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Master of Science / Department of Statistics / Leigh W. Murray / Traditionally the Poisson process is used to model count response variables. However, a problem arises when the particular response variable contains an inordinate number of both zeros and large observations, relative to the mean, for a typical Poisson process. In cases such as these, the variance of the data is greater than the mean and as such the data are over-dispersed with respect to the Poisson distribution due to the fact that the mean equals the variance for the Poisson distribution. This case study looks at several common and uncommon ways to attempt to properly account for this over-dispersion in a specific set of nematode count data using various procedures in SAS 9.2. These methods include but are not limited to a basic linear regression model, a generalized linear (log-linear) model, a zero-inflated Poisson model, a generalized Poisson model, and a Poisson hurdle model. Based on the AIC statistics the generalized log-linear models with the Pearson-scale and deviance-scale corrections perform the best. However, based on residual plots, none of the models appear to fit the data adequately. Further work with non-parametric methods or the negative binomial distribution may yield more ideal results.

The objective of this project is to fit a sequence of increasingly complex zero-inflated censored regression models to a known data set. It is quite common to find censored count data in statistical analyses of health-related data. Modeling such data while ignoring the censoring, zero-inflation, and overdispersion often results in biased parameter estimates. This project develops various regression models that can be used to predict a count response variable that is affected by various predictor variables. The regression parameters are estimated with Bayesian analysis using a Markov chain Monte Carlo (MCMC) algorithm. The tests for model adequacy are discussed and the models are applied to an observed data set.

Рассматривается обобщенное ядро Пуассона, представляющее собой линейную комбинацию ядра Пуассона и сопряженного ядра Пуассона. Найдены величины наилучшего интегрального приближения снизу и сверху этого ядра тригонометрическими полиномами порядка не выше заданного и соответствующие полиномы наилучшего одностороннего приближения. / We consider the generalized Poisson kernel,which is a linear combination of the Poisson kernel and the conjugate Poisson kernel. The values of the best integral approximations to this kernel from below and from above by trigonometric polynomials of degree not exceeding a given number has been found. The corresponding polynomials of the best one-sided approximation has been obtained.

世界衛生組織表示「先前有過自殺行為的人，再度自殺的機率比一般人高」，因此如何針對自殺企圖者提供即時的關懷與介入服務，是世界各國重要的自殺防治策略之一。本研究希望針對曾有過自殺企圖的個案，經由統計模型的配適來找出自殺企圖個案的「自殺危險因子」，區別出再度自殺的高危險個案，以方便將人力及醫療資源投入到最需要被協助的個案上。 本研究的反應變項為「重複自殺企圖次數」，但是由於資料中「零值」的人數相當多，此外也呈現出變異數大於平均數的現象，因此我們採用可以同時處理Zero-inflated及Over-dispersion情況的廣義Zero-inflated卜瓦松迴歸模型 (Generalized Zero-inflated Poisson Regression Model)來進行資料的配適。我們得知重複自殺企圖之高風險因子有「65歲以上」、「曾患有精神疾病」、「不確定是否曾患有精神疾病」及「離婚」之個案，而「治癒」可能性較高的因子為「45~64歲」、「因情感因素自殺」、「已婚」之個案。藉由模型也可以進一步估計自殺企圖個案之再企圖機率，並且對自殺企圖個案進行分層，以進行不同程度的關懷與訪視，藉以提昇關懷的即時性及有效性。 / World Health Organization (WHO) has indicated that suicide attempt rate is much higher among those who have ever had suicide attempts. Hence, how to express concerns and provide timely consultations for suicide reattempters has become one of the key issues in suicide prevention. In this study, we try to identify the risk factors associated with suicide reattempters, and predict high-risk cases so that the limited resources can be distributed effectively. The primary variable of interest is the number of repeated suicide attempt for a suicide attempter after his/her index attempt. However, there are more zeros and greater variability in the data than that would be predicted by a Poisson model. We hence fit the data using a zero-inflated generalized Poisson regression model, a model that is frequently used for modeling over-dispersed count data with too many zeros. We find that the risk factors for repeated suicide attempts are those who are 65 or older, those who are classified as psychiatric disorders and those diagnostically uncertain cases, and those who are divorced. We also find that non-repeaters are more likely among those who are between 45 to 65 of age, married, and having a suicide attempt history due to an emotional reason. Through the use of the model, we can also estimate a subject’s reattempt probability, classify them, and provide them with suitable care and attention accordingly.