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Menopause is not an illness but rather an important event as it changes the body physiology and mental cognition via hormonal changes. During data analysis of menopauses incidence data, new bivariate distribution is discovered. Their marginal, conditional distribution and statistical properties including the inter and partial correlations are explored and utilized to interpret menopauses data. A likelihood ratio hypothesis testing procedure is constructed to test the statistical significance of the sample estimate of the chance for menopause and estimate of the chance for operative menopause. The menopause data are analyzed and interpreted in the illustration. Research directions for future work are pointed out.

The model is an abstraction of the reality. The selection of the usual inverse binomial as an underlying model for the number of patients waiting in months for heart and lung transplant is questionable because the data exhibit not the required balance between the dispersion and its functional equivalent in terms of the mean but rather an over or under dispersion. This phenomenon of over/under dispersion has been a challenge to find an appropriate underlying model for the data. This article offers an innovative approach with a new model to resolve the methodological breakdown. The new model is named Imbalanced Inverse Binomial Model (IIBM). A statistical methodology is devised based on IIBM to analyze the collected data. The methodology is illustrated with a real life data on the number of patients waiting in months for heart and lung transplants together. The results in the illustration do convince that the new approach is quite powerful and brings out a lot more information which would have been missed otherwise. In specific, the odds of receiving the organs are higher under an estimated imbalance in the data than under an ideal zero imbalance in all the states except Alabama. The odds are consistently higher under an estimated imbalance in the data than under an ideal zero imbalance across all the age groups waiting in months. Further research work is needed to identify and explain the factors which might have caused the imbalance between the observed dispersion in the data and its functionally equivalent amount according to the underlying inverse binomial model for the data. The contents of this article remains the foundation on which the future research work will be built.

The rape victims are frightened to report with a fear of retaliation or humiliation. Consequently, the number of reported rapes is under-estimated. How should the number of unreported rapes be identified is discussed in this article. For this purpose, the Poisson distribution is modified and it is named Bumped-up Poisson distribution in this article. Related probability-informatics are derived to estimate the unreported rapes and proportion fearing to report. A hypothesis testing procedure is developed to assess the significance of an estimated proportion fearing. Our approach is tried with the reported rapes during the years 2007 and 2008 in a random sample of nations in all the continents. Proximities among the nations are identified in rape incidences.

Motivation for this research work started while helping a hospital administrator to assess whether patient oriented activity duration, X≥0 is reflecting the service’s efficiency? Higher value of the sample mean duration, X implies lesser productivity in the hospital and more healthcare cost. Likewise, larger value of sample dispersion, sx2 in the service durations is an indicator of lesser reliability and inefficiency. Of course, the dispersion, sx2 in a healthcare hospital operation could be due to diverse medical complications among patients or operational inefficiency. Assuming that it is not the diverse medical complications of patients, how should the pertinent information from data be extracted, quantified and interpreted to address inefficient operation? This is the problem statement for discussion in this article. To be specific, in an inefficient hospital operation, the sample dispersion and mean of service durations are likely to be highly correlated. Their correlation is a clue to identify an inefficient operation of a hospital. To compute the correlation, currently there is no appropriate formula in the literature. The aim of this article is, therefore, to derive a working formula to compute the correlation between sample dispersion and mean. The dispersion is too valuable statistical measure to quickly dispense, not only in healthcare operations but also in engineering, economics, business, social or sport applications. The approach starts first in quantifying a general relationship between the dispersion and mean in a given data. This relationship might range from a linear to a quadratic, cubic or higher degree. Suppose that the dispersion, σ2 is a function, f(µ)of the mean, µ of patient oriented activity durations. Specific functionality depends on the frequency pattern of the data. The tangent at a locus of their relationship curve is either declining or inclining line with an angle θ whose cosine value is indeed the correlation between the mean, x and dispersion, sx2. An expression to compute the angle is nowhere seen in the literature. Therefore, this article derives a general expression based on geometric concepts and then obtains specific formula for several count and continuous distributions. These expressions are foundations for further data analyses. To initiate, promote or maintain an efficient service operation for patients in a hospital, practical strategies have to be formulated based on the cluein the form of correlation value. For this purpose, a one-to-one relationship between sample dispersion and mean could be utilized to improve the service efficiency. In this process, a formula is developed to check whether the model parameters are orthogonal. The curvature and the shifting angle in the relationship between dispersion and mean are captured when the mean changes one unit. Both Poisson and exponential distributions are illustrated to comprehend the concepts and the derived expressions of this article. Efficient healthcare service is a necessity not only in USA but also in other nations because of an escalating demand by medical tourists in this era of globalized medical treatment. A reformation to the entire healthcare field could be achievable with the help of biostatical concepts and tools. To extract and comprehend pertinent data information in the patient oriented activity durations, the correlation is a tool. The data information holds the key to make the much needed reformation and operational efficiency. This article illustrates that the correlation between the data mean and dispersion provides clues. The correlation helps to assess healthcare service efficiency as it is demonstrated in this article with data. Similar applications occur in engineering, business and science fields.

A prelude to interpret a pattern in the repeating incidences is to identify the underlying frequency distribution of the collected data. A case in point is the Poisson distribution which is often selected for medical count data such as gene mutations, medication error and number of ambulatory pickups in a day. A requirement for the Poisson distribution is that the variance ought to be equal to the mean. The variance signifies the volatility in the occurrences. An implication is that the volatility becomes more when the average incidence is higher. When this requirement of the functional equivalence of the Poisson mean and variance is breached, the data deviates from a Poisson distribution. How could a data analyst recognize and point out to the medical team the dilution level of the requirement in their data? For this purpose, a simple and easier geometrical approach is developed in this article and illustrated with several historical data sets in the literature.