Interpreting model fit

Generalized Linear Models in Python

Ita Cirovic Donev

Data Science Consultant

Parameter estimation

Maximum likelihood estimation (MLE)
Iteratively reweighted least squares (IRLS)

The response function

Poisson regression model $$ log(\lambda)=\beta_0+\beta_1x_1 $$
The response function: $$ \lambda=exp(\beta_0 + \beta_1x_1) $$ or $$ \lambda=exp(\beta_0) \times exp(\beta_1x_1) $$

The response function

Poisson regression model $$ log(\lambda)=\beta_0+\beta_1x_1 $$
The response function: $$ \lambda=exp(\beta_0 + \beta_1x_1) $$ or $$ \lambda=exp(\beta_0) \color{red}{\times} exp(\beta_1x_1) $$

Interpretation of parameters

$exp(\beta_0)$
- The effect on the mean $\lambda$ when $x=0$
$exp(\beta_1)$
- The multiplicative effect on the mean $\lambda$ for a 1-unit increase in $x$

Interpreting coefficient effect

If $\color{#FF550D}{\beta_1 > 0}$
- $exp(\beta_1)>1$
- $\lambda$ is $\color{#FF550D}{exp(\beta_1)\text{ times larger}}$ than when $x=0$

If $\color{#D04A73}{\beta<0}$
- $exp(\beta_1)<1$
- $\lambda$ is $\color{#D04A73}{exp(\beta_1) \text{ times smaller}}$ than when $x=0$

If $\color{#0099FF}{\beta_1 = 0}$
- $exp(\beta_1)=1$
- $\lambda=exp(\beta_0)$
- Multiplicative factor is 1
- $y$ and $x$ are $\text{\color{#0099FF}{not related}}$

Example

model = glm('sat ~ weight', data = crab, 
            family = sm.families.Poisson()).fit()

                 Generalized Linear Model Regression Results (print cut)                 
=============================================================================
                 coef    std err          z      P>|z|      [0.025     0.975]
-----------------------------------------------------------------------------
Intercept     -0.4284      0.179     -2.394      0.017      -0.779     -0.078
weight         0.5893      0.065      9.064      0.000       0.462      0.717
=============================================================================

Example - interpretation of beta

Extract model coefficients
```
model.params
```

Intercept   -0.428405
weight       0.589304

Compute the effect
```
np.exp(0.589304)
```

1.803

Confidence interval for ...

$\beta_1$

print(model.conf_int())

                  0         1
Intercept -0.779112 -0.077699
weight     0.461873  0.716735

The multiplicative effect on mean

print(np.exp(crab_fit.conf_int()))

                  0         1
Intercept  0.458813  0.925243
weight     1.587044  2.047737

Let's practice!

Generalized Linear Models in Python