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Chapter 12--Multisample Inference

# 2 tss i 2 j sum of square within ssw is a measure

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Unformatted text preview: )2 ¯ TSS = i 2 j Sum of Square Within (SSW): is a measure of variability within samples (yij − yi . )2 ¯ SSW = i 3 j Sum of Squares Between (SSB): is a measure of variability between samples ni (¯i . − y.. )2 y ¯ SSB = i Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Measuring Variabilities It can be proved that TSS can be partitioned as TSS = SSB + SSW Degrees of Freedom: is the amount of information (independent observations) used to estimate variability. Source of Variation Sum of Square Degrees of Freedom Between SSB k −1 Within SSW n−k Total TSS n−1 Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Measuring Variabilities Mean Square: is a sum of square divided by its degrees of freedom. Mean Square adjusts a sum of squares for the number of independent observations used (degrees of freedom). Mean Squares: Mean Square Mean Square Between Mean Square Within 2 sB 2 sW Formula = SSB /(k − 1) = SSW /(n − k ) 2 2 sW is always an estimate of σ 2 . Further, sB can be an estimate of σ 2 when H0 is true. 2 2 However, sB tends to be bigger than sW when H0 is not true. 2 2 Therefore, we can compare sB and sW to test our hypothesis. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Test Statistic The test statistic for testing our hypothesis is: F= 2 sB H0 ∼ Fk −1,n−k 2 sW Our decision rule is to reject H0 when F > F1−α,k −1,n−k . The upper 1 − α percentile F1−α,k −1,n−k of can be computed in R using the command qf (1 − α, k − 1, n − k ). The p-value is computed as p − value = P (Fk −1,n−k > Fcomputed ). In R, 1 − pf (Fcomputed , k − 1, n − k ). Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA ANOVA Table The calculations involved in ANOVA is summarized in the following table known as the ANOVA table. Source Between Samples Within Samples Total Sum of Squares SSB SSW TSS Degrees of Freedom k −1 n−k n−1 Mean Square 2 sB = 2 sW = SSB k −1 SSW n −k The R command for ANOVA (in its simplest use) is aov(formula,data=NULL) Chapter 12: Multisample Inference Stat 491: Biostatistics F 2 2 sB / sW Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA An Example: Pulmonary Disease Twenty-two young asthmatic volunteers were studied to assess the short-term eﬀects of sulfur-dioxide(SO2 ) exposure under various conditions. Bronchial reactivity to SO2 (cm H2 O /s ) grouped by lung function (as deﬁned by forced expiratory volume divided by forced vital capacity) at screening among 22 asthmatic volunteers is given below. FEV/FVC ≤ 74% (Group A): 20.8, 4.1, 30.0, 24.7, 13.8, FEV/FVC 75 − 84% (Group...
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