0
$\begingroup$

Let us assume a generalized linear mixed model with a binary dependent variable $y_{i, t } $ that is explained by a fixed effect matrix X and a simple random intercept for each individual $i$

$y_{i,t} = x_{i,t}'\beta +\alpha_i + \epsilon_{i,t} \hspace{35pt} \alpha \sim N(0, \sigma_{\alpha})$

As far as I know, when we want to estimate the fixed and random effects via maximum likelihood, we have to integrate out the random effects over every individual so that the log likelihood becomes

$ LL(\beta) = \sum_{i=1}^N \log \left(\int_{-\infty}^{\infty} \prod_{t=1}^{T_i}F(y_{i,t}|\alpha_i, \beta) g(\alpha_i) d \alpha_i \right)$

However, if we integrate the random intercepts out, the $\alpha$ disappears and we can only solve for $\beta$. How do we estimate $\alpha$ then?

$\endgroup$
  • $\begingroup$ The parameters of your model are $(\beta, \sigma_\alpha)$ and not $(\beta,\alpha)$. The integrals over the random effects are function of both $\beta$ and $\sigma_\alpha$. $\endgroup$ – winperikle Jul 12 at 14:00
  • $\begingroup$ Thank you for your quick reply. You are right, the random intercepts are not parameters. But still they take on values, right? How do we determine those values? $\endgroup$ – Benkyozamurai Jul 12 at 15:41
  • $\begingroup$ The model should be $y_{i,t} = x_{i,t}'\beta +\alpha_i + \epsilon_{i,t} \hspace{35pt} \alpha_i \sim N(0, \sigma_{\alpha}) \hspace{35pt} \epsilon_{i,t} \sim N(0, \sigma) $ and $\epsilon_{i,t}$ and $\alpha_i$ are independent. $\endgroup$ – user158565 Jul 12 at 17:14
4
$\begingroup$

When you have a binary outcome variable you typically use a link function to connect the probability of a positive response to the linear predictor that includes the random effects, i.e., \begin{equation} \left \{ \begin{array}{l} \log \displaystyle \frac{\pi_{it}}{1 - \pi_{it}} = x_{it}^\top \beta + z_{ij}^\top b_i,\\\\ \pi_{it} = \Pr(y_{it} = 1 \mid b_i),\\\\ b_i \sim \mathcal N(0, D), \end{array} \right. \end{equation} where $x_{it}$ is a design vector for the fixed effects coefficients $\beta$, and $z_{it}$ a design vector for the random effects $b_i$.

When you fit the model under maximum likelihood, the random effects are unobserved variables that are integrated out from the expression of the log-likelihood of the model, i.e., $$\ell(\theta) = \sum_{i = 1}^n \log \int \prod_t p(y_{it} \mid b_i; \beta) \; p(b_i; D) \; db_i,$$ where $\theta^\top = (\beta^\top, \mbox{vech}(D))$ denotes the parameters of the model. The integral does not have a closed-form solution, and is typically approximated using either the Laplace approximation or adaptive Gaussian quadrature (the latter is considered better).

The random effects are estimated in a second step using empirical Bayes methodology, that is, for $i = 1, \ldots, n$ they are estimated as the modes of the posterior distribution $$p(b_i \mid y_i) \propto p(y_i \mid b_i; \hat \beta) \; p(b_i; \hat D),$$ where $\hat \beta$ and $\hat D$ denote the maximum likelihood estimates of $\beta$ and $D$, respectively.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.