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For the mixed effects (random slope) binary logistic regression model the logit can be expressed as (Applied Logistic Regression (2010) - David W,HOSMER, JR.):

$$g(\boldsymbol{x}_{i,j},\alpha_i,\boldsymbol{\beta}_s) = \beta_0 + \alpha_i + \boldsymbol{x}_{i,j} \boldsymbol{\beta}_s $$

Where $\alpha_i \sim N(0,\sigma^2_\alpha)$. and (i,j) refers to the jth observation in the i:th cluster. the subscript s in $\beta_s$ refers to the fact that coefficients refer to a model that is specific to subjects with random effect $\alpha_i$

This model is said to account for correlated observations by adding a random effect term ,specific to the cluster, to the logit.

I dont understand what is the effect of adding this "random term" to the model? Why is this model formultion better when having correlated observations?

'The hetrogenity in the $\alpha_i$'s is controlled by the varaince $\sigma^2_\alpha$.Thus, as $\sigma^2_\alpha$ increases the within cluster correlation increases'

I dont see the relation with the variance $\alpha_i$ and the correlation within each cluster. Why does greater variance imply greater within cluster correlation?

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$\newcommand{\e}{\varepsilon}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\Cov}{\operatorname{Cov}}$It seems a lot of your question can be answered with just a random intercepts model so I'll start there. It also doesn't seem like this being a logistic regression particularly matters so I'll begin by considering the simple random intercepts model $$ y_{ij} = \mu + \alpha_i + \e_{ij} $$ with $\alpha\sim \mathcal N(\mathbf 0, \sigma^2_\alpha I) \perp \e \sim \mathcal N(\mathbf 0, \sigma^2 I)$. $i=1,\dots,m$ will index groups and $j=1,\dots,n_i$ will index observations within a group. There are $n = \sum_{i=1}^m n_i$ observations in all.

Note that $$ \Cov(y_{ij}, y_{kl}) = \begin{cases} \sigma^2 + \sigma^2_\alpha & i=j, k=l\\ \sigma^2_\alpha & i=j, k\neq l \\ 0 & i\neq j \end{cases} $$ so this reveals the effect of including $\alpha_i$, namely that it adds to the within-cluster covariance. As $\sigma^2_\alpha$ increases, especially with $\sigma^2$ fixed, the $y_{ij}$ are dominated by the $\alpha_i$ and this leads to the values varying relatively little within a cluster.

For a GLM, this'll still have the effect of making within-cluster predictions more similar than between-cluster, and the larger $\sigma^2_\alpha$ is, the more similar the predictions for units in the same cluster will be.


Now adding in a random slopes term, we'll have $$ y_{ij} = \mu + \alpha_i + (\beta + \gamma_i) x_{ij} + \e_{ij} $$ where $\gamma_i \stackrel{\text{iid}}\sim \mathcal N(0, \sigma^2_\gamma)$ is independent of $\e$ but I'll let $\Cov(\alpha, \gamma) = \rho\sigma_\gamma\sigma_\alpha I$. $\beta$ is the fixed global slope and $\gamma_i$ represents group-level deviations from that. $\rho$ is the correlation between $\alpha_i$ and $\gamma_i$.

On page 7 of the lme4 manual Bates et al. comment on how $\rho \neq 0$ can be used to make the model invariant to translations of $x$. When we see the form of the covariance we'll see why this is.

It can be shown that $$ \Cov(y_{ij}, y_{kl}) = \begin{cases} \sigma^2\delta_{jl} + \sigma^2_\alpha(1-\rho^2) + (\sigma_\gamma x_{ij} + \rho\sigma_\alpha)(\sigma_\gamma x_{kl} + \rho\sigma_\alpha) & i=j\\ 0 & i\neq j \end{cases} $$ where $\delta_{jl}$ is the Kronecker delta. I've written the covariance this way because it helps interpret what it means in terms of the covariance matrix to assume a random slopes model. It's the same as the random intercepts model except we also get this "similarity" term $(\sigma_\gamma x_{ij} + \rho\sigma_\alpha)(\sigma_\gamma x_{kl} + \rho\sigma_\alpha)$ which gives a large positive covariance when $x_{ij}$ and $x_{kl}$ are large with the same sign and a large negative covariance when they are large with different signs, and the larger the random slope variance $\sigma^2_\gamma$ is, the more the covariance structure is dominated by changes in the $x_{ij}$. Intuitively, when $\sigma^2_\gamma$ is relatively large, we expect to see $y_{ij}$ better explained by its slope term so we'll have a "clustering" around the group level trend lines.

This also shows what Bates et al. were referring to with $\rho$, in that if the $x_{ij}$ are all translated by a constant amount then a non-zero $\rho$ can account for this in the $\sigma_\gamma x_{ij} + \rho\sigma_\alpha$ terms. If $\rho = 0$ there's no way to "undo" the translation and then the estimates will actually change.

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