The LOGISTIC Procedure

Link Functions and the Corresponding Distributions

Five link functions are available in the LOGISTIC procedure. The logit function is the default. To specify a different link function, use the LINK= option in the MODEL statement. The link functions and the corresponding distributions are as follows:

  • The logit function

    g left-parenthesis p right-parenthesis equals log left-parenthesis p slash left-parenthesis 1 minus p right-parenthesis right-parenthesis

    is the inverse of the cumulative logistic distribution function, which is

    upper F left-parenthesis x right-parenthesis equals 1 slash left-parenthesis 1 plus exp left-parenthesis negative x right-parenthesis right-parenthesis equals exp left-parenthesis x right-parenthesis slash left-parenthesis 1 plus exp left-parenthesis x right-parenthesis right-parenthesis

  • The probit (or normit) function

    g left-parenthesis p right-parenthesis equals normal upper Phi Superscript negative 1 Baseline left-parenthesis p right-parenthesis

    is the inverse of the cumulative standard normal distribution function, which is

    upper F left-parenthesis x right-parenthesis equals normal upper Phi left-parenthesis x right-parenthesis equals left-parenthesis 2 pi right-parenthesis Superscript negative 1 slash 2 Baseline integral Subscript negative normal infinity Superscript x Baseline exp left-parenthesis minus z squared slash 2 right-parenthesis d z

    Traditionally, the probit function contains the additive constant 5, but throughout PROC LOGISTIC, the terms probit and normit are used interchangeably.

  • The complementary log-log function

    g left-parenthesis p right-parenthesis equals log left-parenthesis minus log left-parenthesis 1 minus p right-parenthesis right-parenthesis

    is the inverse of the cumulative extreme-value function (also called the Gompertz distribution), which is

    upper F left-parenthesis x right-parenthesis equals 1 minus exp left-parenthesis minus exp left-parenthesis x right-parenthesis right-parenthesis

  • The generalized logit function extends the binary logit link to a vector of levels left-parenthesis p 1 comma ellipsis comma p Subscript k plus 1 Baseline right-parenthesis by contrasting each level with a fixed level:

    g left-parenthesis p Subscript i Baseline right-parenthesis equals log left-parenthesis p Subscript i Baseline slash p Subscript k plus 1 Baseline right-parenthesis i equals 1 comma ellipsis comma k

  • The adjacent-category logit function extends the binary logit link to a vector of levels left-parenthesis p 1 comma ellipsis comma p Subscript k plus 1 Baseline right-parenthesis by contrasting each level with the following level:

    g left-parenthesis p Subscript i Baseline right-parenthesis equals log left-parenthesis p Subscript i Baseline slash p Subscript i plus 1 Baseline right-parenthesis i equals 1 comma ellipsis comma k

The variances of the normal, logistic, and extreme-value distributions are not the same. Their respective means and variances are shown in the following table:

Distribution Mean Variance
Normal 0 1
Logistic 0 pi squared slash 3
Extreme-value negative gamma pi squared slash 6

Here gamma is the Euler constant. In comparing parameter estimates from different link functions, you need to take into account the different scalings of the corresponding distributions and, for the complementary log-log function, a possible shift in location. For example, if the fitted probabilities are in the neighborhood of 0.1 to 0.9, then the parameter estimates from the logit link function should be about pi slash StartRoot 3 EndRoot larger than the estimates from the probit link function.

Last updated: December 09, 2022