In numerous clinical/experimental studies, multiple endpoints are measured on each subject. It is often not clear which of these endpoints should be designated as of primary importance. The desirability function approach is a way of combining multiple responses into a single unitless composite score. The response variables may include multiple types of data: binary, ordinal, count, interval data. Each response variable is transformed to a 0 to1 unitless scale with zero representing a completely undesirable response and one representing the ideal value. In desirability function methodology, weights on individual components can be incorporated to allow different levels of importance to be assigned to different outcomes. The assignment of the weight values are subjective and based on individual or group expert opinion. In this dissertation, it is our goal to find the weights or response variable transformations that optimize an external empirical objective criterion. For example, we find the optimal weights/transformations that minimize the generalized variance of a prediction regression model relating the score and response of an external variable in pre-clinical and clinical data. For application of the weighting/transformation scheme, initial weighting or transformation values must be obtained then calculation of the corresponding value of the composite score follows. Based on the selected empirical model for the analyses, parameter estimates are found using the usual iterative algorithms (e.g., Gauss Newton). A direct search algorithm (e.g., the Nelder-Mead simplex algorithm) is then used for the minimization of a given objective criterion i.e. generalized variance. The finding of optimal weights/transformations can also be viewed as a model building process. Here relative importance levels are given to each variable in the score and less important variables are minimized and essentially eliminated.

Desirability Functions continue to attract attention of scientists and researchers working in the area of multi-response optimization. There are many versions of such functions, differing mainly in formulations of individual and overall desirability functions. Derringer and Suich&rsquo / s desirability functions being used throughout this thesis are still the most preferred ones in practice and many other versions are derived from these. On the other hand, they have a drawback of containing nondifferentiable points and, hence, being nonsmooth. Current approaches to their optimization, which are based on derivative-free search techniques and modification of the functions by higher-degree polynomials, need to be diversified considering opportunities offered by modern nonlinear (global) optimization techniques and related softwares. A first motivation of this work is to develop a new efficient solution strategy for the maximization of overall desirability functions which comes out to be a nonsmooth composite constrained optimization problem by nonsmooth optimization methods. We observe that individual desirability functions used in practical computations are of mintype, a subclass of continuous selection functions. To reveal the mechanism that gives rise to a variation in the piecewise structure of desirability functions used in practice, we concentrate on a component-wise and generically piecewise min-type functions and, later on, max-type functions. It is our second motivation to analyze the structural and topological properties of desirability functions via piecewise max-type functions. In this thesis, we introduce adjusted desirability functions based on a reformulation of the individual desirability functions by a binary integer variable in order to deal with their piecewise definition. We define a constraint on the binary variable to obtain a continuous optimization problem of a nonlinear objective function including nondifferentiable points with the constraints of bounds for factors and responses. After describing the adjusted desirability functions on two well-known problems from the literature, we implement modified subgradient algorithm (MSG) in GAMS incorporating to CONOPT solver of GAMS software for solving the corresponding optimization problems. Moreover, BARON solver of GAMS is used to solve these optimization problems including adjusted desirability functions. Numerical applications with BARON show that this is a more efficient alternative solution strategy than the current desirability maximization approaches. We apply negative logarithm to the desirability functions and consider the properties of the resulting functions when they include more than one nondifferentiable point. With this approach we reveal the structure of the functions and employ the piecewise max-type functions as generalized desirability functions (GDFs). We introduce a suitable finite partitioning procedure of the individual functions over their compact and connected interval that yield our so-called GDFs. Hence, we construct GDFs with piecewise max-type functions which have efficient structural and topological properties. We present the structural stability, optimality and constraint qualification properties of GDFs using that of max-type functions. As a by-product of our GDF study, we develop a new method called two-stage (bilevel) approach for multi-objective optimization problems, based on a separation of the parameters: in y-space (optimization) and in x-space (representation). This approach is about calculating the factor variables corresponding to the ideal solutions of each individual functions in y, and then finding a set of compromised solutions in x by considering the convex hull of the ideal factors. This is an early attempt of a new multi-objective optimization method. Our first results show that global optimum of the overall problem may not be an element of the set of compromised solution. The overall problem in both x and y is extended to a new refined (disjunctive) generalized semi-infinite problem, herewith analyzing the stability and robustness properties of the objective function. In this course, we introduce the so-called robust optimization of desirability functions for the cases when response models contain uncertainty. Throughout this thesis, we give several modifications and extensions of the optimization problem of overall desirability functions.

