The multilevel model (MLM) is easily parameterized to handle partially clustered data (see, for example, Baldwin, Bauer, Stice, & Rohde, 2011). The current study evaluated the performance of this model under various departures from underlying assumptions, including assumptions of normally distributed and homoscedastic Level 1 residuals. Two estimating models – one assuming homoscedasticity, the other allowing for the estimation of unique Level 1 variance components – were compared. Results from a Monte Carlo simulation suggest that the MLM for partially clustered data yields consistently unbiased parameter estimates, except for an underestimation of the Level 2 variance component under heteroscedastic generating conditions. However, this negative parameter bias desisted when the MLM allowed for Level 1 heteroscedasticity. Standard errors for variance component estimates at both levels were underestimated in the presence of nonnormally distributed Level 1 residuals. A discussion of results, as well as suggestions for future research, is provided. / text

Master of Science / Department of Statistics / Weixing Song / The relationship between a random variable and a random vector is often investigated through the regression modeling. Because of its relative simplicity and ease of interpretation, a particular parametric form is often assumed for the regression function. If the pre-specified function form truly reflects the truth, then the resulting estimators and inference procedures would be reliable and efficient. But if the regression function does not represent the true relationship between the response and the predictors, then the inference results might be very misleading. Therefore, lack-of-fit test should be an indispensable part in regression modeling. This report compares the finite sample performance of several classical lack-of-fit tests in regression models via simulation studies. It has three chapters. The conception of the lack-of-fit test, together with its basic setup, is briefly introduced in Chapter 1; then several classical lack-of-fit test procedures are discussed in Chapter 2; finally, thorough simulation studies are conducted in Chapter 3 to assess the finite sample performance of each procedure introduced in Chapter 2. Some conclusions are also summarized in Chapter 3. A list of MATLAB codes that are used for the simulation studies is given in the appendix.

In the purview of efficient communication in MANETs for enhanced data rates and reliable routing of information, this thesis deals with dual polarised directional antenna based communication. This thesis proposes a dual polarised directional communication based cross-layer solution to mitigate the problems of interference, exposed nodes, directional exposed nodes, and deafness, and to achieve efficient routing of information. At the physical layer of network protocol stack, this thesis proposes the use of dual polarised directional antenna for the mitigation of interference. Use of dual polarised directional communication at the physical layer calls for appropriate modifications in the functionality of MAC and network layers. At the MAC layer, the DPDA-MAC protocol proposed in this thesis achieves mitigation of the problems of exposed nodes, directional exposed nodes and deafness, by using dual polarised directional antenna at physical layer. At network layer, the DPDA-MRP protocol presented in this thesis facilitates the discovery of multiple routes between the source and destination nodes to route information in accordance with the desired dual polarised directional communication. To achieve efficient dual polarised directional communication and routing of information, it is essential to maintain well populated Neighbour Table (NT) and Routing Table (RT). This thesis proposes a novel Corruption Detection Pulse (CDP) based technique to handle corruption of broadcast packets such as Link ID and RREQ arising due to hidden node problem. Since the nodes participating in the formation of MANETs have limited battery energy, the protocols proposed in this thesis are featured with a provision for dynamic power control to achieve energy efficient communication. Nodes maintain Received Signal Strength Indicator (RSSI) information in the NT, which along with the information of node location is used in the formulation of decision logic of dynamic power control. Through numerous simulation studies, this thesis demonstrates the benefits of dual polarised directional communication to enhance the performance of MANET. The design principles, benefits and conceptual constraints of proposed DPDA-MAC protocol are analysed with SPDA-MAC and CSMA/CA, while those for DPDA-MRP are analysed with SPDA-MRP and DSR through performance metrics of throughput, Packet Delivery Ratio (PDR) and per hop delay. The thesis also analyses the impact of variations of channel capacity, node density, rate of packet transmission and mobility of nodes on the performance of the proposed and conventional protocols invoked in MANETs.

