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I want to use a Logistic regression and my dependent variable is continuous, it is a percentage. However, the percentage is technically not bounded between $[0,1].$ It can take a value up to $500\%.$

If i'm not mistaken, logistic regression can be used if $Y$ is bounded between $0$ and $1.$ I want to know if there is a transformation that I could apply to my dependent variable so that it becomes bounded?

For example, let's say that my dependent variable's upper limit is $500\%;$ would it be appropriate to divide $Y$ by $5, $ which would make my new upper limit $100\%? $ Then I could use logistic regression and re-transform back my estimated $Y$s by multiplying by $5.$ Does it make any sense?

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  • $\begingroup$ Rank the percentages you have observed and tally. If in the range over 100%, in my opinion, there are just a few observations, capping at 100% may be acceptable, but otherwise, the actual empirical distribution (or a smooth version?), should be preserved. Use the quantile function associated with the logistic distribution to then create the associated log odd data. $\endgroup$
    – AJKOER
    May 29, 2020 at 21:39
  • $\begingroup$ Why do you "want to use Logistic"? If linear does not work, you can just cut the percentages into two levels, "low" and "high", based on some threshold (median is a good bet but you may need another one). And then you estimate logistics as $\Pr(y \in high) = \Lambda(\beta' \cdot x)$. $\endgroup$
    – Ott Toomet
    May 29, 2020 at 23:51
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    $\begingroup$ This is a comment expressing a vview that if percentages can be 500 or so, they are more likely to be percentage changes with no well-defined upper bound. If so, they are deeply unsuitable for logistic regression regardless of whether bounds are enforced by arithmetic scaling to [0, 1]. I don't agree with the advice in either of the previous comments. $\endgroup$
    – Nick Cox
    Dec 17, 2022 at 12:29
  • $\begingroup$ This seems more like a lift metric / ratio. On that case I suggest taking log transform of your ratio. That removes the arbitrary choice of ratio or 1/ratio as dependent variable $\endgroup$ Apr 21, 2023 at 23:58

2 Answers 2

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If your dependent variable is continuous, use a linear regression instead of a logistic regression. The logistic regression is appropriate in case of a binary endpoint (ex; progression Y/N -> coded 1/0), not a continuous variable bounded between 0 and 1.

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    $\begingroup$ Not so. There is no problem with generalized linear models for a [0, 1] bounded continuous variable and a logit link beyond calculating standard errors reasonably. This model has been in the literature for almost 50 years. $\endgroup$
    – Nick Cox
    Jun 2, 2022 at 16:59
  • $\begingroup$ @NickCox While I see your point, calling a model whose response variable is continuous bounded in $[0, 1]$ a "logistic regression" (this is how OP framed its question) seems at least non-conventional, if not incorrect (in addition, a GLM with a "logit link" does not equal to a "logistic regression" in its most widely accepted sense). From this perspective, I think jgilhode's critique to OP's question is reasonable and your comment "Not so" seems a little too harsh. $\endgroup$
    – Zhanxiong
    Apr 22, 2023 at 3:22
  • $\begingroup$ Who is to say what is most widely accepted? My point goes back to Nelder and Weddburn JRSS 1972 and Wedderburn Biometrika 1974. $\endgroup$
    – Nick Cox
    Apr 22, 2023 at 6:27
  • $\begingroup$ @NickCox My point is, when people used the term "logistic regression", they typically meant the response is binary $\{0, 1\}$ -- for example, Chapter 5, 6 in Agresti's book Categorical Data Analysis. The Nelder and Weddburn's paper is titled "Generalized Linear Model", which (correct me if I am wrong) focuses on building the foundation of GLM but does not contain the term "logistic regression" at all. $\endgroup$
    – Zhanxiong
    Apr 22, 2023 at 14:57
  • $\begingroup$ And my point is that a logit link is entirely defensible for a bounded continuous proportion—which this answer denies emphatically. I don’t think that anyone calls that model a linear regression. If you or anybody else prefers a narrower sense of logistic regression that’s your choice. I don’t see the issue as primarily about terminology. If you want to deny GLMs with logit link the title of logistic regression, that’s a choice I don’t share, being happy that the wider sense is defensible. $\endgroup$
    – Nick Cox
    Apr 22, 2023 at 15:41
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I wish I could find a more formal reference for this, but it makes sense to me to apply a logit transformation to your response variable (logit(x) = ln(1 / (1 - x)) to produce a response variable that's akin to the link in logistic regression. Then, can fit with ordinary least squares. Bear in mind, this will only be valid for response values greater than zero and less than 1.

The interpretation of the coefficients will be similar to how you interpret coefficients of a logistic regression. You could transform them to log-odds if you prefer, or you can evaluate their effect on the untransformed response given a baseline value of the link function.

This approach will allow variables with positive coefficients to never produce a response value >= 1, but rather "squeeze" the prediction closer and closer to 1. Vice versa for negative coefficients and zero.

Again, wish I had a better reference. Would be very interested to see if there is more formal guidance on this, and if OLS is valid on the transformed response variable, or if there's some sort of MLE approach that's better.

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  • $\begingroup$ The spirit is right here if the data were proportions or percents bounded by [0, 1] or [0, 100] -- which manifestly is not true in the original question. But the letter is wrong in detail. Continuous proportions and percents may be fitted with logistic models. Values that are equal to 0 and 1 when scaled are not out of order any more than values of 0 and 1 are out of order for a logit regression; in fact for the latter they are essential. Ordinary least squares is sometimes applied to such responses; there is a literature on that under the heading linear probability model. $\endgroup$
    – Nick Cox
    Dec 17, 2022 at 12:32
  • $\begingroup$ But if values at or near 0 or 1 are common it's more likely that a S-shape and its generalizations is a better functional form. $\endgroup$
    – Nick Cox
    Dec 17, 2022 at 12:33
  • $\begingroup$ Re: fitting logistic models with continuous response, what family and link arguments would you use with R glm? I think binomial is only appropriate for a categorical response variable. $\endgroup$
    – Arthur
    Dec 19, 2022 at 14:25
  • $\begingroup$ I don’t use R routinely but I understand the keyword here to br quasi-binomial. $\endgroup$
    – Nick Cox
    Dec 19, 2022 at 14:33
  • $\begingroup$ This is a problem for ordinal logistic regression. $\endgroup$ Dec 26, 2023 at 8:08

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