5 Generalized Linear Models
Generalized linear models are just as easy to fit in R as ordinary linear model. In fact, they require only an additional parameter to specify the variance and link functions.
5.1 Variance and Link Families
The basic tool for fitting generalized linear models is the glm
function,
which has the folllowing general structure:
> glm(formula, family, data, weights, subset, ...)
where ...
stands for more esoteric options. The only parameter that
we have not encountered before is family
, which is a simple way of
specifying a choice of variance and link functions. There are six choices
of family:

As can be seen, each of the first five choices has an associated variance function (for binomial the binomial variance m(1m)), and one or more choices of link functions (for binomial the logit, probit or complementary loglog).
As long as you want the default link, all you have to specify is the
family name. If you want an alternative link, you must add a link
argument. For example to do probits you use
> glm( formula, family=binomial(link=probit))
The last family on the list, quasi
, is there to allow fitting
userdefined models by maximum quasilikelihood.
5.2 Logistic Regression
We will illustrate fitting logistic regression models using the contraceptive use data shown below:
age education wantsMore notUsing using <25 low yes 53 6 <25 low no 10 4 <25 high yes 212 52 <25 high no 50 10 2529 low yes 60 14 2529 low no 19 10 2529 high yes 155 54 2529 high no 65 27 3039 low yes 112 33 3039 low no 77 80 3039 high yes 118 46 3039 high no 68 78 4049 low yes 35 6 4049 low no 46 48 4049 high yes 8 8 4049 high no 12 31 
The data are available from the datasets section of the website for my generalized linear models course. Visit http://data.princeton.edu/wws509/datasets to read a short description and follow the link to cuse.dat.
Of course the data can be downloaded directly from R:
> cuse < read.table("http://data.princeton.edu/wws509/datasets/cuse.dat", + header=TRUE) > cuse age education wantsMore notUsing using 1 <25 low yes 53 6 2 <25 low no 10 4 3 <25 high yes 212 52 4 <25 high no 50 10 5 2529 low yes 60 14 ... output edited ... 16 4049 high no 12 31
I specified the header
parameter as TRUE
,
because otherwise it would not have been obvious that the first line
in the file has the variable names.
There are no row names specified, so the rows will be numbered from 1 to 16.
Print cuse
to make sure you got the data in alright.
Then make it your default dataset:
> attach(cuse)
Let us first try a simple additive model where contraceptive use depends on age, education and wantsMore:
> lrfit < glm( cbind(using, notUsing) ~ + age + education + wantsMore , family = binomial)
There are a few things to explain here. First, the function is
called glm
and I have assigned its value to
an object called lrfit
(for logistic regression fit).
The first argument of the function is a model formula, which defines
the response and linear predictor.
With binomial data the response can be either a vector or a matrix with two columns.
 If the response is a vector it can be numeric with 0 for failure and 1 for
success, or a factor with the first level representing "failure" and all others
representing "success". In these cases R generates a vector of ones to
represent the binomial denominators.
 Alternatively, the response can be a matrix where the first column is the number of "successes" and the second column is the number of "failures". In this case R adds the two columns together to produce the correct binomial denominator.
Because the latter approach is clearly the right one for us
I used the function cbind
to create a matrix
by binding the column vectors containing the numbers using and not
using contraception.
Following the special symbol ~
that separates the response
from the predictors, we have a standard WilkinsonRogers model formula.
In this case we are specifying main effects of age, education and
wantsMore. Because all three predictors are categorical variables,
they are treated automatically as factors, as you can see by
inspecting the results:
> lrfit Call: glm(formula = cbind(using, notUsing) ~ age + education + wantsMore, family = binomial) Coefficients: (Intercept) age2529 age3039 age4049 educationlow 0.8082 0.3894 0.9086 1.1892 0.3250 wantsMoreyes 0.8330 Degrees of Freedom: 15 Total (i.e. Null); 10 Residual Null Deviance: 165.8 Residual Deviance: 29.92 AIC: 113.4
Recall that R sorts the levels of a factor in alphabetical order.
Because <25 comes before 2529, 3039, and 4049, it has been
picked as the reference cell for age
.
Similarly, high is the reference cell for education
because high comes before low!
Finally, R picked no as the base for wantsMore
.
