Health Value Added of Healthcare Entities

Abstract The paper deals with an evaluation of the quality of services provided by healthcare organizations. First, an index representing a patient’s health condition is described, then its changes before and after being treated by a given entity are employed as a criterion to assess the operations of this entity. The index of a patient’s health condition is based on the theory of survival analysis, while a model of random effects is used to determine the quality of services based on health value added.


Introduction
Quantitative research concerning health, healthcare and general medical issues have become more frequent recently. The current literature provides more and more evidence of such studies. A classic example of a book that presents all seminal research trends in this area is that by Shoukri and Cihon (1998). The authors present comprehensively statistical models applied by researchers of health in a broad understanding of this term.
One part of healthcare research deals with evaluating performance of healthcare entities, exemplified by studies using mostly logistic regression models. Logistic regression is used in statistics when a variable under study takes two values. In the case of healthcare entities, the explained variable is equal to 1 or 0 depending on whether the patient was restored to health or not. Logistic regression allows to calculate the probability of the patient's recovery. This type of research is provided by Tay (2002) and Normand and Shahian (2007). Generally, the most popular statistical models employed to assess the performance quality of healthcare entities are discrete models, e.g. Geweke, Gowrisankaran and Town (2003), mostly due to the type of discrete data available for research on hospitals.
The paper aims to put forward a method of evaluating the performance quality of healthcare entities, i.e. a model that will determine the Health Value Added (HVA) based on the survival analysis theory and the random effects model. Health value added has been researched and reported by, e.g. Friedman, Kokia and Shemer (2003) who characterize it as a quantity representing a complex model of healthcare management that aims to maximize the quality of medical services. The authors claim that when measuring hospital performance one should consider not only the final outcomes, but also the process that contributed to those effects. It is much easier to treat a patient who is hospitalized at an early stage of illness than when the illness is more advanced. Therefore, it is necessary to find out how much has the hospital has contributed to the patient's recovery, and to evaluate the hospital's performance against this background.
The idea to employ the random effects model is based on the concept of computing the patient's chance to survive at least one more year. Such a quantity can be computed before and after healthcare treatment. The difference between the two values represents the contribution of the healthcare entity to the improvement of the patient's health condition and may be used as a tool to evaluate the performance quality of the healthcare organization. When defining a patient's health condition, one can employ a survival analysis which is a well-known mathematical and statistical theory allowing to determine the duration of a patient's life.

Survival analysis
Mathematical survival theory aims at estimating the distribution of the survival function based on the characteristics of the population under study. The differences between the investigated individuals result from specific factors such as age, sex, smoking habits, etc. Certainly, other variables should also include some medical tests. Their inclusion should be preceded by consulting specialists, i.e. physicians, and also by testing for the statistical significance of each indicator. These factors, denoted by 1 , , i i k z z for each individual i ∈ {1, 2, …, n}, represent the explanatory variables in a given model.
One may put forward several dozen different measurable medical factors that should be considered when assessing a medical condition. There are many more such characteristics and it is not possible to include all of them whilst estimating a survival Health value added of healthcare entities 53 model. One example of the relevant factors is information about the patient's previous accidents before being admitted to the hospital, that could significantly shorten his or her lifetime, irrespective of the actual medical conditions.
Assuming that T is nonnegative random variable, i.e. P(T ≥ 0) = 1, which represents the duration of life, F and f denote the cumulative distribution function and the probability density function, respectively, of random variable T, one may define the survival function as: where tdenotes the left-handed limit of the cumulative distribution function F at point t. S(t) should be interpreted as a function indicating that an individual's lifespan will be at least t years.
The survival function is related to the function because density function f is continuous. This is the second important function in survival analysis -the hazard (intensity) function. One can demonstrate that and as a consequence obtain S(0) = 1 by assumption. Cox (1972) introduced the Cox proportional hazard modelin the form of and denoting 0 The form of function h(t) does not matter when comparing the life durations of two individuals, because for any i 1 , i 2 ∈{1, 2, …, n} and hence the model is sensitive to the variations of specific attributes 1 , , representing the chances of individual i to survive at least one year given that i had already survived 1 , , i i k z z months. Applying standard formulas for conditional probability, one obtains the following: This paper applies the term health condition as a quantity expressed by formula (10) computed at two different times, i.e. when the individual enters and leaves hospital. The quantity can represent the health condition of i-th individual given health factors 1 , , i i k z z before and after the medical treatment, e.g. at the time of entering hospital and at the time of leaving hospital. Hence by applying the random effects model, one can measure the contribution of the entity to increase the probability of the individual to survive at least one year, and consequently to determine the entity's health value added that will allow to assess the quality of the entity's services.