本研究的主要目的是探究四分位數定義在國中、高中教材不同情形下，所可能衍生出來的差異情況，並統整有關四分位數之八組公式，以第一、第三四分位數及四分位距觀點來探討不同公式適用於何種資料型態。 本研究發現如下： 一、國中、高中在四分位數之計算結果的確有差異，尤其當原始資料之個數為N＝4k + 1(k為正整數)時，藉由高中版本的計算公式所算出來的Q1與Q3，與藉由國中版本所計算出來的25及75百分位數可能有所不同。其餘資料個數情況，似乎並不會發生不一致的現象。 二、假設原始資料分別來自五種右偏程度不一的分配，包括卡方分配 (〖" χ" 〗_"(1)﹐" ^"2" 〖 χ〗_"(5)﹐" ^2 χ_"(10)" ^2 )、指數分配(Exp"(" λ="10)" ﹐Exp"(" λ="15)" )以及平均數為100﹐ 標準差為10的常態分配這一種對稱的型式。在不同樣本數(N＝4k、4k+ 1、4k + 2、4k + 3)及不同資料分配的情況下，透過模擬實驗的方式，並依據四項評估指標（偏誤、標準差、均方根誤差、平均相對誤差百分比）來探討這八組公式的適用性。雖然各家公式的版本不盡相同，但大體上而言，在大樣本時的表現差不多，僅在小樣本時(N=40、41、42、43)呈現出明顯差異。卻因為在不同資料分配下，八組公式的表現不一，所以無法歸納出較具體的結論。相對而言，國中、高中公式的表現儘管稱不上是最好，不過就學生的學習而言，卻是清晰易懂。 關鍵詞：四分位數公式、模擬比較 / The main purpose of this study is to explore the possible implications for different definitions of quartiles taught in high schools. In addition, the two formulae are also compared with six others that are found popular in literature according to their performances under different distributions. The main findings of the study are summarized as follows: 1.Differences are found in the calculation of quartile based on the definition used in junior and senior high schools. Specifically, when the data size is N=4k+1 (k is a positive integer), Q1 and Q3 calculated using the formula taught in senior high schools may not be same as the 25th and 75th percentiles, the way junior high students are taught to calculate Q1 and Q3. 2.Assuming that data come from five right-skewed distributions ( , , ), chi-squared distributions with degrees of freedom 1, 5, and 10, and Exp( ), Exp( ), exponential distributions with means 10 and 15, and a symmetrical distribution (normal distribution with mean 100 and standard deviation 10), simulation studies are carried out to assess the performances of the eight formulae in terms of bias, standard deviation, root mean square error and average relative error. Generally speaking, no differences are found when the sample sizes are large. Noticeable differences, on the other hand, are found under the situations of small sample sizes, particularly when N=40, 41, 42, and 43. However, since the performances varied dramatically, there appears no clear winner among the eight formulae. Although the performances of the two formulae taught in junior high and senior high schools scarcely perform as the “best”, they are easily understood, and clearly appropriate for use in teaching high school students the concept of quartiles. Key words: Quartiles, Simulation studies

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In this study we propose three new distributions and a study with longitudinal data. The first was the Odd log-logistic normal distribution: theory and applications in analysis of experiments, the second was Odd log-logistic t Student: theory and applications, the third was the Odd log-logistic skew normal: the new distribution skew-bimodal with applications in analysis of experiments and the fourth regression model with random effect of the Odd log-logistic skew normal distribution: an application in longitudinal data. Some have been demonstrated such as symmetry, quantile function, some expansions, ordinary incomplete moments, mean deviation and the moment generating function. The estimation of the model parameters were approached by the method of maximum likelihood. In applications were used regression models to data from a completely randomized design (CRD) or designs completely randomized in blocks (DBC). Thus, the models can be used in practical situations for as a completely randomized designs or completely randomized blocks designs, mainly, with evidence of asymmetry, kurtosis and bimodality. / A distribuição normal é uma das mais importantes na área de estatística. Porém, não é adequada para ajustar dados que apresentam características de assimetria ou de bimodalidade, uma vez que tal distribuição possui apenas os dois primeiros momentos, diferentes de zero, ou seja, a média e o desvio-padrão. Por isso, muitos estudos são realizados com a finalidade de criar novas famílias de distribuições que possam modelar ou a assimetria ou a curtose ou a bimodalidade dos dados. Neste sentido, é importante que estas novas distribuições tenham boas propriedades matemáticas e, também, a distribuição normal como um submodelo. Porém, ainda, são poucas as classes de distribuições que incluem a distribuição normal como um modelo encaixado. Dentre essas propostas destacam-se: a skew-normal, a beta-normal, a Kumarassuamy-normal e a gama-normal. Em 2013 foi proposta a nova família X de distribuições Odd log-logística-G com o objetivo de criar novas distribuições de probabildade. Assim, utilizando as distribuições normal e a skew-normal como função base foram propostas três novas distribuições e um quarto estudo com dados longitudinais. A primeira, foi a distribuição Odd log-logística normal: teoria e aplicações em dados de ensaios experimentais; a segunda foi a distribuição Odd log-logística t Student: teoria e aplicações; a terceira foi a distribuição Odd log-logística skew-bimodal com aplicações em dados de ensaios experimentais e o quarto estudo foi o modelo de regressão com efeito aleatório para a distribuição distribuição Odd log-logística skew-bimodal: uma aplicação em dados longitudinais. Estas distribuições apresentam boas propriedades tais como: assimetria, curtose e bimodalidade. Algumas delas foram demonstradas como: simetria, função quantílica, algumas expansões, os momentos incompletos ordinários, desvios médios e a função geradora de momentos. A flexibilidade das novas distrições foram comparada com os modelos: skew-normal, beta-normal, Kumarassuamy-normal e gama-normal. A estimativas dos parâmetros dos modelos foram obtidas pelo método da máxima verossimilhança. Nas aplicações foram utilizados modelos de regressão para dados provenientes de delineamentos inteiramente casualizados (DIC) ou delineamentos casualizados em blocos (DBC). Além disso, para os novos modelos, foram realizados estudos de simulação para verificar as propriedades assintóticas das estimativas de parâmetros. Para verificar a presença de valores extremos e a qualidade dos ajustes foram propostos os resíduos quantílicos e a análise de sensibilidade. Portanto, os novos modelos estão fundamentados em propriedades matemáticas, estudos de simulação computacional e com aplicações para dados de delineamentos experimentais. Podem ser utilizados em ensaios inteiramente casualizados ou em blocos casualizados, principalmente, com dados que apresentem evidências de assimetria, curtose e bimodalidade.

