published_or_final_version / Statistics / Doctoral / Doctor of Philosophy

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in ful lment of the requirements for the degree of Master of Science in Statistics November, 2017 / This study presents survival modelling and evaluation of risk factors of graft survival in the context of kidney transplant data generated in South Africa. Beyond the Kaplan-Meier estimator, the Cox proportional hazard (PH) model is the standard method used in identifying risk factors of graft survival after kidney transplant. The Cox PH model depends on the proportional hazard assumption, which is rarely met. Assessing and accounting for this assumption is necessary before using this model. When the PH assumption is not valid, modi cation of the Cox PH model could o er more insight into parameter estimates and the e ect of time-varying predictors at di erent time points. This study aims to identify the survival model that will e ectively describe the study data by employing the Cox PH and parametric accelerated failure time (AFT) models. To identify the risk factors that mediate graft survival after kidney transplant, secondary data involving 751 adults that received a single kidney transplant in Charlotte Maxeke Johannesburg Academic Hospital between 1984 and 2004 was analysed. The graft survival of these patients was analysed in three phases (overall, short-term and long-term) based on the follow-up times. The Cox PH and AFT models were employed to determine the signi cant risk factors. The purposeful method of variable selection based on the Cox PH model was used for model building. The performance of each model was assessed using the Cox-Snell residuals and the Akaike Information Criterion. The t of the appropriate model was evaluated using deviance residuals and the delta-beta statistics. In order to further assess how appropriately the best model t the study data for each time period, we simulated a right-censored survival data based on the model parameter-estimates. Overall, the PH assumption was violated in this study. By extending the standard Cox PH model, the resulting models out-performed the standard Cox PH model. The evaluation methods suggest that the Weibull model is the most appropriate in describing the overall graft survival, while the log-normal model is more reasonable in describing short-and long-term graft survival. Generally, the AFT models out-performed the standard Cox regression model in all the analyses. The simulation study resulted in parameter estimates comparable with the estimates from the real data. Factors that signi cantly in uenced graft survival are recipient age, donor type, diabetes, delayed graft function, ethnicity, no surgical complications, and interaction between recipient age and diabetes. Statistical inferences made from the appropriate survival model could impact on clinical practices with regards to kidney transplant in South Africa. Finally, limitations of the study are discussed in the context of further studies. / MT 2018

This thesis focuses on estimation and inference for a large class of semiparametric estimands: the class of continuous functionals of regression functions. This class includes a number of estimands derived from causal inference problems, among then the average treatment effect for a binary treatment when treatment assignment is unconfounded and many of its generalizations for non-binary treatments and individualized treatment policies. Chapter 2, based on work with Stefan Wager, introduces the augmented minimax linear es- timator (AMLE), a general approach to the problem of estimating a continuous linear functional of a regression function. In this approach, we estimate the regression function, then subtract from a simple plug-in estimator of the functional a weighted combination of the estimated regression function’s residuals. For this, we use weights chosen to minimize the maximum of the mean squared error of the resulting estimator over regression functions in a chosen neighborhood of our estimated regression function. These weights are shown to be a universally consistent estimator our linear functional’s Riesz representer, the use of which would result in an exact bias correction for our plug- in estimator. While this convergence can be slow, especially when the Riesz representer is highly nonsmooth, the action of these weights on functions in the aforementioned neighborhood imitates that of the Riesz representer accurately even when they are slow to converge in other respects. As a result, we show that under no regularity conditions on the Riesz representer and minimal regularity conditions on the regression function, the proposed estimator is semiparametrically efficient. In simulation, it is shown to perform very well in the context of estimating the average partial effect in the conditional linear model, a simultaneous generalization of the average treatment effect to address continuous-valued treatments and of the partial linear model to address treatment effect heterogeneity. Chapter 3, based on work with Arian Maleki and José Zubizarreta, studies the minimax linear estimator, a simplified version of the AMLE in which the estimated regression function is taken to be zero, for a class of estimands generalizing the mean with outcomes missing at random. We show semiparametric efficiency under conditions that are only slightly stronger than those required for the AMLE. In addition, we bound the deviation of our estimator’s error from the averaged efficient influence function, characterizing the degree to which the first order asymptotic characterization of semiparametric efficiency is meaningful in finite samples. In simulation, this estimator is shown to perform well relative to alternatives in high-noise, small-sample settings with limited overlap between the covariate distribution of missing and nonmissing units, a setting that is challenging for approaches reliant on accurate estimation of either or both of the regression function and the propensity score. Chapter 4 discusses an approach to rounding linear estimators for the targeted average treatment effect into matching estimators. The targeted average treatment effect is a generalization of the average treatment effect and the average treatment effect on the treated units.

