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I have fitted a logistic regression model with a transformed variable $z$, where $z=x/M$. I want to interpret the $\beta$ coefficients in the original scale (in-terms of x).

I started following calculation. First, I wrote the Odds rations in the transformed scale, as follows:

let $\pi_1=logit(p1)=\beta_0+\beta_1z$ and $\pi_2=logit(p2)=\beta_0+\beta_1(z+1)$. Then, $\pi_1-\pi_2=log(\hat{OR})=\beta_1$. And $\hat{OR}=exp(\beta_1)$. This $OR$ was calculated based on z. Now, I want to calculate $OR$ for x.

My friends said that $OR$ based on x would be $exp(\beta_1/M)$. I want to prove it.

So I substitute $z=x/M$ in the above calculation. Then,

$\pi_1=logit(p1)=\beta_0+\beta_1x/M$ and $\pi_2=logit(p2)=\beta_0+\beta_1(x/M+1)$

$\pi_1-\pi_2=log(\hat{OR})=\beta_1$. This is not what I expected. Correct answer should be $\beta_1/M$

Could anyone explain what did I do wrong?

kjetil b halvorsen
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student_R123
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    quick question, why do you want to do this? to see the effect of an increase of 1 for x wrt the predicted probability? – Alberto Jul 22 '22 at 00:31
  • @AlbertoSinigaglia Actually, the reason is the odds ratios in the transformed scale is very large. So, I need to convert it to the original scale. Could you help me? I wanted to show the results like the table 5 in the paper: https://www.sciencedirect.com/science/article/pii/S0376871622002137 – student_R123 Jul 22 '22 at 00:42
  • They used exp(beta\M). But my calculation don't give that. – student_R123 Jul 22 '22 at 00:43
  • I don't quite get your calculation, it's quite late where I am, however, I see that $\beta_1/M$ is the correct answer by the simple fact that affine transformations of convex functions, does not change the minimum... in other words, if $\beta_1$ is optimum for $z$ and $z = x/M$ then $\beta1 / M$ will be the optimum for $x$... however, if you can wait tomorrow morning, I'll try to spot the issue in your reasoning – Alberto Jul 22 '22 at 00:46
  • @AlbertoSinigaglia Yes. I can wait. Thank you very much. – student_R123 Jul 22 '22 at 00:48
  • You are not computing the ratios. For a detailed explanation, see https://stats.stackexchange.com/questions/133623. – whuber Jul 22 '22 at 19:19

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