I've been working in healthcare for a long time now. One of the things I've heard many times over the years goes something like this: "An increase in CMI is worth about $0.63 per hundredth of a point." I've heard that from many people and in many forms. I am guilty of using it too. (I usually simplify it down to $63 per point of CMI.)
I say guilty because I've always had a nagging suspicion that it might not be correct. When I bring it up to people I usually hear things like "It's a conservative estimate" or worse "It's what we've always used." Enough of that. Let's see if we can figure out a better estimate.
Rather than use simple linear regression and getting an r-squared of 21%, I am going to acknowledge that in this case, regression analysis isn't the best tool. I know that there is a distribution of payments for each level of CMI. (I also know that it's possible to calculate the reimbursement exactly if you have enough information. However we're looking for a rule of thumb that's pretty accurate and secondly, we probably don't have the information we need to do the full calculation anyway.)
Based on that, let's try a Bayesian approach, specifically a Monte Carlo simulation. I can create a model that is equivalent to a linear model while at the same time acknowledging that I don't know exactly what some of the parameters are. Then I can run a simulation to find values that are most likely given the data I have.
I am starting in Pennsylvania because I've been doing analysis there recently. Based on every facility in the state for which I have case mix and rates, (>95%) I ran a simulation. Here are the results:
Based on that, you can see that the most likely value for the slope is $77.70 per point of CMI. That's a LOT higher than the traditional value of $63. More importantly, and the reason I used Monte Carlo analysis in the first place, is that I can answer the question: Is $63 per point of CMI even a credible value? The answer, based on this model for this state is yes, but just barely. Only 2.7% of the time is the slope going to be $63 or lower.
Could I have chosen a better model? Absolutely. But we wanted a rule of thumb and most people don't want higher order math in a rule of thumb. (There are a few outliers in the Pennsylvania. I removed a handful prior to my analysis. With those outliers in place the slope would have been around $88 and $63 was not a credible number.)
So, do these results hold in other states? I'm glad you asked. Here are some values for various states.
As you can see, $63 is credible in some states, but it's never ideal. In other states, it isn't even a credible number. (Illinois & Georgia)
If you're trying to estimate changes in PPD using case-mix changes, beware that each state is different. If you want to be conservative in your estimate, you can use the HDI range and quote both numbers. You have to know your state to make a good estimate.