渥太华大学统计英文课件lecture 8.ppt

Université d’Ottawa / University of OttawaLecture 8: Multiple comparisonslWhat are multiple comparisons?lThe problem of experiment-wise a errorlWhen do we do multiple comparisons?lStatistical tests which control aelEstimating treatment effects19991Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaWhat are multiple comparisons?lPair-wise comparisons of different treatmentslThese comparisons may involve group means, medians, variances, etc.lfor means, done after ANOVA lIn all cases, H0 is that the groups in question do not differ.YieldmCmNmN+PControl Experimental (N) Experimental (N+P)mc:mNmN:mN+PmC: mN+PFrequency19992Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaTypes of comparisonslplanned (a priori): independent of ANOVA results; theory predicts which treatments should be different.lunplanned (a posteriori): depend on ANOVA results; unclear which treatments should be different.lTest of significance are very different between the two!YYX1X2X3X4X5X1X2X3X4X5Plannedunplanned19993Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaPlanned comparisons (a priori contrasts): catecholamine levels in stressed fishlComparisons of interest are determined by experimenter beforehand based on theory and do not depend on ANOVA results.lPrediction from theory: catecholamine levels increase above basal levels only after threshold PAO2 = 30 torr is reached.lSo, compare only treatments above and below 30 torr (NT = 12).Predicted thresholdPAO2 (torr)1020304050[Catecholamine]0.00.10.20.30.40.50.60.719994Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaUnplanned comparisons (a posteriori contrasts): catecholamine levels in stressed fishlComparisons are determined by ANOVA results.lPrediction from theory: catecholamine levels increase with increasing PAO2 .lSo, comparisons between any pairs of treatments may be warranted (NT = 21). Predicted relationshipPAO2 (torr)1020304050[Catecholamine]0.00.10.20.30.40.50.60.719995Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaThe problem: controlling experiment -wise a errorlFor k comparisons, the probability of accepting H0 (no difference) is (1 - a)k.lFor 4 treatments, (1 - a)k = (0.95)6 = .735, so experiment- wise a (ae) = 0.265.lThus we would expect to reject H0 for at least one paired comparison about 27% of the time, even if all four treatments are identical.Nominal a = .05Number of treatments0246810Experiment-wise a (ae) 0.00.20.40.60.81.019996Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaControlling experiment-wise a error at nominal a by adjusting by total number of comparisonslTo maintain ae at nominal a, we need to adjust a for each comparison by the total number of comparisons.lIn this manner, ae becomes independent of the number of treatments and/or comparisons.Number of treatments0246810Experiment-wise a (ae) 0.00.20.40.60.81.0Nominal a = .0519997Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaControlling experiment-wise a error at nominal a by using modified test statisticslUse modified test- statistic S for pair-wise comparisons whose distribution depends on the total number of comparisons NT such that p(S) increases with NT.05101520Value of test statistic (S)00.20.3Probability (p)S, NT = 1 S, NT = 2 S, NT = 319998Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaUsing multiple comparisonslUse only after H0 is rejected on the basis of an ANOVA because.l. ANOVA is more robust and reliable than multiple comparisons.lSo, if H0 is accepted in original ANOVA, do not proceed to do multiple comparisons.lNote, however, that there is no universally agreed-upon method for doing multiple comparisons, and.l…results may differ depending on which method you use.lSo, proceed with caution!19999Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaControlling ae by adjusting individual a’s199910Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaControlling ae by adjusting individual a’sp is probability associated with t-test of difference between 1 pair of means; k is total number of comparisons.199911Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaExample: Temporal variation in size of sturgeon (Model II ANOVA)lPrediction: dam construction resulted in loss of large sturgeonlTest: compare sturgeon size before and after dam constructionlH0: mean size is the same for all years1954 195819651966 YEAR35.038.842.646.450.254.0FKLNGTHDam construction199912Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaExample: Temporal variation in size of sturgeon (ANOVA results)Conclusion: reject H0199913Bio 4118 Applied Biostatistics*Université d’Ottawa / University of OttawaMultiple comparison results1954 195819651966 YEAR35.038.842.646.450.254。