Chapter 12--Multisample Inference

That means high type ii error rate and hence large

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Unformatted text preview: : Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Bonferroni’s Method Benferroni Inequality says, αE ≤ mαC . So to control EWER at level α, use Fisher’s method with α/m. Bonferroni’s procedure declares ˆi significant if | ˆi | ≥ t1−α/(2m),n−k ˆ V ( ˆi ). This procedure gets more conservative as m gets large. That means high type II error rate and, hence, large number of false negatives. A 100(1 − α)% simultaneous confidence interval ˆi ± t1−α/(2m),n−k ˆ V ( ˆi ). To apply Bonferroni’s method, we need to know m in advance. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA The Scheffe’s Method For testing the significance of ˆi , , Scheffe’s Method declares ˆi ; i = 1, . . . , m, significant if | ˆi | ≥ ˆ V ( ˆi ). (k − 1)F1−α,k −1,n−k Scheffe’s method controls EWER at a level α. A 100(1 − α)% simultaneous confidence interval for . . ., m is ˆi ± (k − 1)F1−α,k −1,n−k 1, 2, ˆ V ( ˆi ). Notice that Scheffe Method does not depend on m. That is, it is designed for arbitrarily large number of contrasts. In general it tends to be too conservative unless m is large. This method has to be used if you have post-hoc comparisons as it accounts for the after the fact nature of the hypothesis. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Example Consider the blood pressure example, with the three contrasts defined before. ˆ i | ˆi | V ( ˆi ) Fisher Bonferroni Scheffe Cut Off Cut Off Cut Off 1 4.25 1.80 3.80 4.85 6.30 2 1.25 1.80 3.80 4.85 6.30 3 4.75 1.27 2.71 3.42 4.44 Note that Fisher’s Method declares that ˆ1 and ˆ3 significant. Both Bonferroni and Scheffe declare only ˆ3 significant. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Example Cont’d Since Fisher’s does not control EWER, we base our conclusions on Scheffe or Bonferroni. In general, the choice between Scheffe and Bonferoni is made based on which one declares larger number of contrasts significant (has lower cut off point), that procedure would be more powerful or commits less type II error. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Example Cont’d 95% Confidence Interval ˆi i Fisher 1 4.25 0.43 , 8.07 2 1.25 -2.60 , 5.07 3 -4.75 -7.45 , -2.50 Notice that Bonferroni and Fisher’s is individual. Bonferroni Scheffe -0.58 , 9.08 -2.02 , 10.52 -3.58 , 6.08 -5.02 , 7.52 -8.17 , -1.33 -9.18 , -0.32 Scheffe are simultaneous where as We can also construct one sided CI or one sided test by finding the one sided cutoff values. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANO...
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This note was uploaded on 02/03/2014 for the course STAT 491 taught by Professor Solomonharrar during the Fall '12 term at Montana.

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