Stratified Multivariate Mann–Whitney Estimators for the Comparison of Two Treatments with Randomization Based Covariance Adjustment

Abstract

Methodology for comparing two randomly assigned treatments for strictly ordinal response variables has been discussed throughout the literature on multivariate Mann-Whitney estimators with stratification adjustment. Although such estimators can be computed directly as weighted linear combinations of within-stratum Mann-Whitney estimators, consistent estimation of their covariance matrix is done using methods for multivariate U-statistics. The scope of these methods includes ways of managing randomly missing data and ways to invoke randomization-based covariance adjustment for no differences between treatments for background or baseline covariables. The assessment of treatment differences can be done using confidence intervals or statistical tests for the adjusted Mann-Whitney estimators. The methods in this article are illustrated using three examples. The first example is a randomized clinical trial with eight strata and a univariate ordinal response variable. The second example is a randomized clinical trial with four strata, two covariables, and four ordinal response variables. The third example is a randomized two-period crossover clinical trial with four strata, three covariables (as age, screening, first baseline), three response variables (as first period response, second baseline, second period response), and missing data. For these examples, the results are interpretable through the probability of better outcomes for one treatment over the other.

Analyzing multiple endpoints in a confirmatory randomized clinical trial—an approach that addresses stratification, missing values, baseline imbalance and multiplicity for strictly ordinal outcomes

“…However, when the sample sizes within strata are relatively small (eg, <6), the power of the van Elteran test statistic can have limitations. For this reason, adjusting baseline factors with randomization based covariance methods could be more useful than stratification adjustment . Another limitation of the van Elteran test statistic is that it does not provide a corresponding confidence interval.…”

“…Let , with , denote the difference between treatments for the stratification adjusted means for the m th covariate. Let V g denote a consistent estimate for the covariance matrix of g and let V ξ g denote a consistent estimate for the covariances of , and g on the basis of the methods in Appendix A.2 and a previous study . Then the covariance adjusted counterpart for is , and a consistent estimate for its covariance matrix is .…”

“…Excluding the patients with missing values from the analysis can potentially induce biased results and or reduce power. For missing response variables, several approaches have been explored in the applications of multivariate MW estimators assuming MCAR or other paradigms. Besides the approach presented in Section 2.1, which manages missing values as missing and assumes MCAR, other specifications are available in the sanon package.…”

sanon: AnRPackage for Stratified Analysis with Nonparametric Covariable Adjustment

“…For confirmatory randomized clinical trials with stratified designs for comparing two treatments, Kawaguchi et al (2011) propose stratified multivariate Mann-Whitney estimators as a useful structure for the analysis of strictly ordinal response variables. Their scope can address strata with at least minimal sample sizes (e.g., ≥ 16), and randomization-based covariance adjustment is possible.…”

“…The software implementing the method was developed in the R programming environment. The sanon package (Kawaguchi 2015) contains functions to apply the method as well as two example data sets which were used in Kawaguchi et al (2011) and three data sets from other papers. The main function in the package has a similar structure as functions in R for standard statistical methods such as lm for linear models.…”