Is there a Relationship Between Variance and Chi-Square? I am an MBA Student that is taking courses in Statistics.
Up until now, we had only encountered "Chi-Square" in the context of Contingency Tables. That is, how to find if the difference between different groups of subjects is statistically significant or non-significant (e.g. https://web.pdx.edu/~newsomj/uvclass/ho_chisq.pdf).
On the other hand, we have been learning about the concept of "Overdispersion" (related to variance). For example, we are learning about Regression Models for Discrete Count Data (e.g. Poisson GLM) in which an "Overdispersion Parameter" is required to be estimated, and this parameter will serve to "correct" the confidence intervals on the beta-coefficients of the regression model.
I would have thought that this "Overdispersion Parameter" would have been estimated through the Maximum Likelihood Equations - but our prof mentioned that the "Overdispersion Parameter" is estimated independently of the Maximum Likelihood Equations and is in fact calculated using a very similar formula as the "Chi-Square Statistic".
This makes me wonder:

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*Is there some special relationship between "Overdispersion" (i.e. variance) and "Chi-Square" - and why isn't the Overdispersion Parameter estimated from the Maximum Likelihood Equations (as is done for the regression parameters)?


There will be better authorities on CV with respect to this topic, and what follows is mostly what I recall from my actuarial studies.
In the context of a GLM with Poisson link, if there is a need to adjust for overdispersion (which is a contradiction of the assumptions of a Poisson GLM) then often a simple approach is to change the specification of the variance.
$$V(Y_{i})=\phi\mu_{i}\quad\quad\quad(1)$$
where $\phi$ is the dispersion parameter.

I would have thought that this "Overdispersion Parameter" would have been estimated through the Maximum Likelihood Equations

The log-likelihood function for this Poisson GLM cannot incorporate this new information into its form. The Poisson distribution is a single parameter distribution, how can it? We cannot change the variance specification and preserve the mean. By definition, $\mathbb{E}[Y_{i}]=V(Y_{i})$ for a Poisson.
In the case of overdispersion, the general estimation method is Iterative Reweighted Least Squares (IRLS) and not vanilla MLE. The estimates obtained from this are generally referred to as Maximum Quasi-likelihood estimates (MLQE).
In the case of the variance being proportional to the mean in $(1)$, it turns out that the $\hat{\mu}^{\text{MLE}}_{i}$ estimate coincides with the MQLE estimate due to cancellation of the $\phi$ term so it's good-to-go in that regard.
However, we still need to estimate $\hat{\phi}$ but we do this separately because as we just found out, $\hat{\mu}_{i}$ does not depend on $\phi$.
The variance of the estimates is
$$\text{Var}(\hat{\beta})=\phi(X'WX)^{-1}$$
which means we need to know $\phi$ in order to make correct inference about estimated parameters i.e. the standard errors are wrong. It's important to note that we only care about the overdispersion issue because of its effect on the standard errors.
I believe there are a few ways to go about getting $\hat{\phi}$ but the one you noted is to use the Pearson $\chi^{2}$ statistic. I think the general reasoning is that it's a moment-matching exercise. If we define
$$P=\sum_{i=1}^{n}\frac{(y_{i}-\mu_{i})^2}{\text{Var}(y_{i})}=\sum_{i=1}^{n}\frac{(y_{i}-\mu_{i})^2}{\phi\mu_{i}}$$
If the assumptions of the model were correct, then
$$\phi=1\quad\quad\Rightarrow\quad\quad P\sim\chi^{2}_{n-p}\quad\quad\Rightarrow\quad\quad\mathbb{E}[P]=n-p$$
Then we simply solve for the $\hat{\phi}$ that gives us this expectation
$$\hat{\phi}=\sum_{i=1}^{n}\frac{(y_{i}-\mu_{i})^2}{\text{Var}(y_{i})}\bigg/(n-p)$$
So hopefully this answers your other question:

Is there some special relationship between "Overdispersion" (i.e. variance) and "Chi-Square"

