Interpreting model inference

Generalized Linear Models in Python

Ita Cirovic Donev

Data Science Consultant

Estimation of beta coefficient

Likelihood function

Summary output of the fitted model

Summary output of the fitted model with highlighted statistics on coefficient estimation.

Flatter peak
$\rightarrow$ Location of maximum harder to define
$\rightarrow$ Larger SE

Visualization of the likelihood when standard error is larger.

Sharper peak
$\rightarrow$ Location of maximum more clearly defined
$\rightarrow$ Smaller SE

Visualization of the likelihood when standard error is smaller.

# Extract variance-covariance matrix
print(model_GLM.cov_params())

           Intercept    weight
Intercept   0.774762 -0.325087
weight     -0.325087  0.141903

# Compute standard error for weight
std_error = np.sqrt(0.141903)

0.3767

Variance-covariance matrix

Illustration of the variance-covariance matrix

z-statistic $$ \color{#2485F2}{z=\hat\beta/SE} $$
$\color{#2485F2}{z}$ large $\Rightarrow$ coefficient $\ne0$ $\Rightarrow$ variable significant
Rule of thumb: cut-off value of 2

Example: horseshoe crab model
y ~ weight

$z = 1.8151/0.377 = 4.819$

$$ [\color{#5A5AF3}{lower},\color{#D8498E}{upper}] $$

$$ [\color{#5A5AF3}{\hat\beta - 1.96 \times SE},\color{#D8498E}{\hat\beta+1.96 \times SE}] $$

Example: horseshoe crab model

                 coef    std err   
<hr />-------------------------------
Intercept     -3.6947      0.880  
weight         1.8151      0.377

print(model_GLM.conf_int())

                  0         1
Intercept -5.419897 -1.969555
weight     1.076826  2.553463

print(model_GLM.conf_int())

              lower         1
Intercept -5.419897 -1.969555
weight     1.076826  2.553463

print(model_GLM.conf_int())

                  0     upper
Intercept -5.419897 -1.969555
weight     1.076826  2.553463

print(np.exp(model_GLM.conf_int()))

                  0          1
Intercept  0.004428   0.139519
weight     2.935348  12.851533

Generalized Linear Models in Python