Table 2.
Understanding the Hospital Compare Random Effects Model—Some Examples
| Model 1: Model Based on Hospital Compare |
Model 2: Hospital Compare Model Adding in AMI Volume |
Model 3: Hospital Compare Model Adding in Volume and Hospital Characteristics |
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Hosp. | Vol. | O | λ | O/E | 1−λ | F/E | P/E | λ | O/E | 1−λ | F/E | P/E | λ | O/E | 1−λ | F/E | P/E |
| A | 1 | 0 | 0.004 | 0 | 0.996 | 1 | 0.996 | 0.005 | 0 | 0.995 | 1.331 | 1.277 | 0.004 | 0 | 0.996 | 1.282 | 1.277 |
| B | 1 | 1 | 0.008 | 2.775 | 0.992 | 1 | 1.014 | 0.007 | 2.775 | 0.993 | 1.226 | 1.237 | 0.006 | 2.775 | 0.994 | 1.221 | 1.231 |
| C | 8 | 0 | 0.043 | 0 | 0.957 | 1 | 0.957 | 0.042 | 0 | 0.958 | 1.295 | 1.241 | 0.039 | 0 | 0.961 | 1.302 | 1.251 |
| D | 8 | 0.750 | 0.044 | 3.253 | 0.956 | 1 | 1.099 | 0.041 | 3.253 | 0.959 | 1.260 | 1.342 | 0.025 | 3.253 | 0.975 | 1.175 | 1.251 |
| E | 24 | 0.042 | 0.106 | 0.209 | 0.894 | 1 | 0.916 | 0.092 | 0.209 | 0.908 | 1.036 | 0.961 | 0.087 | 0.209 | 0.913 | 1.070 | 0.995 |
| F | 24 | 0.417 | 0.127 | 1.631 | 0.873 | 1 | 1.080 | 0.108 | 1.631 | 0.892 | 1.034 | 1.098 | 0.103 | 1.631 | 0.897 | 1.083 | 1.140 |
| G | 362 | 0.127 | 0.580 | 0.866 | 0.420 | 1 | 0.922 | 0.525 | 0.866 | 0.475 | 0.947 | 0.904 | 0.501 | 0.866 | 0.499 | 0.919 | 0.893 |
| H | 362 | 0.157 | 0.612 | 0.929 | 0.388 | 1 | 0.957 | 0.558 | 0.929 | 0.442 | 0.947 | 0.937 | 0.534 | 0.929 | 0.466 | 0.914 | 0.922 |
Notes. P/E, O/E, F/E are quantities obtained from the random effects model. We can describe P/E as a linear combination of O/E and F/E, where P/E=λ(O/E)+(1−λ)F/E. Model 1 results are based on the present Hospital Compare model without hospital characteristics. Model 2 adds in volume and Model 3 adds in both volume and five other hospital characteristics. We have chosen eight hospitals to illustrate the results: each pair of hospitals has the same reported acute myocardial infarction (AMI) volume. For Model 1, F/E=E/E=1 because there are no hospital characteristics used to estimate F. For Models 2 and 3, F/E is generally different from 1, as a hospital may have better or worse characteristics compared with the typical hospital. For the small hospitals, there are great differences between the P/E estimates for Model 1 versus Models 2 and 3, because Model 1 is shrinking the prediction to the national rate (F/E=1), while Models 2 and 3 shrink toward F/E, which is higher because smaller hospitals as a group perform worse in the data set. As hospital volume increases, the models place increasing emphasis on O/E, with λ values over 0.50 for hospitals G and H.
E, expected; F, forecasted rate (estimated rate based on fixed effects parameters using hospital and patient characteristics); Hosp., hospital name; O, observed rate; P, predicted; vol., volume of Medicare AMI patients.