salary that the number of years spent modelling uniquely explains once the models’ age was taken into account:
Hierarchical regression
A 2.0%
b17.8%
c39.7%
d42.2%
A –>The R square change in step 2 was .020,
Test for multicollinearity (select tolerance and VIF statistics).
Based on this information, what can you conclude about the suitability of your regression model?
AThe VIF statistic is above 10 and the tolerance statistic is below 0.2, indicating that there is no multicolinearity.
BThe VIF statistic is above 10 and the tolerance statistic is below 0.2, indicating that there is a potential problem withmulticolinearity.
CThe VIF statistic is below 10 and the tolerance statistic is above 0.2, indicating that there is nomulticolinearity.
DThe VIF statistic is below 10 and the tolerance statistic is above 0.2, indicating that there is a potential problem withmulticolinearity.
B
The following graph shows:
A. Regression assumptions met
B. Non-linearity = could indicate curve
C. Hetrodasticity + Non-linearity
D. hetrodasticity
A
A consumer researcher was interested in what factors influence people’s fear responses to horror films. She measured gender (0 = female, 1 = male) and how much a person is prone to believe in things that are not real (fantasy proneness) on a scale from 0 to 4 (0 = not at all fantasy prone, 4 = very fantasy prone). Fear responses were measured on a scale from 0 (not at all scared) to 15 (the most scared I have ever felt). What is the likely population value of the parameter describing the relationship between gender and fear?
Somewhere between 3.369 and 0.517
The t-distributed is defined by its
degrees of freedom - related to the sample size.
The t distribution has heavier tails for - (2)
lower degrees of freedom (small N studies)
increased uncertainty and a higher likelihood of observing extreme values than large N studies with less heavy tails as t distribution goes to normal
Formula of calculating t- test statistic (form depend on whether same or different participants used in each experimental condition) in independent/paired
Formula of calculating t-statistic shows obtaining t-test statistic by diving the model/effect by the in independent /apried
error in the model
Expected difference in calculating t-test statistic in most cases is
0 - expect differences between sample group means we colelcted to be different to 0
Diagram of equation of calculating t-statistic from paired t-test and explain - (2)
- Compares mean differences betwen our samples (–D) to the differences we would expect to find between population means (uD) which is divided by standard error of differences (sD / square root N)
- If H0 is ture, then expect no difference between population means hence uD = 0
How to calcuate the standard error of differences in paired-test?(σ –D)
Standard deviation of differences divided by square root of sample size
What does this paired samples correlation show?
people doing well in first exam likely doing well in second exam regardless of condition they are in and significantly correlated (r= 0.664)
What does this SPSS output show? = paired t- test
t(19) = 2.72, p = 0.012
What does negative t-value mean?
paired t-test.
First condition had smaller mean than second condition
What does 95% confiderence interval of difference mean in SPSS output of paired t-test?- (3)
- 95% of the samples (e.g., if we had 100 samples then 95 of those samples..) the constructucted CIs contain true value (population) of the mean difference
- CIs tell us boundaries within which true mean difference is likely to lie
- The true value of mean difference is unlikely to be 0 if Cis does not contain 0
Diagram of calculating Cohen’s D Statistic for sleep vs no sleep for paired
Minus big mean from small mean divided by smallest SD (control group)
Diagram of writing up paired t-test result
To calculate effect size for independent and paired t-tests, beside Cohen’s D, we can also
calculate effect size r (above 0.50 is large effect) by converting t-value to r-value
Equation of independent t-test of equal N sizes for each condition
Equation of independent t-test of equal N sizes becomes like the final form since - (3)
- We are looking at differences between the overall means of 2 samples and compare with differences we would expect to get between means of 2 populations from which sampels come from
- If H0 is true, samples drawn from same population
- Therefore under H0, u1 = u2 therefore u1 - u2 = 0
Equation of independent t-test in numbers for equal N sizes
We use variance of sum law to obtain the estimate of standard error for each … in independent t-test equation for equal N sizes
sample group
This independent t-test standard error formula combines the
standard error for two samples
In independent t-test when we want to compare two groups that contain different number of participants then equation … is not appropriate
Formula of pooled variance estimate t-test - (2)
Each variance of sample is multipled by its DF and added together and divided by the sum of weights (sum of two DFs)
Larger samples better than small ones as close to population
In formula of pooled variance estimate t-test it weights the variance of each sample by the
number of degrees of freedom (N-1)
What does this output show? - in independent t-test - (2)
Sleep condition scored an average exam score of 66.200 and no sleep condition earned an average of 58.73
Effect size (Cohen’s D) = Mean of sleep minus mean of no sleep divided by standard deviation of sleep (control grp) = 66.20-58/73/7.12
What does this output show in independent t-test? - (2)
- Levene’s test is not significant (p = 0.970) so no stats sig differences in variance between two groups
- t(28) = 2.87, p = 0.008
Diagram of reporting independent t-test
What does this SPSS output of independent-test of Levene’s show
homogeneity of variance as assessed by Levene’s Test for Equality of Variances (F = 1.58, p = .219)
We can check if there is no outliers in independent t-test by looking at
boxplots - no outlier here
To check normality of distribution for both independent groups for two-samples independent t-test, we can use..
histogram, q-qplot and tests of normality
Checking normality for independent, is it - (3)
Research question: Which of the two diet formulas is better for puppies?
Dependent variable: Average daily weight gain (ADG, g/day)
We don’t have sig values for either group in the test of normality, histogram and plots look normal
So we have normality of distribution for both independent groups
Inspection of Q-Q Plots and the non-significant Shapiro-Wilk tests (p > .05) indicate that the ADG is normally distributed for both groups
Checking homogenity of variance in this two-sample independent t-test, what does it show?
Research question: Which of the two diet formulas is better for puppies?
Dependent variable: Average daily weight gain (ADG, g/day)
was homogeneity of variance as assessed by Levene’s Test for Equality of Variances (F = 1.58, p = .219)
What does results of two-sample independent results show?
Research question: Which of the two diet formulas is better for puppies?
Dependent variable: Average daily weight gain (ADG, g/day)
This study found that puppies in diet B had statistically significantly higher average daily weight gain (89.29 ± 9.93 g/day) between 12 and 28 weeks of age compared to puppies in diet A (60.20 ± 6.85 g/day), t(27)= -9.24, p < .001.
In Cohen’s D theoretically 3 SDs can be used - (3) which make very little difference
- Pooled SD (over conditions)
2.Averaged SD - Control group SD
How to calculate Cohen’s D for independent two samples t test for this group?
Research question: Which of the two diet formulas is better for puppies?
Dependent variable: Average daily weight gain (ADG, g/day) - (2)
d = (89.29 - 60.20) / 6.85
d = 4.25
Just like t-test can be represented by linear regression equation, ANOVA can be represented by a
multiple regression equation for three means and models acocunt for 3 levels of categorical variable with dummy variables
What does this one-way ANOVA output show?
Leven’s test is non-significant so equal variances are assumed
What does this SPSS output show in one-way ANOVA?
F(2,42) = 5.94, p = 0.005, eta-squared = 0.22
What is the eta-squared/effect size for this SPSS output and what does this value mean? - (2)
830.207/3763.632 = 0.22
22% of the variance in exam scores is accounted for by the model
What happens if the Levene’s test is significant in the one-way ANOVA?
then use statistics in Welch or Brown-Forsythe test
What does this post-hoc table of Bonferroni tests show in one-way ANOVA ? - (3)
- Full sleep vs partial sleep, p = 1.00, not sig
- Full sleep vs no sleep , p = 0.007 so sig
- Partial sleep vs no sleep = p = 0.032 so sig
Diagram of example of grand mean
Mean of all scores regardless pp’s condition
What are the total sum of squares (SST) in one-way ANOVA?
difference of the participant’s score from the grand mean squared and summed over all participants
The F ratio in one-way ANOVA can be calculated by
How does planned contrasts work in SPSS?