In this study we propose three new distributions and a study with longitudinal data. The first was the Odd log-logistic normal distribution: theory and applications in analysis of experiments, the second was Odd log-logistic t Student: theory and applications, the third was the Odd log-logistic skew normal: the new distribution skew-bimodal with applications in analysis of experiments and the fourth regression model with random effect of the Odd log-logistic skew normal distribution: an application in longitudinal data. Some have been demonstrated such as symmetry, quantile function, some expansions, ordinary incomplete moments, mean deviation and the moment generating function. The estimation of the model parameters were approached by the method of maximum likelihood. In applications were used regression models to data from a completely randomized design (CRD) or designs completely randomized in blocks (DBC). Thus, the models can be used in practical situations for as a completely randomized designs or completely randomized blocks designs, mainly, with evidence of asymmetry, kurtosis and bimodality. / A distribuição normal é uma das mais importantes na área de estatística. Porém, não é adequada para ajustar dados que apresentam características de assimetria ou de bimodalidade, uma vez que tal distribuição possui apenas os dois primeiros momentos, diferentes de zero, ou seja, a média e o desvio-padrão. Por isso, muitos estudos são realizados com a finalidade de criar novas famílias de distribuições que possam modelar ou a assimetria ou a curtose ou a bimodalidade dos dados. Neste sentido, é importante que estas novas distribuições tenham boas propriedades matemáticas e, também, a distribuição normal como um submodelo. Porém, ainda, são poucas as classes de distribuições que incluem a distribuição normal como um modelo encaixado. Dentre essas propostas destacam-se: a skew-normal, a beta-normal, a Kumarassuamy-normal e a gama-normal. Em 2013 foi proposta a nova família X de distribuições Odd log-logística-G com o objetivo de criar novas distribuições de probabildade. Assim, utilizando as distribuições normal e a skew-normal como função base foram propostas três novas distribuições e um quarto estudo com dados longitudinais. A primeira, foi a distribuição Odd log-logística normal: teoria e aplicações em dados de ensaios experimentais; a segunda foi a distribuição Odd log-logística t Student: teoria e aplicações; a terceira foi a distribuição Odd log-logística skew-bimodal com aplicações em dados de ensaios experimentais e o quarto estudo foi o modelo de regressão com efeito aleatório para a distribuição distribuição Odd log-logística skew-bimodal: uma aplicação em dados longitudinais. Estas distribuições apresentam boas propriedades tais como: assimetria, curtose e bimodalidade. Algumas delas foram demonstradas como: simetria, função quantílica, algumas expansões, os momentos incompletos ordinários, desvios médios e a função geradora de momentos. A flexibilidade das novas distrições foram comparada com os modelos: skew-normal, beta-normal, Kumarassuamy-normal e gama-normal. A estimativas dos parâmetros dos modelos foram obtidas pelo método da máxima verossimilhança. Nas aplicações foram utilizados modelos de regressão para dados provenientes de delineamentos inteiramente casualizados (DIC) ou delineamentos casualizados em blocos (DBC). Além disso, para os novos modelos, foram realizados estudos de simulação para verificar as propriedades assintóticas das estimativas de parâmetros. Para verificar a presença de valores extremos e a qualidade dos ajustes foram propostos os resíduos quantílicos e a análise de sensibilidade. Portanto, os novos modelos estão fundamentados em propriedades matemáticas, estudos de simulação computacional e com aplicações para dados de delineamentos experimentais. Podem ser utilizados em ensaios inteiramente casualizados ou em blocos casualizados, principalmente, com dados que apresentem evidências de assimetria, curtose e bimodalidade.

