Nonparametric Analysis of Multivariate Data in Factorial Designs with Nondetects: A Case Study with Microbiome Data

Abstract The term “nondetects” describes observations that are not fully observed because the true value is below a detection threshold—and can therefore not be precisely detected. One may also consider them a special case of left-censored data. Nondetects occur frequently, for instance, in life sciences research in medicine or microbiology. This article examines the use of nonparametric inference methods for multivariate data in factorial designs in situations where nondetects are present, and it evaluates their performance. The focus is on testing hypotheses regarding interaction and main factor effects. The nonparametric centerpiece of the methodology is assuming the nonparametric relative effect (probabilistic index) and its generalizations as the functional on which inference is built, along with the respective invariance properties of the resulting tests. On this basis, we apply and evaluate recently proposed nonparametric analogs to the following types of multivariate test statistics: (1) Wald-type statistic (WTS), (2) ANOVA-type statistic (ATS), (3) Lawley–Hotelling trace, (4) Wilks Lambda (Likelihood ratio), (5) Bartlett–Nanda–Pillai trace. Except for the WTS, all the mentioned methods are available through the R-package nparmd . Extensive simulations and a case study from the field of microbiology demonstrate that the proposed methods can handle commonly occurring rates of nondetects without substantial impairment of specificity and sensitivity.