Coefficients add to 0 for each contrast (-2 + 1 +1) and once group used alone in contrast then enxt contrasts set coefficient to 0 (e.g., -2 to 0)|
Polynominal contrasts can also look at more complex trends other than linear such as in one way ANOVA?
quadratic, cubic and quartic
What does this ANOVA error line graph show? - (2)
- Linear trend as dose of Viagra increases so does mean level of libido
- Error bars overlap indicating no between group differences
What does the within groups gives deails of in ANOVA table?
SSR (unsystematci variation)
The between groups label in ANOVA table tells us
SSM (systematic variation)
What does this ANOVA table demonstrate? - (2)
- Linear trend is significant (p = 0.008)
- Quadratic trend is not significant (p = 0.612)
What does this output show if we conduct two planned comparisons of:
one to test whether the control group was different to the two groups which received Viagra, and one to see
whether the two doses of Viagra made a difference to libido
- (2)
the table of weights shows that contrast 1 compares the placebo group against the two experimental groups,
contrast 2 compares the low-dose group to the high-dose group
What does this table show if levene’s test is non significant =equal variances assumed
To test hypothesis that experimental groups would increase libido above the levels seen in the placebo group (one-tailed)
To test another hypothesis that a high dose of Viagra would increase libido significantly more than a low dose
one-way ANOVA
- (3)
Signifiance value given in table is two-tailed and since hypothesis one-tail we divide by 2
for contrast 1, we can say that taking Viagra significantly increased libido compared to the control group (p = .0029/2 = 0.0145)
. The significance of contrast 2 tells us that a high dose of Viagra increased libido significantly more than a low dose (p(one-tailed) = .065/2 = .0325)
A psychologist was interested in whether there was a gender difference in the use of email. She hypothesized that because women are generally better communicators than men, they would spend longer using email than their male counterparts. To test this hypothesis, the researcher sat by the computers in her research methods laboratory and when someone started using email, she noted whether they were male or female and then timed how long they spent using email (in minutes). Based on the output, what should she report?
(NOTE: Check for the assumption of equality of variances).
A Females spent significantly longer using email than males, t(14) = –1.90, p = .079
BFemales and males did not significantly differ in the time spent using email,t(7.18) = –1.90,p= .099
CFemales and males did not significantly differ in the time spent using email, t(7.18) = –1.90, p < .003
DFemales and males did not significantly differ in the time spent using email, t(14) = –1.90, p = .079
BFemales and males did not significantly differ in the time spent using email,t(7.18) = –1.90,p= .099
The student welfare office was interested in trying to enhance students’ exam performance by investigating the effects of various interventions. They took five groups of students before their statistics exams and gave them one of five interventions: (1) a control group just sat in a room contemplating the task ahead; (2) the second group had a yoga class to relax them; (3) the third group were told they would get monetary rewards contingent upon the grade they received in the exam; (4) the fourth group were given beta-blockers to calm their nerves; and (5) the fifth group were encouraged to sit around winding each other up about how much revision they had/hadn’t done (a bit like what usually happens). The final percentage obtained in the exam was the dependent variable. Using the critical values for F, how would you report the result in the table below?
AType of intervention did not have a significant effect on levels of exam performance, F(4, 29) = 12.43, p > .05.
BType of intervention had a significant effect on levels of exam performance, F(4, 29) = 12.43, p < .01.
CType of intervention did not have a significant effect on levels of exam performance, F(4, 33) = 12.43, p > .01.
DType of intervention had a significant effect on levels of exam performance, F(4, 33) = 12.43, p < .01.
Type of intervention had a significant effect on levels of exam performance, F(4, 29) = 12.43, p < .01.
A psychologist was looking at the effects of an intervention on depression levels. Three groups were used: waiting list control, treatment and post-treatment (a group who had had the treatment 6 months before). The SPSS output is below. Based on this output, what should the researcher report?
A. The treatment groups had a significant effect on depression levels,F(2, 45) = 5.11.
B. The treatment groups did not have a significant effect on the change in depression levels,F(2, 35.10) = 5.11.
C. The treatment groups did not have a significant effect on depression levels,F(2, 26.44) = 4.35.
D. The treatment groups had a significant effect on the depression levels,F(2, 26.44) = 4.35.
D. The treatment groups had a significant effect on the depression levels,F(2, 26.44) = 4.35.
The student welfare office was interested in trying to enhance students’ exam performance by investigating the effects of various interventions.
They took five groups of students before their statistics exams and gave them one of five interventions: (1) a control group just sat in a room contemplating the task ahead (Control); (2) the second group had a yoga class to relax them (Yoga); (3) the third group were told they would get monetary rewards contingent upon the grade they received in the exam (Bribes); (4) the fourth group were given beta-blockers to calm their nerves (Beta-Blockers); and (5) the fifth group were encouraged to sit around winding each other up about how much revision they had/hadn’t done (You’re all going to fail).
The student welfare office made four predictions: (1) all interventions should be different from the control; (2) yoga, bribery and beta-blockers should lead to higher exam scores than panic; (3) yoga and bribery should have different effects than the beta-blocker drugs; and (4) yoga and bribery should also differ.
Which of the following planned contrasts (with the appropriate group codings) are correct to test these hypotheses?
ANSWER 1
ANSWER 2
ANSWER 3
ANSWER 4
ANSWER 1 - sum of all weights should be 0
Deciding what post hoc tests to run
What does this one-way ANOVA output show?
Research question: Is there a statistically significant difference in Frisbee throwing distance with respect to education status?
Variables:
IV - Education, which has three levels:
High School, Graduate and PostGrad;
DV - Frisbee Throwing Distance
There was homogeneity of variance as assessed by Levene’s Test for Equality of Variances (F (2,47) = 1.94, p = .155)
What does the results of one-way ANOVA show?
Research question: Is there a statistically significant difference in Frisbee throwing distance with respect to education status?
Variables:
IV - Education, which has three levels:
High School, Graduate and PostGrad;
DV - Frisbee Throwing Distance
There was a statistically significant difference between groups as demonstrated by one-way ANOVA (F(2, 47) = 3.50, p = .038).
What does the results of one-way ANOVA show? –> post hoc
Research question: Is there a statistically significant difference in Frisbee throwing distance with respect to education status?
Variables:
IV - Education, which has three levels:
High School, Graduate and PostGrad;
DV - Frisbee Throwing Distance
A Tukey post hoc test shows that the PostGrad group was able to throw the frisbee statistically significantly further than the High School group (p = .034). There was no statistically significant difference between the Graduate and High School groups (p = . 691) nor between the Graduate and PostGrad groups (p = .099).
Diagram of regression slopes satisfying homogenity of regression slopes in ANCOVA
- exhibits the same slopes for control and 15 minute group
Diagram of a regression slopes not satisfying homogenity of regression slopes in ANOCVA
- 30 minutes of therapy exhibts a different slope compared to others
What is design, variables and test would you use to test this researh scenario? - (5)
- ANCOVA
- Independent samples-design
- One IV , two conditions, interval regime and steady state
- One covariate (age in years)
- One DV (Race time)
What does this ANCOVA output show?
- IV = Regime –> steady or interval
- Covariate = Age
- DV = Racetime- (2)
- Age F(1,27) = 5.36, p = 0.028, partial eta-squared = 0.17 (large and sig main effect)
- Regime F(1,27) = 4.28, p = 0.048, partial eta-squared = 0.14 (large and sig main effect)
What DF do you report from this ANCOVA table for age for example…
DF for age and DF for error
What does this SPSS output for ANCOVA show? - (3)
- Interval has a marginal mean of race times of 56.57
- Steady state has a marginal mean of race times 62.97
- Estimated marginal means partialled out the effects of age and view mean scores of race times in interval and steady state if mean age scores (30.07) across two groups was held constant
What does this output show in terms of homogenity of regression slopes?
age is covariate and regime is IV and DV is race times - (2)
- Interaction effect of regime * age has a p-value of 0.980
- Since p-value is not significant the assumption of homogeneity of regression slopes has been met
What does this SPSS output show in terms of independence of covariate and exp effect (IV)?
age is covariate (treated as DV) , regime is IV - (2)
- P-value is not signifcant (p=0.528) so effect of variable age is not sig difference of age across training regime
- and so independent variable are assumed to be independent.
