# GLM with logit link and Gaussian family to predict a continuous DV between 0 and 1

Can you run a GLM using a logit link with a continuous DV (between 0 and 1)? Generally it's suggested to use a binomial family with a logit link, but I'm guessing that is because the model assumes a binary DV. If we have a continuous DV would we want to use a Gaussian family instead of binomial?

I apologize if this question doesn't make much sense: I have only a very basic knowledge of statistics, and am just trying to recalibrate a model specified by a colleague a number of years ago.

You seem to want to use a fractional logit, i.e. a quasi-likelihood model for a proportion. The key here is that it is a quasi-likelihood model, so the family refers to the variance function and nothing else. In quasi-likelihood that variance is a nuisance parameter, which does not have to be correctly specified in your model if your dataset is large enough. So I would stick with the usual family for a fractional logit model, and use the binomial family.

• +1. Note that with continuous proportions, just like binary (0, 1) variables, there is a variance-mean relationship which necessarily rules out a Gaussian. Consider limiting cases. A mean of 0 implies all values 0 and so variance 0; similarly a mean of 1 implies all values 1 and so variance also 0. Hence variance must be largest for some intermediate mean proportion and the binomial is more nearly right, at least qualitatively. As @Gavin Simpson rightly points out, a beta regression may also be defensible. – Nick Cox Jan 4 '16 at 16:04
• Note that the argument in my comment above is a little hand-waving. For example, it's possible in principle that all values are 0.42 and so variance would then also be 0. But in practice such cases don't need or deserve modelling. – Nick Cox Jan 4 '16 at 16:30

If your data really are continuous proportions (the common example I see is % silt, clay, or sand in sediment samples - only one of these types for beta regression, all three for a Dirichlet regression) then a beta regression would suggest itself. It's not a GLM sensu McCullagh and Nelder, but it is part of the extended family of GLMs that look, walk, and quack like a GLM.

• I (together with Nick) have worked with regression based on the beta and Dirichlet distributions, so I should be partial to them. However, I am slowly being convinced (based on numerious simulations) that a fractional (multinomial) logit tends to be more robust. The variance does no longer have to be correctly specified in a fractional logit, while it has to be correctly specified in beta or Dirichlet regression. If it is the variance that is of substantive interest, then a fractional logit won't do what you want, but otherwise a fractional logit would be my default model for fractional data. – Maarten Buis Jan 4 '16 at 16:08
• @MaartenBuis Indeed; I didn't intend this to be taken as an either/or - I've also used both quasi-binomial and beta regressions. – Reinstate Monica - G. Simpson Jan 4 '16 at 16:51
• Why is beta regression not a GLM sensu strictu, @Gavin? – amoeba says Reinstate Monica Sep 4 '16 at 16:22
• With all the parameters to be estimated I didn't think you could write it down in the form required for GLMs sensu McCullagh & Nelder. In same sense that a negative binomial model doesn't fit the GLM scheme if the theta parameter is to be estimated too. – Reinstate Monica - G. Simpson Sep 4 '16 at 16:32

Yes you can. The model parameters are still log-odds ratios, but they are estimated differently. Your model with such specifications is basically a nonlinear least squares, where a logit "S" curve is being fit to 0/1 outcomes so as to minimize the squared error. However, the contrasts to usual logistic regression are very well known: this approach puts very little weight on 0/1 outcomes since a proportional difference of 0.95 versus 0.96 is much larger when scaled by its binomial variance. Gaussian families do not assume any mean-variance relationship. That's why this approach is not often used.

If the results given you are proportions, then the burning question is: do you have the denominators for these proportions? e.g. is the 0.43 percent calculated out of $n=100$ or $n=200$ participants and/or does this value differ between the various observations you've obtained? If so, weighting the binomial likelihood gives equivalent inference to fully observed 0/1 counts.

In R, for instance, it will still give you warnings that you have used non-binary outcome variables, but the fitting algorithm does not "break" when inputting data of this format. Other software may prevent such approaches altogether so you will have to create product variables.

However, without such counts in place, other robust error estimation methods should be used. Others' suggestions of quasilikelihood seems like a reasonable choice.

• +1. What if the data are probabilities? For example, data come from a psychological experiment where people were estimating probabilities of something; these predictions (between 0 and 1) are the DV. It's like logistic regression but instead of the binomial outcome we have probability itself. What is a reasonable approach then? – amoeba says Reinstate Monica Sep 4 '16 at 16:21
• @amoeba I think the approach is still valid, provided the mean model is correct. – AdamO Feb 20 '18 at 20:49