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Percentiles are used everyday in descriptive statistics and data analysis. In real life, many quantities are normally distributed and normal percentiles are often used to describe those quantities. In life sciences, distributions like exponential, uniform, Weibull and many others are used to model rates, claims, pensions etc. The need to compare two or more independent populations can arise in data analysis. The ratio of percentiles is just one of the many ways of comparing populations. This thesis constructs a large sample confidence interval for the ratio of percentiles whose underlying distributions are known. A simulation study is conducted to evaluate the coverage probability of the proposed interval method. The distributions that are considered in this thesis are the normal, uniform and exponential distributions.

The Method of Moments (MOM) has been extensively used in statistics for obtaining conventional moment-based estimators of various parameters. However, the disadvantage of this method is that the estimates “can be substantially biased, have high variance, or can be influenced by outliers” (Headrick & Pant, 2012). The Method of Percentiles (MOP) provides a useful alternative to the MOM when the distributions are non-normal, specifically being more computationally efficient in terms of estimating population parameters. Examples include the generalized lambda distribution (Karian & Dudewicz, 1999), third order power method (Koran, Headrick & Kuo, 2015) and fifth order power method (Kuo & Headrick, 2017). Further, the HH, HR and HQ distributions, as extensions of the Tukey g-h (GH) family, are of interest for investigation using the MOP in this dissertation. More specifically, closed form solutions are obtained for left-right tail-weight ratio (a skew function) and tail-weight factor (a kurtosis function). A Monte Carlo simulation study which includes the comparison of monotonic and nonmonotonic transformation scenarios is also performed. The effect on Type 1 error and power rates under severely nonmonotonic scenarios are of special interest in the study. Dissimilarities of not strictly monotonic scenarios are discussed. The empirical confirmation that Rank Transform (RT) is appropriate for 2x2 designs is obtained.

The measurement of achievement growth raises many challenges, including how to define "growth" and select or develop a growth measure that captures that definition. Despite these complications, current federal educational policies focus on student growth measures for accountability purposes. Student growth percentiles (SGPs) are one metric developed under these policies. They use quantile regression to produce normative growth interpretations: They describe how much a student has grown relative to students with similar past test scores. SGPs are increasingly popular, but there are gaps in the literature concerning their performance for small sample sizes and the number of prior years of test scores included in the model, as well as their invariance to transformations of the test scale. This study proposes an ordinary least squares analog, the percentile rank of residuals (PRRs). PRRs are the percentile rank of the residuals found by regressing the current grade-level assessment score on past grade-level assessment scores. PRRs may be a more robust alternative to SGPs, especially for small samples. They also stem from a wide array of regression based metrics in education and only require estimation of one regression line, as opposed to the 100 regression lines estimated for SGPs. This dissertation first places the growth metrics of interest in a framework anchored by four key contrasts in growth interpretations: (1) absolute versus normative, (2) unconditional normative versus conditional normative, (3) student- versus group-level, and (4) aggregated individual growth versus growth of aggregated-individuals. SGPs and PRRs afford normative conditional growth interpretations. They are investigated at the student level using simulated multivariate normal data and two statewide empirical datasets. These student-level analyses assess the accuracy of SGPs and PRRs by their recovery of benchmark growth percentiles under multivariate normality, or normal conditional growth percentiles (NCGPs), their robustness to scale transformations, their comparability to each other under varying conditions, and their stability over different sample sizes and numbers of prior years included in the models. SGPs and PRRs are also investigated at the group level by aggregating them with the mean and median functions. The robustness of the aggregated growth percentiles to test scale transformations is also assessed. Finally, the aggregated growth percentiles are contrasted against group effects from a simple layered value-added model (VAM). The analyses found that PRRs better recover expected growth percentiles under multivariate normality and are more accurate and stable for small samples, whereas SGPs are substantially more robust to test scale transformations. However, estimation issues with the SGPs can cause students with extreme initial statuses to obtain substantially different SGPs under transformations of the data. At the aggregate level, there is little distinction in how robust SGPs and PRRs are to scale transformations of the test score data. The mean SGPs and mean PRRs are consistently more robust to scale transformations of the test score data then their median counterparts. They are also the most highly correlated and rank order the groups more similarly to the value-added school effects than the median SGPs and PRRs.

