##### Code for Lecture on Poisson regression


###Example - effects of age and birth location (wild or captive) on litter sizes in red wolves
### Litter size is not normally distribution (skewed to the left) and isn't continuous


###import the data


datum=read.csv(file.choose())


###Analyze the data using glm


help(glm)
#glm is the same as lm, but requires an additional argument - family, which specifies what the error structure is
results=glm(LitterSize~Age+BirthLoc,data=datum,family=poisson) ### runs a Poisson regression
summary(results) #examine the results

#Call:
#glm(formula = LitterSize ~ BirthLoc + Age, family = poisson, 
#    data = datum)
#
#Deviance Residuals: 
#    Min       1Q   Median       3Q      Max  
#-2.2591  -1.1052  -0.1177   0.6177   2.8361  
#
#Coefficients:
#            Estimate Std. Error z value Pr(>|z|)    
#(Intercept) -0.21178    0.15463  -1.370    0.171    
#BirthLoc     1.70471    0.12722  13.400  < 2e-16 ***
#Age          0.10908    0.01627   6.703 2.05e-11 ***
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #
#
#(Dispersion parameter for poisson family taken to be 1)
#
#    Null deviance: 443.85  on 98  degrees of freedom
#Residual deviance: 122.54  on 96  degrees of freedom
#AIC: 411.91
#
#Number of Fisher Scoring iterations: 5

# Note that coefficient estimates cannot be interpreted as is.
# To make conclusions about effect, must take exp to the coefficient power
# Value represents a multiplier effect of dependent variable on average outcome


### Examples


## Effect of Birth Location
exp(1.70471)
# Wild born wolves have 5.5 times as many pups per litter (on average) as captive born wolves
# to calculate confidence limits
exp(confint(results))
# Wild born individuals have 5.5 times (4.26 - 7.09; 95% CL; p<0.0001)
# as many pups as captive born individuals


## Effect of Age
exp(0.10908)
# For each 1 year increase in age, wolves had 1.11 times as many pups
exp(0.10908*(5))
# For each 5-year increase in age, wolves had 1.73 times as many pups
min(datum$Age) #minimum age in data-set was 1
max(datum$Age) #maximum age in data-set was 10
exp(0.10908*(10-1)))
# the oldest wolves had 2.66 times as many pups as the youngest wolves


###Prediction


#How many pups does a wild-born, 5-year-old wolf have, on average?
exp(-0.21178+1.70471+0.10908*5)