What does positive and negative b-value for covariates in ANCOVA parameter estimate box indicate? - (2)
f the b-value for the covariate is positive then it means that the covariate and the outcome variable have a positive relationship
If the b-value is negative it means the opposite: that the covariate and the outcome variable have a negative relationship
What does this table of parameter estimates show for ANCOVA where..
DV = PP’sLibido, IV = Dose of Viagara, Covariate is Partner’sLibido - (3)
- b for covariate is 0.416
- Besides other things being equal, if a a partner’s libido increases by one unit, then the person’s libido should
increase by just 0.416 units - Since b is positive then partner’s libido ahs pos relation with pps’s libido
How is DF calculated for these t-tests in ANCOVA table? - (2)
N - p -1
N is total sample size, p is number of predictors (2 dummy variables and covariate )
What does these planned contrast results show in ANCOVA?
DV = Pp’s Libido, IV = Dose of Viagara, Covariate is Partner’s Libido -
IV Dose: Level 3 = high dose, level 2 = low dose, level 1 = placebo
(3)
- Contrast 1 of comparing level 2 (low dose) against level 1 (placebo) is significant (p = 0.045)
- Contrast 2 of comparing level 3 (high dose) with level 1 (placebo) is significant (p - 0.010)
What does this Sidak correction post-hoc comparison in ANCOVA output show?
DV = Libido, IV = Dose of Viagara, Covariate is Libido -
IV Dose: Level 3 = high dose, level 2 = low dose, level 1 = placebo
- (3)
- The significant difference between the high-dose and placebo groups remains (p = .030)
- high-dose and low-dose groups do not significantly differ (p = .93)
- Low dose and placebo groups do not significantly differ (p value = 0.130)
What do these scatterplot of regression lines show in terms of homogenity of regression slopes?
DV = Libido, IV = Dose of Viagara, Covariate is Libido -
IV Dose: Level 3 = high dose, level 2 = low dose, level 1 = placebo
(3)
For placebo and low dose there appears to be a positive relationship between pp’s libido and that of their partner
However, in the high-dose condition there appears to be no relationship at all between participant’s libido and that of their partner - shows negative relationship
Doubts whether homogenity of regression slopes is satisfied as not all the slopes are the same (go same direction)
How is eta-squared calcuated?
Dividing the effect of interest SSM by total variance in the data SST
How is partial eta-squared calculated for ANCOVA??
SS Effect/ SS Effect + SS Residual
What test is used to investigate this question and how is it conducted? - (2)
We want to know whether or not studying technique (3 levels) has an impact on exam scores,but we want to account for the grade that the student already has in the class.
- ANCOVA
- ANCOVA is conducted to determine i f there is a statistically significant difference between different studying techniques (IV) on exam score (DV) after controlling for current grade (covariate)
In ANCOVA, we partion the total variance into
IV, DV and covariate
In ANCOVA, between subject effects we quote DF such as for dose as…
Quote df for the effect and error, e.g. 2,26
In ANCOVA, adjusted means table in SPSS shows.. - (2)
outcome/DV = happiness measure ranging from 0 to 10 (as happy as I can image) = continous = interval
The fixed factor (IV) which is dose of therapy which is people have 15 minutes of puppy therapy or 30 minutes
Covariate is control group is how much they love puppies = continous = interval
The group means can be recalculated once the effect of the covariate is ‘discounted’ = impact of covariate is taken into account and adjusted into each level of predictor variable in mean column
These values can differ markedly from the original group means and help with interpretation.
What ANCOVA was conducted?
We want to know whether or not studying technique has an impact on exam scores,but we want to account for the grade that the student already has in the class.
A three-way ANCOVA was conducted to determine a statistically significant difference between different study techniques on students exam scores after controlling for their current grades.
In one-way ANOVA we partition the total variance into
IV and DV
A psychologist was interested in the effects of different fear information on children’s beliefs about an animal. Three groups of children were shown a picture of an animal that they had never seen before (a quoll). Then one group was told a negative story (in which the quoll is described as a vicious, disease-ridden bundle of nastiness that eats children’s brains), one group a positive story (in which the quoll is described as a harmless, docile creature who likes nothing more than to be stroked), and a final group weren’t told a story at all. After the story children rated how scared they would be if they met a quoll, on a scale ranging from 1 (not at all scared) to 5 (very scared indeed). To account for the natural anxiousness of each child, a questionnaire measure of trait anxiety was given to the children and used in the analysis
what analysis has been used -
Independent analysis of variance
Repeated-measures analysis of variance
Mixed analysis of variance
Analysis of covariance
Analysis of covariance (ANCOVA)
A psychologist was interested in the effects of different fear information on children’s beliefs about an animal. Three groups of children were shown a picture of an animal that they had never seen before (a quoll). Then one group was told a negative story (in which the quoll is described as a vicious, disease-ridden bundle of nastiness that eats children’s brains), one group a positive story (in which the quoll is described as a harmless, docile creature who likes nothing more than to be stroked), and a final group weren’t told a story at all. After the story children rated how scared they would be if they met a quoll, on a scale ranging from 1 (not at all scared) to 5 (very scared indeed). To account for the natural anxiousness of each child, a questionnaire measure of trait anxiety was given to the children and used in the analysis
what is covariate?
Natural Fear Level
Example of two-way independent factorial ANOVA
The study tested the prediction that subjective perceptions of physical attractiveness become inaccurate after drinking alcohol which is IV, DVs
- What are the IVs, DVs- (3)
IV = Alcohol - 3 levels = Placebo, Low dose, High dose
Iv = face type 2 levels = unattractive, attractive
DV = Physical attractiveness score
The study tested the prediction that subjective perceptions of physical attractiveness become inaccurate after drinking alcohol
IV = Alcohol - 3 levels = Placebo, Low dose, High dose
Iv = face type 2 levels = unattractive, attractive
DV = Physical attractiveness score
Create a linear model for this two-way ANVOA scenario which adds interaction term and explain why is it important - (3)
- The first equation models the two predictors in a way that allows them to account for variance in the outcome separately, much like a multiple regression model
- The second equation adds a term that models how the two predictor variables interact with each other to account for variance in the outcome that neither predictor can account for alone.
- The interaction is important to us because it tests our hypothesis that alcohol will have a stronger effect on the ratings of unattractive than attractive faces
How is two-way INDEPENDENT ANOVA different to one-way INDEPENDENT ANOVA? - (3)
in two-way ANOVA, the variance explained by the experiment is made up of not one experimental manipulation but two.
Therefore, we break the model sum of squares down
into variance explained by the first independent variable (SSA), variance explained by the second independent variable (SSB) and variance explained by the interaction of these two
variables (SSA × B)
How to calculate total sum of squares SST in two-way independent ANOVA?
How to compute model sum of squares SSM in two-way independent ANOVA? - (2)
sum of all grps (pairing each level of IV with another)
n = number of scores in each grp which is multipled by the mean value of each group subtracted by grand mean of all pps regardless of grp squared
How is SSA (face type) computed in two-way independent ANOVA?
IV = Alcohol - 3 levels = Placebo, Low dose, High dose
Iv = face type 2 levels = unattractive, attractive
DV = Physical attractiveness score - (2)
considering only two groups at a time and add together - for first IV variable (SSA) (e.g., grps of pps rated attractive and grp of pps that rated unattractive)
number of pps in that grp multiplied by mean of grp subtracted by grand mean overall of all pps squared
How to compute SSB in two-way independent ANOVA for alcohol type
IV = Alcohol - 3 levels = Placebo, Low dose, High dose
Iv = face type 2 levels = unattractive, attractive
DV = Physical attractiveness score - (2) - (3)
same formula as SSA but for the second IV
added for all grps of pps in second IV
number of pps in one grp of secondIV(mean score of that grp subtract by grand mean of all pps regardless of grp) squared
How to calculate SSR in two-way independent ANOVA?