Background: Serumlipid concentrations are thought to be risk factors for the development of cardiovascular disease. The present study aims to investigate the prevalence of dyslipidemia and provide sex- and age-related reference values for triglycerides, total cholesterol, LDL and HDL cholesterol as well as apolipoproteins A1 and B by using modern analytical approaches. Materials and methods: Venous blood and anthropometric data were collected from 2571 subjects of the LIFE Child study, aged between 0.5 and 16 years. Age- and gender-related reference intervals (3rd and 97th percentiles) were established by using Cole's LMS method. Results: Serumconcentrations of TC, LDL-C, TG and ApoBwere higher in girls than in boys. In girls TC reached peak levels two years earlier than in boys. Triglyceride levels initially declined until the school age.Until early adolescence there was a steady increase. The LDL-C concentrations in girls and boys followed similar patterns to that of TC. Up to the age of 8 years, a continuous increase in HDL levels for both sexeswas found. Due to the strong correlation between HDL-C and ApoA1 (r=0.87) or rather between LDL-C and ApoB (r=0.93), the respective percentiles showed very similar patterns. Dyslipidemia prevalence were as follows: increased TC 7.8%, increased LDL 6.1%, increased TG 0–9 years 22.1%, increased TG 10–16 years 11.7%, and decreased HDL 8.0%. Conclusion: Age- and sex-related trends for all parameters are similar to those of the German KIGGS study. With the exception of HDL cholesterol, the prevalence of dyslipidemias in the German LIFE Child cohort are similar to the US-American prevalence.:I Abkürzungsverzeichnis - 03 - 1 Bibliographische Beschreibung - 04 - 2 Einleitung - 05 - 2.1 Hintergrund - 05 - 2.2 Serumlipide, Apolipoproteine und Dyslipidämien - 05 - 2.3 Referenzintervalle - 08 - 2.4 Soziodemographische Faktoren - 10 - 2.5 Die LIFE-Child Studie - 12 - 2.6 Hypothesen, Frage- und Zielstellungen - 13 - 3 Publikationen - 14 - 3.1 Pediatric reference data of serum lipids and prevalence of dyslipidemia: results from a population-based cohort in Germany - 15 - 3.2 Serum lipid levels were related to socio-demographic characteristics in a German population-based child cohort. Serum lipid levels and social class - 25 - 4 Zusammenfassung - 33 - 5 Literaturverzeichnis - 36 - II Anhang - 45 - III Erklärung über die eigenständige Abfassung der Arbeit - 64 - IV Curriculum vitae - 65 - V Danksagung - 66 - / Aim: Socio-demographic factors affect the development and lives of children and adolescents. We examined links between serum lipids and apolipoproteins and sociodemographic factors in the Leipzig Research Centre for Civilization Diseases Child (LIFE Child) study. Methods: The Winkler index and the Family Affluence Scale were used to define characteristics of the social status of 938 boys and 860 girls aged from birth to 19 years. We then used univariate and multivariate regression analyses to examine the sociodemographic impact on total cholesterol, low-density lipoprotein (LDL) cholesterol, highdensity lipoprotein (HDL), cholesterol triglycerides and apolipoproteins A1 (ApoA1) and B (ApoB). Results: No significant influences on the Winkler index or the Family Affluence Scale were observed regarding the concentrations of serum lipids for total cholesterol or LDL cholesterol. However, and most importantly, children and adolescents with high social status and high family affluence showed significantly higher HDL cholesterol and ApoA1 levels than those with lower individual totals. A higher Winkler index was associated with significantly lower values for triglycerides and ApoB. Conclusion: Adolescents with higher family wealth and social status showed a lower cardiovascular risk profile, as measured by the concentrations of HDL cholesterol and triglycerides as well as ApoA1 and B.:I Abkürzungsverzeichnis - 03 - 1 Bibliographische Beschreibung - 04 - 2 Einleitung - 05 - 2.1 Hintergrund - 05 - 2.2 Serumlipide, Apolipoproteine und Dyslipidämien - 05 - 2.3 Referenzintervalle - 08 - 2.4 Soziodemographische Faktoren - 10 - 2.5 Die LIFE-Child Studie - 12 - 2.6 Hypothesen, Frage- und Zielstellungen - 13 - 3 Publikationen - 14 - 3.1 Pediatric reference data of serum lipids and prevalence of dyslipidemia: results from a population-based cohort in Germany - 15 - 3.2 Serum lipid levels were related to socio-demographic characteristics in a German population-based child cohort. Serum lipid levels and social class - 25 - 4 Zusammenfassung - 33 - 5 Literaturverzeichnis - 36 - II Anhang - 45 - III Erklärung über die eigenständige Abfassung der Arbeit - 64 - IV Curriculum vitae - 65 - V Danksagung - 66 -

The role of topography on the spatial distribution of tornadoes was assessed through geospatial and statistical techniques. A 100-m digital elevation model was used to create slope, aspect, and surface roughness maps; and; tornado beginning and ending points and paths were used to extract terrain information. Tornado touchdowns, liftoffs, paths, and path-land angles were examined to determine whether tornado paths occur more frequently in or along certain terrain or slopes. Statistical analyses, such as bootstrapping, were used to analyze tornado touchdowns, liftoffs and paths and path-relative terrain angles. Results show that tornado paths are more common with downhill-movement. Tornadoes are not as likely to move uphill because the 73.6 percent northeast path bias represents the highest frequencies of path-angles. Tornado touchdowns and paths occur more often in smooth terrain, rather than rough terrain. Complex topographic variability seems to not have an effect on the spatial distribution of tornadoes.