在多組壽命試驗中，為了準確地估計模型的參數，我們必須找出最合適的實驗品數量，以分配給每一個應力水平。近來, Ng, Chan and Balakrishnan(2006)，在完整樣本情況下，利用「極值回歸模型」發展了找尋實驗品數量最合適的分配方法。其後，Ka, Chan, Ng and Balakrishnan (2011)在同一個回歸模型下，研究了對於「II型截尾樣本」最合適的分配方法。因為我們仍未確立對「I型截尾樣本」的最合適分配方法，所以我們將會在本篇論文中探討如何在「I型截尾壽命試驗」中找出最合適的實驗品分配方法。 / 在本論文中，我們會利用最大似然估計的方法去估計模型參數。我們也會計算出「逆費雪訊息矩陣」(「漸近方差協方差矩陣」)I⁻¹，用以量度參數估計值的準確度。以下是三個對最合適分配方法的決定準則: / 1.費雪訊息矩陣的行列式最大化, / 2. ν1估計值的方差最小化, var( ν1)(V -優化準則 ) / 3.漸近方差協方差矩陣的跡最小化, tr(⁻¹)(A-優化準則 ) / 我們也會討論在「極值回歸模型」的特例：「指數回歸模型」之下最合適的分配方法。 / In multi-group life-testing experiment, it is essential to optimize the allocation of the items under test to dierent stress levels in order to estimate the model parameter accurately. Recently Ng, Chan and Balakrishnan(2006) developed the optimal allocation for complete sample case with extreme value regression model, and Ka, Chan, Ng and Balakrishnan (2011) discussed about the optimal allocation for Type -II censoring cases with the same model. The optimal allocation for Type-I censoring scheme has not been established, so in this thesis, we are going to investigate the optimal allocation if Type-I censoring scheme is adopted in life-testing experiment. / Maximum likelihood estimation method will be adopted in this thesis for estimating model parameter. The inverted Fisher information matrix (asymptotic variance -covariance matrix),I⁻¹ , will be derived and used to measure the accuracy of the estimated parameters. The optimal allocation will be determined based on three optimal criteria: / 1. Maximizing the determinant of the expected Fisher Information matrix, / 2. Minimizing the variance of the estimator of ν1, var( ν1) (V -optimality ) / 3. Minimizing the trace of the variance-covariance matrix, tr(I⁻¹) (A-optimality ) / Optimal allocation under the exponential regression model,which is a spe¬cial case of extreme value regression model, will also be discussed. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / So, Hon Yiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 46-48). / Abstracts also in Chinese. / Abstract --- p.i / Acknowledgement --- p.i / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Accelerated Life Test --- p.1 / Chapter 1.2 --- Life-Stress Relationship --- p.1 / Chapter 1.3 --- Type I Censoring --- p.3 / Chapter 1.4 --- Optimal Allocation --- p.3 / Chapter 1.5 --- The Scope of the Thesis --- p.4 / Chapter 2 --- Extreme Value Regression Model --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Model and Maximum Likelihood Estimation --- p.5 / Chapter 2.3 --- Expected Fisher Information --- p.8 / Chapter 3 --- Criteria for Optimization and the Optimal Allocation --- p.12 / Chapter 3.1 --- Introduction --- p.12 / Chapter 3.2 --- Criteria for Optimization --- p.12 / Chapter 3.3 --- Numerical Illustrations and the Optimal Allocation --- p.14 / Chapter 4 --- Sensitivity Analysis --- p.17 / Chapter 4.1 --- Introduction --- p.17 / Chapter 4.2 --- Sensitivity Analysis --- p.17 / Chapter 4.3 --- Numerical Illustrations --- p.19 / Chapter 4.3.1 --- Illustration with McCool (1980) Data --- p.19 / Chapter 4.3.2 --- Further Study --- p.21 / Chapter 5 --- Exponential Regression Estimation --- p.26 / Chapter 5.1 --- Introduction --- p.26 / Chapter 5.2 --- The Model and the Likelihood Inference --- p.27 / Chapter 5.3 --- Optimal Sample Size Allocation for Estimation of Model Pa- rameters --- p.30 / Chapter 5.4 --- Numerical Illustration --- p.33 / Chapter 5.5 --- Sensitivity Analysis --- p.35 / Chapter 5.5.1 --- Parameter Misspeci cation --- p.35 / Chapter 5.5.2 --- Censoring Time --- p.38 / Chapter 5.5.3 --- Further Study --- p.40 / Chapter 6 --- Conclusion and Further Research --- p.44

