Logistic Regression - Finding out information about a particular $X_i$ when all other Xs and parameters are known

Question

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.)?

Answer 1

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 𝐿𝑜𝑔𝑖𝑡(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 𝐿𝑜𝑔𝑖𝑡(𝑝) calculation And that the data of x was not replaced by random noise and thus 𝛽’2 is similar to 𝛽2.