Interpreting model fit

1. Interpreting model fit

In the previous video, you learned about Poisson regression and saw its workings in the exercises where you fitted your first Poisson regression. In this video, you will learn how to interpret the fitted model.

2. Parameter estimation

Similarly as in logistic regression, the maximum likelihood estimation, is used to obtain the values of betas, the parameters, which maximize the log-likelihood function. As a result, the general principles and inference procedures carry over to Poisson regression analysis, like confidence intervals and hypothesis tests.

3. The response function

In the previous video we defined the Poisson regression as follows with the log link function providing for the linear combination in the parameters. Since all the parameters are on the log scale we need to exponentiate to obtain the response function in terms of lambda. Note that the log link exponentiates the linear predictors but does not transform the response variable. Having this form we can interpret the effect of beta on the response.

4. The response function

Note that the effect of x on lambda is multiplicative.

5. Interpretation of parameters

The constant term provides the intercept of the regression model, i.e. the predicted value when the explanatory variable is zero. Note that in some research problems it is not possible to have explanatory variable zero, and in such cases, you wouldn't interpret the intercept effect. Beta_1 is the expected difference in $y$ on a logarithmic scale for a 1-unit increase in x, where exponential of beta is the expected multiplicative effect on the mean lambda for a 1-unit increase in x.

6. Interpreting coefficient effect

To interpret the effect of the coefficient on the mean of the response we consider 3 cases. First, if beta_1 is greater than zero then the exponential of beta_1 is greater than 1. Hence lambda is exponential of beta_1 times larger than when x is zero. If beta_1 is less than zero the exponential is less than one and lambda is exponential of beta_1 time smaller than when x is zero. Finally, if beta_1 is zero then the exponential of beta_1 is 1 leading to no effect on the response and hence the explanatory variable and the response are not related. Note that we are comparing the effect based on 1 given the inherent exponential scale.

7. Example

Let's revisit the crab dataset. Consider the Poisson regression model where we model the number of satellites present given the weight of the female crab.

8. Example - interpretation of beta

We can extract model coefficients using the params function. Since the weight of the crab cannot be zero, we will not interpret the effect of the constant term i.e. the intercept. Coefficient of weight is greater than 0 at 0.5893, meaning the exponential of beta_1 will be greater than one. To compute the multiplicative effect we take the exponential of beta_1 using the numpy exp function. To conclude, for a 1-unit increase in weight the mean response of y, i.e.lambda, will multiply by 1.803, or 80% increase in the estimated mean number of satellites.

9. Confidence interval for ...

Similarly as we did in logistic regression we can extract 95% confidence intervals using the conf_int function to obtain the lower bound denoted as 0 and the upper bound denoted as 1. To obtain the confidence intervals for the multiplicative effect we take the exponential of the confidence intervals for the coefficient. The computed confidence intervals for beta imply that for one unit increase in weight the number of satellites will increase at least by 1.58 times and at most by 2.04 times.

10. Let's practice!

Now let's fit some Poisson regression models!

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