Odds to Quicken Reporting Already Delayed Cases: Acquired Immune Deficiency Syndrome Incidences are Illustrated

Delayed reporting in a medical system complicates efforts to estimate the number of cases that occurred in a time period. A case in point is the government’s difficulty to estimate the number of Acquired Immune Deficiency Syndrome (AIDS) cases. The reporting delays are not intentional but are ongoing due to changing Federal regulations or medical definitions of the case like AIDS. To simplify the complications, this article approaches by modifying the geometric distribution. To be specific, let 0<1-θ<1 is a chance for a case (like AIDS) to be reported in the same time period of its occurrence to a (Federal or other) agency. If the reporting is missed in its occurrence time period, the case gets reported in a next or later time period. Let Y be the number of time periods skipped until its reporting. In this process, the reporting probability in a current period is chained with that of past period with an “odds of quickening” to report. The implication and significance of “quickening odds” are investigated and explained in this article, using the AIDS data with delayed reporting.


INTRODUCTION
Who might have guessed in year 1981 that more than 45 million people would have died and another estimated 75 million people would have suffered worldwide with "Acquired Immune Deficiency Syndrome (AIDS)"? What is the genesis of AIDS? On June 5, 1981, the Center for Disease Control (CDC) first detailed the biopsy of "5 young men with a rare pneumonia". After failed immune system, their vital CD4 + cells were invaded by viruses, bacteria, fungi and parasites. CDC (1985) and Chamberland et al. (1985) for details. The virus was detected by a Polymerase Chain Reaction (PCR). Later in June, 1982the CDC (1985 announced that the world faced "a new, deadly sexually transmitted disease". A month later, the CDC coined the name: "Acquired Immune Deficiency Syndrome (AIDS)" to refer this illness.
The World Health Organization (WHO) estimated that about 33.4 million people were suffering with AIDS and two million people (including 330,000 children) died in 2009 alone. The AIDS has become a major deadly human illness in many parts of the world. A scary fact is that AIDS is spreading. The AIDS diagnosis is based on clinical symptoms which appear to vary with lag effects since an initial infection. The medical community periodically debates and perfects the definition of clinical symptoms and recommends that a person with the virus should be declared as an AIDS case only after the illness progressed enough to pass through benchmarks determined by the CDC (1985). The Federal government regulated that the laboratory evidence of the virus should no more be mandated to report an AIDS case. This cautionary federally imposed tedious approach causes an unavoidable reporting delay of AIDS cases. The CDC (1985) mentions that while 42,670 AIDS cases were diagnosed by physicians as of 31 March 1987 but only 33,350 of them were actually reported. Some AIDS cases are never reported while others are reported in any of the subsequent sixteen quarters (Hay and Wolak (1994) for data and their details).
The reporting delays result in practical difficulties to estimate the actual number of AIDS cases. Harris (1990) noticed that the reporting delays of even 0.6 months shifted the frequency trend of AIDS cases to the right and consequently, the estimated AIDS cases fell far Science Publications IJRN below the actual number. DeGruttola et al. (1992) reported that the number pediatric AIDS cases in New York City were under-estimated because of reporting delays which changed over the chronological time in a non-stationary manner. Lindsey (1996) considered bivariate intensity functions of non-stationary Poisson processes and a non-parametric methodology to undo the under-estimation of AIDS cases due to reporting delays. Pagano et al. (1994) developed a regression methodology to make an adjustment to an under-estimate of the completely unobservable actual number of AIDS cases because of reporting delays. Bacchetti (1996) identified that the 1993 re-definition of AIDS caused reporting delays and also disrupted the interpretations of the death trend of AIDS cases. Gebhardt et al. (1998) noticed based on a Bayesian generalized linear model on reverse-time hazards that many industrial countries including Switzerland and Spain incurred significant deaths because of AIDS but it was understood only much later because of reporting delays. Tabnak et al. (2000) developed a change-point model to correct a biased estimate of AIDS cases because of reporting delays. Cui (1999) developed a nonparametric method to analyze Australian left-censored and right-truncated AIDS data and estimated the impact of the reporting delays.
The reporting delays occur in other topics also. Lawless (1994) mentioned that reporting delays occurred in insurance claims and provided a method to model the random temporal fluctuations to compensate for the under-reported claims. MacArthur et al. (1985) traced the source of under-reporting of tumor and other cancers and found that the hospitals rather than the patients cause reporting delays. Clegg et al. (2002) pointed out that reporting delays occur in cancer reporting medical after informing that the reporting delays actually confused the health officials to comprehend the cancer incidence trend as they contained estimation errors with downwardly biased cancer incidence trends and provided an approach with an appropriate methodology to obtain reporting-erroradjusted cancer incidence rate. Midthune et al. (2005) provided a methodology to make adjustments for an accurate cancer incidence rate in general and melanoma cancer in particular in the U.S. Zou et al. (2009) provided a methodology to capture the effect of reporting year on delay modeling of cancer incidence.
All above mentioned reasons motivate the importance and necessity for an additional statistical methodology to estimate number of cases like AIDS with reporting delays. A new methodology is pursued in this article by modifying geometric distribution to suit the reality in reporting medical system. This modified probability pattern is named Oscillating Geometric Odds Distribution (OGOD). Benefits include not only a way to estimate the actual number of AIDS cases in a given time period but also offer a statistical methodology to assess the significance of the estimated "odds of quickening" to improve reporting of an already delayed reporting. This methodology helps health administrators to prepare budgets and policies based on a better estimate of the cases like AIDS. Healthcare policies emerge from facts and perceptions. Fan (2004) outline the society's fears and phobias because of threat from AIDS illness. Understanding the AIDS prevalence using OGOD might help to reduce the psychological, social, economic fears or to combat the health insurance industry's denials to deserving applicants with AIDS symptoms. The reporting delay is not unique to AIDS illness alone and is suspected to exist in other illnesses as well. The contents of this article are versatile enough to explain the consequences of reporting delays in engineering, economics, public health, business or other disciplines as well.
The statistical properties of OGOD are derived in section 2. In section 3, they are illustrated with the data about reporting delays of AIDS cases in Hay and Wolak (1994). The last section 4 contains conclusive thoughts and recommendations.

