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This volume was digitized and made accessible online due to deterioration of the original print copy.

Bibliography: leaves 18-19

J. Barnette and J. McLean (1996) proposed a method of controlling Type I error in pairwise multiple comparisons after a significant omnibus F test. This procedure, called Alpha-Max, is based on a sequential cumulative probability accounting procedure in line with Bonferroni inequality. A missing element in the discussion of Alpha-Max was the empirical determination of actual probabilities of Type I errors. This paper compares the Type I error rates of Alpha-Max with other commonly used multiple comparison procedures: (1) Fisher's Least Significant Difference (LSD); (2) Dunn-Bonferroni; (3) Tukey's Honestly Significant Difference (HSD); (4) the Student Newman Keuls (SNK) procedure; and (5) the Scheffe approach. Monte Carlo procedures were used to generate 10,000 replications with varied alpha of 0.05 and 0.01; 3, 4, and 5 groups; and 5 sample sizes. Actual Type I error rates were determined for the greatest difference and for total number of Type I errors. Results indicate that in virtually every situation LSD and Alpha-Max had significantly higher probability of Type I errors than the other four methods. SNK and HSD had higher than nominal alpha probabilities for committing Type I errors, with SNK having a lower level than HSD. Dunn-Bonferroni had a level slightly lower than the nominal level, while the Scheffe had a level much lower than the nominal level. Varying sample size had little practical significance. While Alpha-Max did not provide for acceptable experiment-wise control of Type I error, it may provide an alternative for control of Type I error in the planned, nonorthogonal situation or in situations where assumptions of analysis of variance are violated. (Contains 25 tables and 4 references.) (Author/SLD)

Recent research in neuroimaging has focused on assessing associations between genetic variants that are measured on a genomewide scale and brain imaging phenotypes. A large number of works in the area apply massively univariate analyses on a genomewide basis to find single nucleotide polymorphisms that influence brain structure. In this paper, we propose using various dimensionality reduction methods on both brain structural MRI scans and genomic data, motivated by the Alzheimer's Disease Neuroimaging Initiative (ADNI) study. We also consider a new multiple testing adjustment method and compare it with two existing false discovery rate (FDR) adjustment methods. The simulation results suggest an increase in power for the proposed method. The real data analysis suggests that the proposed procedure is able to find associations between genetic variants and brain volume differences that offer potentially new biological insights.

This volume was digitized and made accessible online due to deterioration of the original print copy.

This article is from Biometrical Journal. Biometrische Zeitschrift , volume 53 . Abstract The confirmatory analysis of pre-specified multiple hypotheses has become common in pivotal clinical trials. In the recent past multiple test procedures have been developed that reflect the relative importance of different study objectives, such as fixed sequence, fallback, and gatekeeping procedures. In addition, graphical approaches have been proposed that facilitate the visualization and communication of Bonferroni-based closed test procedures for common multiple test problems, such as comparing several treatments with a control, assessing the benefit of a new drug for more than one endpoint, combined non-inferiority and superiority testing, or testing a treatment at different dose levels in an overall and a subpopulation. In this paper, we focus on extended graphical approaches by dissociating the underlying weighting strategy from the employed test procedure. This allows one to first derive suitable weighting strategies that reflect the given study objectives and subsequently apply appropriate test procedures, such as weighted Bonferroni tests, weighted parametric tests accounting for the correlation between the test statistics, or weighted Simes tests. We illustrate the extended graphical approaches with several examples. In addition, we describe briefly the gMCP package in R, which implements some of the methods described in this paper.

This article is from Biometrical Journal. Biometrische Zeitschrift , volume 53 . Abstract The confirmatory analysis of pre-specified multiple hypotheses has become common in pivotal clinical trials. In the recent past multiple test procedures have been developed that reflect the relative importance of different study objectives, such as fixed sequence, fallback, and gatekeeping procedures. In addition, graphical approaches have been proposed that facilitate the visualization and communication of Bonferroni-based closed test procedures for common multiple test problems, such as comparing several treatments with a control, assessing the benefit of a new drug for more than one endpoint, combined non-inferiority and superiority testing, or testing a treatment at different dose levels in an overall and a subpopulation. In this paper, we focus on extended graphical approaches by dissociating the underlying weighting strategy from the employed test procedure. This allows one to first derive suitable weighting strategies that reflect the given study objectives and subsequently apply appropriate test procedures, such as weighted Bonferroni tests, weighted parametric tests accounting for the correlation between the test statistics, or weighted Simes tests. We illustrate the extended graphical approaches with several examples. In addition, we describe briefly the gMCP package in R, which implements some of the methods described in this paper.

