# Harmony Healthcare Blog Jim "Smuz" Smerczynski is an Analytics and Statistics Consultant at Harmony Healthcare International (HHI). In his role at HHI, Smuz designs databases and establishes protocols for collecting and analyzing vast amounts of data pertinent to patient length of stay in Healthcare Facilities.

Prior to joining HHI, Smuz worked for 40 years as a Radar Systems Engineer at Raytheon Integrated
Defense Systems.  Smuz received his Masters of Science in Electrical Engineering at Yale University.

Harmony Healthcare International has compiled the length of stay days for 9552 Skilled Nursing Facility Medicare Patients. The data was taken from homes in New England and New York State between 2012 and 2014 for the purpose of describing the complexity involved.

Length of stay can be visualized by means of a histogram, such as that shown in Figure 1, which represents the probability of occurrence of the given days. Mathematically the figure is called a probability density function (pdf). Note that the data is not symmetrical like the conventional bell shaped normal density curve that everyone is accustomed to. This probability density function (pdf) is skewed to the high end due to outliers and herein lays the complexity. Mathematically the current data in Figure 1 is represented by the Gamma density function. FIGURE 1

The length of stay emerges as a central factor in healthcare economics. In order to benchmark the LOS with one number, the trend is to use the average length of stay (ALOS). This calculation is biased by the outliers and gives a misleading assessment. A more accurate representation of the central value of an ensemble of points is provided by the geometric mean since it is not as sensitive to outliers. Both values are shown in the figure. The average is simply the arithmetic mean of the data. Given n numbers the geometric mean is the nth root of the product of the values. However the geometric mean calculation is simplified by using logarithms. For the current data set the geometric mean is 17.4 days and the ALOS is 24.4 days. The difference is significant.

Harmony Healthcare employs Clinically Appropriate Stay (CAS), derived from the geometric mean, as an appropriate benchmark and encourages using this figure of merit for Skilled Nursing Facility Medicare patients. Incidentally as a point of information, when CMS applies benchmarks to Hospital DRG categories they compute both the geometric mean and the average length of stay and compare the two.

A more complete description of length of stay follows from the distribution of days aptly called the cumulative distribution function (CDF) presented in Figure 2. Integrating the area under the density function generates this curve. This graph shows that 60.9% of the days are less than the 24.4 day ALOS. In contrast, 45.9% of the days are less than or equal to the 17.4 day GM. These numbers demonstrate that the GM is much more representative of the central value of the ensemble than the ALOS. The GM is never greater than the ALOS. If the density function turns out to be symmetrical the values converge to the 50% point.

The graph presented in Figure 2 allows for a number of planning strategies to be implemented. The steepest part of the curve occurs between 5 and 30 days. These points correspond to the 10% and 70% probability values, respectively. It can be concluded that 60% of the patients will have lengths of stay between 5 and 30 days. Administrators, as the need arises, can derive any number of decision points from this type of visualization. FIGURE 2

Beyond, the Average Length of Stay suggests that when a single value is required for a length of stay benchmark the Clinically Appropriate Stay, calculated from the geometric mean, is superior to ALOS. In addition, the length of stay combinations can be predicted more easily from the cumulative probability density function like the one shown above. ### Posts by Topic  ### Stay connected! 