Extending distributed lag models to higher degrees

Abstract

Distributed lag (DL) models relate lagged covariates to a response and are a popular statistical model used in a wide variety of disciplines to analyze exposure-response data. However, classical DL models do not account for possible interactions between lagged predictors. In the presence of interactions between lagged covariates, the total effect of a change on the response is not merely a sum of lagged effects as is typically assumed. This article proposes a new class of models, called high-degree DL models, that extend basic DL models to incorporate hypothesized interactions between lagged predictors. The modeling strategy utilizes Gaussian processes to counterbalance predictor collinearity and as a dimension reduction tool. To choose the degree and maximum lags used within the models, a computationally manageable model comparison method is proposed based on maximum a posteriori estimators. The models and methods are illustrated via simulation and application to investigating the effect of heat exposure on mortality in Los Angeles and New York.

Keywords: Dimension reduction; Gaussian process; Heat exposure; Lagged interaction; NMMAPS dataset.

Figure 1.

Simulations results as a function of the dimension reduction factor (). The solid, dashed, and dotted lines correspond to the simulations results when estimating the degree (model A), when treating it as fixed (model B), and when fitting the best model (model C), respectively. The number in the top right corner corresponds to the unconstrained maximum likelihood fit (model D). The first, second, and third column of plots correspond to results from the first-, second-, and third-degree DL surface, respectively. Results from model C are excluded from the second and third columns because only the first-degree surface was estimated.

Figure 2.

Estimated first-degree DL surfaces for (a) New York and (b) Los Angeles. Estimated second-degree DL coefficients for (a) New York and (b) Los Angeles. The solid points correspond to the posterior mean while the error bars represent a 95% central credible interval.

Figure 3.

Percent change in mortality as a function of the deviation from a temperature of F on days and . Allowing for interactions between lagged covariates models the curvature of contour lines.