- use individual variances of each grp (e.g., attractiveness face type + placebo) and multiply by one less than number of people within the group (n - in this case 6) and do it for each group and add it together
How to calculate SSR in two-way independent ANOVA?
number of grps you have in study(number of scores you have per group minus 1)
Diagram of calculating mean sums of squares in two-way ANOVA independent
Example of a research scenario of two-way independent ANCOVA
Pick out IVs and DVs - (4)
- Independent samples design
- Two Ivs, both 2 conditions: drug type (A, B) and onset (early, late)
- One DV is cognitive performance
- Two way ANOVA
What does this two-way ANOVA independent design SPSS output show?
- The levene’s test is not significant so assume equal variances
What does this two-way independent ANOVA table show - (4)
- Drug : F(1,24) = 5.58, p = 0.027, partial eta-squared = 0.19 (large effect + sig effect)
- Onset: F(1,24) = 14.43, p = 0.001, partial eta-squared = 0.38 (large effect + sig effect)
- Interaction Drug * Onset: F(1,24) = 9.40, p = 0.005, partial eta-squared = 0.28 (large effect + sig effect)
- We got two sig main effects and sig interaction effect which are all quite large effect sizes
What does this SPSS output show for two-way independent ANOVA? - (3)
drug B has higher score on cognitive test than A and is sig main effect (CI does not contain 0 and also main effect analysis)
early onset scoring higher on average than late onset (CI does not contain 0 and also main effect analysis)
Important of these main effect as main effects ignoring the effec tof other IV so results for drug at top is regardless of whether late/onset for example , does not tell anything for interaction
What does this interaction plot show TWO WAY ANOVA? - (6)
- Blue line is early onset
- Green line is late onset
- For late onset, drug B lead to higher mean scores on test than drug A
- For early onset, drug A led to slightly higher mean scores than drug B
- Drug A more effective then drug b for early onset but different marginal
- Drug B was substantially more effective than Drug A for late
Non-parallel lines in interaction plot indicate an
sig interaction effect
Example of difference of one-way ANOVA vs two-way ANOVA (independent) - (2)
- One-way ANOVA have one IV categorical variable (level of educaiton - college degree, grad degree, high school)
- Two-way ANOVA , you have 2 categorical IV variables - level of education (college degree, grad degree, high school) and zodaic sign (libra, pisces)
In these outputs is there a effect of gender, education or interaction level TWO WAY ANOVA INDEPENDENT
- Is there an effect of gender overall?
No, F(1,54) = 1.63, p = .207
Is there an effect of education level?
Yes, F(2,54) = 147.52, p < .001
Is there an interaction effect?
Yes, F(2,54) = 4.64, p = .014
How to interpret these findings?
- Main effect of Aspirin: Aspirin reduces heart attackes compard to placebo (1)
- Main effect of carotene: Beta carotene reduces heart attack (2)
- Interaction effect: Yes, bigger effect when aspirin and beta carotene taken together (3) - also lines drawn more its an interaction
A music teacher had noticed that some students went to pieces during exams. He wanted to testwhether this performance anxiety was different for people playing different instruments. He tookgroups of guitarists, drummers and pianists (variable = ‘Instru’) and measured their anxiety(variable = ‘Anxiety’) during the exam. He also noted the type of exam they were performing (inthe UK, musical instrument exams are known as ‘grades’ and range from 1 to 8). He wanted tosee whether the type of instrument played affected performance anxiety when accounting for thegrade of the exam. Which of the following statements best reflects what the effect of ‘Instru’ in theoutput table below tells us?
(Hint: ANCOVA looks at the relationship between an independent and dependent variable, takinginto account the effect of a covariate.
A. The type of instrument played in the exam had asignificant effect on the level of anxietyexperienced, even after the effect of the grade ofthe exam had been accounted for
B. The type of instrument played in the exam had asignificant effect on the level of anxietyexperienced
C. The type of instrument played in the exam did nothave a significant effect on the level of anxietyexperienced
A
Question 5A psychologist was interested in the effects of different fear information on children’s beliefs aboutan animal. Three groups of children were shown a picture of an animal that they had never seenbefore (a quoll). Then one group was told a negative story (in which the quoll is described as avicious, disease-ridden bundle of nastiness that eats children’s brains), one group a positive story(in which the quoll is described as a harmless, docile creature who likes nothing more than to bestroked), and a final group weren’t told a story at all. After the story children rated how scared theywould be if they met a quoll, on a scale ranging from 1 (not at all scared) to 5 (very scaredindeed). To account for the natural anxiousness of each child, a questionnaire measure of traitanxiety was given to the children and used in the analysis. The SPSS output is below.
Whatanalysis has been used?
(Hint: The analysis is looking at the effects of fear information on children’s beliefs about an animal, taking into account children’s natural fear levels.)
A. ANCOVA
B. Independent analysis of variance
C. Repeated measures analysis of variance
A
A study was conducted to look at whether caffeine improves productivity at work in different conditions. There were two independent variables. The first independent variable was email, which had two levels: ‘email access’ and ‘no email access’. The second independent variable was caffeine, which also had two levels: ‘caffeinated drink’ and ‘decaffeinated drink’. Different participants took part in each condition. Productivity was recorded at the end of the day on a scale of 0 (I may as well have stayed in bed) to 20 (wow! I got enough work done today to last all year). Looking at the group means in the table below, which of the following statements best describes the data?
A. A significant interaction effect is likely to be present between caffeine consumption and email access.
B. There is likely to be a significant main effect of caffeine.
C. The effect of email is relatively unaffected by whether the drink was caffeinated.
D. The effect of caffeine is about the same regardless of whether the person had email access.
A = for decaffeinated drinks there is little difference between email and no email, but for caffeinated drinks there is
A psychologist was interested in the effects of different fear information on children’s beliefs about an animal. Three groups of children were shown a picture of an animal that they had never seen before (a quoll). Then one group was told a negative story (in which the quoll is described as a vicious, disease-ridden bundle of nastiness that eats children’s brains), one group a positive story (in which the quoll is described as a harmless, docile creature who likes nothing more than to be stroked), and a final group weren’t told a story at all. After the story children rated how scared they would be if they met a quoll, on a scale ranging from 1 (not at all scared) to 5 (very scared indeed). To account for the natural anxiousness of each child, a questionnaire measure of trait anxiety was given to the children and used in the analysis. Which of the following statements best reflects what the ‘pairwise comparisons’ tell us?
A. Fear beliefs were significantly higher after negative information compared to positive information and no information, and fear beliefs were not significantly different after positive information compared to no information.
B. Fear beliefs were significantly lower after positive information compared to negative information and no information; fear beliefs were not significantly different after negative information compared to no information.
C. Fear beliefs were significantly higher after negative information compared to positive information; fear beliefs were significantly lower after positive information compared to no information.
D. Fear beliefs were all about the same after different types of information.
A
a musical instrument and gender on the level of musical ability. A sample of 30 (15 men and 15 women) participants who had never learnt to play a musical instrument before were recruited. Participants were randomly allocated to one of three groups that varied in the number of hours they would spend practising every day for 1 year (0 hours, 1 hours, 2 hours). Men and women were divided equally across groups. All participants had a one-hour lesson each week over the course of the year, after which their level of musical skill was measured on a 10-point scale ranging from 0 (you can’t play for toffee) to 10 (‘Are you Mozart reincarnated?’). An ANOVA was conducted on the data from the experiment. Which of the following sentences best describes the pattern of results shown in the graph?
A. The graph shows that the relationship between musical skill and time spent practising was different for men and women.
