Two-stage sampling in the estimation of growth parameters and percentile norms: sample weights versus auxiliary variable estimation

The use of auxiliary variables with maximum likelihood parameter estimation for surveys that miss data by design is not a widespread approach. Although efficiency gains from the incorporation of Normal auxiliary variables in a model have been recorded in the literature, little is known about the effects of non-Normal auxiliary variables in the parameter estimation. We simulate growth data to mimic SCALES, a two-stage longitudinal survey of language development. We allow a fully observed Poisson stratification criterion to be correlated with the partially observed model responses and develop five models that host the auxiliary information from this criterion. We compare these models with each other and with a weighted model in terms of bias, efficiency, and coverage. We apply our best performing model to SCALES data and show how to obtain growth parameters and population norms. Parameter estimation from a model that incorporates a non-Normal auxiliary variable is unbiased and more efficient than its weighted counterpart. The auxiliary variable method can produce efficient population percentile norms and velocities. When a fully observed variable, which dominates the selection of the sample and which is strongly correlated with the incomplete variable of interest exists, its utilisation appears beneficial.