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Distributed lag non-linear models with spatial effect modification using Laplacian P-splines

Sara Rutten, Thomas Neyens, Elisa Duarte, Antonio Gasparrini, Christel Faes

Abstract

Distributed lag non-linear models (DLNMs) are a popular approach to flexibly model the effect of time-delayed exposures. Classical DLNMs specify a common exposure-lag-response relationship across geographical areas. However, this relationship might be altered by an effect modifier that differs between spatial units. Although some methods have been proposed to account for effect modification, their applicability is context-dependent. For example, a meta-analysis can account for heterogeneity between groups, but this technique requires sufficiently large study groups. This limitation is particularly relevant when working with count data, where small numbers of events are often encountered. In this paper, we review existing methods that allow for spatial effect modification for count-based outcomes and propose a Bayesian DLNM alternative method that accounts for the modifier through flexible interaction effects. Through the use of Laplacian P-splines, we provide a computationally fast estimation procedure by avoiding the use of classical Markov Chain Monte Carlo (MCMC) approaches. The performance of the different methods is evaluated through simulation studies. Moreover, the practical applicability of our proposed method is showcased through a data application, containing daily temperature and mortality count data in 87 Italian cities.

Distributed lag non-linear models with spatial effect modification using Laplacian P-splines

Abstract

Distributed lag non-linear models (DLNMs) are a popular approach to flexibly model the effect of time-delayed exposures. Classical DLNMs specify a common exposure-lag-response relationship across geographical areas. However, this relationship might be altered by an effect modifier that differs between spatial units. Although some methods have been proposed to account for effect modification, their applicability is context-dependent. For example, a meta-analysis can account for heterogeneity between groups, but this technique requires sufficiently large study groups. This limitation is particularly relevant when working with count data, where small numbers of events are often encountered. In this paper, we review existing methods that allow for spatial effect modification for count-based outcomes and propose a Bayesian DLNM alternative method that accounts for the modifier through flexible interaction effects. Through the use of Laplacian P-splines, we provide a computationally fast estimation procedure by avoiding the use of classical Markov Chain Monte Carlo (MCMC) approaches. The performance of the different methods is evaluated through simulation studies. Moreover, the practical applicability of our proposed method is showcased through a data application, containing daily temperature and mortality count data in 87 Italian cities.
Paper Structure (14 sections, 13 equations, 5 figures, 3 tables)

This paper contains 14 sections, 13 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Review existing methods
  3. Two-stage approach
  4. Case time series design
  5. Spatial effect modification in DLNM-LPS
  6. DLNM-LPS with effect modification
  7. Bayesian model formulation
  8. Laplace approximation
  9. Model comparison
  10. Simulation study
  11. Simulation set-up
  12. Simulation results
  13. Data application
  14. Conclusion

Figures (5)

  • Figure 1: Exposure-lag-response relationship for three different levels of the effect modifier $z$ ($5\%, 50\%$ and $95\%$ percentile of $z$) for the linear case (top) and the complex case (bottom).
  • Figure 2: Lag-response relationship for an exposure of $x=8$ and three different levels of the effect modifier $z$ ($5\%, 50\%$ and $95\%$ percentile of $z$) for both linear case (left) and complex case (right).
  • Figure 3: Exposure-response relationship represented by the overall cumulative RR for $3$ deprivation levels, cumulated over 22 days, compared to a temperature percentile of $0.6$. The vertical lines represent the $2.5\%, 50\%$ and $97.5\%$ quantiles of temperature percentiles. The shaded bands indicate the $95\%$ credible intervals.
  • Figure 4: RRR comparing the overall cumulative RR at $90$th and $10$th deprivation percentile for different temperature percentiles, using a reference temperature percentile of $0.6$. The bands are the associated credible intervals.
  • Figure 5: True (red dots) and counterfactual (blue triangle) attributable fraction in $2021$ for the $87$ Italian cities if the deprivation index would be equal to the median for all cities. The darker the colour, the higher the deprivation index.