Impact of Raising Copayment and/or Reducing Reimbursement Benefits on Healthcare according to "Disequilibrium Bivariate Distribution"

Healthcare is a significant part of every country’s economy. In 2011, the healthcare industry consumed about 9.3 percent of the US gross domestic product (GDP), 10 percent of the Canada’s GDP, and 11 percent of the France’s as well as in Germany’s GDP. Based on a random sample of 337 sex offenders, who received treatment in the Ontario region, Canada, during the years 1993 to 1998, Mailloux et al. [1] reported an increase of risk to die due to over-prescription. Ding et al. [2] stated that China’s health reforms in 2009 resulted in significant changes in the prescribing patterns. He [3] warned that over-prescription increased the threats of malpractice/litigation lawsuits. Sacarny et al. [4] announced that over-prescribing threatened Medicare benefits. Based on a study examining 230,800 prescriptions written during the period 2007 and 2009 in 784 community hospitals over 28 cities across China, Parveen et al. [5] mentioned that over-prescribing caused incorrect, ineffective and insecure treatment, exacerbation or continuation of illness, and increased the healthcare cost. Li et al. [6] proved the existence of a substantial overprescribing against the recommendation by the World Health Organization. The German’s healthcare reform of the year 1997 resulted in a significant reduction of the number of visits by the patients [7,8]. They wondered whether the extra reimbursement possibility from the insurance coverage induce the physicians to write more prescriptions? In December 2011, the Centers for Medicare and Medicaid Services announced that about 30% of the healthcare spending is waste. According to an administrator, the first cause for the waste was identified to be over-prescription/overtreatment of the patients. To minimize the waste, would raising the copayment and/or reducing the reimbursement benefits help? The tendencies for overprescription and frequent visit to physician is the central theme of this article.


Motivation
Healthcare is a significant part of every country's economy. In 2011, the healthcare industry consumed about 9.3 percent of the US gross domestic product (GDP), 10 percent of the Canada's GDP, and 11 percent of the France's as well as in Germany's GDP. Based on a random sample of 337 sex offenders, who received treatment in the Ontario region, Canada, during the years 1993 to 1998, Mailloux et al. [1] reported an increase of risk to die due to over-prescription. Ding et al. [2] stated that China's health reforms in 2009 resulted in significant changes in the prescribing patterns. He [3] warned that over-prescription increased the threats of malpractice/litigation lawsuits. Sacarny et al. [4] announced that over-prescribing threatened Medicare benefits. Based on a study examining 230,800 prescriptions written during the period 2007 and 2009 in 784 community hospitals over 28 cities across China, Parveen et al. [5] mentioned that over-prescribing caused incorrect, ineffective and insecure treatment, exacerbation or continuation of illness, and increased the healthcare cost. Li et al. [6] proved the existence of a substantial overprescribing against the recommendation by the World Health Organization. The German's healthcare reform of the year 1997 resulted in a significant reduction of the number of visits by the patients [7,8]. They wondered whether the extra reimbursement possibility from the insurance coverage induce the physicians to write more prescriptions? In December 2011, the Centers for Medicare and Medicaid Services announced that about 30% of the healthcare spending is waste. According to an administrator, the first cause for the waste was identified to be over-prescription/overtreatment of the patients. To minimize the waste, would raising the copayment and/or reducing the reimbursement benefits help? The tendencies for overprescription and frequent visit to physician is the central theme of this article.
Paradoxically, discussions concentrate on minimizing healthcare cost rather than building up healthy population and/or quality healthcare. An ancient recommendation is that an ounce of prevention is worth tons of treatments. This article hypothetically probes the impacts of raising co-payment and/or reducing reimbursement benefits in the healthcare operations. Logically, would rising of the copayment restrict the patient's visits to physician less often than needed. Let 0 1 ξ ≤ ≤ be the impact level of raising copayment on patient's visitation rate, 0 λ > to the physician. Would reducing reimbursement benefits slow down physician's prescription rate. Let 0 the raising copayment onpatient's visit and the reduction of reimbursement benefits with respect to the physician's prescription. For this purpose, an appropriate new model is introduced with its properties. The model is named a disequilibrium bivariate Poisson distribution (DBPD). A data analytic methodology is devised, based on DBPD. Healthcare managers of a hospital/clinic operation and/or health insurance organization could emulate the illustration of the methodology in their pursuits.
The article is organized as follows to understand the reality for formulating strategic policies to adapt for better healthcare management. In Section 2, we derive the disequilibrium bivariate Poisson distribution and its statistical properties. We formulate a hypothesis testing procedure to check whether the impact levels. In Section3, the concepts and all derived algebraic expressions are illustrated and interpreted, using the Australian Health Study data for [1977][1978]. Finally, in Section 4, a few conclusive comments and suggestions for future research direction to attain an efficient healthcare management are stated.

