Chapter 12--Multisample Inference

# 0520 kk 120 chapter 12 multisample inference stat 491

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Unformatted text preview: her the means that are not signiﬁcantly diﬀerent. The groups may overlap. They are identiﬁed by underlining the ordered means belonging to the same groups. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Methods 1. Fisher (only PCER) LSDij (F ) = t1−α/2,n−k 1 1 + n(i ) n(j ) 2 sW 2. Bonferroni (EWER) LSDij (B ) = t1− k (kα 1) ,n−k − 1 1 + n(i ) n(j ) 2 sW 3. Scheﬀe (EWER) LSDij (S ) = (k − 1)F1−α,k −1,n−k Chapter 12: Multisample Inference 1 1 + n(i ) n(j ) Stat 491: Biostatistics 2 sW Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Example: The BP Data apc<-glht(A,linfct=mcp(f="Tukey")) summary(apc,test=adjusted(type=c("bonferroni"))) confint(apc,c_alpha=(1-0.05/20)) #k(k-1)=20 Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Example: Insectology In a research to investigate the attractiveness of ﬁve diﬀerent colors to insects, the number of cereal leaf beetles trapped when six boards of each of the ﬁve colors were placed in a ﬁeld of oats were recorded. The following summary statistics were computed. Color ni yi . ¯ Yellow 6 49.2 Orange 6 34.9 Red 6 26.4 Blue 6 17.2 White 6 15.0 2 = 63.298. sW Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Example Cont’d ... Pair Y vs W Y vs B Y vs R Y vs O O vs W B vs O O vs R R vs W R vs B B vs W Diﬀ 34.2 32 22.8 14.3 19.9 17.7 8.5 11.4 9.2 2.2 F 9.46 9.46 9.46 9.46 9.46 9.46 9.46 9.46 9.46 9.46 B 14.139 14.139 14.139 14.139 14.139 14.139 14.139 14.139 14.139 14.139 Chapter 12: Multisample Inference S 15.259 15.259 15.259 15.259 15.259 15.259 15.259 15.259 15.259 15.259 Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Remarks The cut oﬀ points for diﬀerent pairwise comparisons would be diﬀerent for each of the method if the sample sizes were not equal. We want a procedure that, ideally, not only controls EWER but also has high power to detect true signiﬁcance (has the smallest LSD). Which of Bonferroni or Scheﬀe method should we use? Fisher is not generally recommended because it does not control EWER. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA False Discovery Rate (FDR) Bonferroni’s and Schefe procedures control EWER but they tend to be conservative when m and k are large. Controlling EWER, in general, may not be practical for large-scale inference (large k and m). For large scale inference, EWER results in conservative inferential procedures. An error rate which has gained popularity in genetic studies is the so-called False Discovery Rate. FDR is deﬁned as the expected proportion of falsely declared signiﬁcant among all declared signiﬁcant, FDR = E V R . In general, FDR ≤ αE . Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Benjamini-Hochberg Procedure We ﬁrst compute p -value for each comparison using a method that controls (PCER) such as Fisher’s procedure. Order the p -values from the smallest to the largest...
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