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

    \[ g(p)=\log (p/(1-p)) \]

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

    \[ F(x)=1/(1+\exp (-x))=\exp (x)/(1+\exp (x)) \]

  • The probit (or normit) function

    \[ g(p)=\Phi ^{-1}(p) \]

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

    \[ F(x)=\Phi (x)=(2\pi )^{-1/2}\int _{-\infty }^ x \exp (-z^2/2) dz \]

    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

    \[ g(p)=\log (-\log (1-p)) \]

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

    \[ F(x)=1-\exp (-\exp (x)) \]

  • The generalized logit function extends the binary logit link to a vector of levels $(p_1,\ldots ,p_{k+1})$ by contrasting each level with a fixed level:

    \[ g(p_ i)=\log (p_ i/p_{k+1}) \quad i=1,\ldots ,k \]

  • The adjacent-category logit function extends the binary logit link to a vector of levels $(p_1,\ldots ,p_{k+1})$ by contrasting each level with the following level:

    \[ g(p_ i)=\log (p_ i/p_{i+1}) \quad i=1,\ldots ,k \]

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

$\pi ^2/3$

Extreme-value

$-\gamma $

$\pi ^2/6$

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 $\pi /\sqrt {3}$ larger than the estimates from the probit link function.