A Unified Framework for Multiple Exposure Distributed Lag Non-Linear Models for Air Pollution Epidemiology

Alex Stringer; Glen McGee; Hwashin Hyun Shin; Tianyi Pan

arxiv: 2604.22692 · v1 · submitted 2026-04-24 · 📊 stat.ME · stat.AP

A Unified Framework for Multiple Exposure Distributed Lag Non-Linear Models for Air Pollution Epidemiology

Tianyi Pan , Hwashin Hyun Shin , Alex Stringer , Glen McGee This is my paper

Pith reviewed 2026-05-08 10:47 UTC · model grok-4.3

classification 📊 stat.ME stat.AP

keywords distributed lag non-linear modelsmultiple exposuresair pollution epidemiologymodel selectionmodel stackingrespiratory mortalityPM2.5O3

0 comments

The pith

A unified framework integrates four DLNM structures with estimation, AIC selection and stacking to assess multi-pollutant effects on mortality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unified framework that handles the complexities of time-lagged, non-linear associations across multiple air pollutants by bringing together two additive and two single-index DLNM variants. It supplies a common estimation method, derives an AIC for choosing among the structures, and extends model stacking to combine results without committing to one specification. These tools are demonstrated on Ontario mortality counts, where individual models give different estimates but stacking identifies clear links between respiratory mortality and mixtures of PM2.5, O3 and NO2. The framework applies to general outcome types and scales to datasets with over 100,000 observations.

Core claim

The authors propose a unified framework for multiple-exposure DLNMs that covers four model structures—two additive and two single-index—together with estimation procedures applicable to all, an AIC criterion for selection, and a scalable stacking method for combining inferences. In the Ontario analysis this produces varying exposure-response estimates across the four models, yet the stacked approach detects significant associations between respiratory mortality and the pollutant mixture of PM2.5, O3 and NO2.

What carries the argument

Four DLNM structures (two additive, two single-index) together with AIC-based selection and model stacking.

If this is right

Different DLNM structures applied to the same data can produce noticeably different estimates of pollutant effects.
AIC provides a practical criterion for selecting among the candidate DLNMs.
Model stacking yields combined inferences that remain stable even when individual models disagree.
The approach scales to large observational datasets and applies to count outcomes such as daily mortality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Epidemiological analyses of pollutant mixtures may become less sensitive to arbitrary model choice when stacking is used.
The same machinery could be applied to other health endpoints or to mixtures that include additional pollutants.
Explicit comparison of additive versus single-index structures may highlight which features of the exposure-response surface are most important in a given setting.

Load-bearing premise

The true lagged non-linear relationships among multiple pollutants and mortality are adequately captured by the four candidate DLNM structures or their stack.

What would settle it

A simulation in which data are generated from a known multi-pollutant exposure-response surface outside the four structures, and the stacked estimator fails to recover the true associations or yields misleading significance statements.

Figures

Figures reproduced from arXiv: 2604.22692 by Alex Stringer, Glen McGee, Hwashin Hyun Shin, Tianyi Pan.

**Figure 1.** Figure 1: Results of Simulation C. The RMSE and coverage of the quantity of interest view at source ↗

**Figure 2.** Figure 2: Confidence intervals from the first 50 replicates. In panel (a) the single-index view at source ↗

**Figure 3.** Figure 3: Relative mortality reduction under a 10% exposure reduction for three types view at source ↗

**Figure 4.** Figure 4: CD-specific relative all-cause mortality reduction under a 10% exposure reduction view at source ↗