The explicit use of networks in modelling stochastic processes such as epidemic dynamics has revolutionised research into understanding the impact of contact pattern properties, such as degree heterogeneity, preferential mixing, clustering, weighted and dynamic linkages, on how epidemics invade, spread and how to best control them. In this thesis, I worked on mean-field approximations of stochastic processes on networks with particular focus on weighted and dynamic networks. I mostly used low dimensional ordinary differential equation (ODE) models and explicit network-based stochastic simulations to model and analyse how epidemics become established and spread in weighted and dynamic networks. I begin with a paper presenting the susceptible-infected-susceptible/recovered (SIS, SIR) epidemic models on static weighted networks with different link weight distributions. This work extends the pairwise model paradigm to weighted networks and gives excellent agreement with simulations. The basic reproductive ratio, R0, is formulated for SIR dynamics. The effects of link weight distribution on R0 and on the spread of the disease are investigated in detail. This work is followed by a second paper, which considers weighted networks in which the nodal degree and weights are not independent. Moreover, two approximate models are explored: (i) the pairwise model and (ii) the edge-based compartmental model. These are used to derive important epidemic descriptors, including early growth rate, final epidemic size, basic reproductive ratio and epidemic dynamics. Whilst the first two papers concentrate on static networks, the third paper focuses on dynamic networks, where links can be activated and/or deleted and this process can evolve together with the epidemic dynamics. We consider an adaptive network with a link rewiring process constrained by spatial proximity. This model couples SIS dynamics with that of the network and it investigates the impact of rewiring on the network structure and disease die-out induced by the rewiring process. The fourth paper shows that the generalised master equations approach works well for networks with low degree heterogeneity but it fails to capture networks with modest or high degree heterogeneity. In particular, we show that a recently proposed generalisation performs poorly, except for networks with low heterogeneity and high average degree.

Chen, Hok. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 69-71). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Review of benchmarking methods --- p.8 / Chapter 2.1 --- Regression method --- p.9 / Chapter 2.2 --- Signal extraction method with known autocovariance of signal --- p.11 / Chapter 2.3 --- Signal extraction method with unknown autocovariance of signal --- p.16 / Chapter 3 --- Survey error modelling for MA(1) model --- p.21 / Chapter 3.1 --- A method proposed by Chow and Lin --- p.21 / Chapter 3.2 --- An alternate method proposed by Chen and Wu --- p.28 / Chapter 3.2.1 --- Original sketch for estimating 0 using annual benchmarks --- p.28 / Chapter 3.2.2 --- Nonstationary assumption for η(t) --- p.30 / Chapter 3.2.3 --- Estimating ve*ε(k) from data --- p.34 / Chapter 3.2.4 --- Simulation results --- p.36 / Chapter 4 --- Simulation Studies on Benchmarking --- p.42 / Chapter 4.1 --- Simulation procedure --- p.42 / Chapter 4.2 --- Simulation results --- p.46 / Chapter 5 --- Conclusion --- p.66 / References --- p.68

published_or_final_version / Statistics / Master / Master of Philosophy

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