1. if = 10 and = 0.05 per experiment = 0.5 Type I Error Rates I.Per Comparison II.Per Experiment (frequency) = error rate of any comparison = # of comparisons (frequency) III.Familywise (for independent comparisons) Per ComparisonFamilywisePer Experiment probability of at least one Type I error. Multiple Comparison

2. Complete vs. Restricted H 0 123 4164 749 1010098114196 74986412144 131691419618324 749636864 41 59 8.2 11.8 383393377 =141 Treatments Independent Samples MS Treat MS error Complete H 0

3. Restricted H 0 e.g., A Priori Comparisons Post Hoc Comparisons The role of overall F - might NOT pick up - changes Familywise

4. A Priori Comparisons - replace individual or with MS error Multiple t tests (compare two conditions) - test t with df error Comparing Treatments 1 and 3 n.s.

5. Linear Contrasts Compare:  two conditions  a set of conditions and a condition  two sets of conditions Let for equal n’s

6. (1) Contrasting Treatments 1 & 3 again = = = = SS contrast = SS error = MS contrast = always = 1 MS error = MS contrast = SS contrast = MS error = F (1,12) = (Look at t- test)

7. (2) Contrasting Treatments 1 & 2 with 3 = = = = = = SS contrast = = SS error = MS contrast = = F (1,12) =

8. (3) Contrasting Treatments 1 & 3 with 2 = = = = = = SS contrast = SS error = MS contrast = = F (1,12) =

9. Orthogonal Contrasts = = # of comparisons = df Treat if n’s are equal df Treat = 2 in our example contrasts 1 and 2 = = = 2 and 3 = = = 1 and 3 = = = SS contrast1 SS contrast3 = = SS Treat =

10. Bonferroni’s Control for FW error rate (  ) use  = 0.01 Bonferroni Inequality e.g.: if, per comparison  = 0.05 and if, 4 comparisons are made then, the FW  CANNOT exceed p = 0.02 EW  or FW   c(PC  ) c = # of comparisons Thus, we can set the FW  or the per experiment  to a desired level (e.g., 0.05) and adjust the PC  If we desire a FW  = 0.05 then: 0.05 = PC  (4) 0.0125 = PC 

11. Bonferroni’s (comparing 2 means) using t 2 = F and moving terms This allows us to contrast groups of means. (linear contrasts) if

12. Multistage Procedures Bonferroni: divides  into equal parts Multistage (Holm): divides  into different size portions ifor heterogenous S 2 ’s

13. compare next largest to critical value C = C-1 compare largest s to critical Multistage 1.calculate alls 2.arrange in order of magnitude 3. value (Dunn’s Tables) ONLY if significant 4. 5. C = total # of contrasts to be made and so on FW  is kept at 0.05 (  )

14. Subject X Treatment Design I Weight each observation by its assigned condition weight II Compute D i for each subject III Sum D i across subjects IV Compute SS contrast V Compute SS SsXC(error) Linear Contrast

15. Subjects123 1447 210914 37812 4131418 5768 Treatments 40-7-39 100-14-416 70-12-525 130-18-525 70-81 1876 Contrast 1 with 3 F (1,4) = 23.14 SS con = SS SsXcon = MS con = MS SsXcon = df = 4 DiDi Di2Di2

16. Subjects123 1447 210914 37812 4131418 5768 -48-7-39 -1018-14-636 -716-12-39 -1328-18-39 -712-8-39 1872 Treatments Contrast 1 & 3with 2 DiDi Di2Di2 SS con = SS SsXcon = MS con = MS SsXcon = F (1,4) = 36

17. SS con1 = = SS Treat Total = Orthogonal Contrasts Error term could be SS res or error SS SsXcon1 + SS SsXcon2 = ? 5.61.26.8 ??=+ df con1 + df con2 = df Treat 1+ 4 = 2 df SsXcon1 + df SsXcon2 = df ? 4+=8 SS con2 =