Week 5: Comparing Means: Independent and Paired T-test Flashcards
What are the 3 types of t-tests? - (3)
- One-samples t-test
- Paired t-test
- Independent t-test
Whats a one-sample t-test?
Compares the mean of the sample data to a known value
What is the assumptions of one-sample t-test? - (4)
- DV = Continous (interval or ratio)
- Independent scores (no relation between scores on test variable)
- Normal distribution via frequency histogram (normal shape) and Q-plot (straight line) and non significant Shaprio Wilk
- Homogenity of variances
Example of one-sample t-test RQ
Is the average IQ of Psychology students higher than that of the general population (100)?
What is the assumptions of independent samples t-tests (listing all of them) - (7)
- Independence. – no relationship between the groups
- Normal distribution via frequency histogram (normal shape) and Q-plot (straight line) and non significant Shaprio Wilk
- Equal variances
- Homogeneity of variances (i.e., variances approximately equal across groups) via non significant Levene’s test
- DV = Interval or continuous
- IV = Categorical
- No significant outliers
What is an RQ example of independent samples t-tesT?
Do dog owners in the country spend more time walking their
dogs than dog owners in the city?
What is the assumptions of paired t-test? (listing all) - 3
DV is continuous
Related samples: The subjects in each sample, or group, are the same. This means that the subjects in the first group are also in the second group
Normal distribution via frequency histogram (normal shape) and Q-plot (straight line) and non significant Shaprio Wilk
What is an example of RQ of paired t-test?
Do cats learn more tricks when given food or praise as positive feedback?
What is the decision framework for choosing a paired-sample (dependent) t-test? - (5)
- What sort of measurement = continous
- How many predictor variables = one
- What type of predictor variables = categorical
- How many levels of categorical predictor = two
- Same or different participants for each predictor level = same
What is the decision framework for choosing independent-t-test? (5)
- What sort of measurement = continous
- How many predictor variables = one
- What type of predictor variables = categorical
- How many levels of categorical predictor = two
- Same or different participants for each predictor level = different
If we are comparing differences between means of two groups in independent/paired t-test then all we are doing is
predicting an outcome based on membership of two groups
Indepdnent and paired t-tests can fit into an ideal of a
linear model
What is coding with dummy variables and an example?
E.g., coding absence of cloak in terms of numbers (like 0) and pps with clock as 1 even though it is a categorical variable
Independent and paired t-tests (comparing difference of two means) fit into idea of general linear model
What does b0 and b1 represent in this general linear model? - (2)
*b0 is equal to the mean of group coded as 0 (in this case no cloak)
* b1 is difference between group means
The t-distributed is defined by its
degrees of freedom - related to the sample size.
The t distribution has heavier tails for
lower degrees of freedom (small N studies)
What are the two different t-tests? - (2)
- Independent-means t-test
- Dependent-means test
Independent and Paired T-tests have one predictor (X) variable with 2 levels and only …. outcome variable (Y)
one
When is an independent-means t-test used?
When 2 experimental conditions and different participants are assigned to each condition
What is independent-means t-test sometimes called as well?
independent-samples t-test
When is a dependent-means t-test used?
Used when there are 2 experimental conditions and same participants took part in both conditions of the experiment
What is dependent-means t-test sometimes referred to?
Matched pairs or paired samples t-test
For independent and paired t-tests we compare between the sample means that we collected to the difference between sample means that we would expect if
there was no effect (i.e., null hypothesis was true)
In independent and paired t-tests if standard error was small then suggests that sample means of two groups are quite
similar
In independent and paired t-tests If standard error is large then large differences in sample means more likely then assume one of two things - (2)
- No effect and sample means in population flcutuate by chance and collected two samples that are atypical from population
- Difference between samples represent genuine difference and typical of respective population - so null hypothesis si incorrect
Formula of calculating t- test statistic (form depend on whether same or different participants used in each experimental condition)
Formula of calculating t-statistic shows obtaining t-test statistic by diving the model/effect by the
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
If observed difference between sample means get larger in t-tests then more confident we become that
null hypothesis is rejected and two sample means differ because of experimental manipulation
Both independent t-test and paired t-test are … tests based on normal distribution
parametric tests
Since independent and paired t-tests are parametric tests they assume that the - (2)
- Sampling distribution is normally distributed - in paired it means sampling distribution of differences of scores is normal not the socres itself!
