Order‐restricted inference for multivariate longitudinal data with applications to the natural history of hearing loss

Abstract

Multivariate outcomes are often measured longitudinally. For example, in hearing loss studies, hearing thresholds for each subject are measured repeatedly over time at several frequencies. Thus, each patient is associated with a multivariate longitudinal outcome. The multivariate mixed-effects model is a useful tool for the analysis of such data. There are situations in which the parameters of the model are subject to some restrictions or constraints. For example, it is known that hearing thresholds, at every frequency, increase with age. Moreover, this age-related threshold elevation is monotone in frequency, that is, the higher the frequency, the higher, on average, is the rate of threshold elevation. This means that there is a natural ordering among the different frequencies in the rate of hearing loss. In practice, this amounts to imposing a set of constraints on the different frequencies' regression coefficients modeling the mean effect of time and age at entry to the study on hearing thresholds. The aforementioned constraints should be accounted for in the analysis. The result is a multivariate longitudinal model with restricted parameters. We propose estimation and testing procedures for such models. We show that ignoring the constraints may lead to misleading inferences regarding the direction and the magnitude of various effects. Moreover, simulations show that incorporating the constraints substantially improves the mean squared error of the estimates and the power of the tests. We used this methodology to analyze a real hearing loss study.

Ordered Regressions

“…In this section, we analyse a longitudinal data set collected by the Israeli Ministry of Health as part of a larger study. These data were previously described by Carel & Brinker () and analysed in a different context by Davidov & Rosen () and Rosen & Davidov (). The data consist of two groups of male participants.…”

CLME: An R Package for Linear Mixed Effects Models under Inequality Constraints

“…CLME is designed to implement two general classes of statistical tests. The likelihood ratio type (LRT) statistic (Rosen and Davidov 2012) is the default setting, but the user may instead choose the Williams' type test statistic (Williams 1971(Williams , 1977Peddada, Prescott, and Conaway 2001;Farnan et al 2014). In both cases, to keep the methodology robust to non-normality and potential heteroscedasticity, the p values are evaluated using the residual bootstrap methodology developed in Farnan et al (2014).…”

“…Thus, although our likelihood ratio type statistic is motivated by the likelihood ratio principle under the normality assumption, it does not use the normal theory based asymptotic distribution for the test statistic. Hence we use the phrase "likelihood ratio type test" rather than "likelihood ratio test", and results from CLME will not always align with those of a direct implementation of Rosen and Davidov (2012). Farnan (2011) and later Farnan et al (2014) investigated the performance of the residual bootstrap based test using the above defined likelihood ratio type test (LRT) statistic and the following Williams' type statistic (W ) under a wide range of distributions and variance structures.…”

“…However, despite the existence of a large body of literature on constrained inference spanning over five decades and three books on testing for order restrictions (Barlow, Bartholomew, Brenner, and Brunk 1972;Robertson, Wright, and Dykstra 1988;Silvapulle and Sen 2005), it was only recently that researchers developed methods for performing constrained inference in linear mixed effects models (Rosen and Davidov 2012;Davidov and Rosen 2011;Farnan, Ivanova, and Peddada 2014). While Rosen and Davidov (2012) and Davidov and Rosen (2011) developed likelihood ratio based methods, Farnan et al (2014) developed a residual bootstrap based method that is designed to be robust to non-normality as well as to heteroscedasticity. Furthermore, Farnan et al (2014)'s methodology allows for modeling categorical as well as continuous covariates.…”

Joint modeling of multivariate hearing thresholds measured longitudinally at multiple frequencies

“…Advanced modeling approaches have been developed and assessed to address the limitations of standard univariate approaches (Davidov and Rosen, 2011; Fieuws and Verbeke, 2004, 2006; Rosen and Davidov, 2012; Verbeke et al, 2014; Verbeke, Spiessens and Lesaffre, 2001), but some lead to loss of efficiency (Fieuws and Verbeke, 2004, 2006) and none deal with inter-ear correlation, nesting, and right-left ear differences. Ignoring any right-left ear differences among individuals can lead to incorrect conclusions (Faraway, 2004).…”