Table 1 Similarities and differences in the methodological properties of the four selected statistical models for analysing CRCTs
S/NO | Feature | GLMM | GEE1 | GEE2 | QIF |
|---|---|---|---|---|---|
1 | Adjustment for clustering | Clustering is accounted for via a random effects term with its coefficient and that of fixed effects term estimated simultaneously using a single mean model equation [25]Â | The structure of clustering is described using a separate working covariance matrix (characterised by the working correlation matrix) which is specified separately from the mean model equation [40]Â | A separate set of estimating equations and link functions are used to model the mean and correlation parameters, thereby explicitly explaining the source of the cluster-level variations [13]Â | Avoids the direct use of the correlation parameters in its algorithm, instead, it uses a linear combination of the product of basis matrices and some constants [6] |
2 | Assumption on the distribution of the cluster-level random effects | Most times in GLMM it is assumed that the cluster-level random effects follow a parametric distribution, and Normal distribution is a common choice | As a semi-parametric method, it does not assume any distribution for the cluster-level random effects | Same as GEE1 | Same as GEE1 |
3 | Multiple forms of clustering | Accommodates multiple forms of correlation to be investigated by incorporating them as random effects in the mean model | Allows multiple forms of correlation but through a complex procedure of including higher forms of clustering as fixed effects in the mean model | Same as GEE1 | Same as GEE1 |
4 | Assumption of missing data mechanism required to obtain consistent parameter estimates | Missing completely at random and missing at random | Missing completely at random [40]Â | Same as GEE1 | Same as GEE1 |
5 | Heterogenous correlation | Flexible in modelling complex correlation structures using multiple random effects variables | Not flexible in modelling data with complex correlation structure | More flexible than GEE1 by using a separate equation, link function, and covariates for the correlation parameter | Same as GEE1 |
6 | Improvement in efficiency (i.e., the treatment effect estimate with a smaller SE) | Gain in efficiency by including random effects components in the mean model to account for correlation among outcomes in a cluster, especially when the correlation is large | Gain in efficiency by using a "working covariance matrix" which accounts for the effect of the correlation among outcomes in a cluster, however, it treats the correlation as a nuisance | More gain in efficiency compared to GEE1 by explicitly modelling the effect of the correlation among outcomes with a separate equation that allows covariates adjustment. This provides some protection against misspecification of the correlation structure | Firstly, it uses a different strategy that protects against the misspecification of the correlation structure. Secondly, it weights the information contributed by each cluster using an empirical weighting matrix, clusters with large variation are given less weight and vice versa. It is acclaimed that these two features increase its gain in efficiency compared to the GEE1 |
7 | Moment specification | First and second-order moments are to be specified | Same as GLMM | The first four order moments1, but the third and fourth can be specified as a function of the first two moments since a working correlation is being used | Same as GLMM |
8 | Approximation technique | Laplace/Adaptive Gauss-Hermite Quadrature2 | Modified Fisher scoring algorithm | Alternate between the Modified Fisher scoring algorithm and the method of the moment | Newton–Raphson algorithm |
9 | Goodness of fit | All the model selection criteria that are based on maximum likelihood theory are applicable, such as the LRT, AIC, and BIC | Uses a modification to the AIC based on a quasi-likelihood theory known as QIC (and QICu3) for model and working correlation selections | Same as GEE1 | Provides an objective function that follows a chi-square distribution (which is analogue to the likelihood ratio test) |
10 | Availability in selected statistical software, function(package) | R = glmer(lme4) and SAS = glimmix(proc) | R = glmgee(geepack) and SAS = genmod(proc) | R = geese(geepack) only | R = qif(qif) and SAS = qif(macro) |