# Generalized Linear Models (GLM) Reference

This document provides comprehensive guidance on generalized linear models in statsmodels, including families, link
functions, and applications.

## Overview

GLMs extend linear regression to non-normal response distributions through:

1. **Distribution family**: Specifies the conditional distribution of the response
2. **Link function**: Transforms the linear predictor to the scale of the mean
3. **Variance function**: Relates variance to the mean

**General form**: g(μ) = Xβ, where g is the link function and μ = E(Y|X)

## When to Use GLM

- **Binary outcomes**: Logistic regression (Binomial family with logit link)
- **Count data**: Poisson or Negative Binomial regression
- **Positive continuous data**: Gamma or Inverse Gaussian
- **Non-normal distributions**: When OLS assumptions violated
- **Link functions**: Need non-linear relationship between predictors and response scale

## Distribution Families

### Binomial Family

For binary outcomes (0/1) or proportions (k/n).

**When to use:**

- Binary classification
- Success/failure outcomes
- Proportions or rates

**Common links:**

- Logit (default): log(μ/(1-μ))
- Probit: Φ⁻¹(μ)
- Log: log(μ)

```python
import statsmodels.api as sm
import statsmodels.formula.api as smf

# Binary logistic regression
model = sm.GLM(y, X, family=sm.families.Binomial())
results = model.fit()

# Formula API
results = smf.glm('success ~ x1 + x2', data=df,
                  family=sm.families.Binomial()).fit()

# Access predictions (probabilities)
probs = results.predict(X_new)

# Classification (0.5 threshold)
predictions = (probs > 0.5).astype(int)
```

**Interpretation:**

```python
import numpy as np

# Odds ratios (for logit link)
odds_ratios = np.exp(results.params)
print("Odds ratios:", odds_ratios)

# For 1-unit increase in x, odds multiply by exp(beta)
```

### Poisson Family

For count data (non-negative integers).

**When to use:**

- Count outcomes (number of events)
- Rare events
- Rate modeling (with offset)

**Common links:**

- Log (default): log(μ)
- Identity: μ
- Sqrt: √μ

```python
# Poisson regression
model = sm.GLM(y, X, family=sm.families.Poisson())
results = model.fit()

# With exposure/offset for rates
# If modeling rate = counts/exposure
model = sm.GLM(y, X, family=sm.families.Poisson(),
               offset=np.log(exposure))
results = model.fit()

# Interpretation: exp(beta) = multiplicative effect on expected count
import numpy as np
rate_ratios = np.exp(results.params)
print("Rate ratios:", rate_ratios)
```

**Overdispersion check:**

```python
# Deviance / df should be ~1 for Poisson
overdispersion = results.deviance / results.df_resid
print(f"Overdispersion: {overdispersion}")

# If >> 1, consider Negative Binomial
if overdispersion > 1.5:
    print("Consider Negative Binomial model for overdispersion")
```

### Negative Binomial Family

For overdispersed count data.

**When to use:**

- Count data with variance > mean
- Excess zeros or large variance
- Poisson model shows overdispersion

```python
# Negative Binomial GLM
model = sm.GLM(y, X, family=sm.families.NegativeBinomial())
results = model.fit()

# Alternative: use discrete choice model with alpha estimation
from statsmodels.discrete.discrete_model import NegativeBinomial
nb_model = NegativeBinomial(y, X)
nb_results = nb_model.fit()

print(f"Dispersion parameter alpha: {nb_results.params[-1]}")
```

### Gaussian Family

Equivalent to OLS but fit via IRLS (Iteratively Reweighted Least Squares).

**When to use:**

- Want GLM framework for consistency
- Need robust standard errors
- Comparing with other GLMs

**Common links:**

- Identity (default): μ
- Log: log(μ)
- Inverse: 1/μ

```python
# Gaussian GLM (equivalent to OLS)
model = sm.GLM(y, X, family=sm.families.Gaussian())
results = model.fit()

# Verify equivalence with OLS
ols_results = sm.OLS(y, X).fit()
print("Parameters close:", np.allclose(results.params, ols_results.params))
```

### Gamma Family

For positive continuous data, often right-skewed.

