Why McFadden's pseudo-R^2?

Question

Why can't we simply use ordinary $R^2$ in logistic regression, as we do in linear regression? Domencich and McFadden seem to imply that heteroskedasticity is an issue:

but I don't understand why. In linear regression, homoskedasticity is an assumption (in max. likelihood the variance is typically not considered a parameter to optimise), but why it is an issue for calculating a measure of fit?

Below is a plot from my simulations, for normally distributed classes of different sizes and variances:

McFadden's pseudo-$R^2$ and ordinary $R^2$ seem to be closely related. I'd appreciate if someone could give an illustrative example for a problem arising from heteroskedasticity or any other justification against ordinary $R^2$.

Update:

To avoid confusion: I do not imply using ordinary linear regression on nominal data, and/or calculating squared distances from the straight line. For the purpose of my question I'd perform logistic regression, which would produce a logistic curve. My question is: Why is ordinary $R^2$, as a goodness-of-fitt measure of that curve bad, or worse than McFadden's pseudo-$R^2$.

P.S. (for all those thinking this question is a duplicate):

Please quote the relevant portion of that allegedly existing answer to show how it answers my question. Also, please consider: If something is obvious to you, it need not be obvious to others.

Answer 1

McFadden's pseudo-$R^2$ is consistent with the log-likelihood model we optimise in logistic regression. The ordinary $R^2$ is consistent with the log-likelihood model for the linear regression.

In linear regression, we maximise the log-likelihood:

$$ - \sum_i (y_i - \beta x_i)^2 $$

Compare this with the definition of $R^2$:

$$ R^2 = 1 - \frac{\sum_i (y_i - \beta x_i)^2}{\sum_i (y_i - \bar y)^2} $$

In the numerator we have the likelihood of our model, and in the denominator the likelihood of the null-model. McFadden's pseudo-$R^2$ is constructed according to the exactly the same schema, just that the log-likelihoods are defined differently for logistic regression. In fact, one can use McFadden's pseudo-$R^2$ for any log-likelihood-based model, just by plugging in the corresponding log-likelihood function. Maybe "generalised $R^2$" would be a more appropriate name.

Homo/heteroskedasticity is not the actual issue here. In linear regression it is implied in the log-likelihood (actually each term in the sum can be thought of as divided by $2 \sigma^2$, but that doesn't matter, since it doesn't change the vector $\beta$). But, in logistic regression, variance doesn't figure at all.*

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*Well, it does on a different level, but we have implicitly accepted that as soon as we agreed on using logistic regression in the first place.