Chapter 12--Multisample Inference

Treatment response mm hg mean i y 1 27 26 21 26 2500

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Unformatted text preview: ltiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA An Example Cont’d There were four replication of each treatment . The reduction in blood pressure (mm Hg) over a period of four weeks observed for the experimental subjects were as tabulated below. Treatment Response (mm Hg) Mean (¯i . ) y 1 27, 26, 21, 26 25.00 2 19, 13, 15, 16 15.75 3 15, 10, 10, 11 11.50 4 22, 15, 21, 18 19.00 5 20, 18, 17, 16 17.75 2 Note SSB = 388.7 and sW = 6.433. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA An Example Cont’d The following are questions the investigator asked about the treatment effects. 1 Is there a difference between the effects of the low and high doses of A? 2 Is there a difference between the effects of the low and high doses of B? 3 Is there a difference between the average of the expected responses for the two doses of A and the average of the expected responses for the two doses of B? Express the above questions in terms of contrasts, compute the sample contrasts and estimated variances. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Error Rates Per-Comparison (Comparison-Wise) Error Rate : The expected proportion of contrasts that will be falsely declared positive. We denote this error rate by αC . Experiment-wise (Family-wise) Error Rate:The probability of at least one false positive. We denote this error rate by αE . Suppose we would like to test m contrasts. For a given multiple comparison procedure, we have the scenarios described in the following table Number Number Number of not rejected rejected Total True Null Hypothesis U V m0 T S m1 False Null Hypotheses Total m−R R m Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Error Rates Cont’d... Notice that αC = E (V ) . m Also that αE = P (V ≥ 1). A multiple comparison procedure is said to control a particular error rate at a level α, if this error rate is less than or equal to α. In general, we have the relationship αC ≤ αE That is, a multiple comparison procedure that controls EWER will automatically control PCER. Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Fisher’s Method Suppose we are interested in testing the significance of ˆ1 , ˆ2 , . . ., ˆm . The Fisher’s MC procedure declares ˆi significant if | ˆi | ≥ t1−α/2,n−k ˆ V ( ˆi ). The Fisher’s MC procedure controls PCER at a level α. A 100(1 − α)% confidence interval for i is ˆi ± t1−α/2,n−k ˆ V ( ˆi ) Note: if a MC procedure controls only PCER then expected number of false positives may increases with m. The m contrasts must be pre-planned comparisons for Fisher’s procedure to have any use. Chapter 12: Multisample Inference Stat 491...
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