Sample size issues in time series regressions of counts on environmental exposures

Abstract

Background: Regression analyses of time series of disease counts on environmental determinants are a prominent component of environmental epidemiology. For planning such studies, it can be useful to predict the precision of estimated coefficients and power to detect associations of given magnitude. Existing generic approaches for this have been found somewhat complex to apply and do not easily extend to multiple series studies analysed in two stages. We have sought a simpler approximate approach which can easily extend to multiple series and give insight into factors determining precision.

Methods: We derive approximate expressions for precision and hence power in single and multiple time series studies of counts from basic statistical theory, compare the precision predicted by these with that estimated by analysis in real data from 51 cities of varying size, and illustrate the use of these estimators in a realistic planning scenario.

Results: In single series studies with Poisson outcome distribution, precision and power depend only on the usable variation of exposure (i.e. that conditional on covariates) and the total number of disease events, regardless of how many days those are spread over. In multiple time series (eg multi-city) studies focusing on the meta-analytic mean coefficient, the usable exposure variation and the total number of events (in all series) are again the sole determinants if there is no between-series heterogeneity or within-series overdispersion. With heterogeneity, its extent and the number of series becomes important. For all but the crudest approximation the estimates of standard errors were on average within + 20% of those estimated in full analysis of actual data.

Conclusions: Predicting precision in coefficients from a planned time series study is possible simply and given limited information. The total number of disease events and usable exposure variation are the dominant factors when overdispersion and between-series heterogeneity are low.

Keywords: Environment; Poisson regression; Power; Sample size; Statistics; Time series regression.

Fig. 1

Approximate power as a function of number of cases, size of risk to be detected in relation to usable exposure spread, and overdispersion. Blue lines (left cluster): coefficient (log(RR) per useable SD(x|z)) = 5%. Green lines (middle-left cluster): coefficient = 2% per SD(x|z). Red lines (middle right cluster): coefficient = 1% per SD(x|z). Black lines (right cluster): coefficient = 0.05% per SD(x|z). Within clusters, from left (top): Solid (left): dispersion = 1.0. Dashed (middle): dispersion = 1.2. Dotted (right): dispersion = 1.5