Browsing by Author "Molenberghs, Geert."

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Connectedness and the hyperspace of metric spaces.
(2015) Rathilal, Cerene.; Matthews, Glenda Beverley.; Molenberghs, Geert.
One of the prime motivations for studying hyperspaces of a metric space is to understand the original space itself. The hyperspace of a metric space X is the space 2X of all non-empty closed bounded subsets of it, endowed with the Hausdorff metric. Our purpose is to study, in particular, connectedness properties of X and its hyperspace. We shall be concerned with knowing if a property P is extensional, that is, if X has property P then so does the hyperspace, or if a property is P is re ective, that is, if the hyperspace has property P then so does X itself. The hyperspace 2X and its subspace C(X) will be the focus of our study. First the Hau- dorff metric, p, is considered and introduced for the hyperspace 2X which is also inherited by C(X). As in (Nadler; [8]), when X is a continuum, the property of compactness is shown to be extensional to 2X and C(X). This is further generalised, when it is shown that each of 2X and C(X) is arcwise connected and hence are each arcwise connected continua, when X is a continuum. The classical results, the Boundary Bumping Theorems (due to Janiszewski [4]), which provide the required conditions under which the component of a set intersects its boundary, is proved using the Cut Wire Theorem (Whyburn; [13]). As an ap- plication, the Boundary Bumping Theorem (for open sets) is used to show the existence of continua arising out of convergence, in the Continuum of Convergence Theorem(Nadler; [8]). Using a construction of Whitney( [12]), the existence of a Whitney map, , for 2X and ! for C(X) are given. Using u, a special function o : [0; 1] -! 2X (due to Kelley [3]) called a segment is considered in the study of the arc structure of 2X and C(X). The equivalence of the existence of an order arc in 2X and the existence of a segment in 2X is also shown. A segment homotopy is then utilised to show that if one of 2X or C(X) is contractible then so is the other. This is presented in the Fundamental Theorem of Contractible Hyperspaces. The relationship between local connectedness and connectedness im kleinen is examined in order to understand the properties of Peano continua. Property S, introduced by Sierpin- ski( [10]), is considered and its connection to local connectedness is examined. Furthermore, a result of Wojdyslawski( [15]), which shows that local connectedness is an extensional prop- erty of a continuum X to the hyperspaces 2X and C(X), is given. Local connectedness is also re ective if either 2X or C(X) is a locally connected metric continuum. Lastly, Property K, by Kelley( [3]) is examined and is shown to be a sufficient condition for a continuum X to have its hyperspaces 2X and C(X) to be contractible. Consequently, if X is a Peano continuum then 2X and C(X) are contractible.
The impact of missing data on clinical trials : a re-analysis of a placebo controlled trial of Hypericum perforatum (St Johns wort) and sertraline in major depressive disorder.
(Springer., 2014) Grobler, Anna Christina.; Matthews, Glenda Beverley.; Molenberghs, Geert.
Rationale and objective Hypericum perforatum (St John's wort) is used to treat depression, but the effectiveness has not been established. Recent guidelines described the analysis of clinical trials with missing data, inspiring the reanalysis of this trial using proper missing data methods. The objective was to determine whether hypericum was superior to placebo in treating major depression. Methods A placebo-controlled, randomized clinical trial was conducted for 8 weeks to determine the effectiveness of hypericum or sertraline in reducing depression, measured using the Hamilton depression scale. We performed sensitivity analyses under different assumptions about the missing data process. Results Three hundred forty participants were randomized, with 28 % lost to follow-up. The missing data mechanism was not missing completely at random. Under missing at random assumptions, some sensitivity analyses found no difference between either treatment arm and placebo, while some sensitivity analyses found a significant difference from baseline to week 8 between sertraline and placebo (−1.28, 95 % credible interval [−2.48; −0.08]), but not between hypericum and placebo (0.56, [−0.64;1.76]). The results were similar when the missing data process was assumed to be missing not at random. Conclusions There is no difference between hypericum and placebo, regardless of the assumption about the missing data process. There is a significant difference between sertraline and placebo with some statistical methods used. It is important to conduct an analysis that takes account of missing data using valid statistically principled methods. The assumptions about the missing data process could influence the results.
A perspective on incomplete data in longitudinal multi-arm clinical trials, with emphasis on pattern-mixture-model based methodology.
(2014) Grobler, Anna Christina.; Matthews, Glenda Beverley.; Molenberghs, Geert.
Missing data are common in longitudinal clinical trials. Rubin described three different missing data mechanisms based on the level of dependence between the missing data process and the measurement process. These are missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR). Data are MCAR when the probability of dropout is independent of both observed and unobserved data. Data are MAR when the probability of data being missing does not depend on the unobserved data, conditional on the observed data. When neither MCAR nor MAR is valid, data are MNAR. The aim of this thesis is to discuss statistical methodology required for analysing missing outcome data and provide valid statistical methods for the MAR, MCAR and MNAR scenarios. This thesis does not focus on data analysis where covariate data are missing. Under MCAR complete and available case analyses are valid. When data are MAR multiple imputation, likelihood-based models, inverse probability weighting and Bayesian models are valid. When data are MNAR pattern-mixture, selection and shared-parameter models are valid. These methods are illustrated by an in depth analysis of two data sets with missing data. The first data set is the SAPiT trial an open label, randomised controlled trial in HIVtuberculosis co-infected patients. Patients were randomised to three arms; each initiating antiretroviral therapy at a different time. CD4+ count, an indication of HIV progression, was measured at baseline and every 6 months for 24 months. The primary question was whether CD4+ count trajectory over time differed for the three treatment arms. The assumption that missing data are MCAR was not supported by the observed data. We performed a range of sensitivity analyses under both MAR and MNAR assumptions. The second data set is a placebo-controlled, randomised clinical trial conducted for 8 weeks to determine the effectiveness of hypericum or sertraline in reducing depression, measured by the Hamilton depression scale. The trial randomised 340 participants, with 28% lost to follow-up before Week 8. We performed a sensitivity analysis under different assumptions about the missing data process. The missing data mechanism was not MCAR. Under MAR assumptions, some of the sensitivity analyses found no difference between either of the treatment arms and placebo, while some found a significant difference between sertraline and placebo, but not between hypericum and placebo. This re-analysis contributed to the literature around the effectiveness of St John’s Wort because it changed the conclusions of the original analysis.