**When to use:**

- Positive outcomes (insurance claims, survival times)
- Right-skewed distributions
- Variance proportional to mean²

**Common links:**

- Inverse (default): 1/μ
- Log: log(μ)
- Identity: μ

```python
# Gamma regression (common for cost data)
model = sm.GLM(y, X, family=sm.families.Gamma())
results = model.fit()

# Log link often preferred for interpretation
model = sm.GLM(y, X, family=sm.families.Gamma(link=sm.families.links.Log()))
results = model.fit()

# With log link, exp(beta) = multiplicative effect
import numpy as np
effects = np.exp(results.params)
```

### Inverse Gaussian Family

For positive continuous data with specific variance structure.

**When to use:**

- Positive skewed outcomes
- Variance proportional to mean³
- Alternative to Gamma

**Common links:**

- Inverse squared (default): 1/μ²
- Log: log(μ)

```python
model = sm.GLM(y, X, family=sm.families.InverseGaussian())
results = model.fit()
```

### Tweedie Family

Flexible family covering multiple distributions.

**When to use:**

- Insurance claims (mixture of zeros and continuous)
- Semi-continuous data
- Need flexible variance function

**Special cases (power parameter p):**

- p=0: Normal
- p=1: Poisson
- p=2: Gamma
- p=3: Inverse Gaussian
- 1<p<2: Compound Poisson-Gamma (common for insurance)

```python
# Tweedie with power=1.5
model = sm.GLM(y, X, family=sm.families.Tweedie(link=sm.families.links.Log(),
                                                 var_power=1.5))
results = model.fit()
```

## Link Functions

Link functions connect the linear predictor to the mean of the response.

### Available Links

```python
from statsmodels.genmod import families

# Identity: g(μ) = μ
link = families.links.Identity()

# Log: g(μ) = log(μ)
link = families.links.Log()

# Logit: g(μ) = log(μ/(1-μ))
link = families.links.Logit()

# Probit: g(μ) = Φ⁻¹(μ)
link = families.links.Probit()

# Complementary log-log: g(μ) = log(-log(1-μ))
link = families.links.CLogLog()

# Inverse: g(μ) = 1/μ
link = families.links.InversePower()

# Inverse squared: g(μ) = 1/μ²
link = families.links.InverseSquared()

# Square root: g(μ) = √μ
link = families.links.Sqrt()

# Power: g(μ) = μ^p
link = families.links.Power(power=2)
```

### Choosing Link Functions

**Canonical links** (default for each family):

- Binomial → Logit
- Poisson → Log
- Gamma → Inverse
- Gaussian → Identity
- Inverse Gaussian → Inverse squared

**When to use non-canonical:**

- **Log link with Binomial**: Risk ratios instead of odds ratios
- **Identity link**: Direct additive effects (when sensible)
- **Probit vs Logit**: Similar results, preference based on field
- **CLogLog**: Asymmetric relationship, common in survival analysis

```python
# Example: Risk ratios with log-binomial model
model = sm.GLM(y, X, family=sm.families.Binomial(link=sm.families.links.Log()))
results = model.fit()

# exp(beta) now gives risk ratios, not odds ratios
risk_ratios = np.exp(results.params)
```

## Model Fitting and Results

### Basic Workflow

```python
import statsmodels.api as sm

# Add constant
X = sm.add_constant(X_data)

# Specify family and link
family = sm.families.Poisson(link=sm.families.links.Log())

# Fit model using IRLS
model = sm.GLM(y, X, family=family)
results = model.fit()

# Summary
print(results.summary())
```

### Results Attributes

```python
# Parameters and inference
results.params              # Coefficients
results.bse                 # Standard errors
results.tvalues            # Z-statistics
results.pvalues            # P-values
results.conf_int()         # Confidence intervals

# Predictions
results.fittedvalues       # Fitted values (μ)
results.predict(X_new)     # Predictions for new data

# Model fit statistics
results.aic                # Akaike Information Criterion
results.bic                # Bayesian Information Criterion
results.deviance           # Deviance
results.null_deviance      # Null model deviance
results.pearson_chi2       # Pearson chi-squared statistic
results.df_resid           # Residual degrees of freedom
results.llf                # Log-likelihood

# Residuals
results.resid_response     # Response residuals (y - μ)
results.resid_pearson      # Pearson residuals
results.resid_deviance     # Deviance residuals
results.resid_anscombe     # Anscombe residuals
results.resid_working      # Working residuals
```

