Statistical tests for latent class in censored data due to detection limit

Abstract

Measures of substance concentration in urine, serum or other biological matrices often have an assay limit of detection. When concentration levels fall below the limit, the exact measures cannot be obtained. Instead, the measures are censored as only partial information that the levels are under the limit is known. Assuming the concentration levels are from a single population with a normal distribution or follow a normal distribution after some transformation, Tobit regression models, or censored normal regression models, are the standard approach for analyzing such data. However, in practice, it is often the case that the data can exhibit more censored observations than what would be expected under the Tobit regression models. One common cause is the heterogeneity of the study population, caused by the existence of a latent group of subjects who lack the substance measured. For such subjects, the measurements will always be under the limit. If a censored normal regression model is appropriate for modeling the subjects with the substance, the whole population follows a mixture of a censored normal regression model and a degenerate distribution of the latent class. While there are some studies on such mixture models, a fundamental question about testing whether such mixture modeling is necessary, i.e. whether such a latent class exists, has not been studied yet. In this paper, three tests including Wald test, likelihood ratio test and score test are developed for testing the existence of such latent class. Simulation studies are conducted to evaluate the performance of the tests, and two real data examples are employed to illustrate the tests.

Keywords: Censored normal regression; Tobit model; Wald test; detection limit; latent class; likelihood ratio test; mixture Tobit model; score test.

Figure 1.

QQ plots of theoretical p-values and the corresponding empirical type I errors for the Tobit model without covariates and sample sizes 50, 100, 200, 500, and 1000.

Figure 2.

QQ plots of theoretical p-values and the corresponding empirical type I errors for the Tobit model with uniformly distributed predictors and sample sizes 50, 100, 200, 500, and 1000.

Figure 3.

QQ plots of theoretical p-values and the corresponding empirical type I errors for the Tobit model with normally distributed predictors and sample sizes 50, 100, 200, 500, and 1000.

Figure 4.

Power of detecting the latent class in mTobit model when there are no covariates for both Tobit model and the latent class.

Figure 5.

Power of detecting the latent class in mTobit model with uniform distributed covariate for Tobit component only.

Figure 6.

Power of detecting the latent class in mTobit model with uniform distributed covariate for both the Tobit component and the latent class component.

Figure 7.

Power of detecting the latent class in mTobit model with normal distributed covariate for Tobit component only.

Figure 8.

Power of detecting the latent class in mTobit model with normal distributed covariate for Tobit component only.