**Figure 5.** Figure 5: Year-specific relative all-cause mortality reduction under a 10% exposure reduction. view at source ↗

read the original abstract

This study quantifies the association between air pollution and mortality in Ontario, Canada. Exposure-response relationships in air pollution epidemiology are complex due to three features: time-lagged associations, non-linear associations, and multiple pollutants. To address the first two features, two distinct classes of distributed lag non-linear model (DLNM) have been proposed, but extending them to multiple exposures and selecting an appropriate model remain challenging. We propose a unified framework for multiple exposure DLNMs, integrating model specification, estimation, selection and stacking. The framework applies to four different model structures: two additive and two proposed single-index DLNMs, all applicable to general outcome types, including the mortality counts in the motivating application. We develop an estimation approach that applies to all four models. Choosing among the candidate DLNMs is challenging a priori, and we derive an AIC to select among them. As an alternative to selecting a single model, we also extend a model stacking approach to combine inferences across the four DLNMs and propose an implementation scalable to our dataset with 106,346 observations. In the motivating analysis, the four DLNMs yield different estimates, and the proposed stacking approach identifies significant associations between respiratory mortality and a mixture of PM2.5, O3 and NO2.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds two single-index DLNM forms plus a stacking pipeline for multi-pollutant lagged effects, but the abstract shows no validation details and leaves misspecification risk unaddressed.

read the letter

The main point is that the authors give a unified way to handle multiple air pollutants with lags and non-linearity by defining four DLNM structures, two of them new single-index versions, and then providing estimation, AIC selection, and stacking to combine them. They apply the whole thing to Ontario mortality counts and report that stacking still finds significant links for the PM2.5-O3-NO2 mix even when the four models disagree on the curves. That addresses a real practical headache for people who cannot pick one structure in advance. The framework is written to work for general outcomes and scales to over 100k observations, which is a clear plus for applied work. The stacking extension and the single-index proposals are the concrete increments over existing DLNM literature. The soft spots are the missing checks. The abstract gives no derivation or simulation support for the AIC, no diagnostics for the single-index assumptions, and no sensitivity results for how stacking behaves under different dependence patterns. The stress-test point is fair: if cross-pollutant interactions or lag-specific synergies sit outside the span of these four forms, both selection and stacking can produce significant results without any internal warning. That makes the claim of improved inference rest on an untested coverage assumption. Readers already using DLNMs in environmental epidemiology will get the most from this and can see exactly where to test or extend it. The work is concrete enough and the problem common enough that it deserves a serious referee, even if the validation sections will need expansion. I would send it for peer review.

Referee Report

3 major / 2 minor

Summary. The paper proposes a unified framework for multiple-exposure distributed lag non-linear models (DLNMs) that integrates model specification for four structures (two additive, two single-index), a common estimation procedure, AIC-based selection, and an extended stacking method for combining inferences. Applied to Ontario air pollution data (106,346 observations) on PM2.5, O3, and NO2 with respiratory mortality, the four DLNMs produce differing estimates while the stacking procedure identifies significant mixture associations.

Significance. If the framework holds, it offers a practical advance for air pollution epidemiology by handling lagged non-linear multi-pollutant effects without forcing a single model choice, with stacking providing robustness. The application demonstrates potential for detecting real associations in large datasets, and the scalable implementation is a strength. However, significance is limited by the absence of explicit validation for the AIC derivation, single-index model assumptions, and sensitivity to unmodeled interactions.

major comments (3)

[§3 and §4] §3 (model structures) and §4 (estimation): the central claim that the two additive and two single-index DLNMs plus AIC/stacking suffice for complex multi-pollutant lagged effects rests on an untested assumption; no simulation study or diagnostic is shown for data-generating processes with cross-pollutant lag interactions outside these four forms, which directly undermines the reliability of the reported significant associations from stacking.
[§4.2] §4.2 (AIC derivation): the AIC is stated to be derived for all four models including single-index DLNMs, but the explicit formula, effective degrees of freedom calculation for the non-linear lag components, and handling of the single-index constraint are not provided; this is load-bearing because model selection and the subsequent stacking weights depend on it.
[Application/results] Application section (results on stacking): the stacking identifies significant associations for the PM2.5/O3/NO2 mixture, yet no sensitivity analysis to the choice of the four candidate structures or to multiple-testing adjustment across lags/exposures is reported; this weakens the inference claim given that the four models already yield different estimates.

