Remark

Let \(y_{it}\) be \(t\)th response in cluster \(i\), \(i=1,\ldots,n\); \(t=1,\ldots,T\). Also, let \(u_i\) be vector of random effects for cluster \(i\).

Conditional on \(u_i\), Generalized Linear Mixed Model(GLMM) resembles GLM s.t \[ \mu_{it}=E(y_{it}|u_i), \] i.e., \(\{Y_1,\ldots,Y_T\}\) are independent with \[ g(\mu_{it})=x_{it}'\beta+z_{it}'u_i. \] Assume \[ u_i\sim N(0,\Sigma) \] with unknown variance components.

e.g. Binary matched pairs \[ \text{logit}P(Y_{it}=1)=\alpha_i+\beta x_{it}\mbox{ }\mbox{ }\mbox{ with }\mbox{ }\mbox{ }x_{i1}=0, x_{i2}=1. \] Write now as \(u_i+\beta x_{it}\) where \(u_i\sim N(\alpha, \sigma^2)\) or \(\alpha+\beta x_{it}+u_i\) where \(u_i\sim N(0,\sigma^2)\).

Here, \(g(\mu_{it})=x_{it}'\beta+z_{it}'u_i\), where \[ \beta={\alpha \choose \beta},\mbox{ } x_{i1}'=(1,0), \mbox{ }x_{i2}'=(1,1), \mbox{ } z_{it}=1. \] \(\mu_{it}=P(Y_{it}=1|u_i)\), \(g=\)logit link, \(u_i\sim N(0,\sigma^2)\).

This is a random intercept model.



Remark(Connection between Conditional Models and Marginal Models)

Conditional Model : \[ E(Y_{it}|u_i)=g^{-1}(x_{it}'\beta+z_{it}'u_i). \] Marginally, \[ E(Y_{it})=E\{ E(Y_{it}|u_i)\}=\int g^{-1}(x_{it}'\beta+z_{it}'u_i) f(u_i;\Sigma)du_i. \] For identity link, \[ E(Y_{it})=\int (x_{it}'\beta+z_{it}'u_i) f(u_i;\Sigma)du_i= x_{it}'\beta. \] has some model form and effects \(\beta\), but not true for other links.


For Probit GLMM for binary data, \[ \Phi^{-1}\{E(Y_{it}|u_i)\}=x_{it}'\beta+z_{it}'u_i, \] where \(\Phi\) is \(N(0,1)\) cdf.

Note that \[\begin{eqnarray*} E(Y_{it}|u_i)&=&\Phi(x_{it}'\beta+z_{it}'u_i|u_i)\\ &=&P(Z\le x_{it}'\beta+z_{it}'u_i|u_i),\\ E(Y_{it})&=& E\{E(Y_{it}|u_i)\}\\ &=&\int P(Z-z_{it}'u_i\le x_{it}'\beta|u_i)dF(u_i)\\ &=&\int \Phi\left(\frac{x_{it}'\beta}{\sqrt{1+z_{it}'\Sigma z_{it}}}\right)dF(u_i)\mbox{ }\mbox{ }(\mbox{Let }Z\perp u_i)\\ &=&\Phi\left(\frac{x_{it}'\beta}{\sqrt{1+z_{it}'\Sigma z_{it}}}\right)\\ &=&\Phi(x_{it}'\beta^*) \end{eqnarray*}\] where \(\beta^*=\beta/\sqrt{1+z_{it}'\Sigma z_{it}}\). Note that the marginal model also is a probit model(\(\beta\)값만 다르다).

For log-linear GLMM (count data), \[\begin{eqnarray*} \log E(Y_{it}|u_i)&=&x_{it}'\beta+z_{it}u_i,\\ E(Y_{it}|u_i)&=& e^{x_{it}'\beta+z_{it}u_i},\\ E(Y_{it})&=&E\{E(Y_{it}|u_i)\}=e^{x_{it}'\beta}E(e^{z_{it}'u_i})\\ &=&e^{x_{it}'\beta}e^{\frac{1}{2}z_{it}'\Sigma z_{it}}=e^{x_{it}'\beta+\frac{1}{2}z_{it}'\Sigma z_{it}}. \end{eqnarray*}\]

Thus, \[ \log\{ E(Y_{it})\}-\frac{1}{2}z_{it}'\Sigma z_{it}=x_{it}'\beta. \] That is the form of a log-linear GLM with offset(오차가 생겼다).

For random intercept model, \[ E(Y_{it})=e^{\frac{\sigma^2}{2}}e^{x_{it}'\beta}=e^{\frac{\sigma^2}{2}+x_{it}'\beta}. \]

If \(Y_{it}|u_i\) is Poisson (i.e., random intercept Poisson GLMM), variance of marginal distribution is \[\begin{eqnarray*} \text{Var}(Y_{it})&=&E\{\text{Var}(Y_{it}|u_i)\}+ \text{Var}\{E(Y_{it}|u_i)\}\\ &=& E(\mu_{it})+\text{Var}(\mu_i)\\ &=& E(e^{x_{it}'\beta+z_{it}u_i})+\text{Var}(e^{x_{it}'\beta+z_{it}u_i})\\ &=& E(e^{x_{it}'\beta+u_i})+\text{Var}(e^{x_{it}'\beta+u_i})\\ &=& e^{x_{it}'\beta+\frac{1}{2}\sigma^2}+e^{2x_{it}'\beta}\left\{E(e^{2u_i})-\{E(e^{u_i})\}^2\right\}\\ &=&E(Y_{it})+e^{2x_{it}'\beta}\left(e^{2\sigma^2}-e^{\sigma^2}\right)\\ &=&E(Y_{it})+e^{2x_{it}'\beta}e^{\sigma^2}\left(e^{2\sigma^2}-1\right)\\ &=&E(Y_{it})+\{E(Y_{it})\}^2\left(e^{2\sigma^2}-1\right). \end{eqnarray*}\]

As in negative binomial GLM, variance is quadratic function of mean. If \(\sigma^2=0\), get ordinal Poisson GLM. If \(\sigma^2>0\), this is alternative to negative binomial GLM for handling overdispersion.



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