Germán Rodríguez
Introducing R Princeton University

# 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.

## 5.2 Logistic Regression

We will illustrate fitting logistic regression models using the contraceptive use data excerpted below (and shown in full further below):

  age education wantsMore notUsing using
<25       low       yes       53     6
<25       low        no       10     4
... lines omitted ...
40-49      high        no       12    31

The data are available from the datasets section of the website for my generalized linear models course. Visit https://data.princeton.edu/wws509/datasets to read a short description and follow the link to cuse.dat.

> cuse = read.table("https://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  25-29       low       yes       60    14
6  25-29       low        no       19    10
7  25-29      high       yes      155    54
8  25-29      high        no       65    27
9  30-39       low       yes      112    33
10 30-39       low        no       77    80
11 30-39      high       yes      118    46
12 30-39      high        no       68    78
13 40-49       low       yes       35     6
14 40-49       low        no       46    48
15 40-49      high       yes        8     8
16 40-49      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. I also printed the data to make sure we got it alright.

Let us try a simple additive model where contraceptive use depends on age, education and whether or not the woman wants more children:

> lrfit = glm( cbind(using, notUsing) ~ age + education + wantsMore,
+   data = cuse, family = binomial)

There are a few things to explain here. First, the function is 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 Wilkinson-Rogers 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, data = cuse)

Coefficients:
(Intercept)      age25-29      age30-39      age40-49  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 25-29, 30-39, and 40-49, it has been picked as the reference cell for age. Similarly, high is the reference cell for education because high comes alphabetically 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 second approach, defining indicators for women with high education and women who want no more children, both added to the cuse data frame:

> cuse$noMore = cuse$wantsMore == "no"
> cuse$hiEduc = cuse$education == "high"

Now try the model with these predictors

> glm(cbind(using, notUsing) ~ age + hiEduc + noMore,
+   data = cuse, family = binomial)

Call:  glm(formula = cbind(using, notUsing) ~ age + hiEduc + noMore,
family = binomial, data = cuse)

Coefficients:
(Intercept)     age25-29     age30-39     age40-49   hiEducTRUE   noMoreTRUE
-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:

> pchisq(29.92, 10, lower.tail = FALSE)
[1] 0.0008828339

To obtain a p-value I specified lower.tail as FALSE. This is more accurate than computing the default lower tail and subtracting from one.

So, we need a better model. One of my favorites for this dataset introduces an interaction between age and wanting no more children, which is easily specified

> lrfit2 = glm( cbind(using, notUsing) ~ age * noMore + hiEduc , data = cuse,
+ family = binomial)
> lrfit2

Call:  glm(formula = cbind(using, notUsing) ~ age * noMore + hiEduc,
family = binomial, data = cuse)

Coefficients:
(Intercept)             age25-29             age30-39
-1.80317              0.39460              0.54666
age40-49           noMoreTRUE           hiEducTRUE
0.57952              0.06622              0.34065
age25-29:noMoreTRUE  age30-39:noMoreTRUE  age40-49:noMoreTRUE
0.25918              1.11266              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(lrfit2)

Call:
glm(formula = cbind(using, notUsing) ~ age * noMore + hiEduc,
family = binomial, data = cuse)

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  < 2e-16 ***
age25-29             0.39460    0.20145   1.959  0.05013 .
age30-39             0.54666    0.19842   2.755  0.00587 **
age40-49             0.57952    0.34742   1.668  0.09530 .
noMoreTRUE           0.06622    0.33071   0.200  0.84130
hiEducTRUE           0.34065    0.12577   2.709  0.00676 **
age25-29:noMoreTRUE  0.25918    0.40975   0.633  0.52704
age30-39:noMoreTRUE  1.11266    0.37404   2.975  0.00293 **
age40-49:noMoreTRUE  1.36167    0.48433   2.811  0.00493 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 165.77  on 15  degrees of freedom
Residual deviance:  12.63  on  7  degrees of freedom
AIC: 102.14

Number of Fisher Scoring iterations: 4

R follows the popular custom of flagging significant coefficients with one, two or three stars depending on their p-values. Try plot(lrfit2). 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() or resid(), for the deviance residuals
• fitted() or fitted.values(), for the fitted values (estimated probabilities)
• predict(), for the linear predictor (estimated logits)
• coef() or coefficients(), for the coefficients, and
• deviance(), for the deviance.

Some of these functions have optional arguments; for example, you can extract five different types of residuals, called “deviance”, “pearson”, “response” (defined as 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(lrfit2, type = "pearson").

## 5.3 Model Updating

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

> lrfit1 = update(lrfit2, ~ . - age:noMore)

The first argument is the result of a fit, and the second an updating formula. The tilde ~ 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 may also 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(lrfit1, lrfit2)
Analysis of Deviance Table

Model 1: cbind(using, notUsing) ~ age + noMore + hiEduc
Model 2: cbind(using, notUsing) ~ age * noMore + hiEduc
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 p-values next to the deviances. In our case

> anova(lrfit2, test = "Chisq")
Analysis of Deviance Table

Response: cbind(using, notUsing)

Terms added sequentially (first to last)

Df Deviance Resid. Df Resid. Dev  Pr(>Chi)
NULL                          15    165.772
age         3   79.192        12     86.581 < 2.2e-16 ***
noMore      1   49.693        11     36.888 1.798e-12 ***
hiEduc      1    6.971        10     29.917 0.0082860 **
age:noMore  3   17.288         7     12.630 0.0006167 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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 hierarchical models, so it will only consider adding interactions 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

AIC = –2 maximized log-likelihood + 2 number of parameters

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 one term from a model. These functions can be very useful in model selection, and both of them accept atestargument just likeanova().

Consider first drop1(). For our logistic regression model,

> drop1(lrfit2, test = "Chisq")
Single term deletions

Model:
cbind(using, notUsing) ~ age * noMore + hiEduc
Df Deviance    AIC     LRT  Pr(>Chi)
<none>          12.630 102.14
hiEduc      1   20.099 107.61  7.4695 0.0062755 **
age:noMore  3   29.917 113.42 17.2877 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 two-factor interactions:

> add1(lrfit2, ~ .^2, test = "Chisq")
Single term additions

Model:
cbind(using, notUsing) ~ age * noMore + hiEduc
Df Deviance    AIC    LRT Pr(>Chi)
<none>            12.6296 102.14
age:hiEduc     3   5.7983 101.31 6.8313  0.07747 .
noMore:hiEduc  1  10.8240 102.33 1.8055  0.17905
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We see that neither of the missing two-factor 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 start from the additive model and search in a scope defined by all two-factor interactions.

> search = step(lrfit1, ~.^2)

The step() function produces detailed trace output that I 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 + noMore:hiEduc -1  3.356777         3   2.441488  99.9494

As you can see, the automated procedure introduced, one by one, all three remaining two-factor 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).

In this example AIC would lead to a model that may be too complex, and BIC would lead to a model that may be too simple. In my opinion, the model with only one interaction is just right.

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