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Personalised medicine is replacing the one-drug-fits-all approach with many prognostic models incorporating biomarkers available for risk stratifying patients. Evidence has been emerging that the effects of biomarkers change over time and therefore violate the assumption of proportional hazards when performing Cox regression. Analysis using the Cox model when the assumptions are invalid can result in misleading conclusions. This thesis reviews existing approaches for the analysis of non-proportional effects with respect to survival data. A number of well-developed approaches were identified but to date their uptake in practice has been limited. There is a need for more widespread use of flexible modelling to move away from standard analysis using a Cox model when the assumption of proportional hazards is violated. Two novel approaches were applied to investigate the impact of follow-up duration on two residual risk models, IHC4 and Mammostrat, for predicting risk in early breast cancers using two studies with different lengths of follow up; the Edinburgh Breast Conservation Series (BCS) and the Tamoxifen versus Exemestane Adjuvant Multinational (TEAM) trial. Similar results were observed between the two approaches that were considered, the multivariable fractional polynomial time (MFPT) approach and Royston-Parmer flexible parametric models, with their respective advantages and disadvantages being discussed. The analyses identified a strong time-varying effect of IHC4 score with the prognostic effect of IHC4 score on time-to distant recurrence decreasing with increasing follow-up time. Mammostrat score identified a group of patients with an increased risk of distant recurrence over full follow-up in the TEAM and Edinburgh BCS cohorts. The results suggest a combined IHC4 and Mammostrat risk score could provide information on the risk of recurrence and warrants further study.

The paper studies structural credit risk models with incomplete information of the asset value. It is shown that the pricing of typical corporate securities such as equity, corporate bonds or CDSs leads to a nonlinear filtering problem. This problem cannot be tackled with standard techniques as the default time does not have an intensity under full information. We therefore transform the problem to a standard filtering problem for a stopped diffusion process. This problem is analyzed via SPDE results from the filtering literature. In particular we are able to characterize the default intensity under incomplete information in terms of the conditional density of the asset value process. Moreover, we give an explicit description of the dynamics of corporate security prices. Finally, we explain how the model can be applied to the pricing of bond and equity options and we present results from a number of numerical experiments.

In this thesis, we consider a generalization of the classical Gerber-Shiu function in various risk models. The generalization involves introduction of two new variables in the original penalty function including the surplus prior to ruin and the deficit at ruin. These new variables are the minimum surplus level before ruin occurs and the surplus immediately after the second last claim before ruin occurs. Although these quantities can not be observed until ruin occurs, we can still identify their distributions in advance because they do not functionally depend on the time of ruin, but only depend on known quantities including the initial surplus allocated to the business. Therefore, some ruin related quantities obtained by incorporating four variables in the generalized Gerber-Shiu function can help our understanding of the analysis of the random walk and the resultant risk management. In Chapter 2, we demonstrate the generalized Gerber-Shiu functions satisfy the defective renewal equation in terms of the compound geometric distribution in the ordinary Sparre Andersen renewal risk models (continuous time). As a result, forms of joint and marginal distributions associated with the variables in the generalized penalty function are derived for an arbitrary distribution of interclaim/interarrival times. Because the identification of the compound geometric components is difficult without any specific conditions on the interclaim times, in Chapter 3 we consider the special case when the interclaim time distribution is from the Coxian class of distribution, as well as the classical compound Poisson models. Note that the analysis of the generalized Gerber-Shiu function involving three (the classical two variables and the surplus after the second last claim) is sufficient to study of four variable. It is shown to be true even in the cases where the interclaim of the first event is assumed to be different from the subsequent interclaims (i.e. delayed renewal risk models) in Chapter 4 or the counting (the number of claims) process is defined in the discrete time (i.e. discrete renewal risk models) in Chapter 5. In Chapter 6 the two-sided bounds for a renewal equation are studied. These results may be used in many cases related to the various ruin quantities from the generalized Gerber-Shiu function analyzed in previous chapters. Note that the larger number of iterations of computing the bound produces the closer result to the exact value. However, for the nonexponential bound the form of bound contains the convolution involving usually heavy-tailed distribution (e.g. heavy-tailed claims, extreme events), we need to find the alternative method to reinforce the convolution computation in this case.