minor comments (2)

[Methods] The abstract and methods mention scalability to 106,346 observations but provide no explicit description of the computational implementation (e.g., optimization routine or cross-validation scheme for stacking weights).
[§3] Notation for the single-index DLNM structures could be clarified with an explicit equation showing how the index combines the multiple exposures.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's comments. We appreciate the detailed feedback, which has helped us identify areas for improvement in the manuscript. Below, we address each major comment point by point, indicating where revisions will be made.

read point-by-point responses

Referee: [§3 and §4] §3 (model structures) and §4 (estimation): the central claim that the two additive and two single-index DLNMs plus AIC/stacking suffice for complex multi-pollutant lagged effects rests on an untested assumption; no simulation study or diagnostic is shown for data-generating processes with cross-pollutant lag interactions outside these four forms, which directly undermines the reliability of the reported significant associations from stacking.

Authors: We thank the referee for highlighting this important point. Our proposed framework is intended to encompass a range of commonly used model structures for multiple-exposure DLNMs, specifically the additive and single-index forms, which allow for flexible modeling of lagged non-linear effects. The stacking method is presented as a way to combine inferences without committing to a single structure a priori. However, we recognize that data-generating processes involving interactions not captured by these forms could affect performance. To address this, we will include a simulation study in the revised manuscript that evaluates the methods under scenarios with cross-pollutant lag interactions beyond the four structures. This will provide empirical support for the reliability of the stacking inferences in the application. revision: yes
Referee: [§4.2] §4.2 (AIC derivation): the AIC is stated to be derived for all four models including single-index DLNMs, but the explicit formula, effective degrees of freedom calculation for the non-linear lag components, and handling of the single-index constraint are not provided; this is load-bearing because model selection and the subsequent stacking weights depend on it.

Authors: We agree that the AIC derivation requires more explicit detail. In the revised version, we will provide the full AIC formula for each of the four models, including the calculation of effective degrees of freedom for the non-linear lag components (based on the trace of the smoother matrix) and the adjustment for the single-index constraint in the penalized likelihood framework. This will ensure transparency in how model selection and stacking weights are determined. revision: yes
Referee: [Application/results] Application section (results on stacking): the stacking identifies significant associations for the PM2.5/O3/NO2 mixture, yet no sensitivity analysis to the choice of the four candidate structures or to multiple-testing adjustment across lags/exposures is reported; this weakens the inference claim given that the four models already yield different estimates.

Authors: We acknowledge the need for sensitivity analyses in the application section. In the revision, we will conduct and report sensitivity checks by varying the inclusion of the four candidate structures in the stacking procedure and examining the stability of the significant mixture associations. Regarding multiple-testing adjustment, while the primary inferences are based on the stacked estimates which integrate across models, we will add a discussion of potential multiplicity issues across lags and exposures and consider appropriate adjustments, such as controlling the family-wise error rate for key findings. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the unified DLNM framework

full rationale

The paper proposes a unified framework integrating specification, estimation, AIC-based selection, and stacking across four DLNM structures (two additive, two single-index) for multiple exposures. These elements are developed as methodological extensions with new derivations for AIC and scalable stacking implementation, applied to real mortality data to detect associations. No steps reduce by construction to fitted inputs, self-citations, or renamings; the central claims rest on independent model proposals and procedures rather than tautological reductions. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The framework implicitly assumes standard DLNM basis functions and smoothness penalties exist and are appropriate, but these are not detailed.

pith-pipeline@v0.9.0 · 5535 in / 1188 out tokens · 31495 ms · 2026-05-08T10:47:28.521226+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Bell, B. M. (2024), ‘Cppad: A c++ algorithmic differentiation package’. Bell, M. L., McDermott, A., Zeger, S. L., Samet, J. M. & Dominici, F. (2004), ‘Ozone and short-term mortality in 95 us urban communities, 1987-2000’,Jama292(19), 2372–2378. Bernardo, J. M., Smith, A. F. & Berliner, M. (1994),Bayesian theory, Vol. 586, Wiley Online Library. Canadian Co...

