Standardized Martingale Residuals Applied to Grouped Left Truncated Observations of Dementia Cases

The use of martingale residuals have been proposed for modelchecking and also to get a non-parametric estimate of the effectof an explanatory variable. We apply this approach to an epidemiologicalproblem which presents two characteristics: the data are lefttruncated due to delayed entry in the cohort; the data are groupedinto geographical units (parishes). This grouping suggests anatural way of smoothing the graph of residuals which is to computethe sum of the residuals for each parish. It is also naturalto present a graph with standardized residuals. We derive thevariances of the estimated residuals for left truncated datawhich allows computing the standardized residuals. This methodis applied to the study of dementia in a cohort of old people,and to the possible effect of the concentration of aluminum andsilica in drinking water on the risk of developing dementia.