### Pseudo R-squared

```python
# McFadden's pseudo R-squared
pseudo_r2 = 1 - (results.deviance / results.null_deviance)
print(f"Pseudo R²: {pseudo_r2:.4f}")

# Adjusted pseudo R-squared
n = len(y)
k = len(results.params)
adj_pseudo_r2 = 1 - ((n-1)/(n-k)) * (results.deviance / results.null_deviance)
print(f"Adjusted Pseudo R²: {adj_pseudo_r2:.4f}")
```

## Diagnostics

### Goodness of Fit

```python
# Deviance should be approximately χ² with df_resid degrees of freedom
from scipy import stats

deviance_pval = 1 - stats.chi2.cdf(results.deviance, results.df_resid)
print(f"Deviance test p-value: {deviance_pval}")

# Pearson chi-squared test
pearson_pval = 1 - stats.chi2.cdf(results.pearson_chi2, results.df_resid)
print(f"Pearson chi² test p-value: {pearson_pval}")

# Check for overdispersion/underdispersion
dispersion = results.pearson_chi2 / results.df_resid
print(f"Dispersion: {dispersion}")
# Should be ~1; >1 suggests overdispersion, <1 underdispersion
```

### Residual Analysis

```python
import matplotlib.pyplot as plt

# Deviance residuals vs fitted
plt.figure(figsize=(10, 6))
plt.scatter(results.fittedvalues, results.resid_deviance, alpha=0.5)
plt.xlabel('Fitted values')
plt.ylabel('Deviance residuals')
plt.axhline(y=0, color='r', linestyle='--')
plt.title('Deviance Residuals vs Fitted')
plt.show()

# Q-Q plot of deviance residuals
from statsmodels.graphics.gofplots import qqplot
qqplot(results.resid_deviance, line='s')
plt.title('Q-Q Plot of Deviance Residuals')
plt.show()

# For binary outcomes: binned residual plot
if isinstance(results.model.family, sm.families.Binomial):
    from statsmodels.graphics.gofplots import qqplot
    # Group predictions and compute average residuals
    # (custom implementation needed)
    pass
```

### Influence and Outliers

```python
from statsmodels.stats.outliers_influence import GLMInfluence

influence = GLMInfluence(results)

# Leverage
leverage = influence.hat_matrix_diag

# Cook's distance
cooks_d = influence.cooks_distance[0]

# DFFITS
dffits = influence.dffits[0]

# Find influential observations
influential = np.where(cooks_d > 4/len(y))[0]
print(f"Influential observations: {influential}")
```

## Hypothesis Testing

```python
# Wald test for single parameter (automatically in summary)

# Likelihood ratio test for nested models
# Fit reduced model
model_reduced = sm.GLM(y, X_reduced, family=family).fit()
model_full = sm.GLM(y, X_full, family=family).fit()

# LR statistic
lr_stat = 2 * (model_full.llf - model_reduced.llf)
df = model_full.df_model - model_reduced.df_model

from scipy import stats
lr_pval = 1 - stats.chi2.cdf(lr_stat, df)
print(f"LR test p-value: {lr_pval}")

# Wald test for multiple parameters
# Test beta_1 = beta_2 = 0
R = [[0, 1, 0, 0], [0, 0, 1, 0]]
wald_test = results.wald_test(R)
print(wald_test)
```

## Robust Standard Errors

```python
# Heteroscedasticity-robust (sandwich estimator)
results_robust = results.get_robustcov_results(cov_type='HC0')

# Cluster-robust
results_cluster = results.get_robustcov_results(cov_type='cluster',
                                                groups=cluster_ids)

# Compare standard errors
print("Regular SE:", results.bse)
print("Robust SE:", results_robust.bse)
```

## Model Comparison

```python
# AIC/BIC for non-nested models
models = [model1_results, model2_results, model3_results]
for i, res in enumerate(models, 1):
    print(f"Model {i}: AIC={res.aic:.2f}, BIC={res.bic:.2f}")

# Likelihood ratio test for nested models (as shown above)

# Cross-validation for predictive performance
from sklearn.model_selection import KFold
from sklearn.metrics import log_loss

kf = KFold(n_splits=5, shuffle=True, random_state=42)
cv_scores = []

for train_idx, val_idx in kf.split(X):
    X_train, X_val = X[train_idx], X[val_idx]
    y_train, y_val = y[train_idx], y[val_idx]

    model_cv = sm.GLM(y_train, X_train, family=family).fit()
    pred_probs = model_cv.predict(X_val)

    score = log_loss(y_val, pred_probs)
    cv_scores.append(score)

print(f"CV Log Loss: {np.mean(cv_scores):.4f} ± {np.std(cv_scores):.4f}")
```

