STAT Notes Flashcards
Define descriptive statistics
Methods used to summarize or describe our observations
Describe inferential statistics
Using observations as a basis for making estimates or predictions
What two methods can be used to ensure random sampling is truly random?
Mechanical
Blind
Define mechanical sampling
Assigning every individual in the population a number and randomly generating numbers
Define stratified random sampling
Selects characteristics of the sample based on proportion of said characteristics in the population
Define dispersion of data
How far it lies from a given average
How is sample variance calculated?
Σ(difference between each value (xi) and the mean (x̄))^2 ÷ 1(n-1) where n is the number of observations
How is standard deviation calculated?
√var
How is standard error calculated?
sx ÷ √n where n is the number of observations and sx is standard deviation of a sample
Define confidence interval
Specific certainty of a predicted population mean with normal distribution
What proportion of the population stands within one standard error?
68%
What proportion of the population stands within two standard errors?
95%
What proportion of the population stands within three standard errors?
99.7%
What function shows perfect normal distribution?
Gaussian function
Define nominal data
Classifies by names
Define ordinal data
Classified in an order (by categories)
What are the two types of variables?
Categorical
Numeric
What are the two sub categories of categorical data?
Ordinal
Nominal
How is categorical data referred to in R?
factor()
What are the two sub categories of numeric data?
Discrete
Continuous
How is discrete data referred to in R?
integer
How is continuous data referred to in R?
numeric
Define skewed distribution
A measure of asymmetry
Define bimodal distribution
There are two modes (can be symmetrical or asymmetrical)
Define a bin
An area in which data is collected
Define central tendency
Central values
Define probability
Proportion of times a particular outcome will occur from a large sample of trials or the likelihood of a particular outcome of an event
What does a P=0 (probability=0) suggest?
Impossible
What does a P=1 (probability=1) suggest?
Certainty
What does it mean if trials are independent?
The actions of one have no impact on the results of the next trial
What is the probability of a OR b where they are both mutually exclusive?
P(a)+P(b)
What is the sum of the probabilities of mutually exclusive outcomes?
1
When we use OR when describing mutually exclusive probabilities how do we combine these values?
Add
What is a probability distribution?
Graphical distribution of theoretical relative probabilities
y=probability, x=potential outcomes
What is true about the area of any sections of a probability distribution graph?
Equivalent to the relative probability
How can we draw probabilities with multiple trials but limited outcomes?
Table
Probability tree
How do we combine mutually exclusive events using AND?
Multiply probabilities together
Define probability distribution
Theoretical probability of each outcome
Define frequency distribution
Observed frequency of each outcome
After more trials what becomes true about the frequency distribution and probability distribution?
Frequency distribution approaches probability distribution
When can we use binomial statistics
Can be used when there are two groups (such as A and B or pass and fail)
NOTE: we can create these groups if we define some outcomes as “success” and the others as “failure” and classify other outcomes beneath these banners
Give examples for which type of questions binomial distribution may be used for
Predict the probability of success in a single trial
Predict the proportion of successes in n trials
What are requirements for binomial statistics?
- 2 outcomes (P(success)=p and P(failure)=q) and p+q=1
- Each trial is independent with equal p
- Fixed no. trials
As number of trials increases what becomes true of discrete data?
Begins to resemble continuous data
How can we approximate binomial distribution?
Probability distribution
How can we find probability up to any point (normal distribution)
Area under the graph up until that point
Rules for hypothesis testing
- Understand the certainty of a hypothesis test
- Don’t base scientific decisions on hypothesis tests alone
- Consider the wider picture and plausibility of results
Which letter denotes significance level?
Alpha
Which two hypothesis are needed for a hypothesis test?
H0: null hypothesis (no change)
HA: alternative hypotheses (covers all other probability)
These hypotheses must be mutually exclusive
What do we assume about H0 in a hypothesis test?
H0=true
What is referred to as the critical region?
Areas above the critical value (above the alpha)
When is the null hypothesis rejected in hypothesis testing?
p<alpha
What is a tail?
Area at the end of the distribution
How do we test both tails?
Two-tailed test
How many critical regions are present in a two tailed test?
