1
$\begingroup$

I know linear regression is on a continuous response variable and logistic regression is on a binary response variable.

But is there any name for regression on response variable between 0 to 1? Is this approach, where set the objective function to $\text{minimize}~ \|\frac 1 {1+e^{-X\beta}} -y \|_2^2$ valid? and had a name?

Improve this question
$\endgroup$
8
  • $\begingroup$ It seems like this would just be regular linear regression, right? If it is allowed to be continuous, but just bounded between 0 and 1, it's just regression. You could normalize any continuous DV to be bounded at 0 and 1. The scale affects the size of the unstandardized regression coefficients, but not the actual process of doing the regression. $\endgroup$
    – Mark White
    Commented Jul 28, 2017 at 15:14
  • $\begingroup$ @MarkWhite how about this? $\endgroup$
    – Haitao Du
    Commented Jul 28, 2017 at 15:15
  • $\begingroup$ @MarkWhite Standardizing the response variable (not predictors) is still a regression problem, but the model would not be linear. $\endgroup$
    – Kevin
    Commented Jul 28, 2017 at 15:31
  • $\begingroup$ Beta regression can be used for a response variable bound by 0 and 1. $\endgroup$
    – Sal Mangiafico
    Commented Jul 28, 2017 at 15:32
  • 2
    $\begingroup$ Your question relies on mistaken premises - "regression on a continuous variable" is not limited to linear regression, and similarly "regression on a 0/1 variable" is not limited to logistic regression (they're examples of, not names for those things); similarly there's more than one regression model for continuous data on (0,1)... $\endgroup$
    – Glen_b
    Commented Jul 29, 2017 at 8:48

3 Answers 3

Reset to default
4
$\begingroup$

If the response variable is between 0 to 1, then you could model using a Beta Regression. The seminal paper is

Ferrari, S.L.P., and Cribari-Neto, F. (2004). Beta Regression for Modeling Rates and Proportions. Journal of Applied Statistics, 31(7), 799–815.

There is also a R-package available called 'betareg'. An example from the documentation:

library(betareg)
data("GasolineYield")
gy <- betareg(yield ~ batch + temp, data = GasolineYield)
summary(gy)
Improve this answer
$\endgroup$
1
  • 3
    $\begingroup$ Important note: if I recall correctly, the methods in this paper require that $y \in (0,1)$, not $y \in [0,1]$. This is of high consequence if you $y$ is something like the estimated probability from a binomial distribution, which has positive probability of being 0 or 1. $\endgroup$
    – Cliff AB
    Commented Jan 8, 2018 at 18:19
2
$\begingroup$

Not sure what the data is you're trying to model, but another option is to use some sort of transformation on your response using a monovariate approach. Something like

$$ y = log(\frac{x - a}{b - x}) $$

where a = lower limit and b = upper limit

This is used a lot in forecasting I believe (check out the article by the always excellent Rob Hyndman: https://robjhyndman.com/hyndsight/forecasting-within-limits/)

Improve this answer
$\endgroup$
2
$\begingroup$

I believe the generic term is fractional response regression. There are logit, probit, and heteroskedastic probit conditional mean versions.

The standard reference for the logit case is:

Papke, L. E., and J. M. Wooldridge. 1996. "Econometric methods for fractional response variables with an application to 401(k) plan participation rates." Journal of Applied Econometrics 11: 619–632.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.