Examples

The following examples illustrate how to use JMP’s generalized linear models platform.

This example uses data from a study of nesting horseshoe crabs. Each female crab had a male crab resident in her nest. This study investigated whether there were other males, called satellites, residing nearby. The data set CrabSatellites.jmp contains a response variable listing the number of satellites, as well as variables describing the female crab’s color, spine condition, weight, and carapace width. The data are shown in Crab Satellite Data.

Crab Satellite Data

To fit the Poisson loglinear model:

•	Select Analyze > Fit Model

•	Assign satell as Y

•	Assign color, spine, width, and weight as Effects

•	Choose the Generalized Linear Model Personality

•	Choose the Poisson Distribution

The Log Link function should be selected for you automatically.

•	Click Run.

The results are shown in Crab Satellite Results.

Crab Satellite Results

The Whole Model Test table gives information to compare the whole-model fit to the model that contains only the intercept parameter. The Reduced model is the model containing only an intercept. The Full model contains all of the effects as well as the intercept. The Difference is the difference of the log likelihoods of the full and reduced models. The Prob>Chisq is analogous to a whole-model F-test.

Second, goodness-of-fit statistics are presented. Analogous to lack-of-fit tests, they test for adequacy of the model. Low p-values for the ChiSquare goodness-of-fit statistics indicate that you may need to add higher-order terms to the model, add more covariates, change the distribution, or (in Poisson and binomial cases especially) consider adding an overdispersion parameter. AICc is also included and is the corrected Akaike’s Information Criterion, where

and k is the number of estimated parameters in the model and n is the number of observations in the data set. This value may be compared with other models to determine the best-fitting model for the data. The model having the smallest value, as discussed in Akaike (1974), is usually the preferred model.

The Effect Tests table shows joint tests that all the parameters for an individual effect are zero. If an effect has only one parameter, as with simple regressors, then the tests are no different from the tests in the Parameter Estimates table.

The Parameter Estimates table shows the estimates of the parameters in the model and a test for the hypothesis that each parameter is zero. Simple continuous regressors have only one parameter. Models with complex classification effects have a parameter for each anticipated degree of freedom. Confidence limits are also displayed.

Poisson Regression with Offset

The sample data table Ship Damage.JMP is adapted from one found in McCullagh and Nelder (1983). It contains information on a certain type of damage caused by waves to the forward section of the hull. Hull construction engineers are interested in the risk of damage associated with three variables: ship Type, the year the ship was constructed (Yr Made) and the block of years the ship saw service (Yr Used).

In this analysis we use the variable Service, the log of the aggregate months of service, as an offset variable. An offset variable is one that is treated like a regression covariate whose parameter is fixed to be 1.0.

These are most often used to scale the modeling of the mean in Poisson regression situations with log link. In this example, we use log(months of service) since one would expect that the number of repairs be proportional to the number of months in service. To see how this works, assume the linear component of the GLM is called eta. Then with a log link function, the model of the mean with the offset included is:

exp[Log(months of service) + eta] = [(months of service) * exp(eta)].

To run this example, assign

•	Generalized Linear Model as the Personality

•	Poisson as the Distribution, which automatically selects the Log link function

•

N to Y

•	Service to Offset

•	Type, Yr Made, Yr Used as effects in the model

•	Overdispersion Tests and Intervals with a check mark

The Fit Model launch window should appear like the one shown in Ship Damage Fit Model Launch Window.

Ship Damage Fit Model Launch Window

When you click Run, you see the report shown in Ship Damage Report. Notice that all three effects (Type, Yr Made, Yr Used) are significant.

Ship Damage Report

Normal Regression, Log Link

Consider the following data set, where x is an explanatory variable and y is the response variable.

Nor.jmp data set

Using Fit Y By X, you can easily see that y varies nonlinearly with x and that the variance is approximately constant (see Y by X Results for Nor.jmp). A normal distribution with a log link function is chosen to model these data; that is, log(μi) = xi'β so that μi = exp(xi'β). The completed Fit Model launch window is shown in Nor Fit Model Launch Window.

Y by X Results for Nor.jmp

Nor Fit Model Launch Window

After clicking Run, you get the following report.

Nor Results

Because the distribution is normal, the Studentized Deviance residuals and the Deviance residuals are the same. To see this, select Diagnostic Plots > Deviance Residuals by Predicted from the platform drop-down menu.