work page 2024
[2]

Environment and Climate Change Canada (2024), ‘National climate data and information archive’

Available at: www.cana 30 da.ca/en/environment-climate-change/services/environmental-indicators/air-health- trends.html. Environment and Climate Change Canada (2024), ‘National climate data and information archive’. Available at: https://climate.weather.gc.ca/. Farhat, N., Ramsay, T., Jerrett, M. & Krewski, D. (2013), ‘Short-term effects of ozone and pm2....

work page doi:10.1093/ije/18.1.269 2024
[3]

Stieb, D

32 Statistics Canada (2023), ‘Canadian Vital Statistics – Death database (CVSD)’, https: //www23.statcan.gc.ca/imdb/p2SV.pl?Function=getSurvey&SDDS=3233. Stieb, D. M., Burnett, R. T., Smith-Doiron, M., Brion, O., Shin, H. H. & Economou, V. (2008), ‘A new multipollutant, no-threshold air quality health index based on short- term associations observed in da...

work page arXiv 2023
[4]

World Health Organization (2021), ‘Who global air quality guidelines: Particulate matter (pm2.5 and pm10), ozone, nitrogen dioxide, sulfur dioxide and carbon monoxide’

33 World Health Organization (2019), ‘International classification of diseases, 10th revision’. World Health Organization (2021), ‘Who global air quality guidelines: Particulate matter (pm2.5 and pm10), ozone, nitrogen dioxide, sulfur dioxide and carbon monoxide’. Yao, Y., Vehtari, A., Simpson, D. & Gelman, A. (2018), ‘Using stacking to average bayesian p...

work page 2019
[5]

Rates are daily counts per 100,000 people

Mortality data for individuals aged under 1 year are not available, and the corresponding population is excluded. Rates are daily counts per 100,000 people. Population values are annual averages, and all other values are daily averages. All-cause Circulatory Respiratory Pollutants Census division Population (×103) No. Rate No. Rate No. Rate PM 2.5 (µgm−3)...

work page 2001
[6]

PM2.5 O3 NO2 2002 2004 2006 2008 2010 2012 2014 2016 2002 2004 2006 2008 2010 2012 2014 2016 2002 2004 2006 2008 2010 2012 2014 2016−1 0 1 Time f(Time) Toronto Peel Y ork Ottawa Durham Hamilton Waterloo Halton Middlesex Simcoe Niagara Essex Greater Sudbury Thunder Bay Brant Peterborough Lambton Algoma Haldimand−Norfolk Nipissing Figure S7: Long-term trend...

work page 2002
[7]

Correlations are calculated using residuals from the generalized additive models after removing long-term trends and seasonality

37 0.470.270.07 0.460.32−0.16 0.440.53−0.01 0.460.390.01 0.490.120.02 0.590.23−0.05 0.450.470.00 0.370.50−0.14 0.420.600.13 0.470.450.12 0.480.23−0.05 0.530.300.04 0.310.51−0.05 0.430.46−0.10 0.520.410.06 0.490.480.11 0.260.44−0.07 0.460.560.09 0.470.41−0.01 0.480.46−0.01 Nipissing Haldimand−Norfolk Algoma Lambton Peterborough Brant Thunder Bay Greater Su...

work page 2002
[8]

Plugging this into Equation S.8, we have α= [ 1,α∗⊤ ]⊤ / ([ 1,α∗⊤ ]⊤[ 1,α∗⊤ ])1/2 which is the reparameterization used in literature

