This fourth installment of the six-part Health Capital Topics series on the application of statistical methods by valuation analysts will provide a brief overview of datasets, samples, and their utilization in various valuation approaches and methodologies. As discussed in Part One of this series, entitled, “Review of Principles and Applications,”1 a strong understanding of commonly used statistical methods is useful to a valuation analyst in creating, defending, and/or critiquing valuation reports, including when engaging in forecasting and benchmarking functions. Further, any improper use of statistical methods by a valuation analyst may lead to erroneous inferences by the analyst or their client that may significantly affect the calculated value indication. It is imperative that valuation analysts possess a working understanding of the collection, structure, and patterns associated with data given the frequent reliance of valuation analysis on the statistical applications built from data sets and samples.
Variables used as inputs in a valuation analysis may be inherently random, i.e., the variable takes on a numerical value previously unknown to the valuation analyst until the value is observed through some measuring process.2 Populations, and the samples drawn from them, may exhibit certain patterns, referred to as statistical distributions, some of which occur regularly in statistics and have special names (e.g., binomial or normal distributions) with known properties that may provide a valuation analyst with convenient analytical tools for understanding the data.3
Random variables and their distributions are typically classified into one of two general categories: (1) discrete (i.e., data that can only take particular values within an interval); or, (2) continuous (i.e., data that is able to take on all possible values within an interval).4
Discrete distributions have been developed to describe observed patterns in certain types of discrete random variables, including: (1) binomial distribution; (2) hypergeometric distribution; (3) negative binomial distribution; and, (4) Poisson distribution.5 The Binomial distribution may be used to estimate the probability of a “success,” or the phenomena for which the researcher is looking, over any number of fixed trials, e.g., the probability of observing eight or more heads if a coin is tossed ten times.6 The hypergeometric distribution is related to the binomial distribution, but gives the probability for the number of successes in a given sample, e.g. the probability of observing exactly eight heads in ten coin tosses.7 The negative binomial distribution fixes the number of successes desired and informs the researcher as to the necessary number of trials to acquire the preferred number of successes, e.g., the number of required coin tosses to observe eight heads.8 The Poisson distribution uses a parameter, measured as a rate per unit time or unit area, which may be used to find the probability that an event will happen within a specified time interval, e.g., if the average number of people visiting an exhibit is ten people per day, the probability that 15 will visit tomorrow.9
Similarly, continuous distributions have been developed to explain the observed patterns in continuous random variables, including: (1) normal distribution; (2) lognormal distribution; and, (3) beta distribution, among others.10 The normal distribution (bell-shaped curve) is symmetrically centered on its mean,11 and is considered “the most important [distribution] in all of probability and statistics.”12 The normal distribution gives the cumulative probability for a range of values leading up to (or away from) a selected data point, typically located on either side of the mean for any real valued number, e.g., half of observations are less than the mean observation.13 Transforming the normal distribution under a logarithmic operator for a random variable greater than zero yields the lognormal distribution, which models data where the percentage change is normally distributed.14 The Beta distribution is a two parameter distribution, whose parameter values allow for a variation in the shape of the curve, and generates probabilities for a random variable that is bounded from above and below over any interval, e.g., perhaps effective for modeling data that is not symmetric but proportionally bounded between zero and one (i.e., the standard beta).15
A valuation analyst’s understanding of statistical distributions has direct application to valuation. One example of a discrete random variable is the condition factor assigned to assets in the valuation of furniture, fixtures, and equipment (FF&E). When conducting a site visit, an appraiser observes the condition of each item, and may assign a number to a piece of FF&E (e.g., one (1) for “good” through seven (7) for “scrap,” with various conditions represented between these bounds).16 If the condition of certain pieces of FF&E were not captured during the site visit, a valuation analyst may utilize the observed attributes about the condition of the collection of assets to make inferences about the missing data, i.e., to impute the most likely condition value for the missing data. However, the ability to use statistical techniques to make such inferences is dependent on the analyst’s assumptions related to the likely distribution of the condition factor. For instance, if a sufficiently large amount of the collected data which is from FF&E is heavily weighted towards one end of the condition factor spectrum, any missing pieces may be inferred to exhibit a similar level of quality.
A second example of a discrete random variable is the procedure volume of a healthcare provider, since providers are not likely to provide half of a procedure, the observed volume will be a discrete number, i.e., an integer. The comparison of a specific hospital’s procedure volume to a broader market may be built from statistically inferred performance metrics for that hospital, e.g., utilizing the Poisson distribution to find the probability of a certain number of procedures within a given month, to determine how to better schedule staff or medical equipment to increase productivity and make efficient use of equipment or staff.
One example of a continuous random variable a healthcare valuation analyst will likely encounter is practice expense. It is reasonable to expect expenses to range from zero (i.e., no expenses) to some measurably, higher amount (e.g., a number with eight decimal places, in millions of dollars, or both). A valuation analyst may use practice expenses for a statistical comparison of a specific practice’s expenses with other comparable entities in an effort to determine profitability. This comparison is predicated on an assumption regarding the distribution of possible practice expenses, e.g., an analyst might assume a lognormal distribution.
