Quasi-Poisson Regression

The Quasi-Poisson Regression is a generalization of the Poisson regression and is used when modeling an overdispersed count variable.

The Poisson model assumes that the variance is equal to the mean, which is not always a fair assumption. When the variance is greater than the mean, a Quasi-Poisson model, which assumes that the variance is a linear function of the mean, is more appropriate.

## quasipoisson. compare with example(glm)
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
d.AD <- data.frame(treatment, outcome, counts)

d.AD

##   treatment outcome counts
## 1         1       1     18
## 2         1       2     17
## 3         1       3     15
## 4         2       1     20
## 5         2       2     10
## 6         2       3     20
## 7         3       1     25
## 8         3       2     13
## 9         3       3     12

var(counts)/mean(counts)

## [1] 1.32

---

Poisson Model

glm.pois <- glm(counts ~ outcome + treatment, family = poisson())
summary(glm.pois)

## 
## Call:
## glm(formula = counts ~ outcome + treatment, family = poisson())
## 
## Deviance Residuals: 
##        1         2         3         4         5         6         7         8  
## -0.67125   0.96272  -0.16965  -0.21999  -0.95552   1.04939   0.84715  -0.09167  
##        9  
## -0.96656  
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  3.045e+00  1.709e-01  17.815   <2e-16 ***
## outcome2    -4.543e-01  2.022e-01  -2.247   0.0246 *  
## outcome3    -2.930e-01  1.927e-01  -1.520   0.1285    
## treatment2   1.338e-15  2.000e-01   0.000   1.0000    
## treatment3   1.421e-15  2.000e-01   0.000   1.0000    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 10.5814  on 8  degrees of freedom
## Residual deviance:  5.1291  on 4  degrees of freedom
## AIC: 56.761
## 
## Number of Fisher Scoring iterations: 4

---

Quasi-Poisson Model

glm.qpois <- glm(counts ~ outcome + treatment, family = quasipoisson())
summary(glm.qpois)

## 
## Call:
## glm(formula = counts ~ outcome + treatment, family = quasipoisson())
## 
## Deviance Residuals: 
##        1         2         3         4         5         6         7         8  
## -0.67125   0.96272  -0.16965  -0.21999  -0.95552   1.04939   0.84715  -0.09167  
##        9  
## -0.96656  
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.045e+00  1.944e-01  15.665  9.7e-05 ***
## outcome2    -4.543e-01  2.299e-01  -1.976    0.119    
## outcome3    -2.930e-01  2.192e-01  -1.337    0.252    
## treatment2   1.338e-15  2.274e-01   0.000    1.000    
## treatment3   1.421e-15  2.274e-01   0.000    1.000    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for quasipoisson family taken to be 1.2933)
## 
##     Null deviance: 10.5814  on 8  degrees of freedom
## Residual deviance:  5.1291  on 4  degrees of freedom
## AIC: NA
## 
## Number of Fisher Scoring iterations: 4

---

Anova

anova(glm.qpois, test = "F")

## Analysis of Deviance Table
## 
## Model: quasipoisson, link: log
## 
## Response: counts
## 
## Terms added sequentially (first to last)
## 
## 
##           Df Deviance Resid. Df Resid. Dev      F Pr(>F)
## NULL                          8    10.5814              
## outcome    2   5.4523         6     5.1291 2.1079  0.237
## treatment  2   0.0000         4     5.1291 0.0000  1.000

---

## for Poisson results use
anova(glm.qpois, dispersion = 1, test = "Chisq")

## Analysis of Deviance Table
## 
## Model: quasipoisson, link: log
## 
## Response: counts
## 
## Terms added sequentially (first to last)
## 
## 
##           Df Deviance Resid. Df Resid. Dev Pr(>Chi)  
## NULL                          8    10.5814           
## outcome    2   5.4523         6     5.1291  0.06547 .
## treatment  2   0.0000         4     5.1291  1.00000  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

---

summary(glm.qpois, dispersion = 1)

## 
## Call:
## glm(formula = counts ~ outcome + treatment, family = quasipoisson())
## 
## Deviance Residuals: 
##        1         2         3         4         5         6         7         8  
## -0.67125   0.96272  -0.16965  -0.21999  -0.95552   1.04939   0.84715  -0.09167  
##        9  
## -0.96656  
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  3.045e+00  1.709e-01  17.815   <2e-16 ***
## outcome2    -4.543e-01  2.022e-01  -2.247   0.0246 *  
## outcome3    -2.930e-01  1.927e-01  -1.520   0.1285    
## treatment2   1.338e-15  2.000e-01   0.000   1.0000    
## treatment3   1.421e-15  2.000e-01   0.000   1.0000    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for quasipoisson family taken to be 1)
## 
##     Null deviance: 10.5814  on 8  degrees of freedom
## Residual deviance:  5.1291  on 4  degrees of freedom
## AIC: NA
## 
## Number of Fisher Scoring iterations: 4