2
If alpha=0.05 and a two-tailed test is performed, what % of values lie outside the critical region?
95%
What is a p value?
The p value assumes the null hypothesis is true and gives the probability of getting a result that extreme or more assuming this
What is a contingency table?
One that shows all possible HA and H0 outcomes
If H0 is true and we reject it, what is true?
False Positive
Type I error
We do not know what is true
If H0 is true and we fail to reject H0 what is true?
There is a true negative
H0 is true
If HA is true and we fail to reject H0 what is true?
False negative
Type II error
HA was true
If HA is true and we reject H0 what is true?
True positive
H0 is untrue, this does not confirm HA
If H0 is true, what are the possible outcomes/errors?
True negative (H0 is true and we fail to reject H0)
Type I error (H0 is true and we reject H0)
At an alpha value of 0.05, how often would we expect a Type I error, if H0 is true?
5% Type I error
(95% true negative)
What does statistical power tell us about a test?
How powerful a test is at detecting true positives when there really is a difference to detect
When the HA is true, when do we fail to reject the null hypothesis? What type of error is this?
When we are outside the critical value (in the direction of the H0)
This is type II error and is shown where the HA graphs overlaps with H0
How is beta defined graphically?
The area of overlap between the H0 and HA graphs (where HA is true)
How is power of a test calculated?
Power=1-beta
If the power of a test is 0.979, how often do you get type II error?
2.1% of the time
If HA is closer to H0, the power is greater or smaller?
Smaller
It is more difficult to identify a true error
If there is a high power, what is true about the error likely?
There will be a lower rate of false negatives (type II error)
How can the power of a test be increased graphically?
Increase effect size:
Separate the curves to be skinnier
Increase distance between peaks
What is true of power if effect size is increased?
Power increases (less type II error)
How is effect size increased?
Increased trials
(decreases curve dispersion)
What must be present for hypothesis testing?
There must be two hypotheses:
H0 - null hypothesis (no change/ effect)
HA - alternative hypothesis (mutually exclusive and covers all other options (different for one and two-tailed tests))
Why is the p value not the probability of a false positive?
It is only the probability of a false positive if the alternative hypothesis is true, we can not know if the alternative hypothesis is true we can only speculate based on evidence
Define power
Proportion of true positives for a particular HA
Define Multiple Testing
Comparing and testing several conditions or treatments
When is a two-sample t test performed?
When comparing two samples with each other (i.e.: control and drug)
When is a one-sample t-test performed?
When comparing a sample to a mean
When may a paired t test be performed?
When samples are closely replated to one another (such as before and after a treatment)
What are the assumptions of a t test?
Outcome variable is continuous dependent variable and experimental variable is bivariate independent variable
Normal distribution
Equal Variance
What is a bivariate variable?
Contains two groups
What is a Q-Q plot?
A normal quantile-quantile plot compares quantiles of your data to theoretical quantiles for a normal distribution (if these match closely the data is normally distributed)
What is the danger of performing many tests?
There is an increase in the probability of false positives
(FWER (family-wise error rate))
Define FWER
Family wise error rate is the probability of getting a false positive if the null hypothesis is true
What calculation gives the probability of not getting a false positive in n tests?
(1-alpha)^n
in n tests
What calculation gives the probability of at least one false positive in n tests?
1-(1-alpha)^n
Define F-test
Compares several samples with each other and compares variance within samples with that between samples
What are other names for an F-test?
Analysis of variance (ANOVA)
Why is an ANOVA done?
Compare means with one another to find statistical difference
Define overall mean
Mean of sample means
(Add all means and divide by number of groups)
What are the two types of studies?
Observational
Experimental
Define observational study
Makes observations without intervention
Define interventional study
A study where an intervention is made to test a hypothesis
Define statistical or scientific variable
Any relevant condition, characteristic, number or quantity that can be measured, assessed or counted
What is another name for independent variable?
Explanatory variable
What is another name for dependent variable?
Response variable
Define a confounding variable
One that could impact the measurement from your dependent variable in addition to your independent variable
Define error
The difference between the result for a whole
population and the result from our sample or experiment.
What are the two main types of error we can control for?
