I am trying to find future projections of "Mortality to Incidence Ratio" (MIR) using a number of variables. I checked for Binomial (quasibinomial) (although MIR is not a count ratio) and Beta regressions but they both require the variable to be in 0-1. MIR can be greater than 1, and is for some of the countries in my analysis. What would you recommend in this case? Edit: Definition of MIR here is mortality/incidence. but please note that mortality is assumed to occur among prevalent cases so mortality can be greater than incidence for a given year.
You probably should not be trying to use a projection of MIR over time as a proxy for health care access.
This recent paper in the ASCO Journal of Global Oncology, titled "The Mortality-to-Incidence Ratio Is Not a Valid Proxy for Cancer Survival," makes a compelling argument that interpreting MIR in terms of cancer survival rates "is mistaken in principle and misleading in practice."
For one, they note that "The Validity of Cancer Mortality Rates May Be Questionable." A valid MIR should only refer to deaths from the disease in question. So except for cancers with very short survival times, deaths from other causes should tend to keep MIR values below 1 in the steady state. It's quite possible that your MIR values > 1 represent some deaths from other causes that were improperly annotated.
They also note that "The Mortality Rate Does Not Refer to the Same Patients as the Incidence Rate." That's particularly important for examining changes over time, as you intend to do.
Consider what would happen to MIR for cervical cancer if screening for cervical pre-cancers improves in a country at some point in time. Incidence of cervical cancer will decrease in later years. But as you point out, mortality applies to prevalent cases. There will still be many prevalent cases that preceded the improvement in screening, leading to deaths from prior prevalent cases even as new incidence is decreasing. There could thus be an increase in MIR for a while after the improvement in screening. So improved access to health care could paradoxically lead to an increase in MIR over time.
As with any ratio, you have to understand both the numerator and the denominator to interpret changes in the ratio. It seems dangerous to model MIR over time without modeling both incidence and mortality.
That doesn't prevent people from using MIR as a proxy for health care quality in other circumstances, as in this recent paper that used MIRs for 5 cancers as indicators for differences in health care quality among OECD countries. You will notice, however, that no MIR values were greater than 1 in that paper. They simply compared MIR values against another ranking of quality of national health care systems. And they made no attempt to do projections of MIR over time as you wish to do.
If your goal is prediction/forecasting and you have a continuous outcome, you can try a variety of machine learning-based methods that don't require assumptions about functional form, unlike generalized linear models. For example, random forests will flexibly find a relationship between the covariates and the outcome, from which you can generate predicted values on a new data set. You should use cross-validation to choose the tuning parameters of the random forest model so that the model doesn't overfit. A benefit of these tree-based methods is that they won't create predicted values outside the range of the data, so if your variable is bounded below by zero but unbounded above, the predicted values will remain above zero.