# Generalized Linear Model ( GLM ) for Data Analysts & Scientists: Part 2

For an initial understanding of GLM Models. Refer to this:: https://readmedium.com/generalized-linear-model-glm-for-data-analysts-scientists-part-1-4ed52cca2f27

Certainly! Let’s dive into more detail about Generalized Linear Models (GLMs) and provide a comprehensive example using Python and the `statsmodels`

library.

# More In-Depth Explanation:

## 1. Random Component:

- The random component of a GLM specifies the probability distribution of the response variable.
*It must be a member of the exponential family of distributions, which includes common distributions like normal, binomial, Poisson, and more.*

## 2. Systematic Component:

- The systematic component is the linear combination of predictor variables that is related to the expected value of the response variable. This is represented as->
*η=Xβ,*where*η*is the linear predictor,*X*is the design matrix of predictor variables, and*β*is the vector of coefficients.

## 3. Link Function:

- The link function is a mathematical function that connects the expected value of the response variable to the linear predictor. It transforms the scale of the response variable to make it suitable for modeling as a linear function of the predictors.
*Common link functions include the identity link (for normal distribution), logit link (for binomial distribution), and log link (for Poisson distribution).*

## 4. Likelihood Function:

- The likelihood function in a GLM is used to estimate the parameters of the model.
*It measures the probability of observing the data given the model and its parameters. The goal is to find the parameter values that maximize this likelihood.*

## 5. Estimation Methods:

- Maximum Likelihood Estimation (MLE) is commonly used to estimate the parameters in GLMs.
*The MLE method seeks to find the parameter values that make the observed data most probable.*

## 6. Model Validation:

- Like any statistical model, it’s important to validate the assumptions and performance of a GLM.
*This can involve techniques like cross-validation, residual analysis, and goodness-of-fit tests.*

# Example Code:

Here’s a more detailed example using Python and the `statsmodels`

library to fit a GLM:

```
import numpy as np
import statsmodels.api as sm
import statsmodels.formula.api as smf
import pandas as pd
# Generate some example data
np.random.seed(0)
X = np.random.rand(100, 2) # Two predictor variables
y = np.random.poisson(5 * np.exp(X[:, 0] + 0.1*X[:, 1])) # Poisson-distributed response variable
# Create a pandas DataFrame for easier handling of the data
df = pd.DataFrame({'X1': X[:, 0], 'X2': X[:, 1], 'y': y})
# Fit a Poisson regression model
model = smf.glm(formula="y ~ X1 + X2", data=df, family=sm.families.Poisson())
result = model.fit()
# Print the summary of the regression results
print(result.summary())
```

In this example, we go one step further by using a Pandas DataFrame to organize our data. We generate example data with two predictor variables (`X1`

and `X2`

) and a response variable (`y`

) that follows a Poisson distribution.

We then define a GLM formula using `smf.glm()`

, specifying the formula (`"y ~ X1 + X2"`

), the data (`df`

), and the family of the distribution (`sm.families.Poisson()`

).

Finally, we fit the model using `model.fit()`

and print a summary of the regression results.

Remember, in a real-world scenario, you would need to carefully preprocess your data, handle missing values, validate the model, and potentially explore more complex GLM formulations depending on your specific use case.