A single-index model with multiple-links

Abstract

In a regression model for treatment outcome in a randomized clinical trial, a treatment effect modifier is a covariate that has an interaction with the treatment variable, implying that the treatment efficacies vary across values of such a covariate. In this paper, we present a method for determining a composite variable from a set of baseline covariates, that can have a nonlinear association with the treatment outcome, and acts as a composite treatment effect modifier. We introduce a parsimonious generalization of the single-index models that targets the effect of the interaction between the treatment conditions and the vector of covariates on the outcome, a single-index model with multiple-links (SIMML) that estimates a single linear combination of the covariates (i.e., a single-index), with treatment-specific nonparametric link functions. The approach emphasizes a focus on the treatment-by-covariates interaction effects on the treatment outcome that are relevant for making optimal treatment decisions. Asymptotic results for estimator are obtained under possible model misspecification. A treatment decision rule based on the derived single-index is defined, and it is compared to other methods for estimating optimal treatment decision rules. An application to a clinical trial for the treatment of depression is presented.

Keywords: Biosignature; Single-index models; Treatment effect modifier.

Figures

Figure 1:

The first panel shows the linear contrast Ct’s (ω = 0), the second panel the moderately nonlinear contrast Ct’s (ω = 0.5), and the third panel displays highly nonlinear contrast Ct’s (ω = 1). Data points are generated from model (11) with δ = 0 and p = 5. The fourth and the fifth panels show the linear (ν = 0) and the nonlinear main effect M (ν = 1), respectively.

Figure 2:

Boxplots of the proportion of correct decisions (PCD) of the treatment decision rules obtained from 200 training datasets for each of the four methods. Each panel corresponds to one of the six combinations of ω ∈ {0, 0.5, 1} and ν ∈ {0, 1}: the shape of the contrast functions Ct’s controlled by ω; the shape of the main effect function M controlled by ν; the number of predictors p ∈ {5, 10}. The sample sizes are n1 = n2 = 40.

Figure 3:. Depression randomized clinical trial:

For each of the 9 baseline covariates individually, treatment-specific spline approximated regression curves with 5 basis functions are overlaid on to the data points; the placebo group is the blue solid curve and the active drug group is the red dotted curve. The associated 95% confidence bands of the regression curves were also plotted.

Figure 4:. Depression randomized clinical trial:

Pair of estimated link functions (g1 and g2) obtained from SIMML with the “main effect adjusted” profile likelihood (first panel), SIMML with the (main effect un-adjusted) profile likelihood (second panel), and the linear GEM model estimated under the criterion maximizing the difference in the linear regression slopes (third panel), respectively, for the placebo group (blue solid curves) and the active drug group (red dotted curves). The 95% confidence bands were constructed conditioning on the single-index coefficient α. For each treatment group, the observed outcomes are plotted against the estimated single-index.

Figure 5:. Depression randomized clinical trial:

Top panel: Violin plots of the estimated values of treatment decision rules based on each of the individual covariates x1, …, x9, determined from univariate nonparametric and linear regressions, respectively, obtained from 500 randomly split testing sets (with higher values preferred). Bottom panel: The estimated index coefficients α1, …, α9, associated with the covariates x1, …, x9, and the 95% confidence intervals for each of the three methods, obtained from BCa bootstrap with 500 replications. An estimated significant coefficient is marked with * on top of each confidence interval.

Figure 6:. Depression randomized clinical trial:

Boxplots of the estimated values of treatment decision rules, obtained from the 500 randomly split testing sets (higher values are preferred). The estimated values (and the standard deviations) are given as follow. SIMML*: 9.34(2.68); SIMML: 8.72 (2.68); K-Index: 8.04 (2.69); K-LR: 8.36 (2.69); linear GEM (linGEM):8.22 (2.67); All placebo (PBO): 6.17 (2.63); All drug (DRG): 7.57 (2.67).