7.6 Quasi-Poisson Regression

Poisson regression assumes that the mean and variance are equal:

$$\text{Var}(Y_i) = E(Y_i) = \mu_i$$

However, many real-world datasets exhibit overdispersion, where the variance exceeds the mean:

$$\text{Var}(Y_i) = \phi \mu_i$$

where $\phi$ (the dispersion parameter) allows the variance to scale beyond the Poisson assumption.

To correct for this, we use Quasi-Poisson regression, which:

• Follows the Generalized Linear Models structure but is not a strict GLM.

• Uses a variance function proportional to the mean: $\text{Var}(Y_i) = \phi \mu_i$.

• Does not assume a specific probability distribution, unlike Poisson or Negative Binomial models.

7.6.1 Is Quasi-Poisson Regression a Generalized Linear Model?

✅ Yes, Quasi-Poisson is GLM-like:

• Linear Predictor: Like Poisson regression, it models the log of the expected count as a function of predictors: $$\log(E(Y)) = X\beta$$

• Canonical Link Function: It typically uses a log link function, just like standard Poisson regression.

• Variance Structure: Unlike standard Poisson, which assumes $\text{Var}(Y) = E(Y)$, Quasi-Poisson allows for overdispersion: $$\text{Var}(Y) = \phi E(Y)$$ where $\phi$ is estimated rather than assumed to be 1.

❌ No, Quasi-Poisson is not a strict GLM because:

• GLMs require a full probability distribution from the exponential family.

  • Standard Poisson regression assumes a Poisson distribution (which belongs to the exponential family).

  • Quasi-Poisson does not assume a full probability distribution, only a mean-variance relationship.

• It does not use Maximum Likelihood Estimation.

  • Standard GLMs use MLE to estimate parameters.

  • Quasi-Poisson uses quasi-likelihood methods, which require specifying only the mean and variance, but not a full likelihood function.

• Likelihood-based inference is not valid.

  • AIC, BIC, and Likelihood Ratio Tests cannot be used with Quasi-Poisson regression.

When to Use Quasi-Poisson:

• When data exhibit overdispersion (variance > mean), making standard Poisson regression inappropriate.

• When Negative Binomial Regression is not preferred, but an alternative is needed to handle overdispersion.

• If overdispersion is present, Negative Binomial Regression is often a better alternative because it is a true GLM with a full likelihood function, whereas Quasi-Poisson is only a quasi-likelihood approach.

7.6.2 Application: Quasi-Poisson Regression

We analyze the bioChemists dataset, modeling the number of published articles (Num_Article) as a function of various predictors.

7.6.2.1 Checking Overdispersion in the Poisson Model

We first fit a Poisson regression model and check for overdispersion using the deviance-to-degrees-of-freedom ratio:

# Fit Poisson regression model
Poisson_Mod <-
    glm(Num_Article ~ ., family = poisson, data = bioChemists)

# Compute dispersion parameter
dispersion_estimate <-
    Poisson_Mod$deviance / Poisson_Mod$df.residual
dispersion_estimate
#> [1] 1.797988

• If $\hat{\phi} > 1$, the Poisson model underestimates variance.

• A large value (>> 1) suggests that Poisson regression is not appropriate.

7.6.2.2 Fitting the Quasi-Poisson Model

Since overdispersion is present, we refit the model using Quasi-Poisson regression, which scales standard errors by $\phi$.

# Fit Quasi-Poisson regression model
quasiPoisson_Mod <-
    glm(Num_Article ~ ., family = quasipoisson, data = bioChemists)

# Summary of the model
summary(quasiPoisson_Mod)
#> 
#> Call:
#> glm(formula = Num_Article ~ ., family = quasipoisson, data = bioChemists)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -3.5672  -1.5398  -0.3660   0.5722   5.4467  
#> 
#> Coefficients:
#>                  Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)      0.304617   0.139273   2.187 0.028983 *  
#> SexWomen        -0.224594   0.073860  -3.041 0.002427 ** 
#> MarriedMarried   0.155243   0.083003   1.870 0.061759 .  
#> Num_Kid5        -0.184883   0.054268  -3.407 0.000686 ***
#> PhD_Quality      0.012823   0.035700   0.359 0.719544    
#> Num_MentArticle  0.025543   0.002713   9.415  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for quasipoisson family taken to be 1.829006)
#> 
#>     Null deviance: 1817.4  on 914  degrees of freedom
#> Residual deviance: 1634.4  on 909  degrees of freedom
#> AIC: NA
#> 
#> Number of Fisher Scoring iterations: 5

Interpretation:

• The coefficients remain the same as in Poisson regression.

• Standard errors are inflated to account for overdispersion.

• P-values increase, leading to more conservative inference.

