Modeling and Multivariate Methods - SAS

182 Fitting Generalized Linear Models Chapter 6
Examples of Generalized Linear Models
Poisson: V(μ) = μ
Exponential: V(μ) = μ 2
When you select Binomial as the distribution, the response variable must be specified in one of the
following ways:
• If your data is not summarized as frequencies of events, specify a single binary column as the response.
The response column must be nominal. If your data is summarized as frequencies of events, specify a
single binary column as the response, along with a frequency variable in the Freq role. The response
column must be nominal, and the frequency variable gives the count of each response level.
• If your data is summarized as frequencies of events and trials, specify two continuous columns in this
order: a count of the number of successes, and a count of the number of trials. Alternatively, you can
specify the number of failures instead of successes.
Model Selection and Deviance
An important aspect of generalized linear modeling is the selection of explanatory variables in the model.
Changes in goodness-of-fit statistics are often used to evaluate the contribution of subsets of explanatory
variables to a particular model. The deviance, defined to be twice the difference between the maximum
attainable log likelihood and the log likelihood at the maximum likelihood estimates of the regression
parameters, is often used as a measure of goodness of fit. The maximum attainable log likelihood is achieved
with a model that has a parameter for every observation. The following table displays the deviance for each
of the probability distributions available in JMP.
Table 6.2 Deviance Functions
Distribution
normal
Poisson
binomial a
exponential
Deviance
w i
( y i
– μ i
) 2
i
y i
2w i
y i
log----
– ( y
μ i
– μ i
)
i
i
y i
1 – y
2 w i
m i
y i ----
i
log + ( 1 – y
μ i
) log-------------
i
1 – μ i
i
y i y 2 w i ----
i
– μ i
– log + --------------
μ i μ i
i
a. In the binomial case, y i = r i /m i , where r i is a binomial count and m i is the binomial number of trials
parameter