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Suppose someone made the following logistic regression:

$Logit(p)$ = $\beta_0 + \beta_1X_1 + \beta_2X_2$

Now, someone else is trying to replicate the model creation, but by mistake the $X_2$ column is replaced by some other values. Hence, the model is coming as:

$Logit(p)$ = $\beta_0^{'} + \beta_1^{'}X_1 + \beta_2^{'}X_2^{'}$

I know all the values of the Xs and the parameters in the two equations except the values of $X_2$

My question is: How can I get back the values of $X_2$? I know that getting the value of $X_2$ for each row may be a challenge. Can I get something at an aggregated level at least (mean, sum, etc.)?

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  • $\begingroup$ Do you have any distributional assumptions for $X_2$ that you are willing to make? For example normal, log normal? $\endgroup$ – Jesper for President Oct 2 '19 at 7:19
  • $\begingroup$ $X_2$ is simply a column in a real life data. Assuming a distribution of it will not be practical I guess $\endgroup$ – SamRoy Oct 2 '19 at 7:21
  • $\begingroup$ The fact that $X_2$ is a column in a real life data seems to go against you not knowing $X_2$? $\endgroup$ – Jesper for President Oct 2 '19 at 7:23
  • $\begingroup$ No no. My issue is the actual column $X_2$ got replaced with something else by mistake. Now I am trying to get some info about the original $X_2$ column $\endgroup$ – SamRoy Oct 2 '19 at 7:26
  • $\begingroup$ Do you know what the variable $X_2$ measures ... blood pressure, income etc. ? And do you know the population of the original data set including $X_2$? $\endgroup$ – Jesper for President Oct 2 '19 at 7:29
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I would use model fitting to assess the likelihood of x2 being a specific value within a given range:

Let’s say that there are 3 samples (i=3) Every sample is 0.31, 0.49 or 0.84 Distribution of x = average of all the unique x combinations ([0.31, 0.31, 0.31], [0.31, 0.31, 0.49] etc until max of [0.84, 0.84, 0.84])

For each possible value of x, compare the output of the regression to the original regression function πΏπ‘œπ‘”π‘–π‘‘(𝑝) and calculate the loglikelihood (which is basically asking what are the odds that that’s the right value of the lost x)

Assumptions: I’m assuming you have the output of the original πΏπ‘œπ‘”π‘–π‘‘(𝑝) calculation And that the data of x was not replaced by random noise and thus 𝛽’2 is similar to 𝛽2.

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