If you are unhappy about these choices you can
(1) use relevel
to change the base category,
or (2) define your own indicator variables. I will use the latter
approach by defining indicators for women with high
education and women who want no more children:
> noMore < wantsMore == "no" > hiEduc < education == "high"
Now try the model again:
> glm( cbind(using,notUsing) ~ age + hiEduc + noMore, family=binomial) Call: glm(formula = cbind(using, notUsing) ~ age + hiEduc + noMore, family = binomial) Coefficients: (Intercept) age2529 age3039 age4049 hiEduc noMore 1.9662 0.3894 0.9086 1.1892 0.3250 0.8330 Degrees of Freedom: 15 Total (i.e. Null); 10 Residual Null Deviance: 165.8 Residual Deviance: 29.92 AIC: 113.4
The residual deviance of 29.92 on 10 d.f. is highly significant:
> 1pchisq(29.92,10) [1] 0.0008828339
so we need a better model. One of my favorites introduces an interaction between age and desire for no more children:
> lrfit < glm( cbind(using,notUsing) ~ age * noMore + hiEduc , family=binomial) > lrfit Call: glm(formula = cbind(using, notUsing) ~ age * noMore + hiEduc, family = binomial) Coefficients: (Intercept) age2529 age3039 age4049 1.80317 0.39460 0.54666 0.57952 noMore hiEduc age2529:noMore age3039:noMore 0.06622 0.34065 0.25918 1.11266 age4049:noMore 1.36167 Degrees of Freedom: 15 Total (i.e. Null); 7 Residual Null Deviance: 165.8 Residual Deviance: 12.63 AIC: 102.1
Note how R built the interaction terms automatically, and even came up with sensible labels for them. The model's deviance of 12.63 on 7 d.f. is not significant at the conventional five per cent level, so we have no evidence against this model.
To obtain more detailed information about this fit try the summary
function:
> summary(lrfit) Call: glm(formula = cbind(using, notUsing) ~ age * noMore + hiEduc, family = binomial) Deviance Residuals: Min 1Q Median 3Q Max 1.30027 0.66163 0.03286 0.81945 1.73851 Coefficients: Estimate Std. Error z value Pr(>z) (Intercept) 1.80317 0.18018 10.008 < 2e16 *** age2529 0.39460 0.20145 1.959 0.05013 . age3039 0.54666 0.19842 2.755 0.00587 ** age4049 0.57952 0.34733 1.669 0.09522 . noMore 0.06622 0.33064 0.200 0.84126 hiEduc 0.34065 0.12576 2.709 0.00676 ** age2529:noMore 0.25918 0.40970 0.633 0.52699 age3039:noMore 1.11266 0.37398 2.975 0.00293 ** age4049:noMore 1.36167 0.48422 2.812 0.00492 **  Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 165.772 on 15 degrees of freedom Residual deviance: 12.630 on 7 degrees of freedom AIC: 102.14 Number of Fisher Scoring iterations: 3
R follows the popular custom of flagging significant coefficients
with one, two or three stars depending on their pvalues.
Try plot(lrfit)
. You get the same plots as in
a linear model, but adapted to a generalized linear model;
for example the residuals plotted are deviance residuals
(the square root of the contribution of an observation to the
deviance, with the same sign as the raw residual).
The functions that can be used to extract results from the fit include
residuals
orresid
, for the deviance residualsfitted
orfitted.values
, for the fitted values (estimated probabilities)predict
, for the linear predictor (estimated logits)coef
orcoefficients
, for the coefficients, anddeviance
, for the deviance.
Some of these functions have optional arguments; for example, you
can extract five different types of residuals, called
"deviance",
"pearson",
"response" (response  fitted value),
"working" (the working dependent variable in the IRLS algorithm  linear
predictor), and
"partial" (a matrix of working residuals formed by omitting
each term in the model).
You specify the one you want using the type
argument,
for example residuals(lrfit,type="pearson")
.
5.3 Updating Models
If you want to modify a model you may consider using the special
function update
. For example to drop the
age:noMore
interaction in our model one could use
> lrfit0 < update(lrfit, ~ .  age:noMore)
The first argument is the result of a fit, and the second an
updating formula.
The place holder ~
separates the response from the
predictors and the dot .
refers to the right hand side
of the original formula, so here we simply remove age:noMore
.
Alternatively, one can give a new formula as the second argument.
The update function can be used to fit the same model to different
datasets, using the argument data
to specify a new data
frame. Another useful argument is subset
, to fit the
model to a different subsample. This function works with linear
models as well as generalized linear models.
If you plan to fit a sequence of models you will find the anova
function useful. Given a series of nested models, it will calculate
the change in deviance between them. Try
> anova(lrfit0,lrfit) Analysis of Deviance Table Model 1: cbind(using, notUsing) ~ age + noMore + hiEduc Model 2: cbind(using, notUsing) ~ age + noMore + hiEduc + age:noMore Resid. Df Resid. Dev Df Deviance 1 10 29.917 2 7 12.630 3 17.288
Adding the interaction has reduced the deviance by 17.288 at the expense of 3 d.f.
