5.2 Methods, estimands, and performance measures
I fitted a set of models for each simulated dataset under each data-generating mechanism. Specifically, for the data generated assuming a Gamma frailty, I compared the following models:
a shared Gamma frailty model with a baseline Weibull hazard using the analytical formulation of the likelihood (method AF);
a shared Gamma frailty model with a baseline Weibull hazard using the likelihood approximated numerically via Gaussian quadrature (specifically, a Gauss-Laguerre quadrature rule) with 15, 35, 75, and 105 nodes (methods GQ15, GQ35, GQ75, GQ105);
a shared Gamma frailty model with a baseline Weibull hazard using the likelihood approximated numerically via Gauss-Kronrod quadrature (as implemented in R’s
integrate()function; method IN).
Then, for data generated assuming a log-normal frailty, I fitted a Weibull model with a random intercept using the likelihood approximated via Gauss-Hermite quadrature using 15, 35, 75, and 105 nodes (methods GQ15, GQ35, GQ75, GQ105).
For each model I compared the estimated parameters of the Weibull distribution \(\hat{\lambda}\) and \(\hat{p}\), the estimated log-treatment effect \(\hat{\beta}\), and the estimated variance of the frailty \(\hat{\theta}\).
In terms of performance measures, I am interested first of all in the performance of the maximum likelihood estimation procedure; that is, how precise is the maximum likelihood estimator. I will assess this by computing bias for each estimand, defined as \(b = E(\hat{\beta}) - \beta\).
Next, I am interested in coverage, i.e. the proportion of times the \(100 \times (1 - \alpha)\%\) confidence interval \(\hat{\beta} \pm Z_{1 - \alpha / 2} \times SE(\hat{\beta})\) includes the true value \(\beta\). This allow to assess whether the empirical coverage rate approaches the nominal coverage rate (\(100 \times (1 - \alpha)\%\)), to properly control the type I error rate for testing a null hypotesis of no effect.
Finally, I am interested in overall accuracy and therefore I will compute the mean squared error, defined as the sum of bias and variability: \((\bar{\hat{\beta}} - \beta) ^ 2 + (SE(\hat{\beta})) ^ 2\).
Summary measures for \(\lambda\), \(p\), and \(\theta\) are computed on the log-scale. For bias and coverage, I will further include Monte Carlo standard errors to quantify the uncertainty in estimating the performance measures (further details in White (2010)).
References
White, Ian R. 2010. “Simsum: Analyses of Simulation Studies Including Monte Carlo Error.” The Stata Journal 10 (3): 369–85.