In this work a numerical approach for the determination of the energy spectra and the calculation of thermodynamic properties of magnetic molecules is presented. The work is focused on the treatment of spin systems which exhibit point-group symmetries. Ring-like and archimedean-type structures are discussed as prominent examples. In each case the underlying spin quantum system is modeled by an isotropic Heisenberg Hamiltonian. Its energy spectrum is calculated either by numerical exact diagonalization or by an approximate diagonalization method introduced here. In order to implement full spin-rotational symmetry the numerical approach at hand is based on the use of irreducible tensor operators. Furthermore, it is shown how an unrestricted use of point-group symmetries in combination with the use of irreducible tensor operators leads to a reduction of the dimensionalities as well as to additional information about the physics of the systems. By exemplarily demonstrating how the theoretical foundations of the irreducible tensor operator technique can be realized within small spin systems the technical aspect of this work is covered. These considerations form the basis of the computational realization that was implemented and used in order to get insight into the investigated systems.

The nature of food insecurity has been changing in the world. While research on food insecurity in developing countries used to focus on undernourishment (i.e. lack of calorie consumption) and related outcomes, today, many developing countries face at least a “double burden of malnutrition” with persistently high rates of undernourishment and increasing rates of overweight, obesity and related non-communicable diseases (NCDs). An important driver of overweight and obesity in developing countries is the „nutrition transition“, i.e. the trend towards the consumption of more energy-dense, highly processed foods and more sedentary lifestyles. Two essays of this Ph.D. thesis analyse drivers and consequences of the nutrition transition in developing countries with a particular focus on the role of supermarkets, which have been rapidly spreading in many countries. We provide evidence that the presence of supermarkets causally affects dietary choices and nutritional outcomes. Data collection for this research was carried out in small Kenyan towns of the kind that accommodate most of the country’s urban population. We designed our sample to be quasi-experimental in nature and employ instrumental variable techniques to allow for endogeneity of supermarket purchases. Kenya’s supermarket landscape is dynamic and so far, it has followed the ‘traditional pattern’ of the so-called supermarket revolution. Supermarket purchases are found to contribute to the nutrition transition by shifting consumption towards processed and away from unprocessed foods. At the same time, calorie availability increases as calories are sourced at lower prices in supermarkets. We find that supermarket purchases increase adult Body Mass Index and their probability of being overweight or obese. Yet, we also find that buying in a supermarket tends to decrease underweight among children and adolescents (age 5-19) in terms of stunting (height-for-age). In a third essay, we use secondary household survey data from Malawi to analyse ‘one of the other faces of malnutrition’. The world food price crisis of 2007/08 and other global and regional price and income shocks that followed have spurred interest in producing timely predictions on their implications for food security. A critical research gap remains with comparing simulation outcomes across studies that use different, established methods on the same subject. This is to establish if and to which extent they might result in different and potentially conflicting policy recommendations. We address this gap building on three simulation studies set in Malawi, which analyse welfare in terms of food security and income effects using the same 2004/05 household survey data but resort to methodologies of different complexity. We harmonize simulation scenarios across methods and systematically modify relevant parameters for our comparative assessment. We find differences between methods to depend on the scenario under consideration and to grow with increasing rates of simulated price changes. The differences we find are driven by differences in conceptualising price changes. In case of Malawi, for a reasonable set of observed price changes, mean outcomes on district levels are fairly robust to underlying methodologies. We illustrate that is it important to improve our understanding of how changes in the underlying methodologies change results and to analyse the sensitivity of simulation outcomes to different model assumptions.