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I have been recommended the logit link for data in (0,1) since it's interpretable... how?

I am using the logit link function for data which is continuous (i.e., this is not logistic regression ... think beta regression.) That is, I have the following model$$EY_i = \mu\\ g(\mu) = X\beta \\g(x) = \text{logit}(x)$$ and $Y_i$ is some suitable distribution, like the beta.

Then, we have $$\mu = \frac{e^{X\beta}}{1 + e^{X\beta}}$$

How is this interpretable? Meaning, if I gave you one particular $\beta_i$ of one particular covariate, how would you interpret it?

Radisz
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1 Answers1

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This is a really rudimentary question. Any book talks about logistic regression would cover for it.

If you are looking for some recommendations, I would this tutorial for logistic regression (UCLA IRDE LOGIT REGRESSION | R DATA ANALYSIS EXAMPLES).

Here are some examples of interpretation from the tutorial:

  • For every one unit change in GRE, the log odds of admission (versus non-admission) increases by 0.002.

  • For a one unit increase in GPA, the log odds of being admitted to graduate school increases by 0.804.

To summarize, the connection is

  • $X\beta$ gives Log Odds
  • $\exp(X\beta)$ gives Odds
  • $\text{logit}^{-1}(X \beta)$ gives Probability

So, if $X$ change, we can calculate how the probability change. Because the change is non-linear, sometimes, it is more intuitive to explain using "Log odds" or "Odds".

Community
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Haitao Du
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  • +1 except you mean logit inverse in the last line. I also would interpret effects on the log odds scale, but at least transform to the odds scale by exponentiating $\beta$. Lastly, I wouldn't say that admission probability increased for a one unit change in GRE because that is a within subject interpretation. A between subject interpretation appropriate for independent data analysis is: "comparing applicants whose GRE scores differed by one unit, the odds ratio of admission was $e^{\hat{\beta}}$." – AdamO Jun 06 '17 at 16:35