# Using t-tests for related samples in Psychology

This course, PSY 340 Statistics for the Social Sciences, covers the use of t-tests for dependent means in the context of research in psychology. This particular type of analysis is useful

## About Using t-tests for related samples in Psychology

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Slide1Using t-tests (related samples)Statistics for the Social Sciences Psychology 340 Spring 2010

Slide2PSY 340Statistics for the Social Sciences t  Test for Dependent Means • Unknown population mean and variance – More realistic, these designs are commonly found in research, – Based on having two related groups of observations • Two situations – One sample, two scores for each person » Repeated measures design – Two samples, but individuals in the samples are related » Related samples, Dependent samples, matched samples

Slide3PSY 340Statistics for the Social Sciences Statistical analysis follows design • The  related-samples t- test  can be used when: – 1 sample – Two scores per subject

Slide4PSY 340Statistics for the Social Sciences participants Pre-test Post-test Test Test    2-levels, All of the participants are in both levels of the IV levels levels    Sometimes called “repeated measures” design  Within-Groups Factor Memory patients before getting the treatment Memory patients after getting the treatment

Slide5PSY 340Statistics for the Social Sciences Statistical analysis follows design • The  related-samples t- test  can be used when: – 1 sample – Two scores per subject – 2 samples – Scores are related - OR -

Slide6PSY 340Statistics for the Social Sciences Matching groups Group A Group B • Matched groups – Trying to create equivalent groups – Also trying to reduce some of the overall variability • Eliminating variability from the variables that you matched people on Red Short 21yrs Blue tall 23yrs Green average 22yrs Brown tall 22yrs Color Height Age matched Red Short 21yrs matched Blue tall 23yrs matched Green average 22yrs matched Brown tall 22yrs

Slide7PSY 340Statistics for the Social Sciences Testing Hypotheses – Step 1 : State your hypotheses – Step 2 : Set your decision criteria – Step 3 : Collect your data – Step 4 : Compute your test statistics • Compute your estimated standard error • Compute your t-statistic • Compute your degrees of freedom – Step 5 : Make a decision about your null hypothesis • Hypothesis testing: a five step program Very similar to one sample t-test from earlier

Slide8PSY 340Statistics for the Social Sciences Performing your statistical test • What are we doing when we test the hypotheses? – Computing a test statistic:  Generic test Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error Compares the differences between groups of related observations to the difference expected by the null hypothesis Because the groups of observations come from same individuals or matched individuals, the variability is typically reduced

Slide9PSY 340Statistics for the Social Sciences Performing your statistical test Diff. Expected by chance Estimated standard error of the  differences Test statistic What are all of these “D’s” referring to? Mean of the differences Number of difference  scores • Difference scores – For each person, subtract one score from the other – Carry out hypothesis testing with the difference scores • H 0  Population of difference scores has a mean = 0

Slide10PSY 340Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Observed (sample) means Related samples t Difference between Observed (sample) means

Slide11PSY 340Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Related samples t Hypothesized population means •   from the Null hypothesis Hypothesized difference between Population means •   from the Null hypothesis

Slide12PSY 340Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Related samples t Hypothesized difference between Population means •   from the Null hypothesis H 0 : Memory performance by the treatment group is equal to memory performance by the no treatment group. So :

Slide13PSY 340Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Related samples t Hypothesized difference between Population means •   from the Null hypothesis H 0 : Memory performance by the treatment group is equal to memory performance by the no treatment group. So : Hypothesized population means •   from the Null hypothesis The numerator’s of both the independent samples and related samples t-tests are numerically identical.  I’ve used notational differences to illustrate that one is based on two sets of observations (from the two samples), while the other is based on difference scores. = 0 = 0 Is equal to The major difference between the two tests comes from how the  estimated standard errors are computed.

Slide14PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Difference scores 2 6 5 9 22 Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 (Pre-test) - (Post-test) H 0 : There is no difference between pre-test and post- test H A : There is a difference between pre-test and post- test   μ D  = 0 μ D  ≠ 0

Slide15PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Difference scores 2 6 5 9 22 = 5.5 Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 (Pre-test) - (Post-test)

Slide16PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test Difference scores 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5

Slide17PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Difference scores 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5  -3.5 - 5.5 =   0.5 - 5.5 = -0.5 - 5.5 =   3.5 - 5.5 = 12.25 0.25 0.25  12.25 25 = SS D D - D (D - D) 2

Slide18PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25

Slide19PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25

Slide20PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D ? Think back to the null hypotheses  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25

Slide21PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D H 0 :  Memory performance at the post-test are equal to memory performance at the pre-test.  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25

Slide22PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D This is our t obs  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25

Slide23PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D t obs α= 0.05 Two-tailed t crit  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25

Slide24PSY 340Statistics for the Social Sciences + 3.18 = t crit - Reject H 0 Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D t obs α= 0.05 Two-tailed t crit  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25 t obs =3.8

Slide25PSY 340Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D t obs α= 0.05 Two-tailed t crit T obs  > t crit  so we reject the H 0  -3.5   0.5 -0.5   3.5 D - D Difference scores 12.25 0.25 0.25  12.25

Slide26PSY 340Statistics for the Social Sciences Statistical Tests Summary Design Statistical test (Estimated) Standard error One sample,  σ known One sample,  σ unknown Two related samples,  σ unknown Two independent samples,  σ unknown

Slide27PSY 340Statistics for the Social Sciences Effect Sizes & Power for  t  Test for Dependent Means Remember we don’t know these 7-11

Slide28PSY 340Statistics for the Social Sciences Approximate Sample Size Needed for 80% Power (.05 significance level) • Using Power and effect sizes to determine how many participants you need 7-12

Slide29PSY 340Statistics for the Social Sciences Using SPSS: Related samples t • Entering the data – Different groups of observations go into separate columns • e.g., pre-test in one column, post-test in a separate column Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 • Performing the analysis – Analyze -> Compare means -> paired samples t-test – Identify which columns are the related samples of obs • Reading the output – Means of the different groups, the mean difference, the computed-t, degrees of freedom, p-value (Sig.)