In Bayesian decision analysis, uncertainty and risk are accounted for with probabilities for the possible states, or states of nature, that affect the outcome of a decision. Application of Bayes’ theorem requires non-informative prior probabilities, which represent the probabilities of states of nature for a decision maker under complete ignorance. These prior probabilities are then subsequently updated with any and all available information in assessing probabilities for making decisions. The conventional approach for the non-informative probability distribution is based on Bernoulli’s principle of insufficient reason. This principle assigns a uniform distribution to uncertain states when a decision maker has no information about the states of nature. The principle of insufficient reason has three difficulties: it may inadvertently provide a biased starting point for decision making, it does not provide a consistent set of probabilities, and it violates reasonable axioms of decision theory. The first objective of this study is to propose and describe a new method to establish non-informative prior probabilities for decision making under uncertainty. The proposed decision-based method is focuses on decision outcomes that include preference in decision alternatives and decision consequences. The second objective is to evaluate the logic and rationality basis of the proposed decision-based method. The decision-based method overcomes the three weaknesses associated with the principle of insufficient reason, and provides an unbiased starting point for decision making. It also produces consistent non-informative probabilities. Finally, the decision-based method satisfies axioms of decision theory that characterize the case of no information (or complete ignorance). The third and final objective is to demonstrate the application of the decision-based method to practical decision making problems in engineering. Four major practical implications are illustrated and discussed with these examples. First, the method is practical because it is feasible in decisions with a large number of decision alternatives and states of nature and it is applicable to both continuous and discrete random variables of finite and infinite ranges. Second, the method provides an objective way to establish non-informative prior probabilities that capture a highly nonlinear relationship between states of nature. Third, we can include any available information through Bayes’ theorem by updating the non-informative probabilities without the need to assume more than is actually contained in the information. Lastly, two different decision making problems with the same states of nature may have different non-informative probabilities. / text

In light of drastically escalating costs for today’s medications, pharmacy benefit managers are seeking a constant balance of effectiveness and cost control. Step Therapy helps to address these concerns with a try medication “A” before medication “B” logic. Like all medical interventions, the possibility of unintended consequences exists. The purpose of this study was to determine if non-adherence results from application of Step Therapy for selected medication classes (antihyperlipidemics (specifically the HMG Co-A reductase inhibitors), angiotensin receptor blockers, uro-selective alpha-blockers, and dipeptidyl peptidase-4 inhibitors) in the Department of Defense. Using a retrospective database analysis, this study examined the primary adherence rate of subjects after they have been denied coverage due to Step Therapy intervention. Additionally, this study examined the association of demographic and service-related factors with the likelihood that a patient will be non-adherent after encountering the intervention. Finally, the study measured the time to adherence after intervention for those who were persistent after a Step Therapy claim rejection. STATA version 10.0 was used to conduct logistic regression analyses to meet the study objectives. After examination of 279,508 claims for 27,202 subjects, the estimated primary non-adherence rate following the Step Therapy intervention for all medication classes combined was 15.1%. Additionally, there was inter-class variability in this rate ranging between 13.1% and 19.5%. A statistical and practical difference was also noted in non-adherence rates between subjects who received care at the retail point of service versus those who received care at the mail order point of service. Subjects who received care through retail were nearly twice as likely to be non-adherent as those who received care in the mail order segment. For those subjects who were persistent with therapy, the median time-to-fill was estimated at 7 days. The occurrence of non-adherence following a Step Therapy intervention was clearly demonstrated through this study. Although this study provides good framework for designing interventions after claim rejection, further research would help to determine the health impact of primary non-adherence as well as the economic consequences of the intervention. / text

We considered a non-centered parameterization of the standard random-effects model, which is based on the Cholesky decomposition of the variance-covariance matrix. The regression type structure of the non-centered parameterization allows to choose a simple, conditionally conjugate normal prior on the Cholesky factor. Based on the non-centered parameterization, we search for a parsimonious variance-covariance matrix by identifying the non-zero elements of the Cholesky factors using Bayesian variable selection methods. With this method we are able to learn from the data for each effect, whether it is random or not, and whether covariances among random effects are zero or not. An application in marketing shows a substantial reduction of the number of free elements of the variance-covariance matrix. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Many prior distributions are suggested for variance parameters in the hierarchical model. The “Non-informative” interval of the conjugate inverse-gamma prior might cause problems. I consider three priors – conjugate inverse-gamma, log-normal and truncated normal for the variance parameters and do the numerical analysis on Gelman’s 8-schools data. Then with the posterior draws, I compare the Bayesian credible intervals of parameters using the three priors. I use predictive distributions to do predictions and then discuss the differences of the three priors suggested. / text

