Ajit C. Tamhane

Multiple comparison procedures

PROCEDURES BASED ON CLASSICAL APPROACHES FOR FIXED--EFFECTS LINEAR MODELS WITH NORMAL HOMOSCEDASTIC INDEPENDENT ERRORS. Some Theory of Multiple Comparisons Procedure Fixed--effects Linear Models. Single--step Procedures for Pairwise and More General Comparisons Among All Treatments. Stepwise Procedures for Pairwise and More General Comparisons Among All Treatments. Procedures for Some Other Nonhierarchical Finite Families of Comparisons. Designing Experiments for Multiple Comparisons. PROCEDURES FOR OTHER MODELS AND PROBLEMS, AND PROCEDURES BASED ON ALTERNATIVE APPROACHES. Procedures for One--way Layouts with Unequal Variances. Procedures for Some Mixed--effects Models. Distribution--free and Robust Procedures. Some Miscellaneous Multiple Comparison Problems. Optimal Procedures Using Decision--theoretic, Bayesian, and Other Approaches. Appendixes. Tables. References. Index.

A Step-Up Multiple Test Procedure

Abstract We consider the problem of simultaneously testing k ≥ 2 hypotheses on parameters θ1, …, θk . In a typical application the θs may be a set of contrasts, for instance, a set of orthogonal contrasts among population means or a set of differences between k treatment means and a standard treatment mean. It is assumed that least squares estimators 1, …, k are available that are jointly normally distributed with a common variance (known up to a scalar, namely the error variance σ 2) and a common known correlation. An independent χ 2-distributed unbiased estimator of σ 2 is also assumed to be available. We propose a step-up multiple test procedure for this problem which tests the t statistics for the k hypotheses in order starting with the least significant one and continues as long as an acceptance occurs. (By contrast, the step-down approach, which is usually used, starts with the most significant and continues as long as a rejection occurs.) Critical constants required by this step-up procedure to co...

A Comparison of Procedures for Multiple Comparisons of Means with Unequal Variances

Abstract Nine procedures for multiple comparisons of means with unequal variances are reviewed. Modifications in some procedures are proposed either for improvement in their performance or easier implementation. A Monte Carlo sampling study is carried out for pair-wise differences as well as a few selected contrasts and the procedures are compared based on the results of this study. Recommendations for the choice of the procedures are given. Robustness of two procedures designed for homogeneous variances under violation of that assumption is also examined in the Monte Carlo study.

Multiple comparisons in model i one-way anova with unequal variances

A fixed effects one-way layout model of analysis of variance is considered where the variances are taken to be possibly unequal. Conservative single-stage procedures based on Banerjee’s method for the solution of the Behrens-Fisher problem are proposed for the following multiple comparisons problems: 1) all pairwise comparisons with a control population mean, and 2) all pairwise comparisons and all linear contrasts among the means. Since these procedures are likely to be very conservative in practice, approximate procedures based on Welch’s method for the solution of the Behrens-Fisher problem are suggested as alternatives. Monte Carlo studies indicate that the latter are much less conservative and hence may be better in practice. Both these sets of procedures need only the tables of the Student’s t-distribution for their application and are very simple to use. Exact two-stage procedures are proposed for the following multiple comparisons problems: 1) all pairwise comparisons and all linear contrasts amon...

Nonlinear partial least squares

We propose a new nonparametric regression method for high-dimensional data, nonlinear partial least squares (NLPLS), which is motivated by projection-based regression methods, e.g. PLS, projection pursuit regression and feedforward neural networks. The model takes the form of a composition of two functions. The first function in the composition projects the predictor variables onto a lower-dimensional curve or surface yielding scores, and the second predicts the response variable from the scores. We implement NLPLS with feedforward neural networks. NLPLS often will produce a more parsimonious model (fewer score vectors) than projection-based methods. We extend the model to multiple response variables and discuss situations when multiple response variables should be modeled simultaneously and when they should be modeled with separate regressions. We provide empirical results that evaluate the performances of NLPLS, projection pursuit, and neural networks on response variable predictions and robustness to starting values.

