Semi-parametric modelling of the distribution of the baseline risk in meta-analysis

In meta-analysis of clinical trials, often meta-regression analyses are performed to explain the heterogeneity in treatment effects that usually exist between trials. A popular explanatory variable is the risk observed in the control group, the baseline risk. The relationship between the treatment effect and the baseline risk is investigated by fitting a linear model that allows randomness on the true baseline risk by assuming a normal distribution with unknown mean and variance. However, the normality assumption could be too strong to adequately describe the underlying distribution. Therefore, we developed a new semi-parametric method that relaxes the normality assumption to a more flexible and general distribution. We applied a penalized Gaussian mixture distribution to represent the baseline risk distribution. Furthermore, a bivariate hierarchical model is formulated in order to take into account the correlation between the baseline and treatment effect. To fit the proposed model, a penalized likelihood function is maximized by an Expectation Maximization (EM) algorithm. We illustrate our method on a number of simulated data sets and on a published meta-analysis data set.