<p>With the development of modern data collection approaches, researchers may collect hundreds to millions of variables, yet may not need to utilize all explanatory variables available in predictive models. Hence, choosing models that consist of a subset of variables often becomes a crucial step. In linear regression, variable selection not only reduces model complexity, but also prevents over-fitting. From a Bayesian perspective, prior specification of model parameters plays an important role in model selection as well as parameter estimation, and often prevents over-fitting through shrinkage and model averaging.</p><p>We develop two novel hierarchical priors for selection and model averaging, for Generalized Linear Models (GLMs) and normal linear regression, respectively. They can be considered as "spike-and-slab" prior distributions or more appropriately "spike- and-bell" distributions. Under these priors we achieve dimension reduction, since their point masses at zero allow predictors to be excluded with positive posterior probability. In addition, these hierarchical priors have heavy tails to provide robust- ness when MLE's are far from zero.</p><p>Zellner's g-prior is widely used in linear models. It preserves correlation structure among predictors in its prior covariance, and yields closed-form marginal likelihoods which leads to huge computational savings by avoiding sampling in the parameter space. Mixtures of g-priors avoid fixing g in advance, and can resolve consistency problems that arise with fixed g. For GLMs, we show that the mixture of g-priors using a Compound Confluent Hypergeometric distribution unifies existing choices in the literature and maintains their good properties such as tractable (approximate) marginal likelihoods and asymptotic consistency for model selection and parameter estimation under specific values of the hyper parameters.</p><p>While the g-prior is invariant under rotation within a model, a potential problem with the g-prior is that it inherits the instability of ordinary least squares (OLS) estimates when predictors are highly correlated. We build a hierarchical prior based on scale mixtures of independent normals, which incorporates invariance under rotations within models like ridge regression and the g-prior, but has heavy tails like the Zeller-Siow Cauchy prior. We find this method out-performs the gold standard mixture of g-priors and other methods in the case of highly correlated predictors in Gaussian linear models. We incorporate a non-parametric structure, the Dirichlet Process (DP) as a hyper prior, to allow more flexibility and adaptivity to the data.</p> / Dissertation

Multivariate generalized linear mixed models (MGLMM) are used for jointly modeling the clustered mixed outcomes obtained when there are two or more responses repeatedly measured on each individual in scientific studies. The relationship among these responses is often of interest. In the clustered mixed data, the correlation could be present between repeated measurements either within the same observer or between different observers on the same subjects. This study proposes a series of in- dices, namely, intra, inter and total correlation coefficients, to measure the correlation under various circumstances of observations from a multivariate generalized linear model, especially for joint modeling of clustered count and continuous outcomes. Bayesian methods are widely used techniques for analyzing MGLMM. The need for noninformative priors arises when there is insufficient prior information on the model parameters. Another aim of this study is to propose an approximate uniform shrinkage prior for the random effect variance components in the Bayesian analysis for the MGLMM. This prior is an extension of the approximate uniform shrinkage prior. This prior is easy to apply and is shown to possess several nice properties. The methods are illustrated in terms of both a simulation study and a case example.

This qualitative multiple case study explored the identity development of three preservice teachers of literacy. The study focused on the prior knowledge of the preservice teachers of literacy and how their knowledge related to their identity development while in field experiences. The primary question that guided this study was: What is the nature of the construction of identity during field experiences for preservice teachers of literacy? Sub questions explored identity in field experiences and the role of prior pedagogical content knowledge to identity development. Findings indicated that an evolving habitus central to their identity as literacy teachers could be deduced that guided the preservice teachers as they interacted in the figured worlds of their field experiences related to literacy teaching. Also, prior knowledge as a component of identity served to help the preservice teachers author themselves in regard to their interactions with their cooperating teachers, students, and with the classroom and school environment. Findings further indicated that the preservice teachers of literacy relied on their prior knowledge to notice, critique, and anticipate. Noticing, critiquing, and anticipating led to further development of their identity as teachers of literacy in a circular manner. A recommendation for practice includes the use of the NCA/WR Identity Guide to help preservice teachers of literacy become aware of their identity during field experiences. Further, providing an opportunity for reflection when standardized tests are administered could lead to metacognition, which is helpful for the identity development of preservice teachers. Recommendations for future research include examining different populations of preservice teachers and further exploring standardized testing related to identity. This study showed that preservice teachers of literacy navigate a path of diverse experiences as they learn to author themselves in the figured worlds of the field experiences. These experiences serve to shape them as future teachers and continued exploration of the specifics of their identity development will assist in creating strong teachers who are equipped to face the challenges of providing quality literacy instruction.