Additionally, a valuation analyst may use the assumption of a normal distribution to investigate how a specific entity’s net profit compares to the net profits of a sample of market comparables, or the lognormal distribution to investigate volatility in “dollars per unit productivity” across states with differing income levels. In either case, understanding whether the data is discrete or continuous, and understanding the implications of the assumed distribution, will lead to more robust inferences required for competent valuation analysis.
Often, the size of a population renders it impossible or impractical to study in its entirety. Under these circumstances, sampling is the preferred method to develop a representation and draw inferences regarding a population.17 Recall, from Part Two of this series, entitled, “Descriptive Statistics,”18 that a relationship exists between sample size, the Central Limit Theorem, and the Law of Large Numbers. Given a random sample from a population, as the number of observations within the sample increases, the calculated sample mean approaches the value of the population mean, and as samples are repeated, the distribution of the sample means approximates a normal distribution.19 Conversely, small sample sizes may vary greatly from the true population mean and distribution.
Looking ahead in Part Five of this series, entitled, “Regression Analyses,” an additional application of these properties will be used to describe how a valuation analyst may estimate any unknown parameter (through regression analysis) and determine if this estimate meets the requirements necessary to convey meaningful information about a sample.
As noted in Part Two of this series, entitled, “Descriptive Statistics,”20 certain unwanted properties (e.g., non-representative samples and sample bias) may plague a sample while hindering a valuation analyst’s ability to accurately describe a population. In light of these properties, a valuation analyst should be familiar with the role that methodologies associated with the collection of sample data from a population play in judging the data’s quality. Two important fundamental assumptions of sample data are: (1) independence of the data; and, (2) whether the data are identically distributed observations.21 Independence within data requires that one observation cannot influence another, i.e., the occurrence of an observation does not influence the measurement of another observation.22 Identically distributed data arise from adherence to an exact collection methodology when creating a sample, ensuring that the measurement and scope of the information are shared by all observations.23 It may be important for a valuation analyst to make assumptions about the collected data in order to sufficiently generalize the data to gather meaningful interpretations of the population through other statistical tests, such as data collected by industry surveys.
Many such statistical tests are built upon an assumption of normality, or that the data is normally distributed, which if wrong, may lead to “inaccurate inferential statements.”24 A valuation analyst may wish to test whether a sample is approximately normally distributed, suggesting that the sample’s properties are closely related to the properties associated with a normal distribution. First, there are simple “quick checks” that may automatically disqualify a sample, e.g., the sample must be continuous, and the sample may take on all sample values in the support for a normal distribution (e.g., a variable restricted to non-negativity cannot be normal).
Next, the sample may be subject to tests of normality that are more rigorous. An example of a normality test is the Chi-Squared Goodness of Fit Test that uses maximum likelihood tests (a method that identifies the necessary parameter values required to make test true) to investigate the probability that the sample is normally distributed.25 A second test for normality is the Jarque-Bera Test that estimates the sample mean and variance to compare the sample to a normal distribution through hypothesis testing.26 A third test for normality is the Shapiro-Wilk Test that utilizes order statistics (a sub-sample of the data through independent observations, e.g., the collection of the smallest values) and their means and variance to compare the sample to a normal distribution through hypothesis testing.27
Occasionally upon examination of a sample, certain members of the sample may appear to differ significantly from the remainder of the data, potentially arising from data collection errors or unexpected characteristics of the data. These points, referred to as outliers, warrant special attention to obviate any deleterious effects that may arise from their inclusion in the analysis.28 A valuation analyst who suspects that there may be outliers in the data may opt to use a statistical test (e.g., Grubb’s test, discussed below)29 to assist in the identification of outliers.30 This process is somewhat subjective, and relies on the ability of a valuation analyst to appropriately identify outliers within the data and develop forecasts that adjust for the outlier effects.31 Depending on the nature of the outlier, it may or may not be wise to exclude the data point. Outliers arising from data collection errors should always be excluded, but other outliers may highlight important factors affecting the random variable of interest.
In valuation analysis, since historical data may be limited to the two or three years prior to the valuation date, the valuation analyst should carefully consider whether to remove an outlier from a data set.32 With such small sample sizes, rigorously identifying outliers may be difficult, and the analyst should seek, as a rule, to conserve as much relevant information as possible.33 A statistical test by Dr. Frank E. Grubb recommends certain criteria for identifying outliers through hypothesis testing based upon sample means and standard deviations, and where the process is adjusted to reflect the number of suspected outliers by the valuation analyst.34 An alternative heuristic for identifying an outlier uses Chebyshev’s inequality, proven by Russian mathematician Pafnuty Chebyshev.35 The inequality can be applied to any sample, regardless of distribution, which states that the probability of the distance between a data point and the sample mean is greater than a certain standard deviation (one, two, … standard deviations) is at most one over the number of standard deviations squared.36 A valuation analyst can conclude that, for any sample, regardless of distribution, the probability of a data point being three standard deviations away from the mean is, at most, one over the square of three, or approximately 11 percent, and may be an outlier.
Understanding the structure of data, its distribution, and its effects of sampling is important for valuation analysts in making inferences about data that statistics alone cannot answer. The fifth installment of this six-part series will shift from an overview of general statistics to a focus on regression analysis, its uses in the valuation analysis, and potential pitfalls or mistakes in its interpretation.