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I want to simulate data of the form:

$$ Z=e^{X_1+X_2}+X_3 + \epsilon $$

for i.i.d. $X_1,X_2,X_3\sim\mathcal{U}(-1,1)$ and $\epsilon\sim\mathcal{N}(0,1)$

I have (using R) generated 10,000 realisations of $Z$ and now want to associate them to a (binary) categorical variable $Y$ which takes values $0$ and $1$.

I've done this by using Logistic Regression - with the $i^{\text{th}}$ realisation denoted as $Z_i$ - I used the realisations of $Z$ as the log-odds by calculating the probability of $Y=1$ as

$$ \frac{e^{Z_i}}{1+e^{Z_i}} $$

which in turn gives values in $[0,1]$.

My question is:

I've only seen Logistic Regression used when the log-odds is a linear function - am I allowed to do what I have done to attain a probability, despite my log-odds is not a linear function?

If not, what other methods are there so that I can simulate such data? (I am using R so I would be grateful if you could provide insight on what R can do)

Also, I have used this method for a data generating process with an interaction i.e. $Y=X_1+X_2+X_1X_2+\epsilon$.

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  • $\begingroup$ What's the goal of the simulation? There are many ways of relating a real valued variable $Z$ to a binary outcome. $\endgroup$
    – CloseToC
    Sep 22, 2019 at 16:54
  • $\begingroup$ @CloseToC I intend to train a GBM on the data and then use tools such as PDPs to then try identify interactions, in this case, between $X_1$ and $X_2$ $\endgroup$
    – Naji
    Sep 22, 2019 at 17:57

1 Answer 1

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You can do whatever you want to simulate data. It's your data. If you have values between 0 and 1, you can choose to treat them as probabilities and generate a Bernoulli (0/1) variable using them. You could do this regardless of how you got those probabilities.

That said, the data-generating model does not correspond to a logistic regression model, so attempting to fit a logistic regression model on $X_1$, $X_2$, and $X_3$ will not allow you to recover the true model. If you fit a logistic regression model on $e^{X_1+X_2}$, $X_3$, and $\epsilon$, you would recover the true model (note that fitting a logistic regression model on just $e^{X_1+X_2}$ and $X_3$ will not recover the true model because of noncollapsibility).

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