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.

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The problem of identifying the lowest dose level for which the mean response differs from that at the zero dose level is considered. A general framework for stepwise testing procedures that use contrasts among the dose level means is proposed. Using this framework, several new procedures are derived. These and some existing procedures, including that of Williams (1971, Biometrics 27, 103-117; 1972, Biometrics 28, 519-531), are compared analytically and by an extensive simulation study for the normal theory balanced one-way layout case. It is pointed out that the procedures based on the so-called step and basin contrasts proposed by Ruberg (1989, Journal of American Statistical Association 84, 816-822) have excessively high type I familywise error rates (FWEs) and, hence, they should not be used. Some findings of the simulation study are as follows: For monotone dose mean configurations, Williams' procedure and two step-down test procedures based on Helmert and linear contrasts offer the best performance. For nonmonotone dose mean configurations, the performance of Williams' procedure does degrade somewhat, but the other two procedures are still the best. For more complex designs, a simple step-down test procedure that uses any alpha-level tests (not necessarily t-tests) to compare each dose level with the zero dose level controls the FWE and is the only alternative available, but its power is rather low, especially under nonmonotone configurations. Step-up procedures are generally dominated by step-down procedures when the same contrasts are used although the differences are not great.

LINEAR RECONCILIATION PROBLEMProcess data reconciliation and rectification and their relationship to process performance monitoring functions have been the subject of many recent publications. [See, for instance, Mah (1981) for a review of these publications.] In this note we shall confine our attention to process data reconciliation subject to linear constraints, and more specifically, to the problem of detecting and identifying the presence of one or more gross errors in the process data.Generally speaking, process measurements are corrupted by two types of errors: Random errors which are commonly assumed to be independently and normally distributed with zero mean, and gross errors which are caused by non-random events such as instrument biases, malfunctioning measuring devices, incomplete or inaccurate process models. Let y be an (n X 1) vector of measured variables, b be a (p X l ) vector of unknown parameters, D be an (n x p) matrix of known constants, for which rank (D) = p 5 n , and E be an (n X 1) vector of errors distributed normally with a zero mean vector and a known variance-covariance matrix Q. Then in the absence of gross errors, the basic model isand the general linear reconciliation problem is the least-squares estimation of b subject to the linear constraintswhere A is a (q X p) matrix of known constants and c is a (q X 1)vector of known constants. The linear reconciliation problem formulated above is a generalization of the reconciliation problems treated by previous investigators. Thus, the reconciliation of flow and inventory data reported by Mah et al. (1976) is a special case in which y is the vector of measured flow rates (v in their paper), D is an identity matrix, b is the vector of true flow rates s(p), A is the incidence matrix (A), p = n and c = 0. Nogita (1972) treated essentially the same problem but considered only the diagonal terms (variances) of the covariance matrix in his minimization. Almasy and Sztano (1975) also studied this problem but they allowed c to be non-zero.On the other hand, the reactor data reconciliation problem reported by Madron et al. (1977) contains no constraints (Eq. 2) on b which corresponds to the vector of extents of chemical reactions (x). For that problem y is the measured vector of increases in the numbers of moles of species (n+), D is their (1 X I) matrix (AT) of stoichiometric coefficients, n = I = number of reactive species, 1 = number of independent chemical reactions, and Q is denoted by F_ in their paper. Madron et al. (1977) actually considered an rsubvector of n+ (denoted by n : in their paper) corresponding to the r d I species for which measurements were made. A similar problem was studied by Murthy (1973Murthy ( ,1974.

We show how a well-known multiple step-down significance testing procedure for comparing treatments with a control in balanced one-way layouts can be applied in unbalanced layouts (unequal sample sizes for the treatments). The method we describe has the advantage that it provides p-values, for each treatment versus control comparison, that take account of the multiple step-down testing nature of the procedure. These joint p-values can be used with any value of alpha, the fixed type I family wise error rate bound, that may be specified by the investigator. To determine the p-values, it is necessary to compute a multivariate Student t integral, for which a computer program is available. This procedure is more powerful than the step-down Bonferroni procedure of Holm and the single-step procedure of Dunnett. An example from the pharmaceutical literature is used to illustrate the procedure.

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This paper discusses a new class of multiple testing procedures, tree-structured gatekeeping procedures, with clinical trial applications. These procedures arise in clinical trials with hierarchically ordered multiple objectives, for example, in the context of multiple dose-control tests with logical restrictions or analysis of multiple endpoints. The proposed approach is based on the principle of closed testing and generalizes the serial and parallel gatekeeping approaches developed by Westfall and Krishen (J. Statist. Planning Infer. 2001; 99:25-41) and Dmitrienko et al. (Statist. Med. 2003; 22:2387-2400). The proposed testing methodology is illustrated using a clinical trial with multiple endpoints (primary, secondary and tertiary) and multiple objectives (superiority and non-inferiority testing) as well as a dose-finding trial with multiple endpoints.

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