Download Type I Errors & Contrasts in One-Way ANOVA: Multiple Comparisons - Prof. Brian C. Dennis and more Exams Statistics in PDF only on Docsity! Multiple comparisons The problem: why not just look at all the pairs of means with t-tests? !: probability of rejecting the null hypothesis, given that it is true ex. 4 means, 6 tests at 0.05! œ test 1 H : 0 (.95)! " #. .% œ test 2 H : 0 (.95)! " $. .% œ test 3 H : 0 (.95)! " %. .% œ ã ã ã test 6 H : 0 (.95)! $ %. .% œ If the test statistics are independent (they are not), the probability that all six tests avoided type I errors (given H! is true for all six tests) is a b a b1 0.05 0.95 0.735% œ œ' ' So, the probability that tests commit a type Ione or more error (given H 's true) is! !E œ % œ1 0.735 0.265 ( is the )!E experimentwise type I error probability Contrasts 6 œ + , + ,â, +" " # # > >. . . is a of the , , ..., .linear contrast population means . . ." # > Here the 's are constants which sum to zero:+3 + , + ,â, + œ" # > 0 ex. 0, 0, 1, 1+ œ + œ + œ + œ %" # $ % 6 œ %. .$ % (pairwise comparison) ex. 1, , , + œ + œ % + œ % + œ %" # $ %1 1 13 3 3 6 œ % , , . . . . . " " # $ % 3 (compares to the others) (usually redefine as 3 , so that all6 œ % % %. . . ." # $ % the 's are integers)+3 Assume the usual model for 1-way AOV: within each of > populations, normal , ; random sample of size] µ3 3 #a b. 5 83 from each population. Parameter estimates: . 5s œ C œ = œ% 8 % >3 3† # # , ^ SSW W Ta b SAS: (suppose model has four means) ã PROC GLM; CLASS TRT; MODEL Y=TRT; CONTRAST '1 VS 2' TRT 1 -1 0 0; CONTRAST '1 VS 3' TRT 1 0 -1 0; CONTRAST 'CTRL VS REST' TRT -3 1 1 1; Orthogonal contrasts Two (estimated) contrasts given by and6 œ + C%" 3 3†D 6 œ , C%# 3 3†D are iforthogonal + , + , + , 8 8 8 , ,â, œ " " # # > > " # > 0 a b+ , , + , ,â, + , œ 8" " # # > > 30 if the 's are equal Sum of squares for a contrast: SSC œ 6s , ,â, # + + 8 8 8 +Š ‹" # ># # " # > # Orthogonal contrasts allow partitioning of SSB (treatment sum of squares) into 1 additive components:> % Source df SS MS test stat __________________________________________ trt 1 SSB > % 0ˆ ‰SSB1>% contrast 1 1 SSC SSC " " "0 contrast 2 1 SSC SSC # # #0 ã ã contrast 1 1 SSC SSC > % 0>%" >%" >%" error SSW 8 % >T SSWŠ ‹8 %>T ex. 5 weed control treatments: control, biol control 1, biol control 2, chem control 1, chem control 2 orthogonal contrasts of possible interest: control biol 1 biol 2 chem 1 chem 2 4 1 1 1 1% % % % 0 1 1 1 1% % 0 1 1 0 0% 0 0 0 1 1% With orthogonal contrasts, not only does the SSB become partitioned into components (in 1-way AOV), but also, the SSC , SSC , ..., SSC are independent random variables." # >%" However, the statistics are 0 0 œŠ ‹3 Î 8 %>SSCSSW 3a bT not independent independent. Experimentwise error rate (EER) control Bonferroni approach E E E T E œ" # 7 3, , ..., are events; a b ! Bonferroni inequality: T E E â E %7 % % %a b" # 7 1and and and ! Choose (say, 0.05)!E 7 œ hypothesis tests: use for each individual test!I !E7 7 confidence intervals: calculate each CI at 100 1 % confidenceˆ ‰% !E7 Bonferroni: fine for small number of comparisons very conservative (individual tests not powerful, CIs too big)
