A simple and exploratory way to determine the mean–variance relationship in generalized linear models

Abstract

This paper introduces an exploratory way to determine how variance relates to the mean in generalized linear models. This novel method employs the robust likelihood technique introduced by Royall and Tsou.A urinary data set collected by Ginsberg et al. and the fabric data set analysed by Lee and Nelder are considered to demonstrate the applicability and simplicity of the proposed technique. Application of the proposed method could easily reveal a mean-variance relationship that would generally be left unnoticed, or that would require more complex modelling to detect.

Determining the mean–variance relationship in generalized linear models—A parametric robust way

“…The naïve model yields a standard error estimate more than twice the size of its robust counterpart. Note that the exploratory analysis in Tsou (2007) on the same data set indicated that the variance is proportional to the 1.5 power of the mean, which is concordant with the findings here. Nevertheless, the uncertainty of the l estimate is also provided with the proposed parametric robust procedure.…”

“…With l=1, a convenient choice would be Poisson. Therefore, a sensible choice of a statistical model requires the knowledge of l. Tsou (2007) proposed an exploratory way of choosing an appropriate l value. The present article adopts a more confirmatory tactic aiming at introducing a parametric robust means to make inference about l in the setting of generalized linear models.…”

Power estimation using simulations for air pollution time-series studies

“…For G*Power, estimates using the enumeration procedure of Lyles, et al [7] are reported. The algorithms in these software packages require specification of the base rate (“Exp(β0)”), the risk ratio, the proportion of the variance of the pollution variable explained by other variables in the model (“R-squared other X”), the distribution of the predictor variable (“distribution of X1”), the sample size, the mean exposure period, alpha, and the number of tails [1,9,11].…”

“…While methods for sample size calculation for studies using multivariate generalized linear models have been developed [2,7,8] and statistical software packages are available for estimating power for such studies (ex. G*Power [1,9] (which is publicly available at no cost), and PASS [10,11]), these calculations generally rely on simplifying assumptions that may not be valid in a given study, and require specification of parameters that may be difficult to estimate based on available information.…”