Data Reconciliation and Gross Error Detection in Chemical Process Networks

Measurements made on stream flows in a chemical process network are expected to satisfy mass and energy balance equations in the steady state. Because of the presence of random and possibly gross errors, these balance equations are not generally satisfied. The problems of how to reconcile the measurements so that they satisfy the constraints and how to use the reconciled values to detect gross errors are considered in this article. Reconciliation of measurements is usually based on weighted least squares estimation under constraints, and detection of gross errors is based on the residuals obtained in the reconciliation step. The constraints resulting from the network structure introduce certain identifiability problems in gross error detection. A thorough review of such methodologies proposed in the chemical engineering literature is given, and those methodologies are illustrated by examples. A number of research problems of potential interest to statisticians are outlined.

General multistage gatekeeping procedures

A general multistage (stepwise) procedure is proposed for dealing with arbitrary gatekeeping problems including parallel and serial gatekeeping. The procedure is very simple to implement since it does not require the application of the closed testing principle and the consequent need to test all nonempty intersections of hypotheses. It is based on the idea of carrying forward the Type I error rate for any rejected hypotheses to test hypotheses in the next ordered family. This requires the use of a so-called separable multiple test procedure (MTP) in the earlier family. The Bonferroni MTP is separable, but other standard MTPs such as Holm, Hochberg, Fallback and Dunnett are not. Their truncated versions are proposed which are separable and more powerful than the Bonferroni MTP. The proposed procedure is illustrated by a clinical trial example.

Incomplete Block Designs for Comparing Treatments Wth a Control: General Theory

In this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control. This class of designs is appropriate for comparing simultaneously p ≥ 2 test treatments with a control treatment (the so-called multiple comparisons with a control (MCC) problem) when the observations are taken in incomplete blocks of common size k < p + 1. For this problem we propose a new general class of incomplete block designs that are balanced with respect to (wrt) test treatments. We shall use the abbreviation BTIB to refer to such designs. We study their structure and give some methods of construction. A procedure for making exact joint confidence statements for this multiple comparisons problem is described. By using a new concept of admissibility of designs, it is shown how “inferior” designs can be eliminated from consideration, and attention limited to a small class of BTIB designs that can be constructed from so-called generator designs in the minimal complete class of such d...

STEP-UP MULTIPLE TESTING OF PARAMETERS WITH UNEQUALLY CORRELATED ESTIMATES

We consider the problem of simultaneously testing k > or = to 2 hypotheses on parameters theta(1), ..., theta(k) using test statistics t(1), ..., t(k) such that a specified familywise error rate alpha is achieved. Dunnett and Tamhane (1992a) proposed a step-up multiple test procedure, in which testing starts with the hypothesis corresponding to the least significant test statistic and proceeds towards the most significant, stopping the first time a significant test result is obtained (and rejecting the hypotheses corresponding to that and any remaining test statistics). The parameter estimates used in the t statistics were assumed to be normally distributed with a common variance, which was a known multiple of an unknown sigma(2), and known correlations which were equal. In the present article, we show how the procedure can be extended to include unequally correlated parameter estimates. Unequal correlations occur, for example, in experiments involving comparisons among treatment groups with unequal sample sizes. We also compare the step-up and step-down multiple testing approaches and discuss applications to some biopharmaceutical testing problems.

A Two-stage minimax procedure with screening for selecting the largest normal mean

The problem of selecting the normal population with the largest population mean when the populations have a common known variance is considered. A two-stage procedure is proposed which guarantees the same probability requirement using the indifference-zone approach as does the single-stage procedure of Bechhofer (1954). The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure, regardless of the configuration of the population means. The saving in expected total number of observations can be substantial, particularly when the configuration of the population means is favorable to the experimenter. The saving is accomplished by screening out “non-contending” populations in the first stage, and concentrating sampling only on “contending” populations in the second stage. The two-stage procedure can be regarded as a composite one whic...