Constrained groupwise additive index models

Abstract

In environmental epidemiology, there is wide interest in creating and using comprehensive indices that can summarize information from different environmental exposures while retaining strong predictive power on a target health outcome. In this context, the present article proposes a model called the constrained groupwise additive index model (CGAIM) to create easy-to-interpret indices predictive of a response variable, from a potentially large list of variables. The CGAIM considers groups of predictors that naturally belong together to yield meaningful indices. It also allows the addition of linear constraints on both the index weights and the form of their relationship with the response variable to represent prior assumptions or operational requirements. We propose an efficient algorithm to estimate the CGAIM, along with index selection and inference procedures. A simulation study shows that the proposed algorithm has good estimation performances, with low bias and variance and is applicable in complex situations with many correlated predictors. It also demonstrates important sensitivity and specificity in index selection, but non-negligible coverage error on constructed confidence intervals. The CGAIM is then illustrated in the construction of heat indices in a health warning system context. We believe the CGAIM could become useful in a wide variety of situations, such as warning systems establishment, and multipollutant or exposome studies.

Keywords: Additive index models; Dimension reduction; Index; Linear constraints; Quadratic programming.

Fig. 1.

Estimated RMSE for different scenarios, varying the sample size (a), the correlation between variables (b), and the noise level (c). Note that the CGAIM and GAIM curves overlap each other at the bottom. Note the log scale.

Fig. 2.

Average sensitivity and specificity of index selection computed on the 1000 simulations for various number of true indices and error level.

Fig. 3.

Estimated coverage for both inference methods and various noise level. Vertical segments indicate 1 standard error of the coverage.

Fig. 4.

Average estimated for according to the true value . Segments indicate 2.5th and 97.5th percentile of estimated .

Fig. 5.

Resulting indices created in Montreal. Top row: weights for each selected index; bottom row: functions . Indices have been standardized over the range [0–1] for ease of comparison. Each column corresponds to one index. Vertical segments and dotted lines represent block bootstrap 95 confidence intervals.