Malaria is a devastating global health problem that affected 219 million people and caused 660,000 deaths in 2010. Inaccurate estimation of the level of infection may have adverse clinical and therapeutic implications for patients, and for epidemiological endpoint measurements. The level of infection, expressed as the parasite density (PD), is classically defined as the number of asexual parasites relative to a microliter of blood. Microscopy of Giemsa-stained thick blood smears (TBSs) is the gold standard for parasite enumeration. Parasites are counted in a predetermined number of high-power fields (HPFs) or against a fixed number of leukocytes. PD estimation methods usually involve threshold values; either the number of leukocytes counted or the number of HPFs read. Most of these methods assume that (1) the distribution of the thickness of the TBS, and hence the distribution of parasites and leukocytes within the TBS, is homogeneous; and that (2) parasites and leukocytes are evenly distributed in TBSs, and thus can be modeled through a Poisson-distribution. The violation of these assumptions commonly results in overdispersion. Firstly, we studied the statistical properties (mean error, coefficient of variation, false negative rates) of PD estimators of commonly used threshold-based counting techniques and assessed the influence of the thresholds on the cost-effectiveness of these methods. Secondly, we constituted and published the first dataset on parasite and leukocyte counts per HPF. Two sources of overdispersion in data were investigated: latent heterogeneity and spatial dependence. We accounted for unobserved heterogeneity in data by considering more flexible models that allow for overdispersion. Of particular interest were the negative binomial model (NB) and mixture models. The dependent structure in data was modeled with hidden Markov models (HMMs). We found evidence that assumptions (1) and (2) are inconsistent with parasite and leukocyte distributions. The NB-HMM is the closest model to the unknown distribution that generates the data. Finally, we devised a reduced reading procedure of the PD that aims to a better operational optimization and a practical assessing of the heterogeneity in the distribution of parasites and leukocytes in TBSs. A patent application process has been launched and a prototype development of the counter is in process.

進行調查時，最後的抽樣單位常是從不同的群集取得的，而同一群集內的樣本對象，因背景類似而對於某些問題常會傾向相同或類似的反應，研究者若忽略這種群內相關性，仍以獨立性樣本進行分析時，因其共變異數矩陣通常會與多項模式的共變異數矩陣相差懸殊，而造成所謂的額外變異或不足變異的現象。本文在不同的情況下，提出了Dirichlet-Multinomial模式(簡稱DM模式)、擴展的DM模式、以及兩種平均數-共變異數矩陣模式，以適當的彙整所有的群集資料。並討論DM與EDM模式中相關之參數及格機率之最大概似估計法，且分別對此兩種平均數-共變異數矩陣模式，提出求導廣義最小平方估計的程序。此外，也針對幾種特殊的二維表及三維表結構，探討對應的參數及格機率之估計方法。並提出計算簡易的Score統計檢定量以判斷群內相關(intra-cluster correlation)之存在性，及判斷資料集具有額外或不足變異，而對於不同母體的群內相關同質性檢定亦提出討論。 / This paper presents a modelling method of analyzing categorical data with overdispersion or underdispersion.　In many studies, data are collected from differ clusters, and members within the same cluster behave similary.　Thus, the responses of members within the same cluster are not independent and the multinomial distribution is not the correct distribution for the observed counts.　Therefore, the covariance matrix of the sample proportion vector tends to be much different from that of the multinomial model.　We discuss four different models to fit counts data with overdispersion or underdispersion feature, witch include Dirichlet-Multinomial model (DM model), extended DM model (EDM model), and two mean-covariance models.　Method of maximum-likelihood estimation is discussed for DM and EDM models.　Procedures to derive generalized least squares estimates are proposed for the two mean-covariance models respectively. As to the cell probabilities, we also discuss how to estimate them under several special structures of　two-way and three-way tables.　More easily evaluated Score test statistics are derived for the DM and EDM models to test the existence of　the intra-cluster correlation.　And the test of homogeneity of intra-cluster correlation among several populations is also derived.

The term aggregation refers to overdispersion and both are used interchangeably in this thesis. In addressing the problem of prevalence of infectious parasite species faced by most rural livestock farmers, we model the distribution of faecal egg counts of 15 parasite species (13 internal parasites and 2 ticks) common in sheep and goats. Aggregation and excess zeroes is addressed through the use of generalised linear models. The abundance of each species was modelled using six different distributions: the Poisson, negative binomial (NB), zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), zero-altered Poisson (ZAP) and zero-altered negative binomial (ZANB) and their fit was later compared. Excess zero models (ZIP, ZINB, ZAP and ZANB) were found to be a better fit compared to standard count models (Poisson and negative binomial) in all 15 cases. We further investigated how distributional assumption a↵ects aggregation and zero inflation. Aggregation and zero inflation (measured by the dispersion parameter k and the zero inflation probability) were found to vary greatly with distributional assumption; this in turn changed the fixed-effects structure. Serial autocorrelation between adjacent observations was later taken into account by fitting observation driven time series models to the data. Simultaneously taking into account autocorrelation, overdispersion and zero inflation proved to be successful as zero inflated autoregressive models performed better than zero inflated models in most cases. Apart from contribution to the knowledge of science, predictability of parasite burden will help farmers with effective disease management interventions. Researchers confronted with the task of analysing count data with excess zeroes can use the findings of this illustrative study as a guideline irrespective of their research discipline. Statistical methods from model selection, quantifying of zero inflation through to accounting for serial autocorrelation are described and illustrated. / Statistics / M.Sc. (Statistics)