logistic regression & saturation

#1
I had 2 statistics classes, the first one was about the very very basics not going into detail about regression analysis. However we had diff. kinds of logistic regression models in that class (though we didnt spend much time on them).

Then I had another stats class which was exclusively about regression analysis and we never mentioned logistic regressions again.

I assume it must have its place in statistics (just read a post about it), but isn't it used to model a saturation effect? Can't this be done using a normal regression function with a non-linear model (not really sure which one right now) in order to model this effect in the data?
 

BioStatMatt

TS Contributor
#2
Logistic regression is used to model probabilities. Say you were interested in an outcome of a binary nature (0,1) whose probability of occurance was p. One way to model this probability is the linear probability model:

p = a + bx + e

in a similar manner to standard linear regression. Note that in this case, a + bx is not bounded (can go from -inf to inf), but clearly our probability must be greater than 0 and less than one. One way to fix this problem is to use the logit of p:

logit(p) = log(p/(1-p)) = a + bx

the logit transforms the probability onto the entire real line, so we dont have the problem with the linear probability model. We can solve for p in this function by:

p = exp(a+bx) / (1+exp(a+bx))

the plot p as this function of x takes on the 'S' shape that I believe you associate with a saturation effect.

Now, If you are not interested in the probability model, but only want to model the assocation between an outcome y and a covariate x whose relationship appears to be 'S' shaped, then any 'S' shaped function may be used for a model such as:

y = c + d / (1+exp(a+bx))

where a,b,c, and d are parameters.

~Matt