It follows thatBα= [rq1 α, Q+ α] =I M. Plugging this into Equation S.8, we have α= [ 1,α∗⊤ ]⊤ / ([ 1,α∗⊤ ]⊤[ 1,α∗⊤ ])1/2 which is the reparameterization used in literature. C Ranges of Index and Cumulative Index This Appendix Section provides details of the ranges of indexEi(t;α)and cumulative index EL i (t;α,w). By the Cauchy-Schwarz inequality, for a gi...

work page 2025
[9]

with a step-halving strategy (Pan et al. 2025). We estimateλand θby maximizing the Laplace approximate marginal likelihood 40 (LAML; Wood et al. 2016). The log-LAML is L∗ LA(λ,θ) =L(ˆϕ(λ,θ),ˆγ(λ,θ) ;λ,θ)−1 2 log { detH (ˆϕ(λ,θ),ˆγ(λ,θ) ;λ,θ )} +1 2 log|Sλ|++C, where H (ˆϕ(λ,θ),ˆγ(λ,θ) ;λ,θ ) is the negative Hessian ofL with respect to(ϕ,γ)eval- uated at (...

work page 2025
[10]

See supplementary Section E.3 for details

in C++ and implicit differentiation. See supplementary Section E.3 for details. Unlike the forward-mode AD used for single-exposure ACE-DLNM Pan et al. (2025), we employ reverse-mode AD to handle the large number of parameters in our models. Finally, we obtain ˆϕ=ˆϕ(ˆλ,ˆθ)andˆγ=ˆγ(ˆλ,ˆθ). E Derivative Computation E.1 Derivative ofL(ϕ,γ;λ,θ) We consider th...

work page 2025
[11]

Therefore, E∗ {∫ log{φ(y;ˆu)}φ∗(y)dy } ≈ ∫ log{φ(y;u∗)}φ∗(y)dy−1 2nE∗ { ( ˆu−u∗)⊤I∗( ˆu−u∗) } , (S.11) where I∗=−n ∫ ∂2 logφ(y;u∗) ∂u∂u⊤ φ∗(y)dy Plugging Equation S.11 into Equation S.10 leads to E∗{D(φˆu,φ∗)}≈D(φu∗,φ∗) + 1 2nE∗ { ( ˆu−u∗)⊤I∗( ˆu−u∗) } (S.12) We aim to choose the candidate modelφminimizingE ∗{D(φˆu,φ∗)}, an unknown 43 quantity to be estim...

work page 2016
[12]

by estimating the second term E∗ { ( ˆu−E∗( ˆu))( ˆu−E∗( ˆu))⊤ } using( ˆI+S λ)−1ˆK( ˆI+S λ)−1, where ˆK= N∑ i=1 ∑ t∈Ti [∂logp(Yit|xit,zit; ˆu)/∂u] [∂logp(Yit|xit,zit; ˆu)/∂u]⊤. Plugging this estimator into Equation S.13 yields the proposed AIC in a simplified form, AICcond =−2l(ˆu) + tr [ ( ˆI+S λ)−1ˆK ] .(S.14) F.3 Adjusted AIC The conditional AIC does ...

work page 2016
[13]

We included the seasonal trendh(t) = 0.5 sin(t/150)

We consider three versions ofw and f shown in Figure S9. We included the seasonal trendh(t) = 0.5 sin(t/150). 0.0 0.2 0.4 0.6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Lag Weight w Type1 Type2 Type3 (a) Truew 0 2 4 6 0 5 Adaptive Cumulative Exposure ACERF f Type1 Type2 Type3 (b) Truef Figure S9: The true weight functionsw and association functionsf used in Simul...

work page 2000
[14]

71 Index 5 10 15 20 25 30 35 40 Lag 0 2 4 6 8 10 12 14 −0.004 −0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Additive DRF−DLNM: All−Cause Mortality Point Estimate: NO2 Index 5 10 15 20 25 30 35 40 Lag 0 2 4 6 8 10 12 14 −0.004 −0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Additive DRF−DLNM: All−Cause Mortality Lower 95% CI: NO...

