Estimating and Interpreting Effects from Nonlinear Exposure-Response Curves in Occupational Cohorts Using Truncated Power Basis Expansions and Penalized Splines

Abstract

Truncated power basis expansions and penalized spline methods are demonstrated for estimating nonlinear exposure-response relationships in the Cox proportional hazards model. R code is provided for fitting models to get point and interval estimates. The method is illustrated using a simulated data set under a known exposure-response relationship and in a data application examining risk of carpal tunnel syndrome in an occupational cohort.

Figure 1

True exposure-response relationship used to simulate data (a). Histogram of the simulated exposure data (b). Kaplan-Meier estimates of the survival functions for five exposure groups (c).

Figure 2

Linear spline (a) and cubic spline (b) basis functions using knots at quartiles of the case exposures (k₁ = 3.0, k₂ = 5.5, and k₃ = 8.3).

Figure 3

Estimated ln(HR) and corresponding pointwise 95% confidence intervals using linear spline (a) and cubic spline (b) basis functions with knots at quartiles of the case exposures (k₁ = 3.0, k₂ = 5.5, and k₃ = 8.3).

Figure 4

Linear B-spline (a) and cubic B-spline (b) basis functions using equally spaced knots.

Figure 5

Estimated exposure-response curves on the natural logarithmic scale (logarithm of the hazard ratio) using truncated power basis functions and B-spline basis functions.

Figure 6

Estimated exposure-response curves on the natural logarithmic scale (logarithm of the hazard ratio) using penalized splines.

Figure 7

Unscaled (a) and scaled (b) plots of the martingale residuals versus exposure (SI) with Loess curves using various degrees of smoothing (0.4 to 2.0) from a Cox proportional hazards model with all covariates excluding the exposure variable. (b) is scaled to focus on the Loess curves. The distribution of the exposure variable is given in the rug plot on the x-axis.

Figure 8

Estimated exposure-response curves for carpal tunnel syndrome and strain index in a cohort of 569 workers. Rug plot is of cases.

Figure 9

p value functions for the risk of carpal tunnel syndrome at an exposure of 13.5 strain index units versus unexposed (a) and for the simulated cohort data at an exposure of x = 4.0 versus unexposed (b).

Figure 10

Estimated exposure-response curves on the natural logarithmic scale (logarithm of the hazard ratio) for simulated data with 41 cases in 500 observations (a, b, c) and with 40 cases in 5000 observations (d, e, f) using linear, linear splines, and linear B-splines (a, d), cubic spline and cubic B-splines (b, e), and penalized splines (c, f).