title: Reducing and meta-analysing estimates from distributed lag non-linear models.
abstract: BACKGROUND
  The two-stage time series design represents a powerful analytical tool in environmental epidemiology. Recently, models for both stages have been extended with the development of distributed lag non-linear models (DLNMs), a methodology for investigating simultaneously non-linear and lagged relationships, and multivariate meta-analysis, a methodology to pool estimates of multi-parameter associations. However, the application of both methods in two-stage analyses is prevented by the high-dimensional definition of DLNMs.
  
  
  METHODS
  In this contribution we propose a method to synthesize DLNMs to simpler summaries, expressed by a reduced set of parameters of one-dimensional functions, which are compatible with current multivariate meta-analytical techniques. The methodology and modelling framework are implemented in R through the packages dlnm and mvmeta.
  
  
  RESULTS
  As an illustrative application, the method is adopted for the two-stage time series analysis of temperature-mortality associations using data from 10 regions in England and Wales. R code and data are available as supplementary online material.
  
  
  CONCLUSIONS
  The methodology proposed here extends the use of DLNMs in two-stage analyses, obtaining meta-analytical estimates of easily interpretable summaries from complex non-linear and delayed associations. The approach relaxes the assumptions and avoids simplifications required by simpler modelling approaches.
authors.affiliation: Department of Medical Statistics, London School of Hygiene and Tropical Medicine, London, UK. antonio.gasparrini@lshtm.co.uk
authors.forename: Antonio
authors.initials: A
authors.lastname: Gasparrini
authors.forename: Ben
authors.initials: B
authors.lastname: Armstrong
mesh_terms: D000393:Air Pollutants; D000397:Air Pollution; D000465:Algorithms; D003627:Data Interpretation, Statistical; D004739:England; D004781:Environmental Exposure; D004784:Environmental Monitoring / Q000379:methods*; D062665:Epidemiological Monitoring; D006801:Humans; D015201:Meta-Analysis as Topic; D008962:Models, Theoretical; D015999:Multivariate Analysis; D017711:Nonlinear Dynamics; D013696:Temperature; D014852:Wales
chemical_list: D000393:Air Pollutants