Derivation of "Disequilibrium Bivariate Poisson Distribution" and its properties
First, let us recognize that the healthcare practices and management do have chance oriented mechanism to deal with. The patient's visitation and physicians prescription rates are examples and they play a vital role to improve healthcare operations. To be specific, let 0 λ > and 0 θ > denote the patient's visitation rate and the physician's prescription rate respectively. Ifthe copayment is raised,how would it impact the visitation rate? Let it's impact level be 0 1.

ξ ≤ ≤
Likewise, if a reduction of reimbursement benefits is implemented, what might be it's impact level on the prescription rate? Let its impact be 0 1.

φ ≤ ≤
The healthcare's chance mechanism is a mix of three scenarios.
First, the scenario 1 is one in which the number,Y of precriptions exceeds the number, X of visits. Both X and Y are random variables. Assume that the odds for the wavering patient (due to raised copayment) to make a visit is , ξλ θ because the prescription rate deflates it ( Figure 1). When 0, ξ → the odds of making a visit is zero, and the scenario is void. The joint probability mass for visits and prescriptions is proportional to with an observable space, The notation , if Secondly, the scenario 2 is one in which the number, of patient's visits exceeds the number, of the precriptions. The odds for the hesitating physician (due to reduction in reimbursement benefits) to write a prescription is because the patient's visitation rate deflates it (Figure 2). When the odds of writing a prescription by a physician is zero, and the scenario-2 is void. The joint probability mass for x visits and y prescriptions is proportional in observable space, 0,1, 2,..., 1; 1, 2,.., .
x y y = − = ∞ The Secondly, the scenario 2 is one in which the number, X of patient's visits exceeds the number, Y of the precriptions.
The odds for the hesitating physician (due to reduction in reimbursement benefits) to write a prescription , φθ λ is because the patient's visitation rate deflates it ( Figure 2). When 0, φ → the odds of writing a prescription by a physician is zero, and the scenario-2 is void. The joint probability mass for x visits and y Thirdly, the scenario 3 is one in which the number, X of patient's visits match the number, Y of physician's precriptions.
Raising copayment and reducing reimbursement benefits have no impact. Thejoint odds of patient's visiting and physician's writing prescription λθ is because of the absences of patient's wavering and the physician's hesitaion ( Figure 3). The joint probability mass for x visits y and prescriptions is proportional  Because the three scenarios are mutually exclusive and exhaustive, the joint probability mass function for the entire healthcare chance mechansim is Where, the normalizer is The bivariate distribution in (1) and (2) is named a "disequilibrium bivariate Poisson distribution (DBPD)". The DBPD is a new addition to the book of several bivariate distributions, and it is helpful to analyze and interpret healthcare data as done in this article. The random variables and are dependent. See Shanmugam & Chattamvelli [9] for various ways of checking the independence among random variables. Shanmugam [10] presented a history of Poisson model in the healthcare data analysis. Shanmugam [11] provided a list of bivariate Poisson models. Shanmugam [12] demonstrated on how to extract data informatics to address the fear to report rapes. Shanmugam [13] constructed methodologies on how the queuing concepts helped to effectively manage the hospitals when the patients are impatient. Shanmugam [14] probed the non-adherence to prescribed medicines by patients with a bivariate model and its information nucleus. Shanmugam [15] derived a "bivariate model" for infrastructures among women with operative, natural, and no menopauses. Shanmugam [16] derived a bivariate model to identify "honesty" versus "cheating" in economic surveys to illustrate the existence of xenophobia. One wonders about the disequilibrium level, π between the impact levels ξ of raising copayment and φ of reducing π π π < = > dependingrespectively on an existence of tilt to more patient's wavering to visit,perfectequilibrium, or to more physician's hesitation to write prescription. Note that the proportion not visiting a physician and nor receiving a prescription is When the impact level of raising copayment vanishes (that is, when the impact level of reducing reimbursement benefits vanishes (that is, The jump rate (that is, (3) and (4), which is free φ of the impact level due to reducing reimbursement benefits to the physician ( Figure 4). In a situation with extremely large prescription rate (that is, θ → ∞ ), the jump rate converges to inflated visits Likewise, the proportion of physicians do not write a prescription for patients visiting and not visiting is The proportion of physicians writing one prescription (under reduced reimbursement benefits) is The jump rate (which is  (5) and (6), which is free of impact level, ξ of raising copayment. In a situation with extremely large visitation rate (that is, λ → ∞ ), the jump rate for the prescription by the physician converges to inflated prescriptions Now, let us look at the marginal stochastic trend (that is, ) of the patient's visits. Note that The marginal probability mass function (7) of the number of visits under the raised copayment is a size biased Poisson distribution. The size bias is When the sampling process does not represent the intended but a different population, it is recognized