B. The graph shows that the relationship between musical skill and time spent practising was the same for men and women.
C. The graph indicates that men and women were most musically skilled when they practised for 2 hours per day.
D. Women were more musically skilled than men.
A
How is sphereicty assumed in this datasetA (USED IN REPEATED ANOVA
How is spherecity calculated? - (2) REPEATED ANOVA
- Calculating differences between between pairs of scores for all treatment levels e.g., A-B, A-C , B-C
- Calculating variances of these differences e.g., variances of A-B, A-C, B-C
What does the data from table show in terms of assumption of spherecity (calculated by hand) REPEATED ANOVA? - (3)
there is some deviation from sphericity because the variance of the differences between conditions A and B (15.7) is greater than the variance of the differences
between A and C (10.3) and between B and C (10.3).
However, these data have local circularity (or local sphericity) because two of the variances of differences are identical.
The deviation from spherecity in the data does not seem too severe (all variances roughly equal) but here assess deviation is serve to warrant an action
In repeated-measures ANOVA, the within-participant variability is made up of
the effect of experimental manipulation SSM and individual differences in performance (random factors outside of our control) - this is error SSR
What is the differences between independent ANOVA and repeated-measures design ANOVA?
repeated-measures ANOVA the model and residual sums of squares are both part of the within-participant variance.
To compute F-ratios we first compute the sum of squares which is the following REPEATED ANOVA… - (5)
- SST
- SSB
- SSW
- SSM
- SSR
How is SST calculated in one-way repeated measures ANOVA? REPEATED ANOVA
SST = grand variance (N-1)
The SSW (within-participant) sum of squares is calculated in one-way repeated ANOVA by…
square of the standard deviation of each participant’s scores multiplied by the number of conditions minus 1, summed over all participants.
How is SSM calculated in one-way repeated ANOVA? - (2)
square of the differences between the mean of the participant scores for each condition and the grand mean multiplied by the number of participants tested, summed over all conditions.
do this for each condition grp
How is SSR calculated in one-way repeated ANOVA?
the difference between the within-participant sum of squares and the sum of squares for the model.
How do we calculated mean squares (MS) and mean residuals (MR) to acalculate F-ratio in one-way repeated ANOVA?
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
what does these SPSS outputs show? - (2)
- Left shows variables represent each level of IV which is animal
- Right shows descriptive statistics - higher mean time to retch when celebrity eating stick insect (8.12)
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - ONE WAY repeated anova
What does this Mauchly’s Test of Spherecity show? - (2)
- P-value is 0.047 which is less than 0.05
- Thus, reject the assumption of spherecity that variances of the differences between levels are equal
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub)
What to do if this Mauchly’s Test of Spherecity shows assumption of sphereicity is violated..? - (3)
one way repeated ANOVA
- Since there are 4 conditions, lower limit of ε^ is 1/(4-1) = 0.333 (lower-bound estimate in table)
- SPSS Output 13.2 shows that the calculated value of ε
^ is 0.533. - 0.533 is closer to the lower limit of 0.33 than it is to the upper limit of 1 and it therefore represents a substantial deviation from sphericity
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What does this main ANOVA table show in terms of spherecity assumed? - (2)
- The value of F = 3.97 which is compared against a critical value for 3 and 21 DF and p-value is 0.026
- conclude there is significant difference between 4 animals in their capacity to induce retching when eaten
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What has changed and kept the same in the table? - (2)
- The F-ratios are the same across the rows
- the D.F is changed as well as critical value the F-statistic is compared with
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
How is adjustments made to DF?
- Adjustment made by multiplying the DF by the estimate of spherecity.
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What does the results show in terms of Greenhouse-Geisser correction and Huynh-Fedt..? - (3)
- Observed F statistic not significant using Greenhouse-Geisser ( p> 0.05)
- Greenhouse-Geisser is quite conservative and miss true effects that exist
- Thus, Huynh-Feldt showend F-statistic is still significant as p-value of 0.048
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What happens if Greenhouse Geisser is not-significant (p>0.05) and Huynh-Feldt is significant in this example? - (2)
- Taking average of two significant values e.g., 0.063+ 0.048/2 = 0.056
- Thus, go with Greenhouse-Geisser correction and conclude F ratio is non-significant
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What happens if two corrections - Greenhouse and Felt give same conclusion then you can choose which one to
report
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
Important to use valid critical value of F - choosing which p-value to report as it potentially makes a difference between making a
Type 1 error (False positive) or not
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
what does this summary table of repeated contrasts show? - (3)
Level 1 vs 2 is stick insect vs kangaroo testicle
Level 2 vs 3 is kangaroo testicle vs fish eyeball
Level 3 vs 4 is fish eyeball vs witchetty grub
- celebrities took significantly longer to retch after
eating the stick insect compared to the kangaroo testicle (Level 1 vs. Level 2) - p-value of 0.002 - Time taken to retch was not significantly different in Level 2 vs 3 and Level 3 vs 4
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
If main effect is not significant in main ANOVA table for this data then significant contrasts in table below should be … but if MANOVA was significant then… - (2)
ignored
inclined to conclude main effects of animal was significant and proceed with further tetss like contrasts
What IV, DV , design and test to use for this research scenario? - (4)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
What does output show? - (3)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
- sales are increasing across the weeks
- Week 1 start at 427.93 and gradually rise by week 4 to 642,28 pounds
- looks like incentives are having an effect and seem to generate higher sales
What does this output show in terms of Maulchys Test of Spherecity? - (2)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
- P-value is not significant ( p = 0.080)
- Assumption of spherecity is satisfied so we got equal variances between differences across conditions
If Maulchy’s test of spherecity is not significant in one-way repeated ANOVA, then which line do we use in main ANOVA table?
If Maulchy’s test of spherecity is significant in one-way repeated ANOVA, then which line do we use in main ANOVA table?
What does this main ANOVA table show? - (3)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
- DF for week is 3 and 57 (spherecity assumed from week and error)
- Week: F(3,57) = 26.30, p < 0.001 (p = 0.000), eta-squared is 0.58 - large effect
- There is an overall effect going on and change across weeks
What does this Sidmak correction table and table of means show you in this output? - (6)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
- No sig difference betwen W1 and W2
- Sig difference between W1 and W3 = ihigher sales in W3 (538.570) compared to W1 (427.933)
- Sig difference between W1 and W4 = ihigher sales in W3 (642.284) compared to W1 (427.933)
*Not sig diff with W2 and W3 - Sig difference between W2 and W4 , higher sales in W4 (642.284) than W2 (481.388)
- Sig difference between W3 and W4 , higher sales in W4 (642.284) than W3 (538.570)
What does this output show in terms of repeated contrasts? - (3)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
- Did sales increase from W1 to W2? = p = 0.010 significant
- Did sales increase from W2 to W3? = p = 0.030
- Did sales increase from W3 to W4? = p = 0.008
What happens if post hoc and contrasts are telling a different story? - contrasts says weekly increase e.g. W1 to W2 increase, W2 to W3 increase , W3 to W4 increase but post-hoc W1 to W3 was increased sig, W1 to W4 was sig increase but W2 to W3 was not - (2)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
- Post hoc has lack of power due to many multiple comparisons
- By limiting comparisons in contradt we get around problem
Diagram of writing up one-way repeated ANOVA
What design, IV, DV and test would you to to investigate the follow scenario? - (4)
- Repeated measures design
- Two IVs: alcohol (3 conditions) and sleep (2 conditions)
- DV: Reaction Times
- Two-way repeated measures ANOVA
What does this two-way repeated ANOVA SPSS output show? - (2)
- Repeated measures design
- Two IVs: alcohol (3 conditions) and sleep (2 conditions)
- DV: Reaction Times
- Two-way repeated measures ANOVA
- large number for RT means slower RT
- Alcohol seem to have an effect on RT but particularly for 2 pints + no sleep
What does this two-way repeated ANOVA SPSS output show for Mauchly’s Test of Sphericity? - (2)
- Repeated measures design
- Two IVs: alcohol (3 conditions) and sleep (2 conditions)
- DV: Reaction Times
- Two-way repeated measures ANOVA
- Two p-values: alcohol ( p = 0.00) and alcohol * sleep [ interaction effect] (p = 0.00) – > sig so assumption of spherecity is violated so report Grenhouse-Geisser values from main ANOVA table
- No p-value for sleep as only 2 conditions and test of sphericity need more than 2
What does this two-way repeated ANOVA main table show? - (3)
- Repeated measures design
- Two IVs: alcohol (3 conditions) and sleep (2 conditions)
- DV: Reaction Times
- Two-way repeated measures ANOVA
- Error DF was 38.
