Effect of blocking on estimation of response surface

The conditions under which experimental trials are performed may not, in general, be homogeneous. However, it may be possible to achieve a greater degree of homogeneity by carying out limited groups—or block—of trials. In this case, when fitting a response surface model, the least squares estimates of the model's parameters will generally depend on how the response surface design is blocked. The purpose of this article is to demonstrate the effects of the blocks on estimating the mean response, on the prediction variance and on the optimum of the response surface. These are all shown to be affected by the sizes of the blocks and the allocation of experimental runs to the blocks. In particular, the prediction variance increases as a result of blocking. In the special case of an orthogonally blocked design, the least squares estimates of the fitted model's parameters remain unchanged by the block effect, except possibly for the intercept. The increase in the prediction variance in this case depends only on the sizes of the blocks. A numerical example is also given.     

Counting strings at height j in Dyck paths

Let τ be an arbitrary lattice path, called in this context string, consisting of two kinds of steps (rises and falls) and let j be a non-negative integer.In this paper, the explicit formula for the generating function F_j associated with the Dyck path statistic “number of occurrences of τ at height j” is evaluated.For the expression of F_j some basic characteristics of the string are used, namely its number of rises, height, depth and periodicity, as well as the generating function of the Catalan numbers.     

A course for clinical trial personnel in clinical study designs, randomization, allocation schedules, and interactive response systems

This continuing education course for professionals involved in all areas of clinical trials integrates concepts related to the role of randomization in the scientific process. The course includes two interactive lecture and discussion sections and a workshop practicum. The first interactive lecture introduces basic clinical trial issues and statistical principles such as bias, blinding, randomization, control groups, and the importance of formulating clear and discriminating clinical and statistical hypotheses. It then focuses on the most commonly used clinical study designs and the corresponding patient randomization schemes. The second interactive lecture focuses on the implementation of randomization of patients and drug supply through allocation and component ID schedules. The workshop practicum, conducted in small groups, enables students to apply the lecture concepts to real clinical studies. Flexibility was built into the workshop practicum materials to allow the course content to be customized to specific audiences, and the interactive lecture sessions can be stretched to cover more advanced topics according to class interest and time availability.     

Analysis of the Bovine Spongiform Encephalopathy Maternal Cohort Study: Evidence for Direct Maternal Transmission

In an initial exploratory analysis of the bovine spongiform encephalopathy (BSE) maternal cohort study data we demonstrate several confounding effects in the study design. Given these effects, we assess a variety of statistical models to determine the relative contributions of direct maternal transmission of the aetiological agent of BSE and of genetic susceptibility to the observed maternally enhanced risk of BSE in the offspring of affected dams. To control for the substantial between-herd variation in the risk of exposure to the BSE agent, it is essential that analyses take into account the matched pair structure of the data. Maternal exposure is estimated to be most important in animals born within 150 days of disease onset in their dams. The analysis of a full survival likelihood model indicates that the hypothesis of maternal transmission with no genetic variation in susceptibility fits the data significantly better than the hypothesis of genetically variable susceptibility with no maternal transmission. However, models incorporating both maternal transmission and genetically variable susceptibility fit the data significantly better than pure maternal transmission models. Although genetic susceptibility cannot be excluded as the cause of the cohort study results in the absence of detailed genotyping, the analysis of these study data suggest that low level maternal transmission of BSE is, at least in part, responsible for the significantly enhanced risk of BSE in the offspring of affected dams. Similar results indicating significant maternal transmission in the later stages of the dam incubation period are obtained from the independent analysis of data on the dam–offspring relationships among confirmed BSE cases.     

Pooling multivariate data under W, LR and LM tests

Two independent random samples are drawn from two multivariate normal populations with mean vectors μ1 and μ2 and a common variance-covariance matrix Σ. Ahmed and Saleh (1990) considered preliminary test maximum likelihood estimator (PMLTE) for estimating μ1 based on the Hotelling's T _N ², when it is suspected that μ1=μ2. In this paper, the PTMLE based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are considered. Using the quadratic risk function, the conditions of superiority of the proposed estimator for departure parameter are derived. A max-min rule for the size of the preliminary test of significance is presented. It is demonstrated that the PTMLE based on W test produces the highest minimum guaranteed efficiencies compared to UMLE among the three test procedures.     

Prior distributions for stratified capture-recapture models

We consider the Arnason-Schwarz model, usually used to estimate survival and movement probabilities from capture-recapture data. A missing data structure of this model is constructed which allows a clear separation of information relative to capture and relative to movement. Extensions of the Arnason-Schwarz model are considered. For example, we consider a model that takes into account both the individual migration history and the individual reproduction history. Biological assumptions of these extensions are summarized via a directed graph. Owing to missing data, the posterior distribution of parameters is numerically intractable. To overcome those computational difficulties we advocate a Gibbs sampling algorithm that takes advantage of the missing data structure inherent in capture-recapture models. Prior information on survival, capture and movement probabilities typically consists of a prior mean and of a prior 95% credible confidence interval. Dirichlet distributions are used to incorporate some prior information on capture, survival probabilities, and movement probabilities. Finally, the influence of the prior on the Bayesian estimates of movement probabilities is examined.     

Expressing Estimators of Expected Quality Adjusted Survival as Functions of Nelson-Aalen Estimators

Quality adjusted survival has been increasingly advocated in clinical trials to be assessed as a synthesis of survival and quality of life. We investigate nonparametric estimation of its expectation for a general multistate process with incomplete follow-up data. Upon establishing a representation of expected quality adjusted survival through marginal distributions of a set of defined events, we propose two estimators for expected quality adjusted survival. Expressed as functions of Nelson-Aalen estimators, the two estimators are strongly consistent and asymptotically normal. We derive their asymptotic variances and propose sample-based variance estimates, along with evaluation of asymptotic relative efficiency. Monte Carlo studies show that these estimation procedures perform well for practical sample sizes. We illustrate the methods using data from a national, multicenter AIDS clinical trial.     

Temptation and Taxation

We study optimal taxation when consumers have temptation and self‐control problems. Embedding the class of preferences developed by Gul and Pesendorfer into a standard macroeconomic setting, we first prove, in a two‐period model, that the optimal policy is to subsidize savings when consumers are tempted by “excessive” impatience. The savings subsidy improves welfare because it makes succumbing to temptation less attractive. We then study an economy with a long but finite horizon which nests, as a special case, the Phelps–Pollak–Laibson multiple‐selves model (thereby providing guidance on how to evaluate welfare in this model). We prove that when period utility is logarithmic, the optimal savings subsidies increase over time for any finite horizon. Moreover, as the horizon grows large, the optimal policy prescribes a constant subsidy, in contrast to the well known Chamley–Judd result.