Bayesian semiparametric mixture Tobit models with left censoring, skewness, and covariate measurement errors

Abstract

Common problems to many longitudinal HIV/AIDS, cancer, vaccine, and environmental exposure studies are the presence of a lower limit of quantification of an outcome with skewness and time-varying covariates with measurement errors. There has been relatively little work published simultaneously dealing with these features of longitudinal data. In particular, left-censored data falling below a limit of detection may sometimes have a proportion larger than expected under a usually assumed log-normal distribution. In such cases, alternative models, which can account for a high proportion of censored data, should be considered. In this article, we present an extension of the Tobit model that incorporates a mixture of true undetectable observations and those values from a skew-normal distribution for an outcome with possible left censoring and skewness, and covariates with substantial measurement error. To quantify the covariate process, we offer a flexible nonparametric mixed-effects model within the Tobit framework. A Bayesian modeling approach is used to assess the simultaneous impact of left censoring, skewness, and measurement error in covariates on inference. The proposed methods are illustrated using real data from an AIDS clinical study. .

Keywords: measurement error; mixed-effects models; mixture Tobit models; skew distributions.

Figure 1

(a) Histogram of viral load on log scale; (b) Spaghetti plot of viral load on log scale measured from 44 patients in an AIDS clinical trial study. The horizontal line represents the detection limit at log(50).

Figure 2

Trace plots of iterations versus generated values for the first 110,000 iterations.

Figure 3

Plots of goodness-of-fit statistics for the three Models: (i) The first column shows the residual against the fitted values, (ii) the second has plots of the fitted against observed log(RNA) values, and (iii) the last column has the Q-Q plots of residuals.

Figure 4

Individual fitted curves of viral loads for three randomly selected patients based on Model I (Normal), Model II (skew-normal) and Model III (skew-t). The dotted sign indicates the observed values. The horizontal line represents the detection limit at log(50).

Figure 5

Histograms of (a) inaccurate raw data below LOD (dotted line), and predicted viral loads under (b) normal model, (c) skew-normal model, and (d) skew-t model.