Confidence intervals for secondary parameters following a sequential test

In sequential studies, formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest. Valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study. However, the completed data set may warrant a more detailed analysis, involving the estimation of parameters corresponding to effects that were not used to determine when to stop, and yet correlated with those that were. This paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used. The method assumes that information accumulates on the primary and secondary parameters at proportional rates. This requirement will be valid in many potential applications, but only in limited situations in survival analysis.     

Generalized empirical likelihood inference in partially linear model for longitudinal data with missing response variables and error-prone covariates

In this article, we consider statistical inference for longitudinal partial linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. A generalized empirical likelihood (GEL) method is proposed by combining correction attenuation and quadratic inference functions. The method that takes into consideration the correlation within groups is used to estimate the regression coefficients. Furthermore, residual-adjusted empirical likelihood (EL) is employed for estimating the baseline function so that undersmoothing is avoided. The empirical log-likelihood ratios are proven to be asymptotically Chi-squared, and the corresponding confidence regions for the parameters of interest are then constructed. Compared with methods based on NAs, the GEL does not require consistent estimators for the asymptotic variance and bias. The numerical study is conducted to compare the performance of the EL and the normal approximation-based method, and a real example is analysed.     

A New Look at the Psychometric Paradigm of Perception of Hazards

The psychometric paradigm has been the most influential model in the field of risk analysis. The "cognitive maps" of hazards produced by the paradigm seem to explain how laypeople perceive the various risks they face. Because most of the studies used aggregated data, analyzed using principal component analysis, it is not known whether the model neglects individual differences in risk perception. There has been much criticism on the fact that few studies have examined individual differences in the cognitive representation of hazards. In order to detect and describe the internal structure of the three-way data, we conducted a three-way component analysis (3MPCA). Data for the present analysis were derived from a mail survey conducted in Switzerland. Participants were asked to judge 9 attributes for 26 hazards. Individual differences in the cognitive representation of hazards were correlated with external variables (e.g., general trust). The results suggest that methods permitting individual differences should be used more frequently and that utilizing different methods could provide greater insight into the cognitive representation of risks.     

Exact sample size determination in a weibull test plan when there is time censoring

Engineers who conduct reliability tests need to choose the sample size when designing a test plan. The model parameters and quantiles are the typical quantities of interest. The large-sample procedure relies on the property that the distribution of the t -like quantities is close to the standard normal in large samples. In this paper, we use a new procedure based on both simulation and asymptotic theory to determine the sample size for a test plan. Unlike the complete data case, the t -like quantities are not pivotal quantities in general when data are time censored. However we show that the distribution of the t -like quantities only depend on the expected proportion failing and obtain the distributions by simulation for both complete and time censoring case when data follow Weibull distribution. We find that the large-sample procedure usually underestimates the sample size even when it is said to be 200 or more. The sample size given by the proposed procedure insures the requested nominal accuracy and confidence of the estimation when the test plan results in complete or time censored data. Some useful figures displaying the required sample size for the new procedure are also presented.     

Improved approaches to interval estimation of effective doses for a logistic dose-response curve

Asymptotic confidence (delta) intervals and intervals based upon the use of Fieller's theorem are alternative methods for constructing intervals for the <$>gamma<$>% effective doses (ED<$>_gamma<$>). Sitter and Wu (1993) provided a comparison of the two approaches for the ED<$>_{50}<$>, for the case in which a logistic dose response curve is assumed. They showed that the Fieller intervals are generally superior. In this paper, we introduce two new families of intervals, both of which include the delta and Fieller intervals as special cases. In addition we consider interval estimation of the ED<$>_{90}<$> as well as the ED<$>_{50}<$>. We provide a comparison of the various methods for the problem of constructing a confidence interval for the ED<$>_gamma<$>.     

Interval estimation for log-linear models with one variable subject to non-ignorable non-response

Summary.  Log-linear models for multiway contingency tables where one variable is subject to non-ignorable non-response will often yield boundary solutions, with the probability of non-respondents being classified in some cells of the table estimated as 0. The paper considers the effect of this non-standard behaviour on two methods of interval estimation based on the distribution of the maximum likelihood estimator. The first method relies on the estimator being approximately normally distributed with variance equal to the inverse of the information matrix. It is shown that the information matrix is singular for boundary solutions, but intervals can be calculated after a simple transformation. For the second method, based on the bootstrap, asymptotic results suggest that the coverage properties may be poor for boundary solutions. Both methods are compared with profile likelihood intervals in a simulation study based on data from the British General Election Panel Study. The results of this study indicate that all three methods perform poorly for a parameter of the non-response model, whereas they all perform well for a parameter of the margin model, irrespective of whether or not there is a boundary solution.     

Confidence intervals for probability density functions under associated samples

In this paper, we study the construction of confidence intervals for a probability density function under a (positively) associated sample by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²-type${\chi }^2\mathrm{-}\mathrm{type}$ distributed. The result is used to obtain EL-based confidence interval for the probability density function.     

Semi-empirical likelihood inference for the ROC curve with missing data

The receiver operating characteristic (ROC) curve is one of the most commonly used methods to compare the diagnostic performance of two or more laboratory or diagnostic tests. In this paper, we propose semi-empirical likelihood based confidence intervals for ROC curves of two populations, where one population is parametric and the other one is non-parametric and both have missing data. After imputing missing values, we derive the semi-empirical likelihood ratio statistic and the corresponding likelihood equations. It is shown that the log-semi-empirical likelihood ratio statistic is asymptotically scaled chi-squared. The estimating equations are solved simultaneously to obtain the estimated lower and upper bounds of semi-empirical likelihood confidence intervals. We conduct extensive simulation studies to evaluate the finite sample performance of the proposed empirical likelihood confidence intervals with various sample sizes and different missing probabilities.