The presented thesis describes the historical evolution of growth and laboratory reference values and the methods for their creation – leading finally to a family of methods applied in the WHO Multicentre Growth Reference Study (MGRS). The following part describes these methods, their assumptions, and model diagnostics. The original article at the beginning of part III combines these methods with resampling to be able to use LMS -type methods on data containing different dependencies like follow-up measures and family relationships. This method has been applied in the estimation of reference values of several laboratory values in the context of the LIFE child study. Three papers are already published and are also presented. In part III. The next section concentrates on the accompanying R package childsds. A short summary and outlook conclude the work.:I Introduction 1 Introduction 1.1 Standardization and Reference Values 1.2 The Scope of this Thesis 2 Historical Aspects 2.1 Development of Basic Concepts 2.2 Growth References and Growth Charts 2.3 Laboratory Reference Values II Methods 3 Method Selection 3.1 Nomenclature 3.2 The Choice of Statistical Method 3.3 GAMLSS 3.4 LMS-type Methods 4 Model Diagnostics 4.1 Normalized Quantile Residuals 4.2 Diagnostic Plots 4.3 Numerical Statistics III Original Articles & R Pakackage 5 Original Article 6 Articles Using the Proposed Method 7 R Package childsds 7.1 The Example Data 7.2 The Fitting Functions 7.3 The Classes 7.4 The Back-transformation Function 7.5 Convenience Functions 7.6 Collection of Reference Values 8 Summary and Outlook 8.1 Summary 8.2 SDS Values and Quality Control 8.3 The R Package / Die vorliegende Arbeit soll die Entwicklung und Bedeutung von Referenzwerten im pädiatrischen Kontext beschreiben. Der zugehörige Artikel beschreibt die Kombination der von der WHO empfohlene Methode und Resampling, um die Herleitung von Referenzwerten auch im Fall von Messwiederholungen und anderen Abhängigkeiten zu ermöglichen. Drei weitere Artikel, in denen die Methode angewendet wurde und eine kurze Vorstellung des zugehörigen, begleitend entwickelten R Pakets (statistische Software) bilden Teil III der Arbeit.:I Introduction 1 Introduction 1.1 Standardization and Reference Values 1.2 The Scope of this Thesis 2 Historical Aspects 2.1 Development of Basic Concepts 2.2 Growth References and Growth Charts 2.3 Laboratory Reference Values II Methods 3 Method Selection 3.1 Nomenclature 3.2 The Choice of Statistical Method 3.3 GAMLSS 3.4 LMS-type Methods 4 Model Diagnostics 4.1 Normalized Quantile Residuals 4.2 Diagnostic Plots 4.3 Numerical Statistics III Original Articles & R Pakackage 5 Original Article 6 Articles Using the Proposed Method 7 R Package childsds 7.1 The Example Data 7.2 The Fitting Functions 7.3 The Classes 7.4 The Back-transformation Function 7.5 Convenience Functions 7.6 Collection of Reference Values 8 Summary and Outlook 8.1 Summary 8.2 SDS Values and Quality Control 8.3 The R Package

Recent decades, growing trends in economic inequality have been observed across the world. While the underlying causes are many, there is a recent and growing field of research discussing the potential effects that changes in policy rates and other kinds of economic policy may have on economic inequality. While results among the existing pool of research varies, there seems to be a common perception that economic policy, and changes in policy rates in particular, does affect economic inequality. By conducting a panel data study of 34 countries over the course of 30 years, this thesis finds significant decreases in economic equality, measured using 3 different inequality measurements, following increases in the policy rate. The effects are similar in magnitude for all inequality measurements, however, due to the estimated effects being extremely small in relation to the context, the economic significance is initially questioned. By further analyzing the effects in terms of standard deviations, the thesis finds that estimated effects are of reasonable magnitude, and thus, the economic significance of the study is confirmed.