If the argument to anova
is a single model, the function will
show the change in deviance obtained by adding each of the terms in the
order listed in the model formula, just as it did for linear models.
Because this requires fitting as many models as there are terms in the formula,
the function may take a while to complete its calculations.
The anova function lets you specify an optional test. The usual choices
will be "F" for linear models and "Chisq" for generalized linear models.
Adding the parameter test="Chisq"
adds pvalues
next to the deviances. In our case
> anova(lrfit,test="Chisq") Analysis of Deviance Table Model: binomial, link: logit Response: cbind(using, notUsing) Terms added sequentially (first to last) Df Deviance Resid. Df Resid. Dev P(>Chi) NULL 15 165.772 age 3 79.192 12 86.581 4.575e17 noMore 1 49.693 11 36.888 1.798e12 hiEduc 1 6.971 10 29.917 0.008 age:noMore 3 17.288 7 12.630 0.001
We can see that all terms were highly significant when they were introduced into the model.
5.4 Model Selection
A very powerful tool in R is a function for stepwise regression that has three remarkable features:
 It works with generalized linear models, so it will do stepwise logistic regression, or stepwise Poisson regression,
 It understand about hierarchical models, so it will only consider adding interactions only after including the corresponding main effects in the models, and
 It understands terms involving more than one degree of freedom, so it it will keep together dummy variables representing the effects of a factor.
The basic idea of the procedure is to start from a given model (which could well be the null model) and take a series of steps by either deleting a term already in the model or adding a term from a list of candidates for inclusion, called the scope of the search and defined, of course, by a model formula.
Selection of terms for deletion or inclusion is based on Akaike's information criterion (AIC). R defines AIC as
–2 maximized loglikelihood + 2 number of parameters
(SPlus defines it as the deviance minus twice the number of parameters in the model. The two definitions differ by a constant, so differences in AIC are the same in the two environments.) The procedure stops when the AIC criterion cannot be improved.
In R all of this work is done by calling a couple of functions,
add1
and drop1
, that consider adding or
dropping a term from a model. These functions can be very useful
in model selection, and both of them accept a test
argument
just like anova
.
Consider first drop1
. For our logistic regression model,
> drop1(lrfit, test="Chisq") Single term deletions Model: cbind(using, notUsing) ~ age + noMore + hiEduc + age:noMore Df Deviance AIC LRT Pr(Chi)12.630 102.137 hiEduc 1 20.099 107.607 7.469 0.0062755 ** age:noMore 3 29.917 113.425 17.288 0.0006167 ***  Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Obviously we can't drop any of these terms. Note that R considered dropping the main effect of education and the age by want no more interaction, but did not examine the main effects of age or want no more, because one would not drop these main effects while retaining the interaction.
The sister function add1
requires a scope to define the
additional terms to be considered. In our example
we will consider all possible twofactor interactions:
> add1(lrfit, ~.^2,test="Chisq") Single term additions Model: cbind(using, notUsing) ~ age + noMore + hiEduc + age:noMore Df Deviance AIC LRT Pr(Chi)12.630 102.137 age:hiEduc 3 5.798 101.306 6.831 0.07747 . noMore:hiEduc 1 10.824 102.332 1.806 0.17905  Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
We see that neither of the missing twofactor interactions is significant by itself at the conventional five percent level. (However, they happen to be jointly significant.) Note that the model with the age by education interaction has a lower AIC than our starting model.
The step
function will do an automatic search. Here
we let it search in a scope defined by all twofactor interactions:
> search < step(additive, ~.^2) ... trace output supressed ...
The step
function produces detailed trace output
that we have supressed. The returned object, however, includes an
anova
component that summarizes the search:
> search$anova Step Df Deviance Resid. Df Resid. Dev AIC 1 NA NA 10 29.917222 113.4251 2 + age:noMore 3 17.287669 7 12.629553 102.1375 3 + age:hiEduc 3 6.831288 4 5.798265 101.3062 4 + hiEduc:noMore 1 3.356777 3 2.441488 99.9494
As you can see, the automated procedure introduced, one by one, all three remaining twofactor interactions, to yield a final AIC of 99.9. This is an example where AIC, by requiring a deviance improvement of only 2 per parameter, may have led to overfitting the data.
Some analysts prefer a higher penalty per parameter. In particular,
using log(n) instead of 2 as a multiplier yields BIC, the
Bayesian Information Criterion. In our example log(1607) = 7.38,
so we would require a deviance reduction of 7.38 per additional
parameter. The step
function accepts k
as an argument, with default 2. You may verify that specifying
k=log(1607)
leads to a much simpler model; not only
are no new interactions introduced, but the main effect of
education is dropped (even though it is significant).
Continue with Conclusion and References