Ranking procedures for matched pairs with missing data — Asymptotic theory and a small sample approximation

Weighted mean difference statistics for paired data in the presence of missing values

Missing data is a common issue in many biomedical studies. Under a paired design, some subjects may have missing values in either one or both of the conditions due to loss of follow-up, insufficient biological samples, etc. Such partially paired data complicate statistical comparison of the distribution of the variable of interest between the two conditions. In this article, we propose a general class of test statistics based on the difference in weighted sample means without imposing any distributional or model assumption. An optimal weight is derived from this class of tests. Simulation studies show that our proposed test with the optimal weight performs well and outperforms existing methods in practical situations. Two cancer biomarker studies are provided for illustration.

Estimation and Testing of Wilcoxon–Mann–Whitney Effects in Factorial Clustered Data Designs

Clustered data arise frequently in many practical applications whenever units are repeatedly observed under a certain condition. One typical example for clustered data are animal experiments, where several animals share the same cage and should not be assumed to be completely independent. Standard methods for the analysis of such data are Linear Mixed Models and Generalized Estimating Equations—however, checking their assumptions is not easy, especially in scenarios with small sample sizes, highly skewed, count, and ordinal or binary data. In such situations, Wilcoxon–Mann–Whitney type effects are suitable alternatives to mean-based or other distributional approaches. Hence, no specific data distribution, symmetric or asymmetric, is required. Within this work, we will present different estimation techniques of such effects in clustered factorial designs and discuss quadratic- and multiple contrast type-testing procedures for hypotheses formulated in terms of Wilcoxon–Mann–Whitney effects. Additionally, the framework allows for the occurrence of missing data: estimation and testing hypotheses are based on all-available data instead of complete-cases. An extensive simulation study investigates the precision of the estimators and the behavior of the test procedures in terms of their type-I error control. One real world dataset exemplifies the applicability of the newly proposed procedures.

Ranking procedures for repeated measures designs with missing data: Estimation, testing and asymptotic theory

We develop purely nonparametric methods for the analysis of repeated measures designs with missing values. Hypotheses are formulated in terms of purely nonparametric treatment effects. In particular, data can have different shapes even under the null hypothesis and therefore, a solution to the nonparametric Behrens-Fisher problem in repeated measures designs will be presented. Moreover, global testing and multiple contrast test procedures as well as simultaneous confidence intervals for the treatment effects of interest will be developed. All methods can be applied for the analysis of metric, discrete, ordinal, and even binary data in a unified way. Extensive simulation studies indicate a satisfactory control of the nominal type-I error rate, even for small sample sizes and a high amount of missing data (up to 30%). We apply the newly developed methodology to a real data set, demonstrating its application and interpretation.

The nonparametric Behrens-Fisher problem in partially complete clustered data

In randomized trials or observational studies involving clustered units, the assumption of independence within clusters is not practical. Existing parametric or semiparametric methods assume specific dependence structures within a cluster. Furthermore, parametric model assumptions may not even be realistic when data are measured in a nonmetric scale as commonly happens, for example, in quality-of-life outcomes. In this paper, nonparametric effect-size measures for clustered data that allow meaningful and interpretable probabilistic comparisons of treatments or intervention programs will be introduced. The dependence among observations within a cluster can be arbitrary. Point estimators along with their asymptotic properties for computing confidence intervals and performing hypothesis test will be discussed. Small sample approximations that retain some of the optimal asymptotic behaviors will be presented. In our setup, some clusters may involve observations coming from both intervention groups (referred to as complete clusters), while others may contain observations from one group only (referred to as incomplete clusters). In deriving the asymptotic theories, we do not impose any relation in the rate of divergence of the numbers of complete and incomplete clusters. Simulations show favorable performance of the methods for arbitrary combinations of complete and incomplete clusters. The developed nonparametric methods are illustrated using data from a randomized trial of indoor wood smoke reduction to improve asthma symptoms and a cluster-randomized trial for smoking cessation.

Rank-based two-sample tests for paired data with missing values

Two-sample location problem is one of the most encountered problems in statistical practice. The two most commonly studied subtypes of two-sample location problem involve observations from two populations that are either independent or completely paired, but a third subtype can oftentimes occur in practice when some observations are paired and some are not. Partially paired two-sample problems, also known as paired two-sample problems with missing data, often arise in biomedical fields when it is difficult for some invasive procedures to collect data from an individual at both conditions we are interested in comparing. Existing rank-based two-sample comparison procedures for partially paired data, however, do not make efficient use of all available data. In order to improve the power of testing procedures for this problem, we propose several new rank-based test statistics and study their asymptotic distributions and, when necessary, exact variances. Through extensive numerical studies, we show that the best overall power come from the proposed tests based on weighted linear combinations of the test statistics comparing paired data and the test statistics comparing independent data, using weights inversely proportional to their variances. We illustrate the proposed methods with a real data example from HIV research for prevention.