Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes

Abstract

Regression models for continuous outcomes frequently require a transformation of the outcome, which is often specified a priori or estimated from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation by treating the continuous outcome as if it is ordered categorically. They thus represent a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where bounds, one lower and one upper, are chosen and the outcomes outside the bounds are set as two ordinal categories. We prove the uniform consistency of the estimated regression coefficients and of the estimated transformation function between the bounds. We also describe their joint asymptotic distribution, and show that the estimated regression coefficients attain the semiparametric efficiency bound. We show with simulations that results from this approach and those from using the CPM on the original data are very similar when a small fraction of the data are modified. We reanalyze a dataset of HIV-positive patients with CPMs to illustrate and compare the approaches.

Keywords: 62G99; asymptotic distribution; cumulative probability model; semiparametric transformation model; uniform consistency.

Figure 1.

Average estimate of $A\left(y\right)$ after fitting properly specified CPMs compared with the true transformation, $\text{log}\left(y\right)$. Gray curve: original data; black curve: modified data. Dashed lines are the diagonal. Top row: $(L,U)=\left(e^{-4},e^4\right)$; middle row: $(L,U)=\left(e^{-2},e^2\right)$; bottom row: $(L,U)=\left(e^{-1/2},e^{1/2}\right)$. Left to right: $n=\mathrm{100,1000,5000}$. Based on 1000 replications.

Figure 1.

Average estimate of $A\left(y\right)$ after fitting properly specified CPMs compared with the true transformation, $\text{log}\left(y\right)$. Gray curve: original data; black curve: modified data. Dashed lines are the diagonal. Top row: $(L,U)=\left(e^{-4},e^4\right)$; middle row: $(L,U)=\left(e^{-2},e^2\right)$; bottom row: $(L,U)=\left(e^{-1/2},e^{1/2}\right)$. Left to right: $n=\mathrm{100,1000,5000}$. Based on 1000 replications.

Figure 2.

Estimates of ${\beta }_1$ using data categorized outside $(L,U)$ compared with those using the original data and to the truth, ${\beta }_1=1$. Gray lines are mean estimates and dashed gray lines are the truth. Top row: $(L,U)=\left(e^{-4},e^4\right)$; middle row: $(L,U)=\left(e^{-2},e^2\right)$; bottom row: $(L,U)=\left(e^{-1/2},e^{1/2}\right)$. Left to right: $n=\mathrm{100,1000,5000}$. Based on 1000 replications.

Figure 2.

Estimates of ${\beta }_1$ using data categorized outside $(L,U)$ compared with those using the original data and to the truth, ${\beta }_1=1$. Gray lines are mean estimates and dashed gray lines are the truth. Top row: $(L,U)=\left(e^{-4},e^4\right)$; middle row: $(L,U)=\left(e^{-2},e^2\right)$; bottom row: $(L,U)=\left(e^{-1/2},e^{1/2}\right)$. Left to right: $n=\mathrm{100,1000,5000}$. Based on 1000 replications.

Figure 3.

(a) Histogram of CD4:CD8 ratio in our dataset. (b-d) Estimated outcome measures and 95% confidence intervals as functions of age, holding other covariates constant at their medians/modes. (b) Median CD4:CD8 ratio; (c) mean CD4:CD8 ratio; (d) probability that CD4:CD8 > 1.