Ziang Zhang, Patrick E. Brown, Meredith Franklin, and Samuel Perreault's contribution to the Discussion of ‘Methods for estimating the exposure–response curve to inform the new safety standards for fine particulate matter’ by Cork et al.

We would like to congratulate the authors on their commendable work investigating various exposure–response curve (ERC) estimation methods under different data-generating mechanisms. Simulations show that for the most part the design-based methods from the causal inference framework outperform their regression-based counterparts, although when the true ERC exhibits sub-linear behaviour (e.g. μsublinear,int⁠) the design-based entropy-GAM performs worse than the original regression splines. The authors conjecture that this exception may stem from an insufficient number of knots placed in the low-exposure region. Regression splines are unpenalized smoothers, where the balance between flexibility and overfitting is managed though the number and placement of knots. Penalized methods manage this trade-off using an estimated smoothing parameter that penalizes basis function coefficients, typically with a quadratic penalty (see Green & Silverman, 1994). As the number of bases increases, penalized splines converge to an Integrated Wiener process, a specific type of Gaussian process (IWP, see Zhang et al., 2024). We repeat the authors’ simulations of GAM and entropy-GAM models using three different smoothing specifications: (1) a penalized regression spline with four basis functions (what authors present); (2) an unpenalized regression spline with four basis functions; and (3) an approximation to a continuous IWP with 100 penalized basis functions. Smoothing parameters were estimated with maximum likelihood. Results are in Figure 1, showing absolute bias and root mean square error in Figures 1a and 1b respectively. Figure 2 from the original manuscript with additional smoothing models: penalized cubic splines with k=4 (leftmost symbols, red); unpenalized regression splines with k=4 (centre, green); and an approximated IWP with k=100 (right, blue). Shown are absolute bias (a) and root mean square error (b). Contrary to the authors’ conjecture, the poor performance of the entropy-based GAM is not caused by the placement of knots. The unpenalized regression splines shown as green symbols are in most cases a modest improvement on the two penalized models, while the 100 basis function penalized model in blue is comparable to (and in some cases slightly better than) the four basis function penalized model in red. Four basis functions appear to be sufficient and penalizing is unnecessary, although the true ERC used to simulate the data is very smooth and different results would be expected when the underlying ERC is rougher. Also, in Perreault et al. (2024), we found that prediction intervals and coverage probabilities can be highly sensitive to the smoothing method, even when point predictions are not. Are the authors advocating for regression splines with a modest number of knots as the preferred method for ERC estimation in air pollution studies? Relative to the other types of model assumptions (i.e. independent observations, accurate exposure assessment) is limiting the curvature of the ERC a minor concern? Are there circumstances where a more flexible model with a principled choice of smoothing parameter is warranted?

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