7.6.2.3 Comparing Poisson and Quasi-Poisson

To see the effect of using Quasi-Poisson, we compare standard errors:

# Extract coefficients and standard errors
poisson_se <- summary(Poisson_Mod)$coefficients[, 2]
quasi_se <- summary(quasiPoisson_Mod)$coefficients[, 2]

# Compare standard errors
se_comparison <- data.frame(Poisson = poisson_se,
                            Quasi_Poisson = quasi_se)
se_comparison
#>                     Poisson Quasi_Poisson
#> (Intercept)     0.102981443   0.139272885
#> SexWomen        0.054613488   0.073859696
#> MarriedMarried  0.061374395   0.083003199
#> Num_Kid5        0.040126898   0.054267922
#> PhD_Quality     0.026397045   0.035699564
#> Num_MentArticle 0.002006073   0.002713028

• Quasi-Poisson has larger standard errors than Poisson.

• This leads to wider confidence intervals, reducing the likelihood of false positives.

7.6.2.4 Model Diagnostics: Checking Residuals

We examine residuals to assess model fit:

# Residual plot
plot(
    quasiPoisson_Mod$fitted.values,
    residuals(quasiPoisson_Mod, type = "pearson"),
    xlab = "Fitted Values",
    ylab = "Pearson Residuals",
    main = "Residuals vs. Fitted Values (Quasi-Poisson)"
)
abline(h = 0, col = "red")

• If residuals show a pattern, additional predictors or transformations may be needed.

• Random scatter around zero suggests a well-fitting model.

7.6.2.5 Alternative: Negative Binomial vs. Quasi-Poisson

If overdispersion is severe, Negative Binomial regression may be preferable because it explicitly models dispersion:

# Fit Negative Binomial model
library(MASS)
NegBinom_Mod <- glm.nb(Num_Article ~ ., data = bioChemists)

# Model summaries
summary(quasiPoisson_Mod)
#> 
#> Call:
#> glm(formula = Num_Article ~ ., family = quasipoisson, data = bioChemists)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -3.5672  -1.5398  -0.3660   0.5722   5.4467  
#> 
#> Coefficients:
#>                  Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)      0.304617   0.139273   2.187 0.028983 *  
#> SexWomen        -0.224594   0.073860  -3.041 0.002427 ** 
#> MarriedMarried   0.155243   0.083003   1.870 0.061759 .  
#> Num_Kid5        -0.184883   0.054268  -3.407 0.000686 ***
#> PhD_Quality      0.012823   0.035700   0.359 0.719544    
#> Num_MentArticle  0.025543   0.002713   9.415  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for quasipoisson family taken to be 1.829006)
#> 
#>     Null deviance: 1817.4  on 914  degrees of freedom
#> Residual deviance: 1634.4  on 909  degrees of freedom
#> AIC: NA
#> 
#> Number of Fisher Scoring iterations: 5
summary(NegBinom_Mod)
#> 
#> Call:
#> glm.nb(formula = Num_Article ~ ., data = bioChemists, init.theta = 2.264387695, 
#>     link = log)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -2.1678  -1.3617  -0.2806   0.4476   3.4524  
#> 
#> Coefficients:
#>                  Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)      0.256144   0.137348   1.865 0.062191 .  
#> SexWomen        -0.216418   0.072636  -2.979 0.002887 ** 
#> MarriedMarried   0.150489   0.082097   1.833 0.066791 .  
#> Num_Kid5        -0.176415   0.052813  -3.340 0.000837 ***
#> PhD_Quality      0.015271   0.035873   0.426 0.670326    
#> Num_MentArticle  0.029082   0.003214   9.048  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for Negative Binomial(2.2644) family taken to be 1)
#> 
#>     Null deviance: 1109.0  on 914  degrees of freedom
#> Residual deviance: 1004.3  on 909  degrees of freedom
#> AIC: 3135.9
#> 
#> Number of Fisher Scoring iterations: 1
#> 
#> 
#>               Theta:  2.264 
#>           Std. Err.:  0.271 
#> 
#>  2 x log-likelihood:  -3121.917

7.6.2.6 Key Differences: Quasi-Poisson vs. Negative Binomial

Feature	Quasi-Poisson	Negative Binomial
Handles Overdispersion?	✅ Yes	✅ Yes
Uses a Full Probability Distribution?	❌ No	✅ Yes
MLE-Based?	❌ No (quasi-likelihood)	✅ Yes
Can Use AIC/BIC for Model Selection?	❌ No	✅ Yes
Better for Model Interpretation?	✅ Yes	✅ Yes
Best for Severe Overdispersion?	❌ No	✅ Yes

When to Choose:

• Use Quasi-Poisson when you only need robust standard errors and do not require model selection via AIC/BIC.

• Use Negative Binomial when overdispersion is large and you want a true likelihood-based model.

While Quasi-Poisson is a quick fix, Negative Binomial is generally the better choice for modeling count data with overdispersion.