Chapter 9 Binomial GLM and proportions

Sometimes, proportion data are more similar to logistic regression than you think.

In discrete counts, we can, for instance, measure the number of presence of individuals in relation to the total number of populations sampled.

We will thus obtain a proportional number of “success” in observing individuals by dividing the counts by the total counts.

In glm(), we have to provide prior weights if the response variable is the proportion of successes.

Proportions can be modelled by providing both the number of “successes” and prior weights in the function

prop.reg <- glm(cbind(Galumna, totalabund - Galumna) ~ Topo + WatrCont,
                data = mites,
                family = binomial)
summary(prop.reg)

## 
## Call:
## glm(formula = cbind(Galumna, totalabund - Galumna) ~ Topo + WatrCont, 
##     family = binomial, data = mites)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.4808  -0.9699  -0.6327  -0.1798   4.1688  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.288925   0.422109  -7.792 6.61e-15 ***
## TopoHummock  0.578332   0.274928   2.104   0.0354 *  
## WatrCont    -0.005886   0.001086  -5.420 5.97e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 140.702  on 69  degrees of freedom
## Residual deviance:  85.905  on 67  degrees of freedom
## AIC: 158.66
## 
## Number of Fisher Scoring iterations: 5

The weights can also be set explicitly in glm():

prop.reg2 <- glm(prop ~ Topo + WatrCont,
                 data = mites,
                 family = binomial,
                 weights = totalabund) # provide prior weights
summary(prop.reg2)

## 
## Call:
## glm(formula = prop ~ Topo + WatrCont, family = binomial, data = mites, 
##     weights = totalabund)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.4808  -0.9699  -0.6327  -0.1798   4.1688  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.288925   0.422109  -7.792 6.61e-15 ***
## TopoHummock  0.578332   0.274928   2.104   0.0354 *  
## WatrCont    -0.005886   0.001086  -5.420 5.97e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 140.702  on 69  degrees of freedom
## Residual deviance:  85.905  on 67  degrees of freedom
## AIC: 158.66
## 
## Number of Fisher Scoring iterations: 5