work page 2002
[15]

We also examine the randomized quantile residual plot, QQ-plot and autocorrelation function plot for each CD, and present the results for Waterloo and Toronto as examples

plots and the QQ- plot across all CDs. We also examine the randomized quantile residual plot, QQ-plot and autocorrelation function plot for each CD, and present the results for Waterloo and Toronto as examples. The diagnostics suggest adequate model fit. The diagnostics for all-cause mortality in Toronto are the least favorable among all outcomes and CDs,...

work page 2013
[16]

It is difficult to define two fixed exposure levels that consistently represent relatively high and low levels over time. We observe that the estimates are sensitive to the choice of quantiles: quantiles based on pooled distributions lead to significant negative associations between air pollution and all-cause mortality (Figure S60), while quantiles based...

work page 2001
[17]

The boxplot for the overall distribution across all years is shown in blue

0 10 20 30 40 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Overall Y ear NO2 Distribution of NO2 Concentrations by Y ear Figure S59: Boxplots of NO2 concentrations by year. The boxplot for the overall distribution across all years is shown in blue. 85 0.925 0.950 0.975 1.000 1.025 1.050 Single−Index ACE−DLNM (0.25) Single−Ind...

work page 2001
[18]

The point estimates and 95% confidence intervals are obtained using model stacking

86 H.5 CD-Specific Associations Nipissing Haldimand−Norfolk Algoma Lambton Peterborough Brant Thunder Bay Greater Sudbury Essex Niagara Simcoe Middlesex Halton Waterloo Hamilton Durham Ottawa Y ork Peel Toronto −1.0% −0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Stacking Estimate PM2.5 O3 NO2 Overall Respiratory Mortality − Stacking Estimate Relative Mortality...

work page 2001

[1] [1]

Bell, B. M. (2024), ‘Cppad: A c++ algorithmic differentiation package’. Bell, M. L., McDermott, A., Zeger, S. L., Samet, J. M. & Dominici, F. (2004), ‘Ozone and short-term mortality in 95 us urban communities, 1987-2000’,Jama292(19), 2372–2378. Bernardo, J. M., Smith, A. F. & Berliner, M. (1994),Bayesian theory, Vol. 586, Wiley Online Library. Canadian Co...

work page 2024

[2] [2]

Environment and Climate Change Canada (2024), ‘National climate data and information archive’

Available at: www.cana 30 da.ca/en/environment-climate-change/services/environmental-indicators/air-health- trends.html. Environment and Climate Change Canada (2024), ‘National climate data and information archive’. Available at: https://climate.weather.gc.ca/. Farhat, N., Ramsay, T., Jerrett, M. & Krewski, D. (2013), ‘Short-term effects of ozone and pm2....

work page doi:10.1093/ije/18.1.269 2024

[3] [3]

Stieb, D

32 Statistics Canada (2023), ‘Canadian Vital Statistics – Death database (CVSD)’, https: //www23.statcan.gc.ca/imdb/p2SV.pl?Function=getSurvey&SDDS=3233. Stieb, D. M., Burnett, R. T., Smith-Doiron, M., Brion, O., Shin, H. H. & Economou, V. (2008), ‘A new multipollutant, no-threshold air quality health index based on short- term associations observed in da...

work page arXiv 2023

[4] [4]

World Health Organization (2021), ‘Who global air quality guidelines: Particulate matter (pm2.5 and pm10), ozone, nitrogen dioxide, sulfur dioxide and carbon monoxide’

33 World Health Organization (2019), ‘International classification of diseases, 10th revision’. World Health Organization (2021), ‘Who global air quality guidelines: Particulate matter (pm2.5 and pm10), ozone, nitrogen dioxide, sulfur dioxide and carbon monoxide’. Yao, Y., Vehtari, A., Simpson, D. & Gelman, A. (2018), ‘Using stacking to average bayesian p...

