Chapter 12--Multisample Inference

2 20 25 61 75 test the hypothesis that there is an

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Unformatted text preview: B): 7.5, 7.5, 11.9, 4.5, 3.1, 8.0, 4.7, 28.1, 10.3, 10.0, 5.1, 2.2 FEV/FVC ≥ 85 (Group C): 9.2, 2.0, 2.5, 6.1, 7.5 Test the hypothesis that there is an overall mean difference in bronchial reactivity among the three lung-function groups. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA An Example: R code BR<-c(20.8, 4.1, 30.0, 24.7, 13.8, 7.5, 7.5, 11.9, 4.5, 3.1, 8.0, 4.7, 28.1, 10.3, 10.0, 5.1, 2.2, 9.2, 2.0, 2.5, 6.1, 7.5) LF<-c(rep("A",5),rep("B",12),rep("C",5)) Group<-factor(LF) fit<-aov(BR~Group) anova(fit) Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA MODEL The assumptions in the ANOVA for a completely randomized design are, 1 The samples are independent random samples (Independence Assumption). 2 Each population is assumed to have a normal distribution (Normality Assumption). 3 The variances for all the populations are the same (Homogeneity of Variance Assumption). Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Mathematical Model For the observation yij yij = µ + τi + εij where µ: is the overall (general) mean (unknown constant). τi : is the effect of the i th population or treatment (unknown constants). εij : is called the error term and, it is a random variation (noise) εij are independently normally distributed with mean 0 and variance σ 2 . Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Mathematical Model Cont’d ... Summary of the assumptions Population Population Population Mean Variance 1 µ + τ1 σ2 2 µ + τ2 σ2 . . . . . . . . . Sample Data y11 , y12 , . . . , y1n1 y21 , y22 , . . . , y2n2 . . . k µ + τk σ2 yk 1 , yk 2 , . . . , yknk In view of this model, the hypotheses of ANOVA can written as, H0 : τi = 0 for i = 1, 2, . . . , k Ha : at least one τi is different from zero Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA How critical are the assumptions? A careful design or sampling can ensure independence. When all of the sample sizes n1 , n2 , . . . , nk are large, normality assumption is not critical unless the populations are severely skewed. When the sample sizes are nearly equal, the homogeneity of variance assumption is not critical. Normality assumption can be tests by using (Shapiro-Wilk’s, Anderson-Darling’s Tests) There are tests to check the homogeneity of variance (Bartlet’s test, Hartley’s F -max, Levenes and Lehmann’s tests). We have seen some visual aids (residual analysis) to assess these assumptions in chapter 11. Transforming the data may alleviate the situation. nonparametric method (such as Kruskal-Wallis Test) can be used in the other cases. Chapter 12: Multisample Inference Stat 491: Biostatistics Anal...
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