# Distribution Parameters for a GLM (Generalized Linear Models) evaluated at a point

I need help answering a problem I have:

I am modeling data set using GLM that has two independent variables $$x_1, x_2$$, and a single response variable $$y$$. I am using $$\verb|statsmodels.api|$$ python library to set these models up. The problem I am trying to get an answer to is, how I get the shape parameters for the distributions that result from the model?

In other words, suppose that $$\verb|glm_model|$$ is the model I trained. Then given a test data point $$x^*$$, $$\verb|glm_model|(x^*)=y^*=\mu.$$ This is the first shape parameter for my distribution (mean). I want to get the variance, $$\sigma$$, so that I can get the distribution $$N(\mu, \sigma)$$. However, there does not seem to be any method in the glm python package that I can easily use to get this parameter.

Visually what I am trying to get is the distributions that you see on these graphs.

There is a distribution associated with each predicted value point. If there is no way to get the variance this way, is there a way to estimate the variance around $$y^*$$ locally? Is there a formal method of doing this?

• The attached figure visualizes a linear regression. Your question refers to generalized linear regression (GLM). These are not the same models; very briefly, in a GLM the response Y is not $N(\mu,\sigma^2)$. Please clarify what kind of variable the response $y$ is and what kind of GLM you are fitting. Aug 7, 2022 at 12:11
• @dipetkov Sorry for the confusion. I edited my question with more example figures. I got these from the following sites: link1, link2. What I was trying to ask is, regardless of what the response variables distribution was, how can you can the distributions found in the figures above? They fit a Poisson regression on one of them and plotted the distributions resulting from the model. Aug 7, 2022 at 14:16
• The variance of the Poisson distribution is equal to the mean. So with Poisson regression, you obtain $E(Y|X=x)$ from the model and then $\operatorname{Var}(Y|X=x) = E(Y|X=x)$. In the notation of the Medium article you link to: $\lambda_i = E(Y_i|X=x_i)$. Aug 7, 2022 at 14:30
• @dipetkov Ah yes Poisson it is easy to get these parameters. How about other distributions? For example, simple linear regression. Aug 7, 2022 at 14:44
• It's easy for all of the regressions mentioned in your questions as long as you understand the regression model. I suggest you do a bit of revision. For linear regression $\sigma^2$ is the error variance. Aug 7, 2022 at 14:55