Correlation between the Sample Mean and Sample Variance

This article obtains a general formula to find the correlation coefficient between the sample mean and variance. Several particular results for major non-normal distributions are extracted to help students in classroom, clients during statistical consulting service

Learning from an early start but late end epidemics via an incidence rate restricted bivariate distribution and data analysis

Background: An ideal expectation of public health administrators or field medical workers is to have a late start and quick ending of any epidemic. Instead, when an epidemic starts early but ends late, it is where much can be learned from the incidences. A case in point for discussion in this article is the pattern of 2009 H1N1 epidemic.Methods: With a parameter to portray an existing health environment as a deterrent for an epidemic like H1N1 to outbreak in any location at a week, a bivariate distribution is created and is used to analyze the data for a learning so that it helps to prevent a too long prevailing future epidemic. This new distribution is named Incidence Rate Restricted Bivariate Distribution (IRRBGD). Statistical properties of IRRBGD are derived and illustrated using 2009 H1N1 incidences in all five continental regions (Africa, Asia, Europe, Americas, and Oceanic) across on earth.Results: The Asian continent, compared to other four continental regions, had most vulnerability for H1N1 incidences. The odds for no H1N1 to occur is lowest only in Oceanic among the four continental regions, namely Africa, Europe, Americas, and Oceanic. Since the beginning of the year 2009 with 52 weeks, the week number, Y in which the H1N1 appeared first and the number, X of weeks the H1N1 continued on in a region are consistently highly correlated in all five continental regions.   Conclusions: From the data analyses of 2009 H1N1 incidences, no continental region is risk free with respect another round of H1N1 epidemic in future. The medical community and public healthcare administrators ought to identify the common and region specific unique deterrents of the epidemic like H1N1. The impact of such deterrents to H1N1 is captured in our model and analysis. By increasing the deterrent level, the outbreak of an epidemic like H1N1 could be delayed, according to our model and data information.

Patients’ over-visit phobia versus physician’s over-prescription phobia

Background: Hospital administrators conduct survey of patients to solicit their satisfactions and/or concerns for accreditation or renewal of license. For the first time in the literature, this article defines and illustrates the existence of patient’s over-visit phobia and the physician’s over-prescription phobia. These phobias pave way to formulate policy to increase hospital’s efficiency.Methods: The number, of times a patient visits the physician (with a visitation rate) and the number, of prescriptions written by a physician (with a prescription rate) are assumed to follow Poisson type probability patterns. This article, in a novel manner, untangles intricacies and inter-relations of these two phobias.Results: An analysis of the Australian Health Survey data, using our model and methodology, estimates visit and prescription rate to be and respectively. The chance for patient’s visit phobia and physician’s prescription phobia is respectively 0.33 (with a reluctance level 2.16 to make additional visits) and 0.46 (with an avoidance level 3.17 to prescribe more medicines).Conclusions: A few comments and suggestions are stated to save service time/cost for the sake of more hospital’s efficiency. With a methodology in this article, level of over-visits by the patients and the level of over-prescriptions by the physicians are estimable to reduce the waste of hospital’s resources.

Never, once, and repeated illness: a geometric view for insights and interpretations

Background: Medical/health researchers depend on data evidence for knowledge discovery. At times, data analysis to capture the data evidence is overwhelming and the process becomes too tedious to give up the attempt. A prudent thing to do is to seek out a simpler visual approach to obtain insights. One visual approach is devised in this article to understand what the data are really revealing to either get an insight first or then confirm what is intuitively configured by the medical concepts. This visual approach is geometric concepts based. In specific, triangle is employed in this new and novel approach.  Methods: A successful treatment of any illness is a consequence of knowledge build-up arising from data mining about the never, once, or repeated episode of a disease incidence in a patient. This article investigates and illustrates a novel and pioneering geometric approach, especially based on the properties of triangle, to extract hidden evidence in the data. New probabilistic expressions are derived utilizing trigonometric relations among the corner points of a triangle. The conceptual contents of this article are versatile enough for different medical/health data analysis.Results: For illustration here, the medical binomial data in Hopper et al. (Genetic Epidemiology, 1990) on the occurrence of asthma or hay fever among the four groups: (1) monozygotic females (MZF), (2) monozygotic males (MZM), (3) di-zygotic females (DZF), and (4) di-zygotic males (DZM) are considered and triangularly interpreted. The results indicate that the angle in the vertex representing one episode is the largest compared to the other two angles in the vertices representing never or repeated episode of an illness among a random sample of twins from these four groups with respect to getting asthma or hay fever. This geometric finding implies that the event of never and the event of repeated incidence of the illness have farthest Euclidean distance in probability sense. In other words, the never and repeated incidences are not in close proximity as probable.Conclusions: This geometric view of this article is versatile enough to be useful in other research studies in drug assessment, clinical trial outcomes, business, marketing, finance, economics, engineering and public health whether the data are Poisson or inverse binomial type as well.

