Open Access Open Badges Research article

Reducing and meta-analysing estimates from distributed lag non-linear models

Antonio Gasparrini1* and Ben Armstrong2

Author Affiliations

1 Department of Medical Statistics, London School of Hygiene and Tropical Medicine, London, UK

2 Department of Social and Environmental Health Research, London School of Hygiene and Tropical Medicine, London, UK

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BMC Medical Research Methodology 2013, 13:1  doi:10.1186/1471-2288-13-1

Published: 9 January 2013



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.


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.


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.

Discussion and 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.

Distributed lag models; Multivariate meta-analysis; Two-stage analysis; Time series