Distribution family for a ratio dependent variable in a generalized estimating equation

Question

I have several dependent variables that are measures of racial disproportionality; I've calculated them as:

% of events caused by racial minority group / % of events caused by racial majority group

I have a dependent variable for each racial minority group in my sample. I am running longitudinal Generalized Estimating Equations (GEE) on these models, however I am somewhat stumped as to which family is appropriate for these dependent variables. The probability range for my ratios are truncated at 0, as it's not possible to have negative values in my DVs. This makes me question the validity of using a Gaussian family for my models.

The idea behind these variables is that a ratio greater than 1 indicates some level of greater burden of events that a given racial minority is bearing compared to the racial majority, and a ratio less than 1 indicates the opposite.

• What would be the most appropriate family to use for my GEE regressions?

EDIT:

I misspoke about the racial disproportionality measure I was using. The correct formula is:

% events by minority / % of total enrollment that is minority OVER % events by non-minority / % of total enrollment that is non-minority

Because they are ratios, the number of observations with value less than 1 is comparable to the number of observations greater than 1, with the lower bound being 0 and the upper bound being non-bounded. Looking at the histograms of my response variables, they definitely seem to fit a negative binomial distribution better than the normal. The QIC (GEE adjustment to AIC) confirms this suspicion. My questions now are:

• Can I trust this evidence to move forward with the negative binomial family?
• If so, how do I possibly interpret the exponentiated coefficients from the resulting models? They don't see to be Incidence Rate Ratios, as one would interpret them to be from count variables...

Answer 1

maximum entropy is a good way to go here. With maximum entropy, you specify the "structure" that your model is to depend on, and it does the rest. It has a very similar form to a generalised estimating equation. So we have an unknown (or "random") variable $x$ that is the object of inference (may be a vector). It can take on $n$ values $x_1,x_2,\dots,x_n$. We have $m$ "model functions" $g_{k}(x)$ $(k=1,\dots,m)$ and $G_{k}$ "constraints" and the two are related by:

$$\sum_{i=1}^{n}p_i g_{k}(x_i)=G_{k}$$

Where $p_i=Pr(x=x_i)$ is the "unknown" (more appropriately "unassigned") probability distribution. By choosing the distribution with maximum entropy $-\sum_i p_i log(p_i)$ you get:

$$Pr(x=x_i||G_{1},\dots,G_{m})=p_i=exp\left(-\lambda_0-\sum_{k=1}^{m}\lambda_{k}g_{k}(x_i)\right)$$

Where $\lambda_{k}$ are chosen so that the above constraints are satisfied, and that the $p_i$ sum to $1$. This requires discrete sample space, for continuous case, it is the same form with 2 exceptions:

1. The summation is replaced by an integral with respect to $dx$
2. The probability is multiplied by the invariant measure $m(x)$, which in your case is the probability transform of the improper $Dir(0,\dots,0)$ into "odds" form.

So you have:

$$p(x|G_{1},\dots,G_{m})=m(x)exp\left(-\lambda_0-\sum_{k=1}^{m}\lambda_{k}g_{k}(x)\right)dx$$

Now what you will find is that in solving for the lagrange multipliers $\lambda_{k}$ you will get a set of equations which look identical to the GEE equations. Thus the MaxEnt algorithm basically tells you "which probability model" is the most likely to be consistent with your GEE equations. You can then use this distribution to make predictions and give an indication of their accuracy.

Note if the constraints $G_{k}$ are not known, this is solved by multiplying the above $p(x||G_{1},\dots,G_{m})$ by a prior density for $G_{k}$ and then integrating out $G_{k}$ from the probability - this allows for the structure induced by $G_{k}$ to be incorporated into the model, as well as the uncertainty about the true value of $G_{k}$.