Oscillating Geometric Odds Distribution
Let 0<1-θ<1 be a probability of reporting a case (like AIDS) in the same period of its occurrence. Delayed reporting in a medical or other system complicates efforts to estimate the actual number of cases that occurred in a time period. Fan (2004) for details. To resolve this difficulty, this article approaches by modifying geometric distribution as follows. The odds of reporting a case in the same period of its occurrence are odds where, y 0,1, 2,..., ;0 The probability of reporting a case in the same period of its occurrence is  (1). What is memory less property? It means the conditional probability of reporting a case in a time period given it has not been reported so far since its occurrence equals its unconditional probability of reporting in its period of occurrence itself. This is translated in probability terminology below in (2). Note from (1) The ongoing delays create a memory in a reporting medical system. Note that odds θ = 0 when the case is reported in the same period of its occurrence and the "odds of quickening" is obsolete. Otherwise, the odds φ is fused into the reporting probability in a chained manner like: with y = 0,1,2,…,∞; 0<θ<1,0<φ<1. Is expression (3) a bona fide probability distribution? The answer is affirmative. Trivially, the expression (3) is non-negative. The sum of their values equals one as it is shown below: where, θ ∂ denotes the derivative with respect to θ. The expression (3) is named "Oscillating Geometric Odds Distribution (OGOD)".
In the absence of "odds of quickening" to report in a system (that is, φ = 0), the OGOD (3) reduces to the geometric distribution in (1) as a particular case. Otherwise, when all cases are reported in the same period, note that Y = 0. The reporting medical system has no lag with a probability Equation 4: The probability for a reporting medical system to be busy with a lag of cases to report is Equation 5: The odds for a case to be reported in a medical system with a lag is Equation 6a,b: , , Signifies an impact of "odds of quickening" on lag. A Taylorization is used to obtain (6a) and it is: where, 0 f ( , )] Interestingly, the mean in an absence of "odds of quickening" to report delayed cases (that is, odds φ = 0 or equivalently, Is an impact of "odds of quickening" on mean. The impact values could be compared over the years to get a clue on how the delayed reporting has improved. This knowledge is useful to health administrators. It is easy to see that Equation 9a-d: The dispersion 2 , φ θ σ of the OGOD is obtained using the relations in (9a through 9d). After algebraic simplifications, it turns out to be Equation 10: In the absence of "odds of quickening" to report (that is, φ = 0), expression (10) Portrays an impact of "odds of quickening" to report on dispersion. Next, the survival function: For the OGOD (3) is derived in terms of the Fdistribution. The table for F-distribution is popularly available. The incomplete beta function in (12) is indeed F-distribution. It is easy to see that Equation 12: 1 y r y r 1 1 r 1 2( r 1),2 (1 ) (r 1) (1 y) dy Hence, the survival function is Equation13: The survival function (13) By substituting r = 0 in (13), it yields the chance for a lag to exist. That is Equation 15: In the absence of "odds of quickening" to report (that is, φ = 0), expression (15) reduces to Equation 16: Science Publications

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So, what is an impact of "odds of quickening" to report on medical system to be "busy"? This is implied in the relation , 0, busy, , where the level of busy is Equation 17: busy, , and it portrays an impact of "odds of quickening" to report on busy. Now, a discussion on how much a memory is created in a reporting medical system because of the lag. Recall that the geometric distribution (1) is known to possess a memory less property as shown in (2). From OGOD (3), note that: where a reporting medical-system's memory is In the absence of "odds of quickening" to report (that is, φ = 0 or odds φ = 0), expression (18) reduces to baseline value one, confirming the memory less property as stated in (2) for geometric distribution (1). Hence, a theorem is stated.

