# OLS vs. logistic regression for exploratory analysis with a binary outcome

In the idealized logistic model, we obtain an S-shaped curve linking each continuous IV to the DV. But in practice this S-shape infrequently occurs, making the logistic approach seem a little less superior for such types of data. Of course predicted probabilities that each observation will be "1" on the DV are usable in logistic and not in OLS regression, since in the latter these probabilities can exceed the bounds of [0,1]. But, for exploratory purposes, and if we don't need predicted probabilities, how sound is it to use OLS to see which IV have strong vs. moderate vs. weak relationships with the DV? Wouldn't this amount to a sort of multivariate version of point-biserial correlation? (Standardized regression coefficients, not to mention collinearity statistics and partial plots, are all I think more easily obtained in OLS than in logistic.)

## 1 Answer

If the explanatory variables have values over the entire real line it makes little sense to express an expectation that is a proportion in $[0,1]$ as a linear function of variable defined over the entire real line. If the sigmoid shape of the logit transformation doesn't describe the shape then perhaps it is best to search for a different transformation that maps $[0,1]$ into $(-∞ , ∞)$.

• +1. To add to the last thing Michael said, probit and complimentary log-log are two other functions that map $(0,1)$ to $(-\infty, \infty)$ that are implemented in many software packages. – Macro May 30 '12 at 12:14
• Note also, that just about any function which corresponds to a CDF for some real value random variable is a candidate. Logistic, Probit, and C-log-log are three such function (hyperbolic secant, normal and extreme value random variables). So you could also "in principle" use a skew-normal link function, or double exponential, or t, etc. etc. T distribution is useful when degrees of freedom is treated as unknown, as you can approximately balance between probit an logit link function. – probabilityislogic May 30 '12 at 12:50
• @probabilityislogic, you've made an important point but nitpick: I think the logistic function is the (inverse) CDF of the logistic distribution, not the hyperbolic secant distribution. – Macro May 30 '12 at 12:58
• Thanks to you all. Does it follow from your answers that you would practically never use point-biserial correlation? – rolando2 May 31 '12 at 23:59
• Happened on the following: "OLS regression. When used with a binary response variable, this model is known as a linear probability model and can be used as a way to describe conditional probabilities. [...] For a more thorough discussion of [...] problems with the linear probability model, see Long (1997, p. 38-40). Long, J. Scott (1997). Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage Publications." ats.ucla.edu/stat/stata/dae/logit.htm – rolando2 Jun 12 '12 at 1:50