Attributable risk from distributed lag models

Abstract

Background: Measures of attributable risk are an integral part of epidemiological analyses, particularly when aimed at the planning and evaluation of public health interventions. However, the current definition of such measures does not consider any temporal relationships between exposure and risk. In this contribution, we propose extended definitions of attributable risk within the framework of distributed lag non-linear models, an approach recently proposed for modelling delayed associations in either linear or non-linear exposure-response associations.

Methods: We classify versions of attributable number and fraction expressed using either a forward or backward perspective. The former specifies the future burden due to a given exposure event, while the latter summarizes the current burden due to the set of exposure events experienced in the past. In addition, we illustrate how the components related to sub-ranges of the exposure can be separated.

Results: We apply these methods for estimating the mortality risk attributable to outdoor temperature in two cities, London and Rome, using time series data for the periods 1993-2006 and 1992-2010, respectively. The analysis provides estimates of the overall mortality burden attributable to temperature, and then computes the components attributable to cold and heat and then mild and extreme temperatures.

Conclusions: These extended definitions of attributable risk account for the additional temporal dimension which characterizes exposure-response associations, providing more appropriate attributable measures in the presence of dependencies characterized by potentially complex temporal patterns.

Figure 1

Conceptual model for the interpretation of exposure-lag-response associations: forward (left panel) and backward (right panel) perspectives.

Figure 2

Association between temperature and all-cause mortality.Left panels: 3-D graphs of the exposure-lag-response risk surface. Right panels: overall cumulative exposure-response associations with temperature distributions. London 1993–2006 and Rome 1992–2010.

Figure 3

Lag structure and harvesting paradox. Left panel: lag-response associations between various temperatures and all-cause mortality. Rome 1992–2010. Right panel: daily number of deaths attributable to heat, computed forward (green circles) and backward (yellow squares), and temperature trend. Rome July-Sept 1992.