## When Can You Fit a Linear Probability Model? More Often Than You Think

##### March 8, 2017 By Paul von Hippel

In July 2015 I pointed out some advantages of the linear probability model over the logistic model. The linear model is much easier to interpret, and the linear model runs much faster, which can be important if the data set is large or the model is complicated. In addition, the linear probability model often fits about as well as the logistic model, since over some ranges the probability p is almost linearly related to the log odds function ln(p(1-p)) that is used in logistic regression.

As a rule of thumb I suggested that the linear probability model could be used whenever the range of modeled probabilities is between .20 and .80. Within that range the relationship between the probability and the log odds is almost linear.

Figure 1. The relationship between probability and log odds over the range of probabilities from .2 to .8.

This is reasonable advice, and similar advice has been given before (Long, 1997). But it doesn’t go far enough. There is a wider range of circumstances where the linear probability model is viable.

For example, in a new paper Joe Workman and I used a multilevel model to analyze obesity among US children in kindergarten through 2nd grade—an age range over which the probability of obesity increases from .09 to .13. Since these probabilities are less than .20, you might guess that we couldn’t use the linear probability model. But we could, and we did. The linear model ran very quickly, whereas the logistic model can be slow when used in a multilevel context. The linear model also gave us very interpretable results; for example, we could write that “during the summer, children’s probability of obesity increases by approximately 1 percentage point per month.” And we didn’t lose any sleep about model fit; the linear model fit practically as well as a logistic model, because over the range of probabilities from .09 to .13 the probability is almost linearly related to the log odds.

Figure 2. The relationship between probability and log odds over the range of probabilities from .09 to .13.

The basic insight is that the linear probability model can be used whenever the relationship between probability and log odds is approximately linear over the range of modeled probabilities. Probabilities between .2 and .8 are one range where approximate linearity holds—but it also holds for some narrow ranges of probabilities that are less than .2 or greater than .8.

I still haven’t gone far enough. When the relationship between probability and log odds is nonlinear, there are still situations where the linear probability model is viable. For example, if your regressors X are categorical variables, then you’re not really modeling a continuous probability function. Instead, you’re modeling the discrete probabilities associated with different categories of X, and this can be done about as well with a linear model as with a logistic model—particularly if your model includes interactions among the X variables (Angrist & Pischke, 2008, chapter 3; Pischke, 2012).

I don’t think that the linear probability model is always viable. I do use the logistic model sometimes. For example, looking at 30 years of data from the Belmont Stakes horse race, I found that the probability of an upset was strongly related to the number of horses that started the race. The more horses started, the more likely it became that one of them would upset the favorite. The relationship looked like this.[1]

Figure 3. The relationship between the number of horses starting the Belmont Stakes and the probability that the favorite will be upset.

On a probability scale the relationship is strongly nonlinear. It almost has to be, because the relationship is strong and the probabilities cover almost the full range from 0 to 1. A linear probability model can’t fit these data easily. When I tried a linear model, out of curiosity, I found that some of the modeled probabilities were out of bounds—greater than 1. I could improve the fit of the linear model by subjecting the X variable to some nonlinear transformation, but finding the right transformation isn’t trivial, and even if I found it the linear model’s ease of interpretation would be lost. It’s simpler to fit a logistic model which naturally keeps the probabilities in bounds.

To check whether your data are candidates for a linear probability model, then, a basic diagnostic is to plot the relationship between probability and log odds over the likely range of probabilities in your data. If the relationship is nearly linear, as in Figures 1 and 2, then a linear probability model will fit about as well as a logistic model, and the linear model will be faster and easier to interpret. But if the relationship is strongly nonlinear, as in Figure 3, then a linear model may fit poorly—unless your Xs are categorical.

The relationship between probability and log odds is easy to plot in a variety of software. In Stata, for example, I plotted the relationship in Figure 1 with a single command, as follows:

twoway function y=ln(x/(1-x)), range(.2 .8) xtitle(“Probability”) ytitle(“Log odds”)

I plotted Figure 2 using the same command, only changing the range to (.09 .13).

In some situations, the relationship between probability and log odds is slightly but not severely nonlinear. Then you face a tradeoff, and your choice of model will depend on your goals. If what you mainly want is a rough but clear summary of the relationships, you might be willing to tolerate a bit of misfit and use a linear model that runs quickly and gives coefficients that are easy to interpret. But if you really want to get the probabilities just right, then you might be willing to sacrifice runtime and interpretability for better probability estimates. For example, I’ve developed financial risk models that predict the probability that a transaction is fraudulent or a borrower will default. In that situation, the coefficients are not the focus. What you really want the model to do is assign accurate probabilities to individual transactions or borrowers, and the linear model often does that poorly over the range of probabilities that are typical for a risk model. Then the logistic model is a more natural choice, although other nonlinear models, such as neural networks or classification trees, are also used.

Paul von Hippel is an Associate Professor in the LBJ School of Public Affairs at the University of Texas, Austin, with affiliations and courtesy appointments in Sociology, Population Research, and Statistics and Data Science.

### References

Angrist, J. D., & Pischke, J.-S. (2008). Mostly Harmless Econometrics: An Empiricist’s Companion (1st ed.). Princeton University Press.

Long, J. S. (1997). Regression Models for Categorical and Limited Dependent Variables (1st ed.). Sage Publications, Inc.

Pischke, J.-S. (2012, July 9). Probit better than LPM? Retrieved from http://www.mostlyharmlesseconometrics.com/2012/07/probit-better-than-lpm/

von Hippel, P.T. & Workman, J. (2016). From Kindergarten Through Second Grade, U.S. Children’s Obesity Prevalence Grows Only During Summer Vacations. Obesity Volume 24, Issue 11, pages 2296–2300. http://onlinelibrary.wiley.com/doi/10.1002/oby.21613/full

[1] The relationship looked a little different in my original article on the Belmont Stakes. In that article I constrained the modeled probabilities to reflect the assumption that the probability of an upset cannot exceed n/(n-1) where n is the number of starters. I also subjected the regressor to a reciprocal transformation, but this made less of a difference to the modeled probabilities.