Random effects model
Before giving a more detailed description of the results the author introduced some notation and formulas. The model's denotations: x ij -health condition of i-th patient at time of entering j-th hospital, y ij -health condition of i-th patient at time of leaving j-th hospital, n j -the number of patients at j-th hospital, m -the number of hospitals j ∈ {1, …, m}, n -the number of all the patients, i.e. n = n 1 + … + n m , x -arithmetic average health condition of all the patients entering hospital, y -arithmetic average health condition of all the patients leaving hospital, j x , j y -arithmetic average score for the health condition when entering and leaving, respectively, j-th hospital.
The proposed model can be applied for the assessment of the service quality of m homogenous entities of healthcare system, e.g. hospitals, hospital departments, etc. The homogeneity of services provided by healthcare entities is required for the sake of comparability. The author assumes that n j out of n patients are served by j-th entity and their health condition before the treatment is defined by quantity ij ij x  = (input health condition). After the treatment, a new value y ij = π ij is computed based on new attributes 1 , , i i k z z that now defines the patient's health condition. As a result, one obtains unbalanced panel data since the number n j of observations for individual hospitals can vary. When modelling an inhomogeneous population, one has to introduce inhomogeneity into the model. As regards the data under study, there may be various relations between the output and input variables for respective entities. The econometric literature refers to this model as the unbalanced one-way error component model with random effects, e.g. Baltagi (2005). The random effects model is also known as a variance components model (VC), cf. Maddala (2001). Kapteyn (1982a, 1982b) where e ij is a random variable following a normal distribution N(0, s 2 ), whereas ξ j follows N(0, s 2 I ). In addition, it is assumed that random components from different entities and different patients are uncorrelated and that individual random term ξ j is uncorrelated with random term e ij , i.e. ( , ) 0 It follows from the form of the model that ξ j represents a deviation of the average score forjth entity from the average score of the entire population. This average score in Figure 1 is shown as a dotted line, while a solid line illustrates the average score of the entire population. If ξ j is positive then one may argue that jth entity improved its quality with respect to the average score of the entire population, while a negative ξ j indicates the deterioration of its score compared to the average score of the population. Therefore, the value of parameter ξ j is called value added or the operational effectiveness of the entity under evaluation. The value e ij on the other hand represents the deviation of the individual patient's score from the average score of j-th entity where he/she was treated.
The above model is estimated by means of the maximum likelihood method (Aitken and Longford, 1986). The formulas for the estimates are given by Baltagi (2005) and Ejsmont (2009), where also a complete algorithm of estimating variance components σ 2 and 2 I s is provided. The obtained random effects are tested for significance by means of the Breusch-Pagan test (e.g. Baltagi, 2005).  (Skrondal and Rabe-Hesketh, 2008, p. 96). In order to estimate the value of ξ j , one can use the mean squared error theorem (Jakubowski and Sztencel, 2004, p. 135). Since both terms σ 2 and 2 I s are available before the model estimation, thus one can use them as the a priori information. Next, one determines the conditional distribution of random variable ξ j given j y . The mean of j-th hospital has the form: As a result, a comparison of the operational quality of healthcare entities will be based on comparing the mean values from the conditional distribution, i.e.
where b a, ,  respectively are estimates of: correlation ρ, intercept a and slope b.

Conclusion
The presented model enables the evaluation of the operational quality of healthcare entities. The evaluation is relative and shows the position of a given entity against the background of the average performance of other similar organizations. Consequently, the ranking of entities providing similar healthcare services can be obtained. It is worth emphasizing that the model includes both the direct effect of the treatment, i.e. whether the patient recovered or not, and also the entity's contribution to the recovery of the patient. Full recovery from a serious health hazard condition is evaluated differently than the provision of standard treatment when the patient's general state of health is good. By monitoring the patient's current health condition with certain medical characteristics that determine one indicator, i.e. the probability of the patient surviving one year, it is possible to determine by how far the provision of the medical service at a given entity contributed to the change of this index. By averaging those quantities for each entity under evaluation, one can determine health value added that is generated at this entity. The article is theoretical but it can be used to assess health value added in terms of data.