- Test of Spherecity was sig –> assumption violated
- Main sig effect of alcohol: F(1.16,22.06) = 51.38, p < 0.001, partial eta-squared = 0.73
- Main sig effect of sleep: F(1,19) = 88.61, p < 0.001, partial-eta-squared = 0.82
- Interaction effect: F(1.15,21.91) = 23.36, p < 0.001, partial-eta squared = 0.55
What does this two-way repeated ANOVA output show in post hocs? - Sidmak correction - (4)
- Repeated measures design
- Two IVs: alcohol (3 conditions) and sleep (2 conditions)
- DV: Reaction Times
- Two-way repeated measures ANOVA
- condition 1 and condition 2 which was significant
- Condition 1 vs 3 which was significant
- Condition 2 with Condition 3 was significant
- So all groups differing significantly from each other so interpret from that higher does of alcohol has more impact on RT
What does this two-way repeated ANOVA interaction plot show? - (3)
- Repeated measures design
- Two IVs: alcohol (3 conditions) and sleep (2 conditions)
- DV: Reaction Times
- Two-way repeated measures ANOVA
- Interaction effect is there = as line continue they cross
- Most pronouned effect was in alcohol grp 3 (2 pints)
- When alcohol grp 3 had full nights sleep (2), impairs their RT very slightly
- When alcohol grp 3 had sleep deprivation (1) in combination with 2 pints, it impairs RT by a lot –> use simple effect analysis as well as two-way independent ANVOA to see if difference in grp 3 of blue and green line is sig
What does significant Mauchly’s test signify in repeated measures? - (2)
A significant effect means that corrections need to be made later on
Those corrections are listed in the main ANOVA output table
Equation of variance
Leven’s test tests if the variances in independent groups are similar, would levene’s test be significant in this case?
Levene’s test would likely be significant as the variance between the two groups are quite different.
If p-value significant for checking for spherecity ten - (3)
If GG < 0.75 THEN USE GG
IF GG > 0.75 THEN USE HF
Since GG less than 0.75 report adjusted F, DF and sig which is F(1.24, 21.00) = 212.32 , p < 0.001
Homogenity of variance is
distribution of groups are similar?
Spherecity is asking are the disttibution of differences between groups are
similar
What does this 2-way mixed ANOVA show? - (3)
- Independent variable(s)
- Is there more than 1 IV?
- The levels the independent variable(s)
- Dependent variable
- Between(BS) or within-subjects (WS)?
- What type of design is being used?
1) is there an effect overall = Yes (green)
2) Is the effect bread = Yes (red)
3) Is there an interaction = Yes (blue)
Partioning of variance of one way vs two way independent
Contrast coding example SPSS how to read
An experiment was done to compare the effect of having a conversation via a hands-free mobile phone, having a conversation with an in-car passenger, and no distraction (baseline) on driving accuracy. Twenty participants from two different age groups (18–25 years and 26–40 years) took part. All participants in both age groups took part in all three conditions of the experiment (in counterbalanced order), and their driving accuracy was measured by a layperson who remained unaware of the experimental hypothesis.
How do we interpret the main effect of distraction from the SPSS table (next slide)? - (2)
The assumption of sphericity has been met, indicated by Mauchly’s test (p > .05).
There was a significant main effect of distraction (F(2, 36) = 45.95, p < .001). This effect tells us that if we ignore the effect of age, driving accuracy was significantly different in at least two of the distraction groups.
The table shows hypothetical data from 3 conditions
For these data, spherecity will hold when
(Hint: Sphericity refers to the equality of variances of the differences between treatment levels.)
A.The variances of the differences between treatment levels are roughly equal
B. The variance of each condition is roughly equal
C. The variance of each condition is not equal
D. The variances of the differences between treatment levels are not equal
A
Imagine we were interested in the effect of supporters singing on the number of goals scored by soccer teams. We took 10 groups of supporters of 10 different soccer teams and asked them to attend three home games, one at which they were instructed to sing in support of their team (e.g., ‘Come on, you Reds!’), one at which they were instructed to sing negative songs towards the opposition (e.g., ‘You’re getting sacked in the morning!’) and one at which they were instructed to sit quietly. The order of chanting was counterbalanced across groups. Looking at the output below, which of the following sentences is correct?#
A.The results showed that the number of goals scored was significantly affected by the type of singing from the supporters, F(2, 18) = 11.24, p = .001.
B. The results showed that the number of goals scored was significantly affected by the type of singing from the supporters, F(1.58, 14.19) = 11.24, p = .002.
C. The results showed that the number of goals scored was significantly affected by the type of singing from the supporters, F(2, 12.4) = 11.24, p = .001.
D. The results showed that the number of goals scored was significantly higher when supporters sang positive songs towards their team than when they sat quietly, F(2, 18) = 11.24, p = .001.
A = Mauchly’s test was non-significant, so we can report the result in the row labelled ‘sphericity assumed’
Imagine we were interested in the effect of supporters singing on the number of goals scored by soccer teams. We took 10 groups of supporters of 10 different soccer teams and asked them to attend three home games, one at which they were instructed to sing in support of their team (e.g., ‘Come on, you Reds!’), one at which they were instructed to sing negative songs towards the opposition (e.g., ‘You’re getting sacked in the morning!’) and one at which they were instructed to sit quietly. The order of chanting was counterbalanced across groups. An ANOVA with a simple contrasts using the last category as a reference was conducted. Looking at the output tables below, which of the following sentences regarding the contrasts is correct?
a.The first contrast revealed that soccer teams scored significantly more goals when their supporters sang positive songs compared to when they did not sing. The second contrast revealed that soccer teams scored significantly fewer goals when their supporters sang negative songs compared to when they did not sing.
b. The first contrast revealed that soccer teams scored significantly fewer goals when their supporters did not sing compared to when they sang negative songs. The second contrast revealed that soccer teams scored a similar amount of goals when their supporters sang positive songs compared to when they did not sing.
c. The first contrast revealed that soccer teams scored significantly more goals when their supporters sang positive songs compared to when they did not sing. The second contrast revealed that soccer teams scored significantly fewer goals when their supporters sang negative songs compared to when they sang positive songs.
d. The first contrast revealed that soccer teams scored significantly more goals when their supporters sang positive songs compared to when they did not sing. The second contrast revealed that soccer teams did not significantly differ in the number of goals scored when their supporters sang negative songs compared to when they did not sing.
a = see from the means in the Descriptive Statistics table that positive singing resulted in the highest number of goals scored and negative singing resulted in the least number of goals score
An experiment was done to compare the effect of having a conversation via a hands-free mobile phone, having a conversation with an in-car passenger, and no distraction (baseline) on driving accuracy. Twenty participants from two different age groups (18–25 years and 26–40 years) took part. All participants in both age groups took part in all three conditions of the experiment (in counterbalanced order), and their driving accuracy was measured by a layperson who remained unaware of the experimental hypothesis.
Which of the following sentences is the correct interpretation of the main effect of distraction?
AThere was a significant main effect of distraction, F(2, 36) = 45.95, p < .001. This effect tells us that if we ignore the effect of age, driving accuracy was significantly different in at least two of the distraction groups.
B. There was no significant main effect of distraction, F(2, 36) = 45.95, p = .719. This effect tells us that if we ignore the effect of age, driving accuracy was the same for no distraction, hands-free conversation and in-car passenger conversation.
C. There was a significant main effect of distraction, F(2, 36) = 45.95, p < .001. This effect tells us that driving accuracy was different for no distraction, hands-free conversation and in-car passenger conversation in the two age groups.