In this thesis, we consider a generalization of the classical Gerber-Shiu function in various risk models. The generalization involves introduction of two new variables in the original penalty function including the surplus prior to ruin and the deficit at ruin. These new variables are the minimum surplus level before ruin occurs and the surplus immediately after the second last claim before ruin occurs. Although these quantities can not be observed until ruin occurs, we can still identify their distributions in advance because they do not functionally depend on the time of ruin, but only depend on known quantities including the initial surplus allocated to the business. Therefore, some ruin related quantities obtained by incorporating four variables in the generalized Gerber-Shiu function can help our understanding of the analysis of the random walk and the resultant risk management. In Chapter 2, we demonstrate the generalized Gerber-Shiu functions satisfy the defective renewal equation in terms of the compound geometric distribution in the ordinary Sparre Andersen renewal risk models (continuous time). As a result, forms of joint and marginal distributions associated with the variables in the generalized penalty function are derived for an arbitrary distribution of interclaim/interarrival times. Because the identification of the compound geometric components is difficult without any specific conditions on the interclaim times, in Chapter 3 we consider the special case when the interclaim time distribution is from the Coxian class of distribution, as well as the classical compound Poisson models. Note that the analysis of the generalized Gerber-Shiu function involving three (the classical two variables and the surplus after the second last claim) is sufficient to study of four variable. It is shown to be true even in the cases where the interclaim of the first event is assumed to be different from the subsequent interclaims (i.e. delayed renewal risk models) in Chapter 4 or the counting (the number of claims) process is defined in the discrete time (i.e. discrete renewal risk models) in Chapter 5. In Chapter 6 the two-sided bounds for a renewal equation are studied. These results may be used in many cases related to the various ruin quantities from the generalized Gerber-Shiu function analyzed in previous chapters. Note that the larger number of iterations of computing the bound produces the closer result to the exact value. However, for the nonexponential bound the form of bound contains the convolution involving usually heavy-tailed distribution (e.g. heavy-tailed claims, extreme events), we need to find the alternative method to reinforce the convolution computation in this case.

Pøedlo¾ená práce se zabývá dvìma pøístupy øe¹ení problému volby portfolia. Prvním jsou čmean-riskÿ modely, které minimalizují riziko pro pøedem zvolený výnos nebo maximalizují výnos pro pevnì stanovené riziko. Druhým je princip stochastické dominance, úzce související s teorií u¾itku. Cílem této diplomové práce je zkoumat vztah mezi mno¾inami e cientních portfolií, které jsou øe¹e- ním v obou pøístupech. Pro kvanti kaci rizika se kromì základních mìr jako jsou rozptyl, V aR nebo CV aR v práci uva¾ují i spektrální míry, zohledòující sub- jektivní postoj investora k riziku. Uká¾eme, za platnosti jakých podmínek jsou modely minimalizující spektrální míry konzistentní se stochastickou dominancí druhého øádu (SSD). Aplikujeme Kopa-Postùv test, který je jedním z více testù na SSD e cienci portfolia, na reálná data z americké burzy cenných papírù a SSD e cientní portfolia porovnáme s e cientními portfoliami získanými minimalizací CV aR-u uva¾ovaného na rùznych hladinách spolehlivosti. 1

Cholera made an unforeseen appearance on the eastern coast of South Africa in the province of KwaZulu-Natal (KZN) in August 2000. Having started from the more urban centres of the coastal region of the province, cholera proceeded unabated to the interior of the province where no community was spared from the scourge. Despite prompt medical intervention, health education and media awareness campaigns, cholera continued to spread throughout KZN. By March 2004, the official statistics of cholera cases in KZN as per the Cholera Database records, stood at 158 895 cases (Dept-KZN Health, 2000). The death toll as reported in the Cholera Database was 575 persons that translated to a percentage case fatality rate of 0.36%; the lowest when compared to the previous epidemics recorded in South African (Kustner et al., 1981; Küstner and du Plessis, G. 1991). An interesting feature of the epidemic was that 99% of the cases recorded by the central and provincial Departments of Health during the height of the epidemic were all from KZN. The question then was, what factors played a role in the cholera epidemic of KZN? This study sought to understand the outbreak and the factors that possibly contributed to the spread of the 2000-2004 cholera epidemic in KZN. The drivers of disease associated with the communities affected by cholera were also explored by analysing the complex and dynamic interaction of their biological, socio economic, and environmental nature over time and space. The nature of the study was such that it called for a multi faceted design to involve not just understanding the societal aspect of the disease but its demographic, ecological and spatial characteristics as well. Thus GIS was used as a research tool to facilitate the comparison of the disease trends and risk factors on a spatial level in order to determine the possible role(s) played by the different environmental and socio-economic drivers. The objective of the study was to investigate the possible role of the natural environment i.e. temperature, rainfall and humidity as the primary factors that influence cholera outbreaks in KZN; on the basis of its uniqueness in climatic conditions as compared to other areas of the Republic of South Africa (RSA). The other socio-economical and demographic factors were considered as factors that enhance the spread of the disease. As such, the exploration of the Cholera Database by use of spreadsheet, statistical correlations and spatial mapping using GIS technology mutually investigated the relationships between the different variables that came up as important factors in the spread of cholera. Results indicated that 52% of the total cholera cases in KZN were reported from DC28 (Uthungulu), making it the focal point of the epidemic. In general, all the age groups were represented in the cholera database though the age groups 15-19 years and 0-4 years featured more prominently in the overall epidemic picture. On average the male to female case ratio was 1:1.5 respectively. The major cholera peak was experienced in 2001 and a minor peak in 2002. Both the peaks appeared during the summer months, which are also characterised by heavy rains. The issues that were statistically proven to be associated with the spread of the disease were related to issues highlighting the inefficiencies in the provision of water and sanitation, which go hand in hand with poverty. Thus poverty was indirectly reflected in the data as an issue that compounded the cholera epidemic. There was no statistical correlation between the incidence of cholera and the climatic variables of rainfall, humidity and temperature. Notwithstanding, there was an overall seasonality revealed by the data, as seen with the cases peaking and waning between the summers and the winters respectively. Furthermore, GIS mapping revealed a concurrence between the incidence of cholera and the climatic variables of rainfall, humidity and maximum temperature. At the spatial level, the characteristics of the epidemic as revealed by the GIS maps and spatial modelling highlighted possible relationships between the incidence of cholera and the various socio-economic and climatic variables (Chapter 6: 6.2.3; 6.2.3). The spatial disease picture displayed a link between climatic seasons and the incidence of cholera. Spatial modelling offered more insight that the statistically supported climatic and socio-economic aspects were indeed important factors in guiding cholera outbreak predictions in the future. The cholera model illustrated this as it selected for areas considered to be at high risk for cholera (Map 34). The results give an altogether holistic portrayal of the cholera epidemic from all perspectives and also supported to the hypothesis that cholera is a function of social and environmental factors. The results from this study further confirm the negative health effects of inadequacies in basic services delivery. The study made use of data resources to understand the relationships between the incidence of cholera and the different demographic, socio-economic and climatic variables implicated in the spread of cholera epidemics (Chapter 3: 3.3.3). It also emphasizes the importance of using reliable data as a management tool to model various scenarios in order to obtain information that could be used in the prediction and management of diseases like cholera at the community level in the future. / Thesis (PhD (Water Resource Management))--University of Pretoria, 2006. / Microbiology and Plant Pathology / unrestricted

