You construct a generalized linear model by deciding on response and explanatory variables for your data and choosing an appropriate link function and response probability distribution. Explanatory variables can be any combination of continuous variables, classification variables, and interactions.
identity, g(μ) = μ
log, g(μ) = log(μ)
JMP fits a generalized linear model to the data by maximum likelihood estimation of the parameter vector. There is, in general, no closed form solution for the maximum likelihood estimates of the parameters. JMP estimates the parameters of the model numerically through an iterative fitting process. The dispersion parameter φ is also estimated by dividing the Pearson goodness-of-fit statistic by its degrees of freedom. Covariances, standard errors, and confidence limits are computed for the estimated parameters based on the asymptotic normality of maximum likelihood estimators.
A number of link functions and probability distributions are available in JMP. The built-in link functions are
identity: g(μ) = μ
probit: g(μ) = Φ-1(μ), where Φ is the standard normal cumulative distribution function
log: g(μ) = log(μ)
reciprocal: g(μ) =
When you select the Power link function, a number box appears enabling you to enter the desired power.
normal: V(μ) = 1
binomial (proportion): V(μ) = μ(1 – μ)
Poisson: V(μ) = μ
Exponential: V(μ) = μ2
When you select Binomial as the distribution, the response variable must be specified in one of the following ways:
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.
In the binomial case, yi = ri /mi, where ri is a binomial count and mi is the binomial number of trials parameter
where yi is the ith response, μi is the corresponding predicted mean, V(μi) is the variance function, and wi is a known weight for the ith observation. If no weight is known, wi = 1 for all observations.
One strategy for variable selection is to fit a sequence of models, beginning with a simple model with only an intercept term, and then include one additional explanatory variable in each successive model. You can measure the importance of the additional explanatory variable by the difference in deviances or fitted log likelihoods between successive models. Asymptotic tests computed by JMP enable you to assess the statistical significance of the additional term.