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4.3 Estimation of Dispersion Parameters

We now discuss the situation in which the dispersion parameters are unknown. In analogy with generalized linear models, we adopt the following adjusted Pearson estimator for the dispersion parameter $\sigma^2$:

$${\widehat{\sigma }}^2=\frac{1}{m}\sum _{i=1}^m\{(U_i-1{)}^2+c_i\}\mathrm{.}\backslash hat\{\backslash sigma\}^2\; =\; \backslash frac\{1\}\{m\}\; \backslash sum\_\{i=1\}^\{m\}\; \backslash \{(U\_i\; -\; 1)^2\; +\; c\_i\backslash \}.$$

The first term is the Pearson estimator, with the second term being a bias correction term. The corresponding adjusted Pearson estimator for $\omega^2$ is:

$${\widehat{\omega }}^2=\frac{1}{m}\sum _{i=1}^m\frac{1}{J_i}\sum _{j=1}^{J_i}\{(U_{ij}-{\widehat{U}}_i{)}^2+c_{ij}+c_i-2c_i{w}_{ij}\}\mathrm{.}\backslash hat\{\backslash omega\}^2\; =\; \backslash frac\{1\}\{m\}\; \backslash sum\_\{i=1\}^\{m\}\; \backslash frac\{1\}\{J\_i\}\; \backslash sum\_\{j=1\}^\{J\_i\}\; \backslash \{(U\_\{ij\}\; -\; \backslash hat\{U\}\_i)^2\; +\; c\_\{ij\}\; +\; c\_i\; -\; 2c\_i\; w\_\{ij\}\backslash \}.$$

Again, the first term is the Pearson estimator, whereas the remaining terms are bias correction terms. These dispersion parameter estimates can also be shown to be consistent as $m \to \infty$ (Ma 1999). Unlike most other approaches in the literature, our asymptotic variance of the regression parameter estimator is not affected by the variability in the dispersion parameter estimators.

In fact, this orthodox BLUP approach depends on the random effects only via the first and second moments of the sub-cluster random effects. It has been shown to be robust, to a certain extent, against mispecification of the random effects distributions (Ma 1999), and thus covers non-Tweedie random effects such as log-normal random effects.

4.4 Computational Procedures

Initial values for the regression parameters are taken as the regression parameter estimates obtained from standard Poisson regression techniques assuming independent responses. Initial random effects predictions $\hat{U}i$ and $\hat{U}{ij}$ are given by the average of the responses within cluster $i$ divided by the average of all responses and the average of the responses within sub-cluster $(i, j)$ divided by the average of all responses, respectively. The initial dispersion parameter estimates are calculated from the adjusted Pearson estimators, omitting the bias-correction terms.

The algorithm then iterates between updating the regression parameter estimates via the Newton scoring algorithm, updating random effect predictors via the orthodox BLUP, and updating dispersion parameter estimates via the adjusted Pearson estimators.

5 An Illustrative Example

We illustrate the application of our approach to the random effects Cox model using data from an animal carcinogenesis experiment originally re-