Sampling error
Bias
Define sampling error
The possibility that the sample is not a perfect representation of the population
What distribution is shown by sampling error?
Normal (allowing for statistical testing)
What are the main techniques for controlling error?
Replication
Balance
Blocking
Why is replication a method of decreasing sampling error?
The more data we collect he more insignificant errors become
What are the two types of replicates?
Technical
Biological
Define technical replicates
These are additional measurements or analyses taken from the same sample. They help account for variability introduced by the measurement process itself.
Define biological replicates
These involve separate samples that are independently manipulated or tested under identical conditions
Define Blocking
Grouping experimental units with similar properties
Define Balance
This is the process of comparing groups of similar sizes
Define bias
Error caused by a systematic difference in the estimation of the sample and the whole population
In what stages of an investigation may bias occur?
Any
(Design, data collection, analysis, publication etc…)
How can bias be controlled for?
Simultaneous control groups
Blinding
Randomisation
Define a simultaneous control group
A group of subjects not exposed to the experimental treatment but are treater the same in all other ways
What are the two types of control treatments?
Untreated control
Vehicle control
What is an untreated control?
Subject in it’s native state with no treatment
What is a vehicle control?
Subject undergoes treatment with everything but the exact thing being tested (e.g.: the drug)
What is a best-available therapy control?
Testing against a pre-existing drug as opposed to a vehicle control
Define a positive control
A control which defines what a positive result looks like
Define a negative control
Result which defines what a negative result looks like
Describe blinding
The process of obscuring whom has which treatment to limit the placebo effect
Define randomisation
Assigning random places to random individuals such to not introduce further sampling bias
What methods are used to investigate the relationships between 2 continuous variables?
Correlation
Regression
What may correlation tell us about a relationship?
It’s strength and direction
What is denoted by “r”?
Correlation coefficient
What is the range of “r”?
-1 to +1
What would be the “r” value of a perfect positive linear correlation?
+1
What would be the “r” value of a perfect negative linear correlation?
-1
What does an “r” value between +/- 0-0.2 suggest?
Very weak correlation or negligible between the two variables
What does an “r” value between +/- 0.2-0.4 suggest?
Weak or low correlation between the two variables
What does an “r” value between +/- 0.4-0.7 suggest?
Moderate correlation between the two variables
What does an “r” value between +/- 0.7-0.9 suggest?
Strong, high and marked correlation between the two variables
What does an “r” value between +/- 1.0-0.9 suggest?
Very strong and very high correlation between the two variables
What does the r^2 value tell us?
How much of the variation in one variable can be explained by the other
In which types of experiments do we compare continuous variables?
- Looking for an association between variables where neither is experimentally manipulated
- Experimentally manipulating one variable and looking to see whether the other variable changes too
What can we use to predict the value of a variable when we know it’s correlation to another?
Regression
What makes a regression prediction more confident?
A higher correlation coefficient
What is true of values around the line of best fit when there is a strong correlation?
There is little variability about the line of best fit
When can a y=mx+c regression line be drawn?
When there is a linear correlation
What is goodness of fit
Assessment of how well a linear regression line fits data
How can we judge how well a regression equation fits data?
Using the r^2 value
Looking at the residuals
How is a linear regression drawn?
As a straight line through the data points
Define fitted value
The point (y) a dataset at a given is expected to be seen on a regression line
Define the residual
The distance between a given point and it’s fitted value
How can we use the residual to check the goodness of fit of a linear regression?
Plot a residual plot - residual against fitted value - and observe if there are any patterns
What does a pattern on a residual plot suggest?
A linear equation may not be appropriate for the data presented
What does a residual plot look like where the liner relationship was the best possible fit?
Plots are evenly scattered about the line on either side with even distribution
If a is explained by b, with a known value of b, can we predict a?
Yes using the linear regression
If a is explained by b, with a known value of a, can we predict b?
No, we need to create a regression in the other direction to describe b in terms of a
Define questionable research practices (QRPs)
Refers to a number of activities, often related to the misinterpretation of statistics, that occur in published scientific work
Define cherry picking as a QPR
The practice of cherry picking refers broadly to only presenting one side of the story. Specifically in relation to statistics, this translates as choosing not to report parts of your analysis which do not agree with the story you are trying to tell.
