Understanding Random effects model - Binary Logistic regression

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?

$$\newcommand{\e}{\varepsilon}\newcommand{\one}{\mathbf 1}\newcommand{\Var}{\operatorname{Var}}\newcommand{\Cov}{\operatorname{Cov}}$$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.