Imputation of missing variance data comparing Bayesian and Classical non- linear mixed effect modelling to enable a precision weighted meta-analysis.
Objectives: When carrying out a meta-analysis of summary level data, the recommended approach is to weight each observation (e.g. arithmetic mean, treatment effect) by its associated precision. Often there are missing standard deviations in the literature and hence an attempt should be made to impute these missing values. The aim was to compare Classical and Bayesian methods of missing data imputation using a non linear mixed effects model.
Methods: Internal and external reports were searched to find randomised double blind placebo controlled studies where naproxen had been used to treat subjects with osteoarthritis (OA) pain for knee or hip. The endpoint of interest was mean WOMAC pain score which is typically measured at several time points post dose during a study, the desire being to model the time course for placebo and naproxen. However 30% of the WOMAC pain scores did not have a standard deviation reported.
The time course of standard deviations was described using a longitudinal mixed effects Emax model with parameters for baseline, maximum effect over baseline and the time to get to 50% of that maximum. A 2 stage approach was taken using both a Classical (maximum likelihood) and a Bayesian approach. This involved fitting a model to the standard deviations, using the model predictions to impute missing values and then merging these missing values to the original dataset prior to modelling WOMAC pain scores. A Bayesian approach was also used to simultaneously model standard deviations and WOMAC pain scores. The three approaches were then compared.
Results: A visual examination of the standard deviation data across studies revealed a non-linear relationship over time post dose. An Emax model was hence chosen as a suitable model. For the Bayesian and Classical two stage approaches, the resulting predicted standard deviations for the missing observations were almost identical. However for the simultaneous fit of standard deviation and WOMAC pain data, there was less agreement compared to the 2 stage approaches. This suggested that there might be feedback from the WOMAC pain model to the standard deviation model. The use of a ‘CUT' function in WinBUGS was successful in preventing this feedback. The parameter estimates for the WOMAC pain model were comparable across all three approaches.
Conclusions: Simultaneously modelling WOMAC pain score and standard deviations was an efficient way to impute missing data and carry out a meta analysis but care is needed in terms of feedback. The ‘CUT' function takes away this feedback but thought should be given to the reasons for this feedback (e.g. model misspecification). This work was done in a data rich situation where 70% of standard deviations were present. Future work should look into sparse data situations where the assumptions being made may be much greater. It would also be of interest to assess how these approaches compare to other methods such as weighting by sample size or using a common imputed standard deviation across all missing observations.