Imputation of missing variance data using non-linear mixed effects modelling to enable an inverse variance weighted meta-analysis of summary-level longitudinal data: a case study.

Missing variances, on the basis of the summary-level data, can be a problem when an inverse variance weighted meta-analysis is undertaken. A wide range of approaches in dealing with this issue exist, such as excluding data without a variance measure, using a function of sample size as a weight and imputing the missing standard errors/deviations. A non-linear mixed effects modelling approach was taken to describe the time-course of standard deviations across 14 studies. The model was then used to make predictions of the missing standard deviations, thus, enabling a precision weighted model-based meta-analysis of a mean pain endpoint over time. Maximum likelihood and Bayesian approaches were implemented with example code to illustrate how this imputation can be carried out and to compare the output from each method. The resultant imputations were nearly identical for the two approaches. This modelling approach acknowledges the fact that standard deviations are not necessarily constant over time and can differ between treatments and across studies in a predictable way.