- Data measured at least interval level
Since independent-tests used to test different groups of people it also assumes - (2)
- Variances in populations are roughly equal (homegenity of variance) = Leven’s test
- Scores are independent since they come from different people
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
A small standard error of differences tells us that - (3) in paired-t-test
pairs of samples from a population have similar means
(i.e.., the differences between sample means should be very small and big difference between them is unlikely)
sampling distribution of differences is very narrow and centred around 0
A large standard error of differences tells us that in paired t-test - (2)
that sample means can deviate quite a lot from the populatio mean and so differences between pairs of samples can be quite large by chance alone
sampling distribution of differences is more spread out
The average difference between person’s socre in condition 1 and condition 2 -(¯D) in paired t-test is an indicator of
systematic variation in the data (represents experimental effect)
By dividing the average differences in person’s score in condition 1 and 2 in equation of paried t-test statistic it does 2 things- (2)
standardizing the average difference between conditions (this just means that we can compare values of t without having to worry about the scale of measurement used to measure the outcome variable);
contrasting the difference between means that we have against the difference that we could expect to get based on how well the samples represent the populations from which they came.
If average differences (–D) between our samples is large and standard error of differences is small in paired-t test then we can be confident that
the difference we observed in our sample is not a chance result and caused by experimental manipulation
How do we normally calculate the standard error?
SD divided by square root of sample size
How to calcuate the standard error of differences in paired-test?(σ –D)
Standard deviation of differences divided by square root of sample size
the t-statistic in paired t-test is
ratio of systematic variation in experiment (average difference D) and unsystematic variation (standard erro of differences)
When would we expect t statistic greater than 1 in paired-t-test equation?
If the experimental manipulation creates any kind of effect,
When would we expect t statistic less than 1 in paired t-test equation?
If the experimental manipulation is unsuccessful then we might expect the variation caused by individual differences to be much greater than that caused by the
experiment
In pairered and generally independent t-tests we can compare the obtainee value of t against thmaximum value we would expect to get by chance alone in t distribution with same DFs and if value we obtain exceeds the
critical value then conflict if reflects an effect in our IV
What does this output mean?
Sleep condition scored mean of 65.38 and no sleep had mean of 60.22
In Paired Samples Correlation box here we would expect that
someone’s score for first condition would be associated in second condition
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?
t(19) = 2.72, p = 0.012
What does negative t-value mean?
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
How to calculate effect size for independent and paired t-tests?
Using cohen’s D
Diagram of calculating Cohen’s D Statistic for sleep vs no sleep
Minus big mean from small mean divided by smallest SD (control group)
What is interpretation of Cohen’s D? - (3)
- Around 0.20 = a small effect
- Around 0.50 = a medium effect
- Around 0.80 & above = a large effect
What does Cohen’s d of 0.20 represent
difference between groups is a 1/5 of SD
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
With independent t-test there are two different equations that can be used depending on whether the samples
contain an equal number of people
With independent t-test since different participants participate in different condition, the pairs of scores will differ not just of experimental manipulation but also because of
other sources of variance (such as individual differences between participants’ motivation, IQ etc..)
With dependent t-test we look at differences between pairs of scores because
scores came from same participant and so individual differences were eliminated
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
What does variance sum of law state?
variance of the sampling distribution is equal to the sum of the variances of the two populations from which the samples
were taken
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
For comparing two groups with unequal number of participants in independent t-test then we use the
pooled variance estimate t-test
The pooled variance estimate t-test is used which takes into account of the
differnece in sample size by weighting the variance of each sample
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)
As with dependent t-test we compare obtained value of t in independent sample against the
maximum value we would expect to get by chance alone in t distribution with same DFs
As with the dependent t-test we can compare the obtained value of t against the maximum
value we would expect to get by chance alone in a t-distribution with the same degrees of freedom
if value we obtain exceeds this critical value then
we can be confident that this reflects an effect of our independent variable
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
In independent samples t-test we check for Levene’s test for quality of variances which determine whether
we got equal variance across the groups or whether the variances are unequal
In independent t-test, Levene’s test we are looking for a non-significant p-value which shows that
no statistically significant difference in variances between the two groups - report results from equal variances assumed
In independent t-test if Levene’s test was significant then it means that
variances between the 2 groups are different and they are statistically significantly different - report data from equal variances not assumed
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
Paired vs independent t-tests - who has better power?
Paired t-t ests
Since paired-t-tests use same participants across conditions the … is reduced dramatically compared to independent t-test
unsystematic variance
The non-parametric counterpart of dependent t-test is called
Wilcoxon signed rank test
The non-parametric tests of the independent t-test is
Wilcoxon rank sum test and Mank Whitney 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)
What does this independent samples t-test output show? (DV = puppes avergae weight gain between 12 and 28 weeks of age and IV= diet A, B)
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 3 types of SD can be used in the formula … but… - (4)
- Pooled SD (over conditions)
Averaged SD
Control group SD
But they make very small difference
Cohen’s d for diet was 4.25
Is this a:
Small effect
Medium effect
Large effect
Large effect
