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

Longitudinal designs often change at critical times based on available funding, staffing, scientific opportunities, and subjects. This article presents three levels of investigation into missingness by design in a partially completed 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 that 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 longitudinal design is described in detail. We then present a general design and analytical strategy for protecting against MN data midway into a longitudinal study. The new design involves stratified sampling for follow-up based on the pattern of missing data already obtained, and the corresponding estimator 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 effect of lifetime lead exposure on neuropsychological measures of attention. Our conclusion in this example is that a design exists that has sufficiently high statistical power to address the main scientific questions and also provides protection against a broad class of nonignorably missing data.