work page 2019

[5] [5]

Rates are daily counts per 100,000 people

Mortality data for individuals aged under 1 year are not available, and the corresponding population is excluded. Rates are daily counts per 100,000 people. Population values are annual averages, and all other values are daily averages. All-cause Circulatory Respiratory Pollutants Census division Population (×103) No. Rate No. Rate No. Rate PM 2.5 (µgm−3)...

work page 2001

[6] [6]

PM2.5 O3 NO2 2002 2004 2006 2008 2010 2012 2014 2016 2002 2004 2006 2008 2010 2012 2014 2016 2002 2004 2006 2008 2010 2012 2014 2016−1 0 1 Time f(Time) Toronto Peel Y ork Ottawa Durham Hamilton Waterloo Halton Middlesex Simcoe Niagara Essex Greater Sudbury Thunder Bay Brant Peterborough Lambton Algoma Haldimand−Norfolk Nipissing Figure S7: Long-term trend...

work page 2002

[7] [7]

Correlations are calculated using residuals from the generalized additive models after removing long-term trends and seasonality

37 0.470.270.07 0.460.32−0.16 0.440.53−0.01 0.460.390.01 0.490.120.02 0.590.23−0.05 0.450.470.00 0.370.50−0.14 0.420.600.13 0.470.450.12 0.480.23−0.05 0.530.300.04 0.310.51−0.05 0.430.46−0.10 0.520.410.06 0.490.480.11 0.260.44−0.07 0.460.560.09 0.470.41−0.01 0.480.46−0.01 Nipissing Haldimand−Norfolk Algoma Lambton Peterborough Brant Thunder Bay Greater Su...

work page 2002

[8] [8]

Plugging this into Equation S.8, we have α= [ 1,α∗⊤ ]⊤ / ([ 1,α∗⊤ ]⊤[ 1,α∗⊤ ])1/2 which is the reparameterization used in literature

It follows thatBα= [rq1 α, Q+ α] =I M. Plugging this into Equation S.8, we have α= [ 1,α∗⊤ ]⊤ / ([ 1,α∗⊤ ]⊤[ 1,α∗⊤ ])1/2 which is the reparameterization used in literature. C Ranges of Index and Cumulative Index This Appendix Section provides details of the ranges of indexEi(t;α)and cumulative index EL i (t;α,w). By the Cauchy-Schwarz inequality, for a gi...

work page 2025

[9] [9]

with a step-halving strategy (Pan et al. 2025). We estimateλand θby maximizing the Laplace approximate marginal likelihood 40 (LAML; Wood et al. 2016). The log-LAML is L∗ LA(λ,θ) =L(ˆϕ(λ,θ),ˆγ(λ,θ) ;λ,θ)−1 2 log { detH (ˆϕ(λ,θ),ˆγ(λ,θ) ;λ,θ )} +1 2 log|Sλ|++C, where H (ˆϕ(λ,θ),ˆγ(λ,θ) ;λ,θ ) is the negative Hessian ofL with respect to(ϕ,γ)eval- uated at (...

work page 2025

[10] [10]

See supplementary Section E.3 for details

in C++ and implicit differentiation. See supplementary Section E.3 for details. Unlike the forward-mode AD used for single-exposure ACE-DLNM Pan et al. (2025), we employ reverse-mode AD to handle the large number of parameters in our models. Finally, we obtain ˆϕ=ˆϕ(ˆλ,ˆθ)andˆγ=ˆγ(ˆλ,ˆθ). E Derivative Computation E.1 Derivative ofL(ϕ,γ;λ,θ) We consider th...

work page 2025

[11] [11]

