5.3 Results

I selected a single scenario for each aim of this simulation study (out of 54) to present, for conciseness. Specifically, I will present results for the setting of a small frailty variance (0.25) with a negative regression coefficient (-0.50). The full results can be explored online interactively at https://ag475.shinyapps.io/PRR-SiReX/.

5.3.1 Aim 1: accuracy compared to analytical formulae

Bias, coverage, and mean squared error are presented in Tables A.1, A.2, A.3 and Figures B.1, B.2, B.3. Bias, coverage, and overall accuracy were optimal for all methods and across all sample sizes for the scale parameter of the Weibull distribution \(p\) and the regression coefficient \(\beta\); conversely, the methods performed quite differently for the shape parameter \(\lambda\) and the frailty variance \(\theta\). The shape parameter estimated using analytical formulae or Gauss-Kronrod quadrature was generally unbiased, with good coverage and accuracy; vice versa, using Gauss-Laguerre quadrature produced underestimated coefficients when using a small number of nodes and required at least 75 nodes to yield unbiased results. As the number of nodes increased, coverage and mean squared error improved considerably. Also, sample sizes with a higher number of clusters generally yielded better estimates for the shape parameter in terms of bias, coverage, and mean squared error. The frailty variance \(\theta\) was the parameter estimated with the greatest variability in the results. Analytical formulae required a high number of clusters to produce unbiased results (50 or 1000), yielding underestimated coefficients otherwise. Gauss-Kronrod performed similarly to analytical formulae, as did Gauss-Laguerre quadrature with a sufficiently high number of nodes. Coverage was generally good, above 90% (except Gauss-Laguerre with 15 nodes, where coverage fell to 60-70% in some settings), symptom of overestimated standard errors for the frailty variance; this inflation of the standard errors was reflected in the mean squared error, which was generally greater than the other estimated parameters for all methods under all sample sizes explored in this scenario.

5.3.2 Aim 2: accuracy when analytical formulae are not available

Bias, coverage, and mean squared error are presented in Tables A.4, A.5, A.6, and Figures B.4, B.5, B.6. Bias is generally negligible for the parameters of the Weibull distribution \(\lambda\) and \(p\) and the regression coefficient \(\beta\): between 0.0059 and 0.0193 for \(\lambda\), between -0.0424 and -0.0332 for \(p\), between 0.0040 and 0.0867 for \(\beta\). Conversely, estimates for \(\sigma\) were negatively biased for a sample size of 15 clusters - 100 individuals, 1000 clusters - 2 individuals, 15 clusters - 30 individuals (between -0.3057 and -0.0854) and positively biased for a sample size of 15 clusters - 500 individuals (between and 0.2427 and 0.4020). Bias was negligible for a sample size of 50 clusters - 30 individuals and 50 clusters - 100 individuals (between -0.0536 and -0.0095). Coverage of all estimated coefficients was poor (< 75%) for a sample size of 15 clusters - 500 individuals. For the regression coefficient \(\beta\) and the frailty variance \(\sigma\) coverage was good or superoptimal for the remaining sample sizes, with the exception of \(\sigma\) estimated using Gauss-Hermite quadrature with 15 nodes that resulted in slight undercoverage for sample sizes of 15 clusters - 100 individuals and 50 clusters - 100 individuals. The parameters of the Weibull distribution were generally undercovered (< 80%) across sample sizes, except \(\lambda\) with a sample size of 1000 clusters - 2 individuals and \(p\) with a sample size of 15 clusters - 30 individuals for which coverage was in the range 90-95%. Finally, mean squared error was low for \(\lambda\), \(p\), and \(\beta\), comparable for \(\sigma\) with a sample size of 50 clusters - 30 individuals and 50 clusters - 100 individuals, much higher for \(\sigma\) with all the remaining sample sizes (i.e. overall accuracy was lower in these settings).