In the present study the cross sectional areas of individual muscle fibers were investigated with focus on statistical methodology. This thesis includes data from two studies; Resistance Study and Method Study. The Resistance Study analyzes the effect of exercise by comparing muscle fiber area before and after eight weeks of resistance training. Muscle biopsies from the vastus lateralis muscle were obtained from young male participants. The purpose of the Method Study was to examine the variation between right and left leg. Contrary to previous studies, this thesis focuses on individual data rather than on group-based data, and therefore takes a different approach than the previously published articles. This is proven to be successful since information is lost when analyzing group-wise, as the increase in small muscle fibers did not show when analyzing as a group. The results of the Resistance Study is similar to the results of the Method Study. Means and standard deviations have a wide spread both between subjects and between biopsies taken from the same subject. Inference on the 10th and 90th percentiles shows a positive pattern in the Resistance Study, in the sense that both the smallest and the largest muscle fibers have grown as a result of the resistance training. If muscle fiber area is used as a proxy for training effect, the conclusion is that many people seem to have responded well to the training.

Thesis (MScEng (Industrial Engineering))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The purpose of this study is to investigate the capacity of a proposed bus rapid transit (BRT) station in Cape Town. A bus rapid transit system is a high-capacity public transportation system that carries passengers from one point to another, providing a service that is faster and more efficient than an ordinary bus line. The implementation of these systems is increasing rapidly worldwide, serving as a solution to decrease traffic congestion. The capacity of the proposed bus station, known as the Thibault Station, is investigated in the study by developing a simulation model. The aim is to develop a stochastic simulation model, which represents the flow of passengers throughout the station so that the station capacity can be investigated. By developing a stochastic model as opposed to a deterministic model, elements of uncertainty can be included into the model, thereby representing a system that is closer to the real-life situation under investigation. The majority of BRT systems, as well as past studies undertaken on the Thibault Station, are designed using deterministic calculations. The study commences by researching literature on BRT systems and focuses on the current methods used to calculate station capacity requirements. Thereafter, the concept of simulation is introduced. Simulation is the method chosen to model and evaluate the passenger and bus operations at the Thibault Station. The study presents the methods used to build and verify the simulation model. This is done to familiarise the user with a number of aspects of the model. The model can then be used as a tool to investigate capacity parameters and alternative designs or scenarios. Using the results of these investigations, decisions can ultimately be made regarding the planning and design components of any bus rapid transit station given that the model is adapted. Scenario results, as well as interpretations of performance measurements, are presented at the end of the study. The results can be used for more realistic design of BRT stations using stochastic modelling. / AFRIKAANSE OPSOMMING: Die doel van die studie is om ondersoek in te stel na die kapasiteit van „n hoëspoed bus vervoer stelsel (BRT). Die ondersoek is gebaseer op „n voorgestelde bus stelsel vir Kaapstad. „n BRT-stelsel is „n hoë-kapasiteit publieke vervoerstelsel wat passasiers van een punt na „n ander vervoer, deur „n diens te verskaf wat vinniger en meer doeltreffend is as „n gewone bus stelsel. Die implementering van hierdie tipe stelsels neem wêreldwyd toe en dien as „n oplossing om verkeersopeenhopings te verminder. Die spesifieke busstasie wat ondersoek word staan bekend as die Thibault Stasie van Kaapstad se BRT-stelsel. Die kapasiteit van die stasie word ondersoek deur middel van simulasiemodellering. Die doel is om „n stogastiese simulasiemodel wat die vloei van passasiers modelleer te ontwikkel ten einde die kapasiteit van die stasie te ondersoek. Deur „n stogastiese model in plaas van „n deterministiese model te gebruik, kan elemente van onsekerheid ingesluit word. Dit verteenwoordig dus „n stelsel wat nader aan die werklikheid is. Tans word meeste BRT-stelsels ontwerpe gebaseer op deterministiese berekeninge, asook historiese studies wat onderneem is oor die Thibault Stasie. Die studie begin deur literatuur oor BRT-stelsels te bestudeer en fokus daarna op die huidige metodes wat gebruik word om die vereiste kapasiteit van „n busstasie te bepaal. Die konsep van simulasie word daarna voorgestel. Simulasie is die metode wat in die studie gebruik word om die passasier- en busaktiwiteite van die Thibault Stasie te modelleer en te evalueer. Die studie verskaf die metodes wat gebruik word vir die ontwikkeling en geldigmaak van die simulasiemodel. Gebruikers word op dié manier blootgestel aan die verskillende aspekte van die model. Nadat die gebruikers vertroud is met sekere aspekte van die model, word die model verder uiteengesit en word daar verduidelik hoe dit as „n instrument om kapasiteit parameters en alternatiewe ontwerpe van busstasies te ondersoek kan dien. Die resultate van die model kan gebruik word om beplannings- en ontwerpbesluite van enige busstasie te neem. Aan die einde van die studie word scenarioresultate bekendgestel, asook die interpretasie daarvan. Die resultate kan gebruik word vir meer realistiese ontwerp van BRT-stasies met behulp van stogastiese simulasie modellering.