Michael G. Akritas

The Rank Transform Method in Some Two-Factor Designs

Abstract The use of the rank transform for testing problems in some two-factor designs is considered. The rank transform procedure consists of replacing the observations by their ranks in the combined sample and performing one of the standard analysis of variance (ANOVA) procedures on these ranks. The asymptotic version of the rank transform is introduced, and its usefulness as a means for understanding the nature of the rank transform is examined. It is demonstrated that the asymptotic version of the transformation helps identify the testing problems where the rank transform method works and helps suggest which ANOVA procedure should be used on the ranks. This approach is applied on the balanced and unbalanced nested model and the balanced and unbalanced two-way layout with and without interaction. The extension of the rank transform method to some of these models is new. The proposed procedures do not share the known simplicity of existing rank transform statistics, but they do allow heteroscedasticity ...

Nonparametric Hypotheses and Rank Statistics for Unbalanced Factorial Designs

Abstract Factorial designs are studied with independent observations, fixed number of levels, and possibly unequal number of observations per factor level combination. In this context, the nonparametric null hypotheses introduced by Akritas and Arnold are considered. New rank statistics are derived for testing the nonparametric hypotheses of no main effects, no interaction, and no factor effects in unbalanced crossed classifications. The formulation of all results includes tied observations. Extensions of these procedures to higher-way layouts are given, and the efficacies of the test statistics against nonparametric alternatives are derived. A modification of the test statistics and approximations to their finite-sample distributions are also given. The small-sample performance of the procedures for two factors is examined in a simulation study. As an illustration, a real dataset with ordinal data is analyzed.

Non-parametric estimation of the residual distribution

Consider a heteroscedastic regression model Y=m(X) +σ(X)e, where the functions m and σ are “smooth”, and e is independent of X. An estimator of the distribution of e based on non-parametric regression residuals is proposed and its weak convergence is obtained. Applications to prediction intervals and goodness-of-fit tests are discussed.

Fully Nonparametric Hypotheses for Factorial Designs I: Multivariate Repeated Measures Designs

Abstract We introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models, such as the hypotheses of no main effects, no interaction effects, and no factor effects. These natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations. We concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing these hypotheses. These statistics are the rank transform (RT) versions of the classical statistics for testing hypotheses in repeated measures designs. We emphasize that even though recent research has demonstrated the inappropriateness of the RT method for many parametric hypotheses, the RT procedure is always valid for testing our nonparametric hypotheses. We show that the rank statistics converge in distribution to central chi-squared distributions under their respective nonparametric null hypotheses. The noncentrality paramet...

Bootstrapping the Kaplan-Meier estimator

Abstract Randomly censored data consist of iid pairs of observations (Xi, δi), i = 1, …, n; if δ i = 0, Xi denotes a censored observation, and if δ i = 1, Xi denotes an exact “survival” time, which is the variable of interest. For estimating the distribution F of the survival times, the product-limit estimator proposed by Kaplan and Meier (1958) has been studied extensively and it has been shown to enjoy a number of optimality properties (Wellner 1982). See Gill (1980, chaps. 1–4; 1983) for a modern treatment and for references. With censored data, bootstrapping can be carried out using two different resampling plans introduced by Efron (1981) and Reid (1981), respectively. With Efrons plan one takes a random sample with replacement from (X 1), …, (Xn , δ n ), whereas with Reids plan one takes a random sample from the Kaplan—Meier estimator. The purpose of this article is to study the asymptotic behavior of the bootstrapped Kaplan—Meier estimator with both resampling plans. The approach adopted uses the...

A unified approach to rank tests for mixed models

Abstract The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.

Limitations of the Rank Transform Procedure: A Study of Repeated Measures Designs, Part I

Abstract The applicability of the rank transform procedure to two-way repeated measures designs with equicorrelated errors is examined. All of the common testing problems with these models are examined. It is demonstrated that, with the exception of one testing problem, the rank transform procedure is not generally applicable. This complements the findings in Akritas (1990) for models with independent errors. The one valid rank transform statistic presented here does have the interesting feature that it allows the covariance of the equicorrelated errors to depend on the column treatment. This is due to the robustness of the corresponding F statistic to this kind of departure from the assumption of a common covariance matrix. It is shown, however, that the efficacy of this rank transform procedure has the undesirable property of depending on the model parameters.

Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests

Recent articles have considered the asymptotic behavior of the one-way analysis of variance (ANOVA) F statistic when the number of levels or groups is large. In these articles, the results were obtained under the assumption of homoscedasticity and for the case when the sample or group sizes ni remain fixed as the number of groups, a, tends to infinity. In this article, we study both weighted and unweighted test statistics in the heteroscedastic case. The unweighted statistic is new and can be used even with small group sizes. We demonstrate that an asymptotic approximation to the distribution of the weighted statistic is possible only if the group sizes tend to infinity suitably fast in relation to a. Our investigation of local alternatives reveals a similarity between lack-of-fit tests for constant regression in the present case of replicated observations and the case of no replications, which uses smoothing techniques. The asymptotic theory uses a novel application of the projection principle to obtain the asymptotic distribution of quadratic forms.

Asymptotics for Analysis of Variance When the Number of Levels is Large

Abstract We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.

Estimation of bivariate and marginal distributions with censored data

Consider a pair of random variables, both subject to random right censoring. New estimators for the bivariate and marginal distributions of these variables are proposed. The estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution. Both estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring. Such a method of estimation is proposed. The weak convergence of the estimators proposed is obtained. A small simulation study suggests that the estimators of the marginal and bivariate distributions perform well relatively to respectively the Kaplan-Meier estimator for the marginal distribution and the estimators of Pruitt and van der Laan for the bivariate distribution. The use of the estimators in practice is illustrated by the analysis of a data set.