Theorem 1:
The chance mechanism which is governed by the oscillating geometric odds distribution (3) has a finite memory m φ,θ in (18). Now, the Maximum Likelihood Estimate (MLE) of the parameters φ and θ are obtained. A reason for choosing the MLE is that it is invariant. The MLE helps to perform a data analysis. Consider a random sample y 1 , y 2 , y 3 , …., y n from OGOD (3) The MLE mlê And: In the absence of "odds of quickening" to report, note that   of the geometric distribution as a particular case in the absence of "odds of quickening" to report.
A health administrator is interested in a "ratio" where m and T are respectively a specified threshold level and a total number of reported cases in a year. The ratio is the odds of the number of reported versus unreported number cases in a year. The epidemiologists are fond of this kind of odds. To make a probability assessment about the odds, the expected and variance values of R are needed. To find them, the formulas Equation 23a,b: are used (Stuart and Ord (1994) for details), where E j and V j denote the mean and variance respectively of j = W or V. That is Equation 24 and 25: , , And: Under the null hypothesis, Which follows a non-central chi-squared distribution with one degrees of freedom (df) and the non-centrality parameter The non-central chi squared distribution with one df and non-centrality parameter δ approximately follows

Reporting Aids Cases for Illustration
The results of the section 2 are illustrated using delayed reporting of AIDS cases in Hay and Wolak (1994)  believe that "odds of quickening" to report indeed existed. The medical administrators had been quite consciously trying to quicken the reporting of already delayed cases. With the total, T number of AIDS cases, the power of accepting H 1 : φ * = 0.5 using the given data is higher in a quarter during 1982 through 1990 ( Table 2). The Fig. 1 suggests that the chance for reporting an AIDS case at a later quarter had been increasing over the years though the total number of AIDS cases grew according to Fig. 2. As can be seen in Fig. 3, the chance for majority of the AIDS cases get reported in the same quarter of its occurrence oscillated over the years but it became phenomenal in the later time period. The odds of reporting an AIDS case in the same quarter of its occurrence increased more in the beginning than in later period during years 1982-1990 due to "odds of quickening" to report, according to the Fig. 4. The Fig. 5 indicates that the impact of "odds of quickening" to report on busy probability to report an AIDS case in the absence of "odds of quickening". Notice in Fig. 3 that those chances have been oscillating over the years during 1982 through 1990. The probability for an existence of "quickening" attitude to report a case is mlê φ if the case was reported in the same quarter of its occurrence. The chance for reporting an AIDS case in the same quarter of its occurrence is mlê mle, 1 φ − θ in the presence of "quickening odds" to report. The impact, a φ,θ of "quickening odds" is displayed in Table 2. Their values suggest that the "quickening odds" changed over the years during [1982][1983][1984][1985][1986][1987][1988][1989][1990]. The impact, a mean,φ,θ of "quickening odds" on mean has been reducing over the years during 1982 through 1990 (Table 2). Likewise, the dispersion, , a φ θ captures the impact of "quickening odds" to report on dispersion ( Table 2). According to their values, the impact has been increasing over the years during 1982 through 1990.  1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9997 1987, Q1 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9998 1987, Q2 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1987, Q3 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1987, Q4 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1988, Q1 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1988, Q2 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1988, Q3 1 1.000 1.000 1.000 1.000 1.000 1.000 1988, Q4 1 1.000 1.000 1.000 1.000 1.000 1989, Q1 1 1.000 1.000 1.000 1.000 1989, Q2 1 1.000 1.000 1.000 1989, Q3 1 1.000 1.000 1989, Q4 1 1.000 1990, Q1 1 The busy, , a φ θ captures the impact of "quickening odds" to report on system's busy level ( Table 2). According to their values, the impact has been increasing from 1.84 to 1.99 over the years during 1982 through 1990. With notations Y and T-Y denoting respectively the number of reported, non-reported cases with m = 0.5 indicating the reported cases is 50% more than the non-reported cases in the same quarter of its occurrence, the chance Y Pr[ R m] T Y = > − is displayed in Table 2. Interestingly, their values suggest they had been more than 55% but oscillated over the years during 1982 through 1990. The p-value in Table 2 indicates the chance for rejecting the true null hypothesis H°: φ = 0 and it confirms the existence of a significant "odds of quickening" to report. The power in Table 2 implies the chance of accepting a true alternative hypothesis H 1 : φ* = 0.5 with the level of significance α = 0.05. Its oscillation hints the existence of varying administrative efforts to quickly report already delayed cases in the reporting medical system. It is worth examining how the reporting system's memory had been.
In a system with the absence of "quickening odds" to report an AIDS case, the system is recognized to follow a geometric probability distribution with no memory and the system's memory level m φ = 0,θ is just one. With an existence of "quickening odds" to report already delayed AIDS cases, the reporting system possesses a finite memory. The Table 3 displays its memory level for the period 1982 through 1990.

CONCLUSION
This methodology is applicable to any delayed reporting system in other disciplines. In engineering, sports, e-marketing, healthcare insurance, stockmarketing, economic outcomes, cyber-crimes reporting with delay and the existence of efforts to quicken the reporting of already delayed cases are common. The contents of this article would help to discover non trivial impacts in those disciplines. Of course, many covariates are likely to influence the level of quickening efforts. A regression methodology is necessary to address the relevance of the covariates in a given investigation and it