D. There was no significant main effect of distraction, F(2, 36) = 45.95, p > .05. This effect tells us that none of the distraction groups significantly distracted participants across both age groups.
A = We can read the results in the row labelled ‘sphericity assumed’, as we can see from the output of Mauchly’s test that the assumption of sphericity has been met, p > .05. However, we would need to do some follow-up tests to investigate exactly where the differences between groups lie
Field and Lawson (2003) reported the effects of giving children aged 7–9 years positive, negative or no information about novel animals (Australian marsupials). This variable was called ‘Infotype’. The gender of the child was also examined. The outcome was the time taken for the children to put their hand in a box in which they believed either the positive, negative, or no information animal was housed (positive values = longer than average approach times, negative values = shorter than average approach times). Based on the output below, what could you conclude?
A. Approach times were significantly different for the boxes containing the different animals, but the pattern of results was unaffected by gender.
B. Approach times were significantly different for the boxes containing the different animals, and the pattern of results was affected by gender.
C. Approach times were not significantly different for the boxes containing the different animals, but the pattern of results was affected by gender.
D.Approach times were not significantly different for the boxes containing the different animals, but the pattern of results was unaffected by gender.
A
Table scenario in which cats can be trained to dance more effectively with food or affection at reward - chi-squared test
Table scenario in which cats can be trained to dance more effectively with food or affection at reward - chi-squared test
what are the four categories? - (4)
- could they dance - yes
- could they dance - no
- food as reward
- affection as reward
Table scenario in which cats can be trained to dance more effectively with food or affection at reward - chi-squared test
highlight the frequencies for four categories
Table scenario in which cats can be trained to dance more effectively with food or affection at reward - chi-squared test
what do the rows give?
Row totals give frequencies of dancing and non-dancing cats
Table scenario in which cats can be trained to dance more effectively with food or affection at reward - chi-squared test
what do the columns give? - (2)
The column totals give frequencies of food and affection as reward
These are the numbers in each group
We have a list of movie genres; this is our first variable. Our second variable is whether or not the patrons of those genres bought snacks at the theatre. Our idea (or, in statistical terms, our null hypothesis) is that the type of movie and whether or not people bought snacks are unrelated. The owner of the movie theatre wants to estimate how many snacks to buy. If movie type and snack purchases are unrelated, estimating will be simpler than if the movie types impact snack sales.
Diagram of contigency table in Chi-square and calculating row totals and colum and grand total - (7)
50 + 125 + 90 +45 = 310
75 + 175 + 30 + 10 = 290
50 + 75 = 125
125 + 175 = 300
90 + 30 = 120
45 + 10 = 55
310 + 290 = 600
We have a list of movie genres; this is our first variable. Our second variable is whether or not the patrons of those genres bought snacks at the theatre. Our idea (or, in statistical terms, our null hypothesis) is that the type of movie and whether or not people bought snacks are unrelated. The owner of the movie theatre wants to estimate how many snacks to buy. If movie type and snack purchases are unrelated, estimating will be simpler than if the movie types impact snack sales.
Diagram of contigency table in Chi-square of calculating eexpected counts
e.g., for action and snacks it would be column total (310) * row total (125) divided by grand total of 600 = 65
Example of calculating chi-square from table
For this it would be 65.03
How to understand your test statistic from chi-squared? - (5) if you have test statistic of 65.03
- Set your significance level = .05
- Calculate the test statistic -> 65.03
- Find your critical value from chi-squared distribution table based on df & significance level
- Degrees of freedom: df (r – 1) x (C-1)
For the movie example this is; Df = (4-1) x (2-1) = 3 -> 7.815 - compare test statistic with critical level
65.03 > 7.82 so reject the idea that movie type and snack purchases are independent
Does the area of psychology that a person prefers depend on whether they would select a cat or a dog as a pet? - chi-square test of independence
Chi-square example we need to check the assumptions of The expected frequencies should be greater than 5.
What does it show? - (4)
Here we see that all the expected counts in the cat group and one expected count in the dog group are below 5.
We also have one in the cat group that is below 1.
So, SPSS has flagged that we have 60% of the expected counts falling below 5.
So assmption of expected frequencies greater than 5 is not assumed
Does the area of psychology that a person prefers depend on whether they would select a cat or a dog as a pet? - - chi-square test of independence
If assumptions were met (expected frequencies greater than 5) then.. report - (2)
A chi-square independence test was performed to examine whether there was a relationship between their area of studies in psychology and their preference for cats or dogs.
The relationship between these variables was not significant, χ²(4, N = 46) = 1.46, p = .834, so we fail to reject H0.
Example situations you can do chi-square directional and not possible - (5)
If we are just comparing pet preferences between males and females, we can make a directional hypothesis (2 x 2 – male/female, cats/dogs).
Males prefer cats or females prefer dogs.
However, when we start adding variables to the design it gets complicated.
If we wanted to compare drink preferences at different times of the day for students/lecturers, we couldn’t form a directional hypothesis.
This is because we have 3 main effects and several interactions to consider. We need to use loglinear analyses to do this.
Research question: Is the new treatment associated with improvements in health in cats and dogs?
Checking assumption of expected counts - (3):
Here we have 3 things we are comparing: animal (cat/dog), treatment (yes/no) and improvement (yes/no) all of which are categorical.
We look and see that all of the expected counts are above 5.
So met assumption of independence and expected counts
Research question: Is the new treatment associated with improvements in health in cats and dogs?
Here we have 3 things we are comparing: animal (cat/dog), treatment (yes/no) and improvement (yes/no) all of which are categorical.
In loglinear model selection after including all main effects and interactions then it - (4)
Remove a term and compares the new model with the one in which the term was present.
Starts with the highest-order interaction (including max number of variables/categories)
Uses the likelihood ratio to ‘compare’ models below:
If the new model is no worse than the old, then the term is removed and the next highest-order interactions are examined, and so on.
Model selection of loglinear - what does it show?
Research question: Is the new treatment associated with improvements in health in cats and dogs?
Here we have 3 things we are comparing: animal (cat/dog), treatment (yes/no) and improvement (yes/no) all of which are categorical. - (3)
We can see that the model selection worked in a way that it first tried to remove the 3-way interaction.
However, we can see here that it * affected the fit of the model, so it was left in.
Since removing the highest-order interaction made a * difference to the fit of the model, we get a final model that is the saturated model (it contains all main effects and interactions).
Loglinear SPSS K way and Higher order effects what does it show?
Research question: Is the new treatment associated with improvements in health in cats and dogs?
Here we have 3 things we are comparing: animal (cat/dog), treatment (yes/no) and improvement (yes/no) all of which are categorical. - (2)
we are using the likelihood ratio here because that’s how we compare the models to find the best fit
. We see that all main effects and interactions are significantly contributing to explaining the variance in the data
loglinear what does K represent and what does K = 1,2 and 3 represent? - (4)
K represents the level of the terms.
For example, K=1 would be the main effects,
K=2 would be our 2-way interactions and
K=3 is our 3-way interaction.
Loglinear SPSS - what does parameter estimates show?
Research question: Is the new treatment associated with improvements in health in cats and dogs?
Here we have 3 things we are comparing: animal (cat/dog), treatment (yes/no) and improvement (yes/no) all of which are categorical. - (3)
There is a significant three-way interaction between animal, treatment and improvement, as well as two significant two-way interactions between animal and improvement and treatment and improvement (p < .001)
a * 3-way interaction between animal, treatment and improvement as well as two * 2-way interaction between animal/improvement and treatment/improvement.
Like our post-hoc tests, this is telling us where the * differences are.
Loglinear after seeing statistical tests we go to raw data showing that…
Based on the raw data, there seems to be indication that the cats responded better to treatment than dogs, this should be followed up by chi-square tests separately for cats and dogs to determine whether the association between treatment and improvement is present in both cats and dogs
A recent story in the media has claimed that women who eat breakfast every day are more likely to have boy babies than girl babies. Imagine you conducted a study to investigate this in women from two different age groups (18–30 and 31–43 years).
