Introduction to Categorical Data Analysis

3.3 GENERALIZED LINEAR MODELS FOR BINARY DATA 73
the form
π(x) = F(x) (3.4)
where F is a cdf for some continuous probability distribution.
When F is the cdf of a normal distribution, model type (3.4) is equivalent to the
probit model (3.3). The probit link function transforms π(x) so that the regression
curve for π(x) [or for 1 − π(x), when β0
corresponds to a different normal distribution.
For the snoring and heart disease data, probit[ˆπ(x)]=−2.061 + 0.188x. This
probit fit corresponds to a normal cdf having mean −ˆα/ ˆβ = 2.061/0.188 = 11.0
and standard deviation 1/| ˆβ| =1/0.188 = 5.3. The estimated probability of heart
disease equals 1/2 at snoring level x = 11.0. That is, x = 11.0 has a fitted probit of
−2.061 + 0.188(11) = 0, which is the z-score corresponding to a left-tail probability
of 1/2. Since snoring level is restricted to the range 0–5 for these data, well below
11, the fitted probabilities over this range are quite small.
The logistic regression curve also has form (3.4). When β>0 in model (3.2), the
curve for π(x) has the shape of the cdf F(x)of a two-parameter logistic distribution.
The logistic cdf corresponds to a probability distribution with a symmetric, bell shape.
It looks similar to a normal distribution but with slightly thicker tails.
When both models fit well, parameter estimates in probit models have smaller
magnitude than those in logistic regression models. This is because their link functions
transform probabilities to scores from standard versions of the normal and
logistic distribution, but those two distributions have different spread. The standard
normal distribution has a mean of 0 and standard deviation of 1. The standard logistic
distribution has a mean of 0 and standard deviation of 1.8. When both models fit well,
parameter estimates in logistic regression models are approximately 1.8 times those
in probit models.
The probit model and the cdf model form (3.4) were introduced in the mid1930s in
toxicology studies. A typical experiment exposes animals (typically insects or mice) to
various dosages of some potentially toxic substance. For each subject, the response is
whether it dies. It is natural to assume a tolerance distribution for subjects’responses.
For example, each insect may have a certain tolerance to an insecticide, such that it
dies if the dosage level exceeds its tolerance and survives if the dosage level is less
than its tolerance. Tolerances would vary among insects. If a cdf F describes the
distribution of tolerances, then the model for the probability π(x) of death at dosage
level x has form (3.4). If the tolerances vary among insects according to a normal
distribution, then π(x) has the shape of a normal cdf . With sample data, model fitting
determines which normal cdf best applies.
Logistic regression was not developed until the mid 1940s and not used much until
the 1970s, but it is now more popular than the probit model. We will see in the next
chapter that the logistic model parameters relate to odds ratios. Thus, one can fit the
model to data from case–control studies, because one can estimate odds ratios for
such data (Section 2.3.5).