Problems with students’ performance in Mathematics at tertiary level are common in South Africa − as it is worldwide. Pass rates at the university of technology where the researcher is a lecturer, are only about 50%. At many universities it has become common practice to refer students who do not have a reasonable chance to succeed at university level, for additional support to try to rectify this situation. However, the question is which students need such support? Because the Grade 12 marks are often not perceived as dependable, it has become common practice at universities to re-test students by way of an entrance exam or the "National Benchmark Test"- project. The question arises whether such re-testing is necessary, since it costs time and money and practical issues make it difficult to complete timeously. Many factors have an influence on performance in Mathematics. School-level factors include articulation of the curriculum at different levels, insufficiently qualified teachers, not enough teaching time and language problems. However, these factors also affect performance in most other subjects, but it is Mathematics and other subjects based on Mathematics that are generally more problematic. Therefore this study focused on the unique properties of the subject Mathematics. The determining role of prior knowledge, the step-by-step development of mathematical thinking, and conative factors such as motivation and perseverance were explored. Based on the belief that these factors would already have been reflected sufficiently in the Grade 12 marks, the correlation between the marks for Mathematics in Grade 12 and the Mathematics marks at tertiary level was investigated to assess whether it was strong enough for the marks in Grade 12 Mathematics to be used as a reliable predictor of success or failure at university level. It was found that the correlation between the marks for Mathematics Grade 12 and Mathematics I especially, was strong (r = 0,61). The Mathematics marks for Grade 12 and those for Mathematics II produced a correlation coefficient of rs = 0,52. It also became apparent that failure in particular could be predicted fairly accurately on the basis of the Grade 12 marks for Mathematics. No student with a Grade 12 Mathematics mark below 60% succeeded in completing Mathematics I and II in the prescribed two semesters, and only about 11% successfully completed it after one repetition. The conclusion was that the reliability of the prediction based on the marks for Grade 12 Mathematics was sufficient to refer students with a mark of less than 60% to receive some form of additional support. / MEd, Learning and Teaching, North-West University, Vaal Triangle Campus, 2011

This study explored the content of courtroom conversations about children's prior discussions regarding sexual abuse. Sixty felony child abuse trial transcripts including child testimony and reviewing court opinions were collected from the Court of Appeal and from court reporters. Information was obtained from under Section 288 of the California Penal code (sexual abuse of a child under 14 years of age) filed in Los Angeles County from 1997 to 2001. For this study, transcript testimony was transcribed, extracted for the necessary information, coded, assessed for reliability, and analyzed. The findings indicate that conversations about children's prior disclosure conversations, non-disclosure conversations, and conversations with perpetrators are present in nearly all cases of alleged child sexual abuse, although they only represent about 8% of questions asked of children. These courtroom conversations appear to mimic effects found throughout other child testimony research: children are often limited in their responsiveness unless open ended questions are asked and they rarely provide detailed content unless prompted to do so. The findings revealed that overt accusations, references to children's motives for telling or not telling, and conversations with the perpetrator about abuse were infrequently discussed by attorneys when interviewing child witnesses about their alleged sexual abuse during trial testimony. This was surprising as these topics are often discussed in the empirical literature as important factors to consider when assessing children's credibility. In the present study, children were often asked about what they disclosed generally, what was said during abusive acts, and what was (or was not) disclosed during specific prior conversations. Further, our results reflect that children's ultimate credibility assessment, as assessed by the outcome of the trial, related to the presence of non-disclosure questions and not the presence of disclosure questions or conversations between the perpetrator and child; cases without non-disclosure questions consistently resulted in a conviction. This study provided a first step in assessing the content of courtroom conversations about children's prior discussions about sexual abuse. Implications and future directions for research are discussed.

This study investigated the 9th grade students&#039 / achievement regarding diffusion and osmosis in relation to reasoning ability, prior knowledge and gender. A total of 397 ninth grade students participated in the study. The Test of logical thinking (TOLT) and the Diffusion and Osmosis Diagnostic Test (DODT) were administered to determine students&#039 / reasoning ability and achievement in diffusion and osmosis, respectively. DODT results showed that the range of correct answers for the first tier was 41 % to 91%. When both tiers were combined, the correct responses were reduced to a range of 21% to 61%. This result reveals that students have enough content knowledge but they don&rsquo / t know the underlying reason of their choice in diffusion and osmosis concepts. Pearson Product Moment correlations showed a statistically significant positive correlation between achievement and students&#039 / prior knowledge &amp / reasoning ability. MRC Analysis was conducted to determine the contribution of prior knowledge, reasoning ability and gender to the achievement. Prior knowledge and reasoning ability, but not gender, made a statistically significant contribution to the variation on achievement. Prior knowledge and reasoning ability together predicted 37 % of the variation on achievement. Stepwise multiple regression analysis was computed to determine the variables were best predicting students&rsquo / achievement. While prior knowledge explains 33 % of the variation in achievement, reasoning ability explains only 4 % of the variation in achievement. Results indicate that prior knowledge is a better predictor than reasoning ability in students&rsquo / achievement.