京都大学 / 新制・論文博士 / 博士(医学) / 乙第13554号 / 論医博第2283号 / 新制||医||1067(附属図書館) / (主査)教授 村井 俊哉, 教授 永井 洋士, 教授 井上 治久 / 学位規則第4条第2項該当 / Doctor of Medical Science / Kyoto University / DFAM

My dissertation considers solving of linear programming problems with p-order conic constraints that are related to a class of stochastic optimization models with risk objective or constraints that involve higher moments of loss distributions. The general proposed approach is based on construction of polyhedral approximations for p-order cones, thereby approximating the non-linear convex p-order conic programming problems using linear programming models. It is shown that the resulting LP problems possess a special structure that makes them amenable to efficient decomposition techniques. The developed algorithms are tested on the example of portfolio optimization problem with higher moment coherent risk measures that reduces to a p-order conic programming problem. The conducted case studies on real financial data demonstrate that the proposed computational techniques compare favorably against a number of benchmark methods, including second-order conic programming methods.

Objectives: Cardiovascular disease is the leading cause of mortality and morbidity in the developed world. Surgery can improve prognosis and relieve symptoms. Risk prediction models are increasingly being used to inform clinicians and patients about the risks of surgery, to facilitate clinical decision making and for the risk-adjustment of surgical outcome data. The importance of risk prediction models in cardiovascular surgery has been highlighted by the publication of cardiovascular surgery outcome data and the need for risk-adjustment. The overall objective of this thesis is to advance risk prediction modelling in cardiovascular surgery with a focus on the development of models for elective AAA repair and assessment of models for cardiac surgery. Methods: Three large clinical databases (two elective AAA repair and one cardiac surgery) were utilised. Each database was cleaned prior to analysis. Logistic regression was used to develop both regional and national risk prediction models for mortality following elective AAA repair. A regional model to identify the risk of developing renal failure following elective AAA repair was also developed. The performance of a widely used cardiac surgery risk prediction model (the logistic EuroSCORE) over time was evaluated using a national cardiac database. In addition an updated model version (EuroSCORE II) was validated and both models’ performance in emergency cardiac surgery was evaluated. Results: Regional risk models for mortality following elective AAA repair (VGNW model) and a model to predict post-operative renal failure were developed. Validation of the model for mortality using a national dataset demonstrated good performance compared to other available risk models. To improve generalisability a national model (the BAR score) with better discriminatory ability was developed. In a prospective validation of both models using regional data, the BAR score demonstrated excellent discrimination overall and good discrimination in procedural sub-groups. The EuroSCORE was found to have lost calibration over time due to a fall in observed mortality despite an increase in the predicted mortality of patients undergoing cardiac surgery. The EuroSCORE II demonstrated good performance for contemporary cardiac surgery. Both EuroSCORE models demonstrated inadequate performance for emergency cardiac surgery. Conclusions: Risk prediction models play an important role in cardiovascular surgery. Two accurate risk prediction models for mortality following elective AAA repair have been developed and can be used to risk-adjust surgical outcomes and facilitate clinical decision making. As surgical practice changes over time risk prediction models may lose accuracy which has implications for their application. Cardiac risk models may not be sufficiently accurate for high-risk patient groups such as those undergoing emergency surgery and specific emergency models may be required. Continuing research into new risk factors and model outcomes is needed and risk prediction models may play an increasing role in clinical decision making in the future.