as length biased. Shanmugam [17] demonstrated the effectiveness of the sample size in length biased data. Shanmugam [18] constructed a goodness-of-fit test for length-biased data with discussions on its prevalence, sensitivity, and specificity. Shanmugam [19] derived a significance testing procedure for size-biased income data. Shanmugam & Singh [20] derived an urgency biased beta distribution (which is a length-biased version) with application in drinking water data analysis. The marginal expected number of visits to physician is When the prescription rate becomes infinitely large (that is, θ → ∞ ), the size bias attains the baseline value one and the stochastic pattern of the visits by the patients is In the absence of size-bias (that is, ( ) 0 under a convergence θ → ∞ ), the mean in (8) approaches λ and Var X E X = It means the mean number of visits in (8) increased by an amount also wonders whether patients would make lesser visits. For this purpose, we define lesser visits using the mode of the size-biased marginal distribution in (7). In a size-biased healthcare chance mechanism, the mode of its stochastic pattern occurs at the greatest least integer of its parameter. The cumulative Poisson probability is cumulative chi-squared probability. That is, Theorem 1: A patient hesitates to visit physician, if s/he makes lesser than the most probable number of the Poisson distribution with a probability 2 2 Pr (  2 ), using the relationship between the cumulative Poisson distribution and the chi-squared probability, where [] and df denote the greatest least integer and the degrees of freedom respectively.
The intensity rate of the patient's visit to the physician in this size-biased marginal distribution (7) is then Using the link between cumulative Poisson and cumulative chisquared distribution, the intensity rate is quantified as In this situation, for a given threshold 0 τ > number of visits to a physician, the expected excessive visit to physician ( x EEVisit ) by a patient in (11) is In particular with the threshold 2 τ = number of visits to a physician in (11), the expected excessive visit to physician ( x EEVisit ) by a patient is The tail value at risk (13) is a financial term to refer the gain or loss in trading stocks over a period of time. It is a risk for making more visits beyond the threshold in the size-biased chance healthcare mechanism is The conditional probability mass function (14) of the physician's prescription is then Where the indicator function (15) to project the number of prescriptions to be written by a physician for a given number X x = of visits made by a patient is with the squared sampling error  (17) can be addressed. Note that The marginal probability mass function (17) of the number of prescriptions under the reduced reimbursement benefits is also a size biased Poisson distribution. The size bias is ( ) The marginal When the visitation rate by the patients becomes infinitely large (that is, λ → ∞ ), the size bias becomes the baseline value one and the stochastic pattern of the prescriptions by the In the absence of size-bias (that is, ( ) 0 w y = under a convergence λ → ∞ ), the mean in (18)θ approaches and the variance in (19) is It means the mean number of prescriptions in (18) increased by an amount Would physicians write lesser prescriptions? For this purpose, we define lesser prescriptions using the mode of the size-biased marginal distribution in (17). In a size-based healthcare chance mechanism, the mode of the pattern occurs at the greatest least integer of its parameter. The cumulative Poisson probability is linked to chi-squared cumulative probability for easy computations. Hence, Theorem 2: A physician hesitates to prescribe, if s/he makes lesser than the most probable number When the threshold 2 ψ = in (21) denoting the number of prescriptions by a physician, the expected excessive prescription Pr Pr( 2 ( )) The tail value at risk (23) is for making more prescriptions beyond the threshold in the size-biased chance healthcare mechanism is However, the conditional probability mass function (24) of the patient's visitation is then Where, the indicator functions Noticing a patient hesitates to visit physician with a probability 2 2 Pr (  2 ), connecting the four quadrants and DeMorgan's probability laws in the bivariate probability theory [9]. Is there reciprocity? Would the patient react ( ), patient ℜ after noticing the physician hesitates to write prescriptions due to reduction of reimbursements? If so, how probable it is? The chance (28) for the patient's reaction is where, Using (31), their covariance is (8) and (18) The correlation (32) depends on the impact levels of increasing copayment (that is, ξ ) and reducing reimbursement benefits (that is, φ ). When both impact levels approach zero, their correlation, xy ρ becomes zero indicating X that Y and are independent Poisson random variables. Now, we need to examine whether a sample estimateˆx y ρ of the correlation coefficient is significant at a specified confidence level 1 That is, the null hypothesis With the mle of the parameters, note that the projected regression (33) of the number of written prescriptions by the physician [22]. We now turn to the task of estimating the parameters , , λ θ ξ andφ of the disequilibrium bivariate Poisson distribution in (1). For this purpose, suppose a bivariate random sample We now develop score functions based testing procedure [21] to check the validity of the hypothesis about the impact level, 0 1 ξ ≤ ≤ of raising the copayment and/or the impact level

