The extension of the classical linear models to generalized linear models has two parts: a distribution from the exponential family and a link function.
The first part extends the linear model to response variables that are members of a large family of distributions called the exponential family. Members of the exponential family of distributions have probability distribution functions for an observed response in this general form:
where a(∙), b(∙), and c(∙) depend on the distribution of the response variable. The parameter θ is a location parameter that is often called the canonical parameter, and ϕ is called the dispersion parameter. The function a(ϕ) is usually of the form a(ϕ)= ϕ/ ω, where ω is a known constant or weight that may vary from one observation to another. (In Minitab, when weights are given the function a(ϕ), is adjusted accordingly.)
Members of the exponential family can be discrete distributions or continuous distributions. Examples of continuous distributions that are members of the exponential family are the normal and the gamma distributions. Examples of discrete distributions that are members of the exponential family are the binomial and the Poisson distributions. The following table gives the characteristics of some of these distributions.
Distribution | ϕ | b(θ) | a(φ) | c(y, ϕ) |
Normal | σ^{2} | θ^{2}/2 | φω | |
Binomial | 1 | φ/ω | -ln(y!) | |
Poisson | 1 | exp(θ) | φ/ω |
The second part is the link function. The link function relates the mean of the response in the i^{th} observation to a linear predictor in this form:
The classical linear model is a special case of this general formulation where the link function is the identity function.
The choice of the link function in the second part depends upon the specific distribution of the exponential family of the first part. In particular, each distribution in the exponential family has a special link function called the canonical link function. This link function satisfies the equation g (μ_{i}) = X_{i}'β= θ, where θ is the canonical parameter. The canonical link function results in some desirable statistical properties of the model. Goodness-of-fit statistics can be used to compare fits using different link functions. Certain link functions may be used for historical reasons or because they have a special meaning in a discipline. For example, an advantage of the logit link function is that it provides an estimate of the odds ratios. Another example is that the normit link function assumes that there is an underlying variable that follows a normal distribution that is classified into binary categories.
Minitab provides three link functions. The different link functions make it possible to find models that adequately fit a wider variety of data. The link functions are logit, normit (also called probit), and gompit (also called complementary log-log). These are the inverse of the standard cumulative logistic distribution function (logit), the inverse of the standard cumulative normal distribution function (normit), and the inverse of the Gompertz distribution function (gompit). The logit is the canonical link function for binomial models, thus the logit is the default link function.
Model | Name | Link Function, g(μ_{i}) |
Binomial | logit | |
Binomial | normit (probit) | |
Binomial | gompit (complementary log-log) |
Term | Description |
---|---|
μ_{i} | the mean response of the i^{th} row |
g(μ_{i}) | the link function |
X | the vector of predictor variables |
β | the vector of coefficients associated with the predictors |
the inverse cumulative distribution function of the normal distribution |
Describes a single set of factor/covariate values in a data set. Minitab calculates event probabilities, residuals, and other diagnostic measures for each factor/covariate pattern.
For example, if a data set includes the factors gender and race and the covariate age, the combination of these predictors may contain as many different covariate patterns as subjects. If a data set only includes the factors race and sex, each coded at two levels, there are only four possible factor/covariate patterns. If you enter your data as frequencies, or as successes, trials, or failures, each row contains one factor/covariate pattern.
Minitab generates a design matrix for each design. The first column is a column of one's for the constant term. If the design was blocked into k blocks, there are (k − 1) columns for blocks. Minitab uses the same method of coding blocks as in Factorial Models. This is followed by one column for each main effect. Terms with categorical factors can have more than one column. If the model has squared terms, there is one column for each squared term. The column for a squared term is the product of the corresponding factor with itself. If the model has interaction terms, then there is one column for each interaction term. Interactions that include categorical factors can have more than one column. The column for an interaction term is the product of the two columns that are crossed.
If Minitab removes some terms because the data cannot support them, these terms are not in the stored design matrix. The columns stored match the coefficients that are displayed.