python - Calculating scale/dispersion of Gamma GLM using statsmodels

Question

I'm having trouble obtaining the dispersion parameter of simulated data using statsmodels' GLM function.

import statsmodels.api as sm
import matplotlib.pyplot as plt 
import scipy.stats as stats 
import numpy as np

np.random.seed(1)

# Generate data
x=np.random.uniform(0, 100,50000)
x2 = sm.add_constant(x)
a = 0.5
b = 0.2
y_true = 1/(a+(b*x))
# Add error 
scale = 2 # the scale parameter I'm trying to obtain
shape = y_true/scale # given that, for Gamma, mu = scale*shape
y = np.random.gamma(shape=shape, scale=scale)

# Run model
model = sm.GLM(y, x2, family=sm.families.Gamma()).fit() 

model.summary()

Here's the summary from above:

Note that the coefficient estimates are correct (0.5 and 0.2), but the scale (21.995) is way off the scale I set (2).

Can someone point out what it is I'm misunderstanding/doing wrong? Thanks!

Answer 1

As Josef noted in the comments, statsmodels uses a different kind of parameterization.