Background: Antimicrobial-resistant infections result in hospital stays costing between $18,000 and $29,000. As of 2009, Centers for Medicare and Medicaid Services no longer upgrade payments for hospital-acquired infections. Hospital epidemiologists monitor and document rates of individual resistant microbes in antibiogram reports. Overall summary measures capturing resistance within a hospital may be useful. Objectives: We applied four techniques (L1- and L2-principal component analysis (PCA), desirability functions, and simple summary) to create summary measures of resistance and described the four summary measures with respect to reliability, proportion of variance explained, and clinical utility. Methods: We requested antibiograms from hospitals participating in the University HealthSystem Consortium for the years 2002–2008 (n=40). A clinical team selected organism-drug resistant pairs (as resistant isolates per 1,000 patient days) based on 1) virulence, 2) complicated or toxic therapies, 3) transmissibility, and 4) high incidence with increasing levels of resistance. Four methods were used to create summary scores: 1) L1- and L2-PCA: derived multipliers so that the variance explained is maximized; 2) desirability function: transformed resistance data to be between 0 and 1; 3) simple sum: each resistance rate was added and divided by the square root of the total number of microbes summed. Simple correlation analyses between time and each summary score evaluated reliability. For each year, we calculated the proportion of explained variance by dividing each summary score’s variance by the variance in the original data. Clinical utility was checked by comparing the trends for all of the individual microbe’s resistance rates to the trends seen in the summary scores for each hospital. Results: Proportion of variance explained by L1- and L2-PCA and the simple sum was 0.61, 0.62, and 0.29 respectively. Simple sum and L1- and L2-PCA summary scores best followed the trends seen in the individual antimicrobial resistance rates; trends in desirability function scores deviated from those seen in individual trends of antimicrobial resistance. L1- and L2-PCA summary scores were more influenced by MRSA rates, and the simple sum score was less influenced. Pearson correlation coefficients revealed good reliability through time. Conclusion: Deriving summary measures of antimicrobial resistance can be reliable over time and explain a high proportion of variance. Infection control practitioners and hospital epidemiologists may find the inclusion of a summary score of antimicrobial resistance beneficial in describing the trends of overall resistance in their yearly antibiogram reports.

Orientadores: Roy Edward Bruns, Isabel Cristina S. F. Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-09T16:30:39Z (GMT). No. of bitstreams: 1 Breitkreitz_MarciaCristina_M.pdf: 914312 bytes, checksum: 69f8afa4f7adf7b6e40961bef5aed1ee (MD5) Previous issue date: 2007 / Resumo: Este trabalho teve como objetivo o desenvolvimento de modelos combinados considerando simultaneamente o tipo de Fase Estacionária (FE) como variável de processo e diferentes composições de Fase Móvel (FM) como variáveis de mistura para descrever a influência de cada uma destas variáveis, bem como a interação entre elas, na separação de diversos compostos presentes em duas amostras: uma mistura de compostos neutros e uma mistura de agrotóxicos. Os experimentos necessários para a determinação dos coeficientes dos modelos foram realizados de acordo com um planejamento split-plot, no qual os tipos de FE, C 8 e C 18 foram considerados main-plots e as composições de FM, sub-plots. Os resultados foram tratados de duas maneiras: de acordo com a estrutura split-plot do planejamento e supondo completa aleatorização na realização dos experimentos. Para descrever a qualidade da separação, foi utilizada uma função objetiva e o procedimento de otimização simultânea de várias respostas, descrito por Derringer e Suich, empregando, neste caso, critérios elementares como fator de retenção, resolução e fator de separação como respostas. Os modelos foram avaliados empregando-se Análise da Variância quanto à significância dos tratamentos e falta de ajuste. Na descrição da qualidade da separação dos compostos nas duas misturas, o procedimento de Derringer e Suich se mostrou superior às funções objetivas, pois permitiu a construção de modelos tendo como respostas parâmetros cromatográficos, os quais são, de fato, funções da composição da FM e da FE. Estes modelos, quando combinados através da desejabilidade global, permitiram que as condições para a melhor separação de todos os compostos em cada mistura fossem alcançadas, sem perda de informações sobre a separação individual dos pares de picos. Todos os modelos apresentaram capacidade preditiva para as respostas avaliadas ¿ fatores de retenção/resolução/fatores de retenção relativos - nas duas misturas, sem ou com pequena falta de ajuste. Embora todos os planejamentos tenham sido realizados de acordo com um procedimento split-plot, não foram verificadas diferenças nos valores dos erros dos coeficientes dos modelos matemáticos nos cálculos split-plot e supondo completa aleatorização e isto se deveu ao fato do erro main-plot ter sido muito menor que o erro sub-plot / Abstract: The aim of this work was to develop combined statistical models including the stationary phase (SP) as process variables and different compositions of the mobile phase (MP) as mixture variables in order to describe the influence of each type of variable as well as their interactions for the separation of compounds in two samples sets: one containing ten neutral compounds and another containing eleven pesticides. The experiments required to determine the coefficients of the models were carried out according to a split-plot approach, in which the stationary phases, C 8 or C 18 were considered as main-plots and the mobile phase compositions as sub-plots. The results were treated according to the split-plot approach and also supposing a completely random setup. The results provided by an objective function were compared to those obtained by Derringer¿s desirability functions constructed with simple chromatographic criteria such as resolution and relative retention factors as responses. The models were evaluated by means of Analysis of Variance, regarding regression significance and lack of fit. In order to describe the quality of the separation of the compounds in the two mixtures, the desirability procedure was preferred over the objective functions because the responses used in the latter were, in fact, functions of the stationary and mobile phases. The models combined into a global desirability function allowed the best conditions to the separation of all compounds to be found, without loss of information on the individual peak separation. All models presented predictive capabilities for the responses evaluated with none or little lack of fit. Although the experiments were carried out according to a split-plot approach, no significant differences were found in coefficient errors comparing to the complete random approach, which can be explained based on the low main-plot error / Mestrado / Quimica Analitica / Mestre em Química