SRS Part 5: How much more can we lower readmission rates?

I hope you enjoyed my previous post; this is a sequel. Just to recap, the initial analysis of the data suggested a (possible) readmission rate equilibrium. But maybe the real question should be: How much more can we lower readmission rates in the next year? Or technically speaking, what is my expected readmission rate in the next year?

Readmissions are complex beasts, affected by many different factors. To truly understand readmissions, you should model all the policies, regulations, incentives, and processes. For example, CMS uses a sophisticated risk-adjustment model to assess the rate of excessive readmissions for a given population. Patient-level readmission prediction is a even harder problem. Many data scientists tried to solve the problem, but most of the suggested models perform very poorly in practice (more details here).

But, don’t be intimated. What I am trying to do here is, in fact, slightly different. Yes, I will talk about a much simpler model than those. I will take the star rating data, and analyze their “macroscopic” trends. I will be using a simple yet powerful model that anybody can understand. So, are you ready?

The pitfall of our linear regression (in the previous post, SRS Part 4) was that it did not capture the differences of the plans. Each plan has a different population, and thus may have a different initial rate (i.e. readmission rate in 2012). However, fitting an individual statistical model per plan gives you 4 data points at most. This is the classical example of “overfitting”.

Enter Mixed-Effects Modeling! A linear mixed-effects model, also known as a hierarchical model, is a generalization of linear regression models. It can model both global (national) and local (individual MA plan) trends in data, by leveraging the information from both sides (global and local). For more details, see Gelman (theory) and Bates (R package).

First, I present the simplest linear mixed-effects model. I am going to allow each plan to have its own initial intercept (b0) and slope (b1) except that they should be “similar” to each other (I should admit there is a formal and rigorous definition for mixed-effects which I don’t want to repeat here).

Blue model:

$$ Y_i = \beta_{0 j} + \beta_{1 j} X_i + \epsilon_i $$

$$ \beta_{0 j} = \beta_{00} + \eta_{0j} $$

$$ \beta_{1 j } = \beta_{10} + \eta_{1j} $$

where $$Y_i$$ and $$X_i$$ represent readmission rate and year, and $$\beta_{0j}$$ and $$\beta_{1j}$$ are the intercept and slope for MA plan $$j$$.

Each line represents the estimated readmission trend per plan. Note that most of the plans are approaching a readmission rate between 10% to 15%. To understand the fit of the model, I calculated Root Mean Squared Error (RMSE). This is one measure to determine the goodness of the fit to the original data. Even though a RMSE of 1.868 is quite good (my model is off by 1.9% in general), the estimated trends do not quite match the raw data. What we have observed in the previous post is a convergence of readmission rates. So why not see if that’s true?

To do this, I need to transform the independent variable (covariate X). The logarithm function has a nice property that I want to utilize. Primarily, it captures the decreasing slope of the raw data. The mathematical equation for the model is as follows.

Green model:

$$ Y_i = \beta_{0 j} + \beta_{1 j} \log(X_i — \text{offset}) + \epsilon_i $$

where all the other conditions remain the same except for the $$\log$$ on $$X_i$$.

The estimated trends look much more realistic, and of course, the model gives a lower RMSE (1.585) as well. However, there is again a downside of this model. The model does not not capture the outliers (e.g. extreme performers such as Kaiser). To capture these outliers, in fact, there is a simple yet elegant approach. Are you curious? Sorry, this is my secret sauce. With some tweaks, I have this Orange model:

Almost the same as the raw data trajectories (see the last post), and even lower RMSE than the Green model! The Orange model provides both global (national) and local (plan-level) predictions. Let’s take a look at the global predictions:

• 2012 to 2013: -0.59% decrease at nation-level
• 2013 to 2014: -0.47% decrease at nation-level
• 2014 to 2015 (projection): -0.39% decrease at nation-level
• 2015 to 2016 (projection): -0.34% decrease at nation-level
• …

It seems there exists a pseudo-equilibrium for readmission rates. It will become harder and harder to decrease readmission rates in the following years. Some MA plans may achieve very low readmission rates, but most of them would remain at the range of 10% to 15%. So, should I put more resource to reduce the readmission rate? Wait for my next posts. More insights coming up!

- Yubin Park, PhD
Chief Technology Officer