1. Within subjects t tests Related samplesRelated samples Difference scoresDifference scores t tests on difference scorest tests on difference scores Advantages and disadvantagesAdvantages and disadvantages

2. Related Samples The same participants give us data on two measuresThe same participants give us data on two measures Xe. g. Before and After treatment XUsability problems before training on PP and after training With related samples, someone high on one measure probably high on other(individual variability).With related samples, someone high on one measure probably high on other(individual variability). Cont.

3. Related Samples--cont. Correlation between before and after scoresCorrelation between before and after scores XCauses a change in the statistic we can use Sometimes called matched samples or repeated measuresSometimes called matched samples or repeated measures

4. Difference Scores Calculate difference between first and second scoreCalculate difference between first and second score Xe. g. Difference = Before - After Base subsequent analysis on difference scoresBase subsequent analysis on difference scores XIgnoring Before and After data

5. Effect of training

6. Results The training decreased the number of problems with PowerpointThe training decreased the number of problems with Powerpoint Was this enough of a change to be significant?Was this enough of a change to be significant? Before and After scores are not independent.Before and After scores are not independent. XSee raw data Xr =.64 Cont.

7. Results--cont. If no change, mean of differences should be zeroIf no change, mean of differences should be zero  So, test the obtained mean of difference scores against  = 0. XUse same test as in one sample test

8. t test D and s D = mean and standard deviation of differences. df = n - 1 = 9 - 1 = 8 Cont.

9. t test--cont. With 8 df, t.025 = +2.306 (Table E.6)With 8 df, t.025 = +2.306 (Table E.6) We calculated t = 6.85We calculated t = 6.85 Since 6.85 > 2.306, reject H 0Since 6.85 > 2.306, reject H 0 Conclude that the mean number of problems after training was less than mean number before trainingConclude that the mean number of problems after training was less than mean number before training

10. Advantages of Related Samples Eliminate subject-to-subject variabilityEliminate subject-to-subject variability Control for extraneous variablesControl for extraneous variables Need fewer subjectsNeed fewer subjects

11. Disadvantages of Related Samples Order effectsOrder effects Carry-over effectsCarry-over effects Subjects no longer naïveSubjects no longer naïve Change may just be a function of timeChange may just be a function of time Sometimes not logically possibleSometimes not logically possible

12. Between subjects t test Distribution of differences between meansDistribution of differences between means Heterogeneity of VarianceHeterogeneity of Variance NonnormalityNonnormality

13. Powerpoint training again Effect of training on problems using PowerpointEffect of training on problems using Powerpoint XSame study as before --almost Now we have two independent groupsNow we have two independent groups XTrained versus untrained users XWe want to compare mean number of problems between groups

14. Effect of training

15. Differences from within subjects test Cannot compute pairwise differences, since we cannot compare two random people We want to test differences between the two sample means (not between a sample and population)

16. Analysis How are sample means distributed if H 0 is true?How are sample means distributed if H 0 is true? Need sampling distribution of differences between meansNeed sampling distribution of differences between means XSame idea as before, except statistic is (X 1 - X 2 ) (mean 1 – mean2)

17. Sampling Distribution of Mean Differences Mean of sampling distribution =  1 -  2Mean of sampling distribution =  1 -  2 Standard deviation of sampling distribution (standard error of mean differences) =Standard deviation of sampling distribution (standard error of mean differences) = Cont.

18. Sampling Distribution--cont. Distribution approaches normal as n increases.Distribution approaches normal as n increases. Later we will modify this to “pool” variances.Later we will modify this to “pool” variances.

19. Analysis--cont. Same basic formula as before, but with accommodation to 2 groups.Same basic formula as before, but with accommodation to 2 groups. Note parallels with earlier tNote parallels with earlier t

20. Degrees of Freedom Each group has 6 subjects.Each group has 6 subjects. Each group has n - 1 = 9 - 1 = 8 dfEach group has n - 1 = 9 - 1 = 8 df Total df = n 1 - 1 + n 2 - 1 = n 1 + n 2 - 2 9 + 9 - 2 = 16 dfTotal df = n 1 - 1 + n 2 - 1 = n 1 + n 2 - 2 9 + 9 - 2 = 16 df t.025 (16) = +2.12 (approx.)t.025 (16) = +2.12 (approx.)

21. Conclusions T = 4.13T = 4.13 Critical t = 2.12Critical t = 2.12 Since 4.13 > 2.12, reject H 0.Since 4.13 > 2.12, reject H 0. Conclude that those who get training have less problems than those without trainingConclude that those who get training have less problems than those without training

22. Assumptions Two major assumptionsTwo major assumptions XBoth groups are sampled from populations with the same variance “homogeneity of variance”“homogeneity of variance” XBoth groups are sampled from normal populations Assumption of normalityAssumption of normality XFrequently violated with little harm.

23. Heterogeneous Variances Refers to case of unequal population variances.Refers to case of unequal population variances. We don’t pool the sample variances.We don’t pool the sample variances. We adjust df and look t up in tables for adjusted df.We adjust df and look t up in tables for adjusted df. Minimum df = smaller n - 1.Minimum df = smaller n - 1. XMost software calculates optimal df.