## 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:

Family | Variance | Link |
---|---|---|

gaussian | gaussian | identity |

binomial | binomial | logit, probit or cloglog |

poisson | poisson | log, identity or sqrt |

Gamma | Gamma | inverse, identity or log |

inverse.gaussian | inverse.gaussian | 1/mu^2 |

quasi | user-defined | user-defined |

As can be seen, each of the first five choices has an associated variance function (for binomial the binomial variance m(1-m)), and one or more choices of link functions (for binomial the logit, probit or complementary log-log).

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
user-defined models by maximum quasi-likelihood.

### 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 |

25-29 | low | yes | 60 | 14 |

25-29 | low | no | 19 | 10 |

25-29 | high | yes | 155 | 54 |

25-29 | high | no | 65 | 27 |

30-39 | low | yes | 112 | 33 |

30-39 | low | no | 77 | 46 |

30-39 | high | no | 68 | 78 |

40-49 | low | no | 46 | 48 |

40-49 | high | yes | 8 | 8 |

40-49 | 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 25-29 low yes 60 14 ... output edited ... 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.
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 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) 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 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) age25-29 age30-39 age40-49 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:

> 1-pchisq(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) age25-29 age30-39 age40-49 -1.80317 0.39460 0.54666 0.57952 noMore hiEduc age25-29:noMore age30-39:noMore 0.06622 0.34065 0.25918 1.11266 age40-49: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 < 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 0.34733 1.669 0.09522 . noMore 0.06622 0.33064 0.200 0.84126 hiEduc 0.34065 0.12576 2.709 0.00676 ** age25-29:noMore 0.25918 0.40970 0.633 0.52699 age30-39:noMore 1.11266 0.37398 2.975 0.00293 ** age40-49: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 p-values.
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`

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" (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 p-values
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 4.575e-17 noMore 1 49.693 11 36.888 1.798e-12 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

(S-Plus 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 two-factor 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 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 search in a scope defined by all two-factor 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 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).

Continue with Conclusion and References