Two consistent estimators for the non-null variance of Wil-coxon-Mann-Whitney's statistic applied to grouped ordered data, are considered. The first is based on U-statistics and the sec-ond is ...obtained by the Delta method. Some examples are given to demonstrate the extent of error when using a null variance esti-mate for constructing confidence intervals. It appears that the two consistent estimates are very close, but may both be disting-uishably larger or smaller than the null variance estimate.
Simes' (1986) improved Bonferroni test is verified by simulations
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to control the α-level when testing the overall homogeneity hypothesis with all pairwise t statistics in a balanced parallel group ...design. Similarly, this result was found to hold (for practical purposes) in various underlying distributions other than the normal and in some unbalanced designs. To allow the use of step-up procedures based on pairwise t statistics, simulations were used to verify that Simes' test, when applied to testing multiple subset homogeneity hypotheses with pairwise t statistics also keeps the level ♣ α. Some robustness as above was found here too. Tables of the simulation results are provided and an example of a step-up Hommel-Shaffer type procedure with pairwise comparisons is given.
1. Introduction and Summary
Among the various procedures for simultaneous interval estimation, Tukey's T‐method gives the shortest confidence intervals for pairwise contrasts when it is applicable. ...(Cf. Scheffé, 1959, Ch. 3; Miller, 1966, Ch. 2.) However, the T‐method in its exact formulation (excluding one way anova, see Spjøtvoll and Stoline, 1973) is restricted to well‐balanced designs.
In this paper we discuss a generalized T‐type procedure for simultaneous interval estimation of all estimable parametric functions in general fixed effects, univariate linear models. The new method, which we subsequently abbreviate as the GT method, is an extension of a generalized T‐procedure for the unbalanced one‐way anova model given in Spjøtvoll and Stoline (1973).
In Section 2 we derive the new procedure in full‐rank and in less than full‐rank models. Some important features of the procedure are discussed in Section 3. In order to gain insight into the types of models where the new procedure is expected to produce satisfactory confidence intervals for pairwise contrasts, we illustrate, in Section 4, the application of the GT method in some unbalanced designs. In Section 5 we discuss a modified GT procedure which constitutes a combination of the GT1 in Hochberg (1974) and the GT methods. The proofs of some of the theorems are given in an Appendix for purposes of a convenient reading.
Procedures for multiple comparisons among treatment means, in analysis of covariance with a random concomitant, are considered. A Tukey-Kramer (TK)-type procedure is introduced for a conditional ...analysis and is compared with Thigpen and Paulson's (1974) unconditional procedure. It is first established (by simulation) that the TK procedure controls the unconditional familywise error rate at the nominal level set for the conditional procedure. In the second part of this work, the confidence interval lengths obtained by the two procedures are compared. It is found that the expected ratio of the confidence interval length obtained by the conditional method to that obtained by the unconditional method is generally less than 1. Moreover, the probability that the length of a TK interval will be larger than that of the Thigpen and Paulson procedure is (roughly) between .2 and .3.
A general procedure including a simple computational algorithm is presented for the construction of approximate (conservative) simultaneous prediction intervals for L ≥ 2 forecasts generated by an ...autoregressive integrated moving average (ARIMA) model. For models with a monotone non-increasing autocorrelation function of forecast errors (absolute value) further simplification in computations is obtained. Details are given for two important special cases, an AR(l) model and an exponential smoothing model. These two cases are also illustrated with data on hourly measures of viscosity from a chemical process.
A one factor design is considered where each of the experimental units is repeatedly measured under the same set of various factor levels. A simultaneous procedure is con¬sidered based on the maximal ...Student-t statistic for all pairwise comparisons. The goal of this work is to investigate the possibility of obtaining simple approximations of the quantiles of that statistic without any assumptions on the variance matrix of the estimated effects. Two approximations are considered which are more radical than the Bonferroni method. The results of our simulations clearly indicate that one of these two approximations does not control the family-wise error rate. The other approximation can not be rejected on such grounds. In fact, on extending a conjecture due to Tukey (1953) (and which is known to be true for some special cases) the second approximation will control the required error rate (if the generalized conjecture was true). Finally, the required critical points for the 'extended Tukey' procedure are shown (by simulation) to be nicely approximated by quantiles from the Studentized Maximum Modulus distribution which are well documented.
In this note we give a simple proof of the result that, in the symmetric version of Scheffe's mixed model for a balanced two-way layout, an exact T procedure for pair-wise comparisons between the ...levels of the fixed factor can be based on the interaction mean square as the Studentizing factor. Such a proof does not seem to be available in the literature and will hopefully remove the confusion caused by some textbooks that incorrectly prescribe the error mean square as the Studentizing factor.
