Within subjects t tests

1. Within subjects t tests • Related samples • Difference scores • t tests on difference scores • Advantages and disadvantages

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

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

4. Difference Scores • Calculate difference between first and second score • e. g. Difference = Before - After • Base subsequent analysis on difference scores • Ignoring Before and After data

5. Effect of training

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

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

8. t test D and sD = 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) • We calculated t = 6.85 • Since 6.85 > 2.306, reject H0 • Conclude that the mean number of problems after training was less than mean number before training

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

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

12. Between subjects t test • Distribution of differences between means • Heterogeneity of Variance • Nonnormality

13. Powerpoint training again • Effect of training on problems using Powerpoint • Same study as before --almost • Now we have two independent groups • Trained versus untrained users • We 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 H0 is true? • Need sampling distribution of differences between means • Same idea as before, except statistic is (X1 - X2) (mean 1 – mean2)

17. Sampling Distribution of Mean Differences • Mean of sampling distribution = m1 - m2 • Standard deviation of sampling distribution (standard error of mean differences) = Cont.

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

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

20. Degrees of Freedom • Each group has 6 subjects. • Each group has n - 1 = 9 - 1 = 8 df • Total df = n1- 1 + n2 - 1 = n1 + n2 - 2 9 + 9 - 2 = 16 df • t.025(16) = +2.12 (approx.)

21. Conclusions • T = 4.13 • Critical t = 2.12 • Since 4.13 > 2.12, reject H0. • Conclude that those who get training have less problems than those without training

22. Assumptions • Two major assumptions • Both groups are sampled from populations with the same variance • “homogeneity of variance” • Both groups are sampled from normal populations • Assumption of normality • Frequently violated with little harm.

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