This is often used to “tidy up” or create a “convincing” story
Define P-hacking as a QPR
Ultimately manipulating your data or analysis to result in a significant p value
Give examples of P-hacking
- check the statistical significance before deciding whether to collect more data
- stopping data collection as soon as results reflect those desired
- excluding data after checking impact on significance
- adjust models on the basis of whether or not a significant result is obtained without proper justification
- rounding a p-value to the threshold
- hidden multiple testing and therefore no p value adjustments
Define HARKing as a QPR
Hypothesis after results are known is presenting results that have been discovered as if they were expected or as if they were the main study aim (overstating prior knowledge of the study).
Presenting ad hoc or unexpected results in this way is misleading
Define ad hoc
An unplanned or supplementary analyses conducted to explore specific aspects of data that weren’t the primary focus of the study. This is done on an as-needed basis to investigate particular comparisons or relationships not initially accounted for in the main analysis.
Are QPRs evidence of academic misconduct?
No, they are questionable but not misconduct
What are the two main forms of research misconduct?
Fabrication and falsification
Define fabrication
Making up data or results
Define falsification
The manipulation of research materials, data or results
What are the assumptions of an ANOVA?
Data needs to be normally distributed
Data should be from independent observations, which means that there is no relationship between the observations in each group or between the groups themselves.
Equal variances between groups (Homogeneity of variances, Homoscedasticity)
Define homoscedastic
The fundamental assumption that the variance of the errors (or residuals) should be constant across all levels of the independent variable(s)
(Violated homoscedasticity is known as heteroscedasticity)
Define homogeneity
Refers to the similarity or uniformity of certain characteristics within a group or between groups.
When doing an ANOVA how do you find the degrees of freedom (DF) between groups?
K-1
Where K is the number of groups being compared
When doing an ANOVA how do you find the degrees of freedom (DF) within groups?
N-K
Where K is the number of groups being compared and N is the total number of observations/data points collected.
Define the sum of squares
Quantifies variability between the groups of interest and within groups of interest in separate rows
What is the overall sum of sqaures?
The square of the difference between each datapoint and the overall mean, also called SST, for sum of squares (total).
Define SSW
The sum of squares within the groups is defined as the square of the difference between each datapoint and the mean of the group it belongs to. This shows the variation among each single groups.
Define SSB
The sum of squares within the groups is defined as the square of the difference between each mean of the groups and the overall mean for each datapoint. This shows the variation among between the groups.
What is the maximum value of a datapoint before it is considered an outlier
Q3+1.5 IQR
What is the minimum value of a datapoint before it is considered an outlier
Q1-1.5 IQR
What is the nature of a binomial distribution?
The binomial distribution is discrete, dealing with the number of successes in a fixed number of trials.
What is the nature of a normal distribution?
The normal distribution is continuous and is often associated with the distribution of measurements in a population.
Which parameters are common in binomial distribution?
The binomial distribution is characterized by the number of trials (n) and the probability of success (p).
Which parameters are common in normal distribution?
The normal distribution is characterized by the mean (μ) and standard deviation (σ).
What must be born in mind when calculating the critical value for a 2 tailed test?
Use 1-(alpha/2) at each end
What is the difference in the analysis of variance between a boxplot and an ANOVA?
A boxplot is a qualitative analysis whilst an ANOVA is quantitative
What is the Mean sq. and how is it calculated (ANOVA)?
ANOVA output
This is a variance estimate and what is used to calculate the F-statistic, the next column.
Calculated by taking the Sum of Squares divided by DF on the same row
What is an F-statistic (ANOVA) and how is it calculated?
This is defined as the ratio between the Mean Squares between and within.
Calculated by Mean squares of row 1/mean squares of row 2.
How do we use an F-statistic (ANOVA)?
If it is below a threshold value, the NULL hypothesis can be rejected
What does a high F-statistic mean (ANOVA)?
More likely to be a statistically relevant difference between groups.
How do we report an ANOVA?
F(dfbetween, dfwithin) = F Statistic, p =
What test is used to determine where the differences between two groups lies?
post-hoc tests such as the Tukey Honest Significance test
