Beta regression

Question

I have a data set where the response variable Y is a rate between 0 and 1, where the histogram of Y is bimodal. So I feel the linear regression is not suitable.s I have been reading papers about inflated beta regression.

My question is, just as in OLS, we check the plot of Y at a given level of X to se if it is normal, do I need to ensure that the conditional plots of Y given X for each of my predictors is also bimodal?

Also, I have read that R-square isn't applicable for GLMs such as beta regression, and instead I need to use deviance residuals. If I want to ensure that the model fits well(all variance is explained), should I be looking for a deviance residual plot that is normally distributed?

A preliminary run in SAS showed strong hetero-skedasticity in my model's residuals (the raw residuals, i.e., Y - Yhat) ----however, I want to confirm that this is okay, since in non-linear regression the variance depends on the mean (specifically okay for beta regression) ?

Answer 1

There's no reason that inflated beta has to look conditionally bimodal; the shape will change across the $X$'s anyway.

Deviance residuals (or other forms of residuals, assuming you have any of them available) wouldn't necessarily look very normal either. The residuals for observations away from the ends may look reasonably near normal though (but non-normality isn't of itself necessarily an indication of a problem even then).

You would still look at diagnostic displays to check whether they're at least consistent with what you might see with inflated beta models, but they're harder to interpret.

Raw residuals would be expected to be heteroskedastic.

since in non-linear regression the variance depends on the mean

I think you may be slightly confused by terminology there. The phrase "nonlinear regression" usually implies nonlinear least squares, which in the model has constant variance. The 'nonlinear' part refers to the fact that (at least some) parameters enter the model nonlinearly. (They also are generally nonlinear in the variables, because otherwise a reparameterization would often be sufficient.)

GLMs (and beta regression, and a variety of other "non-normal" regression models) - and their inflated companions - do have variance that change with the mean, but they aren't called nonlinear - indeed, they're called linear models (that's what the $L$ in GLM stands for), because they're linear in parameters - the models don't even have to be nonlinear in the relationship between $y$ and $x$, but they can still be heteroskedastic (e.g. a Poisson GLM with linear link will still be heteroskedastic).

Answer 2

You don't want your plot of Y conditional on X to be bimodal, as that suggests the regression isn't meaningful: the predicted response will be in the middle of the two modes, corresponding to neither. In other words, there is systematic variation remaining in the response after the effect of the X's has been removed.

The exception is logistic regression with a binary response (event vs non-event), because the middle is meaningful in this case: it's the probability that an event occurs. But your data doesn't seem to fit this description.