Table 2.

Beta Regression Estimates for the Impact of Hospital Operative Volume on Hospital Mortality Rate, by Operation

Operation	Multivariable risk-adjusted model
Operation	Estimate ^a	95% CI	p Value
Appendectomy	0.39	0.36-0.42	<0.001
Cholecystectomy	0.51	0.48-0.56	<0.001
Colectomy	0.86	0.78-0.96	<0.001
Inguinal and femoral hernia repair	0.45	0.40-0.50	<0.001
Lysis of adhesions	0.61	0.54-0.68	<0.001
Necrotizing soft tissue infection excision	0.53	0.40-0.69	<0.001
Repair of perforated peptic ulcer disease	0.64	0.50-0.82	<0.001
Small bowel resection	0.74	0.65-0.84	<0.001
Umbilical hernia repair	0.40	0.32-0.50	<0.001
Ventral hernia repair	0.46	0.41-0.53	<0.001

These are beta regression coefficient estimates for hospital volume (aka procedure count) They quantitatively demonstrate the decrease in hospital mortality proportion for each operation type when the natural log of hospital volume is increased by +1. For example, a +1 unit change in natural log volume (meaning an integer increase from 1-->2 or 2-->3 or 3-->4) for colectomy will decrease morality by 14% (as defined in the above table). This 14% predicted decrease in mortality proportion occurs at each increase in integer interval, meaning that if a hospital increases colectomy natural log volume by 2-->3 they can expect a 14% decrease in mortality, and if they increase natural log operative volume by 3-->4 they can expect another 14% decrease in mortality proportion. Note that by comparison, a +1 unit change in natural log volume from 1-->2 or 2-->3 or 3-->4 for small bowel resection will decrease morality by 26% over each interval. However, natural log volumes are difficult to conceptualize, so it is helpful to transform these natural log volume integers (such as 1, 2, 3, 4, etc) back into actual hospital operative volumes. The natural log integer can be back-converted to an actual operative volume like this: for the natural log volume integer 2: 2 = ln(x) --> x = e^2 = 7.4 operations; for the volume integer 3: 3 = ln(x) --> x = e^3 = 20.1 operations; for the natural log volume integer 4: 4 = ln(x) --> x = e^4= 54.6 operations; etc. Therefore, in terms of actual operative volume, a +1 unit change in natural log volume integer from 2-->3 (7.4 operations vs 20.1 operations = +12.7 operations) is not equivalent to the change from 3-->4 (20.1 operations vs 54.6 operations = +34.5 operations) – this highlights the exponential function of the natural log. Please see Figure I to appreciate this visually, as there are graphs for mortality proportion (on the y axis) plotted against both actual operative volume as well as natural log volume integer (on the x axis).