Power calculations for data missing by design: Applications to a follow-upstudy of lead exposure and attention

Longitudinal designs often change at critical times based on available fund ing, staffing, scientific opportunities, and subjects. This article present s three levels of investigation into missingness by design in a partially c ompleted longitudinal study: missingness that is completely at random (MCAR ), at random (MAR), and nonignorable (MN). We first derive new expressions for the asymptotic variance and power based on multivariate normal data tha t are either MCAR or missing by design (MAR). These formulas allow for any and all patterns of missing data. The special case of a two-stage longitudi nal design is described in detail. We then present a general design and ana lytical strategy for protecting against MN data midway into a longitudinal study. The new design involves stratified sampling for follow-up based on t he pattern of missing data already obtained, and the corresponding estimato r is based on an approximate likelihood. The methodology for MCAR, MAR, and MN are in turn applied to the design of a follow-up study to examine the e ffect of lifetime lead exposure on neuropsychological measures of attention . Our conclusion in this example is that a design exists that has sufficien tly high statistical power to address the main scientific questions and als o provides protection against a broad class of nonignorably missing data.