This paper studies multi-sample cluster analysis, the problem of grouping samples, as an alternative to multiple comparison procedures through the development and the introduction of model-selection criteria such as those: Akaike's Information criterion (AIC) and Schwarz's Criterion (SC), as new procedures for comparing means, groups, or samples, and so forth, in identifying and selecting the homogeneous groups or samples from the heterogeneous ones in multi-sample data analysis problems. An enumerative clustering technique is presented to generate all possible choices of clustering alternatives of groups, or samples on the computer using efficient combinatorial algorithms without forcing an arbitrary choice among the clustering alternatives, and to find all sufficiently simple groups or samples consistent with the data and identify the best clustering among the alternative clusterings. Numerical examples are carried out and presented on a real data set on grouping the samples into fewer than K groups. Through a Monte Carlo study, an application of multi-sample cluster analysis is shown in designing optimal decision tree classifiers in reducing the dimensionality of remotely sensed heterogenous data sets to achieve a parsimonious grouping of samples. The results obtained demonstrate the utility and versatility of model-selection criteria which avoid the notorious choice of levels of significance and which are free from the ambiguities inherent in the application of conventional hypothesis testing procedures. Originator suggested keywords include: Multi-Sample Cluster Analysis; Multiple Comparison Procedures; Model Selection Criteria; Akaike's Information Criterion; Schwarz's Criterion. (Author)

A review of the literature on multiple comparison procedures suggests several alternative approaches for comparing means when population variances differ. These include: (1) the approach of P. A. Games and J. F. Howell (1976); (2) C. W. Dunnett's C confidence interval (1980); and (3) Dunnett's T3 solution (1980). These procedures control the overall risk of a Type I error experimentwise at approximately the nominal significance level and have the best statistical power among alternative solutions. The two-stage multiple comparison procedures of Y. Hochberg and A. C. Tamhane (1987), and R. R. Wilcox (1987) are also discussed. These procedures and the Tukey-Kramer procedure were applied to data from a study of the effects of exercise on psychological and physiological variables with a total of 36 subjects. Textbooks and a sample of research studies were reviewed to determine the most frequently taught and used multiple comparison procedures. This review indicates that most applied researchers are not aware of the alternative solutions when variances differ. It is suggested that the Games-Howell procedure will provide a valid test for most purposes and should be included in statistical methods textbooks and classes. Four tables present data from the application and the reviews. (SLD)

The two purposes of this investigation were to study the effects of variance heterogeneity on three selected multiple comparison procedures and to determine if either of two nonstandard methods would be superior to the conventional methods based on mean square within. The three procedures studied were the Wholly Significant Difference Test (WSD), The "S" test, and a simple multiple "t" test (MTT) procedure. The investigation was a computer simulation consisting of 1000 experiments with four independent samples of five data points. Six pairwise contrasts were considered. The four variance conditions (VC) constituted one factor of the design. Each of the six contrasts were tested using three methods. The three methods constituted a second factor in the two-factor design with VC crossed with method. Results are tabulated and discussed. (DB)

Conventional wisdom suggests the omnibus F-test needs to be significant before conducting post-hoc pairwise multiple comparisons. However, there is little empirical evidence supporting this practice. Protected tests are conducted only after a significant omnibus F-test while unprotected tests are conducted without regard to the significance of the omnibus F-test. Monte Carlo methods were used to generate replications expected to provide 0.95 confidence intervals of +/- 0.001 around the nominal alphas of 0.10, 0.05, and 0.01 for 42 combinations of "n" (5, 10, 15, 20, 30, 60, and 100) and numbers of groups (3, 4, 5, 6, 8, and 10). Unprotected and protected tests were conducted using the Dunn-Bonferroni, Dunn-Sidak, Holm, and Tukey's Honestly Significant Differences (HSD) procedures. Means and standard deviations of observed per-experiment Type I errors rates and percentages of observed per-experiment Type I error falling below, within, and above the 0.95 confidence intervals were determined for total number of Type I errors. Differences in observed Type I errors for sample size and number of groups was minimal. However, there were differences in Type I error control among the four multiple comparison procedures and when the tests were conducted as protected or unprotected. The Dunn-Bonferroni had the best control of Type I error as an unprotected test with 96.0% of the observed Type I errors falling within the 0.95 confidence interval while 87.3% of the observed Type I errors fell below the 0.95 confidence interval when used as a protected test, thus being very conservative. As unprotected tests, the Dunn-Sidak and Holm tended to be liberal, but were conservative as protected tests. The HSD was liberal in both situations, but much more so as an unprotected test. These results, combined with the ease of using the Dunn-Bonferroni, suggest this method may provide the most accurate and easiest control of per-experiment Type I error when used in an unprotected mode. (Contains 4 tables and 13 references.) (Author/SLD)

A resurgence of interest in multiple hypothesis testing has occurred in the last decade. Motivated by studies in genomics, microarrays, DNA sequencing, drug screening, clinical trials, bioassays, education and psychology, statisticians have been devoting considerable research energy in an effort to properly analyze multiple endpoint data. In response to new applications, new criteria and new methodology, many ad hoc procedures have emerged. The classical requirement has been to use procedures which control the strong familywise error rate (FWE) at some predetermined level \alpha. That is, the probability of any false rejection of a true null hypothesis should be less than or equal to \alpha. Finding desirable and powerful multiple test procedures is difficult under this requirement. One of the more recent ideas is concerned with controlling the false discovery rate (FDR), that is, the expected proportion of rejected hypotheses which are, in fact, true. Many multiple test procedures do control the FDR. A much earlier approach to multiple testing was formulated by Lehmann [Ann. Math. Statist. 23 (1952) 541-552 and 28 (1957) 1-25]. Lehmann's approach is decision theoretic and he treats the multiple endpoints problem as a 2^k finite action problem when there are k endpoints. This approach is appealing since unlike the FWE and FDR criteria, the finite action approach pays attention to false acceptances as well as false rejections.