Sample Size Calculation and Blinded Sample Size Recalculation in Clinical Trials Where the Treatment Effect is Measured by the Relative Risk

In clinical trials with binary endpoints, the required sample size does not depend only on the specified type I error rate, the desired power and the treatment effect but also on the overall event rate which, however, is usually uncertain. The internal pilot study design has been proposed to overcome this difficulty. Here, nuisance parameters required for sample size calculation are re-estimated during the ongoing trial and the sample size is recalculated accordingly. We performed extensive simulation studies to investigate the characteristics of the internal pilot study design for two-group superiority trials where the treatment effect is captured by the relative risk. As the performance of the sample size recalculation procedure crucially depends on the accuracy of the applied sample size formula, we firstly explored the precision of three approximate sample size formulae proposed in the literature for this situation. It turned out that the unequal variance asymptotic normal formula outperforms the other two, especially in case of unbalanced sample size allocation. Using this formula for sample size recalculation in the internal pilot study design assures that the desired power is achieved even if the overall rate is mis-specified in the planning phase. The maximum inflation of the type I error rate observed for the internal pilot study design is small and lies below the maximum excess that occurred for the fixed sample size design.     

An efficient cross-validation algorithm for window width selection for nonparametric kernel regression

This paper presents an approach to cross-validated window width choice which greatly reduces computation time, which can be used regardless of the nature of the kernel function, and which avoids the use of the Fast Fourier Transform. This approach is developed for window width selection in the context of kernel estimation of an unknown conditional mean.     

Kernel estimation of a distribution function

A distribution function is estimated by a kernel method with

a poinrwise mean squared error criterion at a point x. Relation- ships between the mean squared error, the point x, the sample size and the required kernel smoothing parazeter are investigated for several distributions treated by Azzaiini (1981). In particular it is noted that at a centre of symmetry or near a mode of the distribution the kernei method breaks down. Point- wise estimation of a distribution function is motivated as a more useful technique than a reference range for preliminary medical diagnosis.     

A stepwise approach for the purely periodic interaction spline model

The basic idea of an interaction spline model was presented in Barry (1983). The general interaction spline models were proposed by Wahba (1986). The purely periodic spline model, a special case of the general interaction spline models, is considered in this paper. A stepwise approach using generalized cross validation (GCV) for fitting the model is proposed. Based on the nice orthogonality properties of the purely periodic functions, the stepwise approach is a promising method for the interaction spline model. The approach can also be generalized to the non-purely-periodic spline models. But this is no done here.     

Mean intergrated squared error properties and optimal kernels when estimating a diatribution function

Let X₁,X₂,… X_n be a sample of independent identically distributed (i.i.d)random variables having an unknown absolutely continuous distribution function f with density f the twofold aim of his paper consists in, firstly deriving asymptotic expressions of the mean intergrated squared error (MISE) of a kernel estimator of F when f is either assumed to be continuous everywhere or problem of finding optimal kernels in these two cases is studied in detail.     

Bayesian credibility limits for small finite population proportions:stratified and cluster sampling

Bayes credibility limits for small proportions from stratified and fixed size cluster samples are discussed. Ericson’s (JRSS B (1969)) Beta Binomial and Dirichlet-Multinomial priors are used. Approximate limits that are appropriate for large samples and small proportions are derived in both cases. These allow asymptotic comparisons of the efficacy of stratified and cluster sampling relative to simple random sampling for estimating small proportions. Procedures for the selection of hyper parameters are also presented.     

Considerations for optimal nonparametric regression under a generalizederror structure

Nonparametric smoothing, such as kernel or spline estimation, has been examined extensively under the assumption of uncorrelated errors. This paper addresses the effects of potential correlation on consistency and other asymptotic properties in a repeated-measures model, using directly optimized linear smoothers of the replicate means. Unrestricted optimal weights, with respect to squared error loss, are used to confirm a lack of consistency for all linear estimators in an autocorrelated errors model. The results indicate kernel methods that work well for an uncorrelated errors model may not have the ability to perform satisfactorily when correlation is introduced, due to an asymmetry in the optimal weights, which disappears for an uncorrelated errors model. These would include data-driven bandwidth selection methods, adjustments of the bandwidth to accommodate correlation, higher-order kernels, and related bias reduction techniques. The analytic results suggest alternative approaches, not considered here in detail, which have shown merit.     

Cross-Validated Assessments of Test Scoring Models

On a multiple choice test in which each item has r alternative options, a given number c of which are correct, various scoring models have been proposed. In one case the test-taker is allowed to choose any size solution subset and he/she is graded according to whether the subset is small and according to how many correct answers the subset contains. In a second case the test-taker is allowed to select only solution subsets of a prespecified maximum size and is graded as above. The first case is analogous to the situation where the test-taker is given a set of r options with each question; each question calls for a solution which consists of selecting that subset of the r responses which he/she believes to be correct. In the second case, when the prespecified solution subset is restricted to be of size at most one, the resulting scoring model corresponds to the usual model, referred to below as standard. The number c of correct options per item is usually known to the test-taker in this case.

Scoring models are evaluated according to how well they correctly identify the total scores of the individuals in the class of test-takers. Loss functions are constructed which penalize scoring models resulting in student scores which are not associated with the students true (or average) total score on the exam. Scoring models are compared on the basis of cross-validated assessments of the loss incurred by using each of the given models. It is shown that in many cases the assessment of the loss for scoring models which allow students the opportunity to choose more than one option for each question are smaller than the assessment of the loss for the standard scoring model.     

A plug-in technique in nonparametric regression with dependence

The problem addressed is that of smoothing parameter selection in kernel nonparametric regression in the fixed design regression model with dependent noise. An asymptotic expression of the optimum bandwidth parameter has been obtained in recent studies, where this takes the form h = C ₀ n ^?1/5. This paper proposes to use a plug-in methodology, in order to obtain an optimum estimation of the bandwidth parameter, through preliminary estimation of the unknown value of C ₀.