## Prediction

```python
# Point predictions
predictions = results.predict(X_new)

# For classification: get probabilities and convert
if isinstance(family, sm.families.Binomial):
    probs = predictions
    class_predictions = (probs > 0.5).astype(int)

# For counts: predictions are expected counts
if isinstance(family, sm.families.Poisson):
    expected_counts = predictions

# Prediction intervals via bootstrap
n_boot = 1000
boot_preds = np.zeros((n_boot, len(X_new)))

for i in range(n_boot):
    # Bootstrap resample
    boot_idx = np.random.choice(len(y), size=len(y), replace=True)
    X_boot, y_boot = X[boot_idx], y[boot_idx]

    # Fit and predict
    boot_model = sm.GLM(y_boot, X_boot, family=family).fit()
    boot_preds[i] = boot_model.predict(X_new)

# 95% prediction intervals
pred_lower = np.percentile(boot_preds, 2.5, axis=0)
pred_upper = np.percentile(boot_preds, 97.5, axis=0)
```

## Common Applications

### Logistic Regression (Binary Classification)

```python
import statsmodels.api as sm

# Fit logistic regression
X = sm.add_constant(X_data)
model = sm.GLM(y, X, family=sm.families.Binomial())
results = model.fit()

# Odds ratios
odds_ratios = np.exp(results.params)
odds_ci = np.exp(results.conf_int())

# Classification metrics
from sklearn.metrics import classification_report, roc_auc_score

probs = results.predict(X)
predictions = (probs > 0.5).astype(int)

print(classification_report(y, predictions))
print(f"AUC: {roc_auc_score(y, probs):.4f}")

# ROC curve
from sklearn.metrics import roc_curve
import matplotlib.pyplot as plt

fpr, tpr, thresholds = roc_curve(y, probs)
plt.plot(fpr, tpr)
plt.plot([0, 1], [0, 1], 'k--')
plt.xlabel('False Positive Rate')
plt.ylabel('True Positive Rate')
plt.title('ROC Curve')
plt.show()
```

### Poisson Regression (Count Data)

```python
# Fit Poisson model
X = sm.add_constant(X_data)
model = sm.GLM(y_counts, X, family=sm.families.Poisson())
results = model.fit()

# Rate ratios
rate_ratios = np.exp(results.params)
print("Rate ratios:", rate_ratios)

# Check overdispersion
dispersion = results.pearson_chi2 / results.df_resid
if dispersion > 1.5:
    print(f"Overdispersion detected ({dispersion:.2f}). Consider Negative Binomial.")
```

### Gamma Regression (Cost/Duration Data)

```python
# Fit Gamma model with log link
X = sm.add_constant(X_data)
model = sm.GLM(y_cost, X,
               family=sm.families.Gamma(link=sm.families.links.Log()))
results = model.fit()

# Multiplicative effects
effects = np.exp(results.params)
print("Multiplicative effects on mean:", effects)
```

## Best Practices

1. **Check distribution assumptions**: Plot histograms and Q-Q plots of response
2. **Verify link function**: Use canonical links unless there's a reason not to
3. **Examine residuals**: Deviance residuals should be approximately normal
4. **Test for overdispersion**: Especially for Poisson models
5. **Use offsets appropriately**: For rate modeling with varying exposure
6. **Consider robust SEs**: When variance assumptions questionable
7. **Compare models**: Use AIC/BIC for non-nested, LR test for nested
8. **Interpret on original scale**: Transform coefficients (e.g., exp for log link)
9. **Check influential observations**: Use Cook's distance
10. **Validate predictions**: Use cross-validation or holdout set

## Common Pitfalls

1. **Forgetting to add constant**: No intercept term
2. **Using wrong family**: Check distribution of response
3. **Ignoring overdispersion**: Use Negative Binomial instead of Poisson
4. **Misinterpreting coefficients**: Remember link function transformation
5. **Not checking convergence**: IRLS may not converge; check warnings
6. **Complete separation in logistic**: Some categories perfectly predict outcome
7. **Using identity link with bounded outcomes**: May predict outside valid range
8. **Comparing models with different samples**: Use same observations
9. **Forgetting offset in rate models**: Must use log(exposure) as offset
10. **Not considering alternatives**: Mixed models, zero-inflation for complex data