Revelation of shrunken or stretched binomial dispersion and public perception of situations which might spread AIDS or HIV

Background:In 1985, the center for disease control coined the name: “Acquired Immune Deficiency Syndrome (AIDS)” to refer a deadly 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 in many parts of the world. A scary fact is that the public worry about situations which might spread AIDS according to reported survey result in Meulders et al. (2001). This article develops and illustrates an appropriate statistical methodology to understand the meanings of the data.Methods: While the binomial model is a suitable underlying model for their responses, the data mean and dispersion violates the model’s required functional balance between them. This violation is called over-under dispersion. This article creates an innovative approach to assess whether the functional imbalance is too strong to reject the binomial model for the data. In a case of rejecting the model, what is a correct way of warning the public about the spreads of AIDS in a specified situation? This question is answered.Results: In the survey data about how AIDS/HIV might spread according to fifty respondents in thirteen nations, the functional balance exists only in three cases: “needle”, “blood” and “sex” justifying using the usual binomial model (1). In all other seven cases: “glass”, “eating”, “object”, “toilet”, “hands”, “kissing”, and “care” of an AIDS or HIV patient, there is a significant imbalance between the dispersion and its functional equivalence in terms of the mean suggesting that the new binomial called imbalanced binomial distribution (6) of this article should be used. The statistical power of this methodology is indeed excellent and hence the practitioners should make use of it.   Conclusion:The new model called imbalanced binomial distribution (6) of this article is versatile enough to be useful in other research topics in the disciplines such as medicine, drug assessment, clinical trial outcomes, business, marketing, finance, economics, engineering and public health.

An assessment of nurses’ sufficient immunity when treating infectious patients using bumped-up binomial model

Background: In times of an outbreak of a contagious deadly epidemic1-4 such as severe acute respiratory syndrome (SARS), the patients are quarantined and rushed to an emergency department of a hospital for treatment. Paradoxically, the nurses who treat them to become healthy get infected in spite of the nurses’ precautionary defensive alertness. This is so unfortunate because the nurses might not have been in close contact with the virus otherwise in their life. The nurses’ sufficient immunity level is a key factor to avert hospital site infection. Is it possible to quantify informatics about the nurses’ immunity from the virus? Methods: The above question is answered with a development of an appropriate new model and methodology. This new frequency trend is named Bumped-up Binomial Distribution (BBD). Several useful properties of the BBD are derived, applied, and explained using SARS data5 in the literature. Though SARS data are considered in the illustration, the contents of the article are versatile enough to analyze and interpret data from other contagious diseases.Results: With the help of BBD (3) and the Toronto data in Table 1, we have identified the informatics about the attending nurses’ sufficient immunity level. There were 32 nurses providing 16 patient care services. Though the nurses were precautionary to avoid infection, not all of them were immune to infection in those activities. Using the new methodology of this article, their sufficient immunity level is estimated to be only 0.25 in a scale of zero to one with a p-value of 0.001. It suggests that the nurses’ sufficient immunity level is statistically significant. The power of accepting the true alternative hypothesis of 0.50 immunity level, if it occurs, is calculated to be 0.948 in a scale of zero to one. It suggests that the methodology is powerful. Conclusions: The estimate of nurse’s sufficient immunity level is a helpful factor for the hospital administrators in the time of making work schedules and assignments of the nurses to highly contagious patients who come in to the emergency or regular wings of the hospital for treatment. When the approach and methodology of this article are applied, it would reduce if not a total elimination of the hospital site infections among the nurses and physicians.

