Addressing Multiple Detection Limits with Semiparametric Cumulative Probability Models

Abstract

Detection limits (DLs), where a variable cannot be measured outside of a certain range, are common in research. DLs may vary across study sites or over time. Most approaches to handling DLs in response variables implicitly make strong parametric assumptions on the distribution of data outside DLs. We propose a new approach to deal with multiple DLs based on a widely used ordinal regression model, the cumulative probability model (CPM). The CPM is a rank-based, semiparametric linear transformation model that can handle mixed distributions of continuous and discrete outcome variables. These features are key for analyzing data with DLs because while observations inside DLs are continuous, those outside DLs are censored and generally put into discrete categories. With a single lower DL, CPMs assign values below the DL as having the lowest rank. With multiple DLs, the CPM likelihood can be modified to appropriately distribute probability mass. We demonstrate the use of CPMs with DLs via simulations and a data example. This work is motivated by a study investigating factors associated with HIV viral load 6 months after starting antiretroviral therapy in Latin America; 56% of observations are below lower DLs that vary across study sites and over time.

Keywords: HIV; Limit of detection; ordinal regression model; transformation model.

Fig. 1

The changes of most frequent DL values every year at each study site over time.

Fig. 2

Illustration of three approaches for conditional quantiles. The data set has a lower DL 0.5, an upper DL 2, and five observed values of y: 0.7, 0.86, 1, 1.5, 1.8. Thus S = {` < 0.5′, 0.7,0.86,1,1.5,1.8, `> 2′}. The dashed lines are for ${\hat{Q}}_1\left(p\right)$, the dotted lines are for ${\hat{Q}}_2\left(p\right)$, the solid black lines are for $\hat{Q}\left(p\right)$, and the solid gray lines are for the empirical CDF. Here, $\hat{Q}\left(p\right)={\hat{Q}}_1\left(p\right)=\mathrm{‘}<{0.5}^{\mathrm{\text{'}}}$ when $p<\hat{F}(0.5\mid x)$, and $\hat{Q}\left(p\right)={\hat{Q}}_2\left(p\right)=\mathrm{‘}>2^{\mathrm{\text{'}}}$ when $p>\hat{F}(2\mid x)$.

Fig. 3

The estimated conditional 50th and 90th percentiles of 6-month VL and the conditional probability of 6-month VL being greater than 1000 and 20 as functions of age (top row), prior AIDS events (middle row), and baseline VL (bottom row) while keeping other covariates at their medians (for continuous variables) or modes (for categorical variables) based on our method.