Looking at the output tables below, which of the following sentences best describes the results? = chi-square
A. Women who ate breakfast were significantly more likely to give birth to baby boys than girls.
B. There was a significant two-way interaction between eating breakfast and age group of the mother.
C. Whether or not a woman eats breakfast significantly affects the gender of her baby at any age.
D. The model is a poor fit of the data.
C
Logic behind Wilcoxon’s rank sum test, what does SPSS do? - (3)
Step 1: Get some not normally distributed data
Step 2: Rank it (regardless of group)
Step 3: Significance testing
Does one of the groups have more of the higher ranking scores than the other?
For Mann-Whitney, the statistic U use the
sum of ranks for group 1, R1, as follows
Example of table where comparing 2 independent conditions of Wilcoxon rank sum or Mann Whitney U test
Here we have data for two groups; one taking alcohol, the other ecstasy. The scores for a measure of depression. Scores were obtained on two days; Sunday and Wednesday. The drugs were administered on Saturday.
Example of table where comparing 2 independent conditions of Wilcoxon rank sum or Mann Whitney U test
Here we have data for two groups; one taking alcohol, the other ecstasy. The scores for a measure of depression. Scores were obtained on two days; Sunday and Wednesday. The drugs were administered on Saturday.
Two steps for both statistics:
Rank all the data on the the basis of the scores irrespective of the group
compute the sum of ranks of each group - (5)
The graphic here shows how we can list the scores in order and as a result assign each score a rank.
When scores tie, we give them the average of the ranks.
If we ensure we keep track of the group the scores came from we can relatively easily add the ranks up for each group.
Note, that if there was little difference between the groups the sums of their ranks would be similar, as they are for the data shown her for Sunday.
However, the sum of ranks differ considerably for the data obtained on Wednesday.
For Wilcoxon sum of ranks = comparing 2 independent groups the W s statistic,
the group sizes of n1 and n2 the mean of W2 is given:
For Wilcoxon sum of ranks = comparing 2 independent groups the W s statistic,
the standard error of Ws is given
For Wilcoxon sum of ranks = comparing 2 independent groups the z score of Ws can be calculated
For Mann Whitney , the statistic U use the the sum of ranks for group 1, R1, as follows
For Mann Whitney , the statistic U use the the sum of ranks for group 1, R1, as follows
Specificy equation - (2)
The first terms involving n1 and n2 actually compute the maximum possible sum of ranks for group 1.
U is zero when all those in group one have scores that exceed the scores of those in group 2.
In Mann Whitney U there is a standardised test statistic which is z score that can allow you to compute
effect size so r = z / square root of N (number of pps
Example of Wilcoxon signed rank test carrying out steps - (9)
The table shown here has the Depression Scores taken on Sunday and Wednesday for those taking ecstasy on Saturday.
Data for Sunday are in the first column and Wednesday in the second column.
The third column shows the difference between scores obtained on Sunday and Wednesday.
NOte some could be negative, some positive. In this example however the difference is always positive apart from two values when the difference is zero.
The fourth column notes the sign of the difference or notes it is going to be excluded because the difference was zero.
The fifth column ranks the differences in terms of their size, but not sign.
The sixth and seventh column list the ranks that were for positive and negative differences, respectively.
It is these two columns that are summed to get the relevant statistics, called T+ and T-.
Because T+ and T- are not independent, we take only the T+ value.
For Wilcoxon signed rank test the group size n the mean T is given
For Wilcoxon signed rank test thestandard error fo T is given:
For Wilcoxon signed rank test compute z score of T by
For Kruskal-Wallis, the statistic H is as follows
For Friedman, the statistic F is as follows
K = conditions
N = number of pps
Example when using chi-square test - (3)
- In this example, they wanted to look at whether attendance at lectures had an impact on their exam performance on whether they passed or failed
- Attendence was coded as 1 if participants generally attended lectures , barirng illness, and 2 if they did not attend
- Exam was scored as 1 = Pass and 2 = Fail
- In this example, they wanted to look at whether attendance at lectures had an impact on their exam performance on whether they passed or failed - chi square
What does it show? - (4)
- Attendence, Attended Lectures , Count = this is people who attended lecture and number of people who passed was 84 and people who failed was 29
- % Within attendence give same info so 74.3% passed and 25.7% failed when attended lectures
-Going to didn’t attend lectures, 22 people passed and 35 failed and below is in percentages: - Easier using percentages writing up
- In this example, they wanted to look at whether attendance at lectures had an impact on their exam performance on whether they passed or failed - chi square
What does it show? - (2)
- At top row, pearson chi-squared which chi-square statistic which was 20.617, DF which is 1 and p-value was 0.000
- 0 cells have a count less than 5 met assumption of chi-square test that expected counts greater than 5
- In this example, they wanted to look at whether attendance at lectures had an impact on their exam performance on whether they passed or failed - chi square
What does it this effect size show? - (2)
- x^2 (1) = 20.62, p < 0.001
- Cramer’s V = 0.35 , indicating a medium effect size
Example of calculating odds ratio for chi-square - (3)
odds of passing/failing for students who attended lecture = no. of students who attended and passed (84) / no. of students who attedned and failed (29) = 2.897
odds of passing/failing for students who did not attend = no. of students who did not attend lectures and passed (22) / no of students who did not attended and fail (35)
Odds ratio = odds of P/F of attended/ odds of P/F of not attended = 2.897/ 0.629 = 4.606 saying for an individual who attended lectures lead them to be more likely to pass exam
Example research scenario of Mann Whitney - (4)
- Independent sample design
- One IV, two conditions = existing vs new medication
- One DV (symptoms) but this time on ordinal (scale from 1 to 5) and got combination of non-normally distributed data and small sample size (very problematic for t-tests)
- Mann Whitney U Test
Example of using Mann Whitney U = skew
What does this Mann Whitney U show?
- Independent sample design
- One IV, two conditions = existing vs new medication
- One DV (symptoms) but this time on ordinal (scale from 1 to 5) and got combination of non-normally distributed data and small sample size (very problematic for t-tests)
- Mann Whitney U Test
- This box summarises the p-value ( p = 0.026) and tells you whether to accept or reject the null hypothesis.
What does this output show of Mann Whitney U? - (3)
- Mann Whitney U test statistic is 166.000 to report and also people report standardised test statistic 2.292 which is z score so handy to report as know if its above +/-1.96 then p-value we get out of test is significant
- P-value of exact significant is p = 0.026
- This is significant difference between the 2 groups
- Next we would want to look at the median scores to see which group is scoring highest and lowest after sig Mann Whitney U test
What does this output show? - (3)
- Independent sample design
- One IV, two conditions = existing vs new medication
- One DV (symptoms) but this time on ordinal (scale from 1 to 5) and got combination of non-normally distributed data and small sample size (very problematic for t-tests)
- Mann Whitney U Test
- For existing treatment, median score was 3.
And new treatment the median score was 4.
It suggests new treatment was more effective in reducing symptons than the existing treatment
Example research scenario of Friedman ANOVA - (5)
- Again we got ordinal data for DV not sure distances between levels is going to be the same
- Related design
- One IV, 3 conditions
- One DV (level reached to video game)
- Friedman’s ANOVA = more than 2 groups in related design
What does this Friedman ANOVA output show?
- We got total sample size which is 30 and test statistic which is 21.788, DF = 2 and p value is 0.000 so significant difference between the 3 groups
A psychologist was interested in whether there was a gender difference in the use of email. She hypothesized that because women are generally better communicators than men, they would spend longer using email than their male counterparts. To test this hypothesis, the researcher sat by the computers in her research methods laboratory and when someone started using email, she noted whether they were male or female and then timed how long they spent using email (in minutes). How should she analyse the differences in males and females (use the output below to help you decide)?
A. Mann–Whitney test
B. Paired t-test
C.Wilcoxon signed-rank test
D. Independent t-test
Distribution of z
