Semiparametric estimation of the attributable fraction when there are interactions under monotonicity constraints

Abstract

Background The population attributable fraction (PAF) is the fraction of disease cases in a sample that can be attributed to an exposure. Estimating the PAF often involves the estimation of the probability of having the disease given the exposure while adjusting for confounders. In many settings, the exposure can interact with confounders. Additionally, the exposure may have a monotone effect on the probability of having the disease, and this effect is not necessarily linear. Methods We develop a semiparametric approach for estimating the probability of having the disease and, consequently, for estimating the PAF, controlling for the interaction between the exposure and a confounder. We use a tensor product of univariate B-splines to model the interaction under the monotonicity constraint. The model fitting procedure is formulated as a quadratic programming problem, and, thus, can be easily solved using standard optimization packages. We conduct simulations to compare the performance of the developed approach with the conventional B-splines approach without the monotonicity constraint, and with the logistic regression approach. To illustrate our method, we estimate the PAF of hopelessness and depression for suicidal ideation among elderly depressed patients. Results The proposed estimator exhibited better performance than the other two approaches in the simulation settings we tried. The estimated PAF attributable to hopelessness is 67.99% with 95% confidence interval: 42.10% to 97.42%, and is 22.36% with 95% confidence interval: 12.77% to 56.49% due to depression. Conclusions The developed approach is easy to implement and supports flexible modeling of possible non-linear relationships between a disease and an exposure of interest.

Benchmarking the Observed Best Predictor

“…The second one is Benchmarked EBLUP according to You et al [11] . The third one is Benchmarked EBLUP using an augmented model according to Wang et al [7] . The fourth method is the original OBP by Jiang et al [1] The fifth and sixth method are the two benchmarked OBP’s, namely viz.…”

“…Study 1: The first simulation study is similar to the one used in Wang et al [7] . The study is designed to simulate a large national survey in which state estimates are of interest.…”

“…Study 2: As one can see, in case of the previous simulation, the benchmarked EBLUP by You and Rao [3] did not perform well. This was the exact scenario in case of the simulation study carried out by Wang et al [7] Now, in defense of their method, You et al [11] argued that it is so because of the way the Wang et al [7] use an augmented model in their simulation study. In the simulation study (Wang et al [7] ) the true model has the co-variate zi=popi0.2−popi0.2true¯ which is then missing in the misspecified fitted model.…”

“…You and Rao [6] discusses benchmarking methods for a unit level model. Wang et al [7] proposed two different ways of developing self-benchmarked predictors. We discuss these two methods here briefly.…”

“…1. Wang et al [7] proposed benchmarking to minimizing the overall loss function: Q(trueθ^ia)=ψiE(trueθ^ia−θi)2, where ψi’s are a chosen set of positive weights. When A is known, under the Fay-Herriot model consider the adjusted BLUP : …”