Case 2:
The null hypothesis to be tested is

Case 3:
The null hypothesis to be tested is In the next section, all the derived expressions are illustrated, using the Australian Health Survey data of 1977-1978.

Illustration with Australian Health Survey data analysis
In this section, the Cameron's [23] data of 5,194 Table 1. These data are analyzed and interpreted using the analytic expressions in Section 2. Note that the sample size for scenario-1 is  . According to (15) and (16) and it is 0.978. The power curve is sketched in Figure 5 for different values of a ξ in the horizontal axis. and it is 0.951. The power curve is sketched in Figure 6 for different values of a φ in the horizontal axis.

Comments and Conclusions
The "disequilibrium bivariate Poisson distribution" model of this article constructs probabilistic interpretation of the data evidence about the impact of hypothetical raising copayment and/or reducing reimbursement benefits in healthcare system. The significance of the estimated correlation coefficient between the number of visits made by patients and the number of prescriptions written by the physicians under this hypothetical scenario is also assessed. The chance for the patient's reaction to visit to the physician is captured, estimated and interpreted. It is worthwhile to extend this breakthrough healthcare managerial approach to discover reasons and circumstances in which the physicians lesser-prescribing tendency and/or the patients lesser visitation tendency might exist.
The healthcare managers would benefit a lot by collecting pertinent data and scrutinizing evidence data using the methodology in this article for better quality healthcare system. For the discovery to become reality, data on related covariates including the cost details, tax allowances for the prescribing physicians and the visiting patients need to be collected. The healthcare professionals ought to pay extra attention to collect such data. The analysts ought to build a multivariate regression methodology to make projections of when and how many lesser prescriptions and/or lesser visitations are possible. A discovery of reasons for such disequilibrium scenarios (quite different from an ideal situation in which one prescription per single visit of the patient occurs) is a necessity and it ought to be the future goal in this 21st century of intensive efforts to reform the healthcare practice towards cost-effectiveness and efficiencies. This article is a seminal step upward to attain such goal.

Funding
This study was not funded by anyone. The contents are the sole opinion of the author.