Therefore, E∗ {∫ log{φ(y;ˆu)}φ∗(y)dy } ≈ ∫ log{φ(y;u∗)}φ∗(y)dy−1 2nE∗ { ( ˆu−u∗)⊤I∗( ˆu−u∗) } , (S.11) where I∗=−n ∫ ∂2 logφ(y;u∗) ∂u∂u⊤ φ∗(y)dy Plugging Equation S.11 into Equation S.10 leads to E∗{D(φˆu,φ∗)}≈D(φu∗,φ∗) + 1 2nE∗ { ( ˆu−u∗)⊤I∗( ˆu−u∗) } (S.12) We aim to choose the candidate modelφminimizingE ∗{D(φˆu,φ∗)}, an unknown 43 quantity to be estim...

work page 2016

[12] [12]

by estimating the second term E∗ { ( ˆu−E∗( ˆu))( ˆu−E∗( ˆu))⊤ } using( ˆI+S λ)−1ˆK( ˆI+S λ)−1, where ˆK= N∑ i=1 ∑ t∈Ti [∂logp(Yit|xit,zit; ˆu)/∂u] [∂logp(Yit|xit,zit; ˆu)/∂u]⊤. Plugging this estimator into Equation S.13 yields the proposed AIC in a simplified form, AICcond =−2l(ˆu) + tr [ ( ˆI+S λ)−1ˆK ] .(S.14) F.3 Adjusted AIC The conditional AIC does ...

work page 2016

[13] [13]

We included the seasonal trendh(t) = 0.5 sin(t/150)

We consider three versions ofw and f shown in Figure S9. We included the seasonal trendh(t) = 0.5 sin(t/150). 0.0 0.2 0.4 0.6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Lag Weight w Type1 Type2 Type3 (a) Truew 0 2 4 6 0 5 Adaptive Cumulative Exposure ACERF f Type1 Type2 Type3 (b) Truef Figure S9: The true weight functionsw and association functionsf used in Simul...

work page 2000

[14] [14]

71 Index 5 10 15 20 25 30 35 40 Lag 0 2 4 6 8 10 12 14 −0.004 −0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Additive DRF−DLNM: All−Cause Mortality Point Estimate: NO2 Index 5 10 15 20 25 30 35 40 Lag 0 2 4 6 8 10 12 14 −0.004 −0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Additive DRF−DLNM: All−Cause Mortality Lower 95% CI: NO...

work page 2002

[15] [15]

We also examine the randomized quantile residual plot, QQ-plot and autocorrelation function plot for each CD, and present the results for Waterloo and Toronto as examples

plots and the QQ- plot across all CDs. We also examine the randomized quantile residual plot, QQ-plot and autocorrelation function plot for each CD, and present the results for Waterloo and Toronto as examples. The diagnostics suggest adequate model fit. The diagnostics for all-cause mortality in Toronto are the least favorable among all outcomes and CDs,...

work page 2013

[16] [16]

It is difficult to define two fixed exposure levels that consistently represent relatively high and low levels over time. We observe that the estimates are sensitive to the choice of quantiles: quantiles based on pooled distributions lead to significant negative associations between air pollution and all-cause mortality (Figure S60), while quantiles based...

work page 2001

[17] [17]

The boxplot for the overall distribution across all years is shown in blue

0 10 20 30 40 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Overall Y ear NO2 Distribution of NO2 Concentrations by Y ear Figure S59: Boxplots of NO2 concentrations by year. The boxplot for the overall distribution across all years is shown in blue. 85 0.925 0.950 0.975 1.000 1.025 1.050 Single−Index ACE−DLNM (0.25) Single−Ind...

work page 2001

[18] [18]

The point estimates and 95% confidence intervals are obtained using model stacking

86 H.5 CD-Specific Associations Nipissing Haldimand−Norfolk Algoma Lambton Peterborough Brant Thunder Bay Greater Sudbury Essex Niagara Simcoe Middlesex Halton Waterloo Hamilton Durham Ottawa Y ork Peel Toronto −1.0% −0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Stacking Estimate PM2.5 O3 NO2 Overall Respiratory Mortality − Stacking Estimate Relative Mortality...

work page 2001