Data directed root cause analyses of hospital adversities and their proximities

Background: In this current era of healthcare reformations, medical professionals, patients, governments, and insurance agencies seek zero tolerance with respect to adverse outcomes. When adversities occur, hospital administrators more often than not perform root cause analysis (RCA) to avoid future reoccurrence. There exist three types of RCA. They are divergent, serial, and convergent root causes. Which adversity type exists in a situation is not medically or intuitivcly trivial. This article develops a data directed new methodology to characterize the type and interpret it.Methods: Because tracing root causes of medical adversity is a necessity, pertinent data are collected. Patterns in correlation are examined to check whether it is a divergent or serial type. When it is not either, it is concluded to be convergent. This practice is too elementary to convince professionals. For this purpose, this article innovatively develops a new methodology using inverted correlation matrix and Mahalanobis distances to sort out causes of adversity as serial, divergent, or convergent type. Their proximities are quantifiable due to new expressions in the article. These expressions are not seen in the literature and hence, would benefit practioners.Results: A new methodology of this article is illustrated using medical adversities that existed in hospitals during 2006 through 2014 in Indiana state. Data consist of number of surgeries, cases with ulcer acquired in hospital, cases with foreign objects in patient after surgery, cases with wrong part surgery, deaths due to medication error, and disability cases due to fall during hospital treatment. Their correlations ranged from -0.87 to 0.79.  Conclusions: This article has developed expressions to quantify non-equilibrium level in serial and divergent RCA and has demonstrated their use to identify a convergent RCA. The Mahalanobis distance of attained diversities from an ideal scenario is obtained. A formula to make and interpret safety index is developed and demonstrated using adversities that occurred in India State during 2006-2014. These concepts and analytic expressions would enrich the practice of RCA which is a necessity in the current era of healthcare reforms.

Distracted multinomial model for corona screening at entry ports

Background: On 24 January 2020, 1287 corona cases were noticed in Wuhan, China, causing 41 deaths. Its incubation period is at least 14 days. Now, this deadly virus has spread to other foreign countries. The prevalence of corona cases is changing daily. See www.who.org for daily reports. The corona cases are mystic and nightmare to the public, health professionals, and governing agencies globally.Methods: The Center for Disease Control (CDC) compiled in their webpage (www.cdc.org) the number of confirmed, number of healthy, and the number of pending cases at the port of entries in United States of America (USA). These numbers are perhaps under-estimates because of inappropriate diagnostics and imprecise incubation period. To resolve the under estimating, this article, introduces a distracted multinomial model to refine the imprecise corona screening process and interpret the probability of detecting a corona case in US entry gates.Results: An alternate expression (2) for the correlation between the corona ill and corona free cases at the USA ports of entry reveals that it was rising since 31st January 2020, reached its maximum on 5th February 2020, then declined to hit a bottom on 7th February 2020 only to rise again.Conclusions: Most desirable is an accurate predictability of a traveller with the corona virus at the portal entry to minimize its spread. To make such prediction, a regression is necessary with involvement of covariates like age, body’s immunity level, comorbidity, and precise understanding of its incubation period. The model in this article is the starting point for further future research work

How do queuing concepts and tools help to efficiently manage hospitals when the patients are impatient? A demonstration

Background: Due to severe pain, patients are impatient in several wings sporadically and more frequently in emergency wing of the hospitals. To efficiently administer in such environment and the hospital management seeks helpful strategies. The queuing concepts and related methodologies can help as this article has demonstrated by an analysis and interpretation of real data from a hospital in Malta. Methods: The queuing concepts are probabilistic and statistical ideas based approach. They require configuration of the rate and pattern of arriving patients, the rate and pattern of the service, the number of channels serving, the capacity of the waiting room, and the criterion for selecting patients for service etc. New ideas are presented in this article to manage in various scenarios of real life emergency operations. The pertinent queuing concepts and tools are made easier for the readers to comprehend and practice in their own situations in which they notice that the patients are impatient in their waiting.Results:Using the new ideas and formulas of this article, the data in the emergency wing of a hospital in Malta (a largest island of an archipelago situated in the center of the Mediterranean with a total population of a million) are analyzed and interpreted. The results clearly explain why there were a prolonged waiting times at the emergency department creating public dissatisfaction and patients were leaving without waiting to be seen. The total time spent by non-urgent patients with nurse and casualty officer is more in the second shift and lesser and lesser in the third and fourth shifts. The interactive time with a nurse by patient is statistically same in all three types: life-threatening, non-life threatening but urgent, and non-urgent. Very strikingly, the patients in all three groups wait longer to be seen by the nurse in shift three and lesser time in shifts two or four.Conclusion: In 21st century with flourishing globalized medical tourism, a standardized approach to minimize efficiently the waiting time in emergency and other wings of the hospitals in developing as much as in developed nations is a necessity as this auricle has pointed out. The impediments and the remedies for an efficient standardization are overdue.