Confidence intervals for variance components in unbalanced one-way random effects model using non-normal distributions

In scenarios where the variance of a response variable can be attributed to two sources of variation, a confidence interval for a ratio of variance components gives information about the relative importance of the two sources. For example, if measurements taken from different laboratories are nine times more variable than the measurements taken from within the laboratories, then 90% of the variance in the responses is due to the variability amongst the laboratories and 10% of the variance in the responses is due to the variability within the laboratories. Assuming normally distributed sources of variation, confidence intervals for variance components are readily available. In this paper, however, simulation studies are conducted to evaluate the performance of confidence intervals under non-normal distribution assumptions. Confidence intervals based on the pivotal quantity method, fiducial inference, and the large-sample properties of the restricted maximum likelihood (REML) estimator are considered. Simulation results and an empirical example suggest that the REML-based confidence interval is favored over the other two procedures in unbalanced one-way random effects model.     

Reducing confidence bands for simulated impulse responses

It is emphasized that the shocks in structural vector autoregressions are only identified up to sign and it is pointed out that this feature can result in very misleading confidence intervals for impulse responses if simulation methods such as Bayesian or bootstrap methods are used. The confidence intervals heavily depend on which variable is used for fixing the signs of the responses. In particular, when the shocks are identified via long-run restrictions the problem can be severe. It is pointed out that a suitable choice of variable for fixing the signs of the responses and, hence, of the shocks, can result in substantial reductions in the confidence bands for impulse responses.     

Confidence intervals for survival quantiles in the Cox regression model

Median survival times and their associated confidence intervals are often used to summarize the survival outcome of a group of patients in clinical trials with failure-time endpoints. Although there is an extensive literature on this topic for the case in which the patients come from a homogeneous population, few papers have dealt with the case in which covariates are present as in the proportional hazards model. In this paper we propose a new approach to this problem and demonstrate its advantages over existing methods, not only for the proportional hazards model but also for the widely studied cases where covariates are absent and where there is no censoring. As an illustration, we apply it to the Stanford Heart Transplant data. Asymptotic theory and simulation studies show that the proposed method indeed yields confidence intervals and bands with accurate coverage errors.     

Semiparametric inference on the absolute risk reduction and the restricted mean survival difference

For time-to-event data, when the hazards are non-proportional, in addition to the hazard ratio, the absolute risk reduction and the restricted mean survival difference can be used to describe the time-dependent treatment effect. The absolute risk reduction measures the direct impact of the treatment on event rate or survival, and the restricted mean survival difference provides a way to evaluate the cumulative treatment effect. However, in the literature, available methods are limited for flexibly estimating these measures and making inference on them. In this article, point estimates, pointwise confidence intervals and simultaneous confidence bands of the absolute risk reduction and the restricted mean survival difference are established under a semiparametric model that can be used in a sufficiently wide range of applications. These methods are motivated by and illustrated for data from the Women’s Health Initiative estrogen plus progestin clinical trial.     

Confidence intervals for a two-parameter exponential distribution: One- and two-sample problems

The problems of interval estimating the mean, quantiles, and survival probability in a two-parameter exponential distribution are addressed. Distribution function of a pivotal quantity whose percentiles can be used to construct confidence limits for the mean and quantiles is derived. A simple approximate method of finding confidence intervals for the difference between two means and for the difference between two location parameters is also proposed. Monte Carlo evaluation studies indicate that the approximate confidence intervals are accurate even for small samples. The methods are illustrated using two examples.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

Confidence intervals for the difference between two median survival times for clustered survival data

Clustered survival data arise often in clinical trial design, where the correlated subunits from the same cluster are randomized to different treatment groups. Under such design, we consider the problem of constructing confidence interval for the difference of two median survival time given the covariates. We use Cox gamma frailty model to account for the within-cluster correlation. Based on the conditional confidence intervals, we can identify the possible range of covariates over which the two groups would provide different median survival times. The associated coverage probability and the expected length of the proposed interval are investigated via a simulation study. The implementation of the confidence intervals is illustrated using a real data set.     

A new generalized confidence interval for the among-group variance in the heteroscedastic one-way random effects model

Based on the generalized inference idea, a new kind of generalized confidence intervals is derived for the among-group variance component in the heteroscedastic one-way random effects model. We construct structure equations of all variance components in the model based on their minimal sufficient statistics; meanwhile, the fiducial generalized pivotal quantity (FGPQ) can be obtained through solving an implicit equation of the parameter of interest. Then, the confidence interval is derived naturally from the FGPQ. Simulation results demonstrate that the new procedure performs very well in terms of both empirical coverage probability and average interval length.     

The Finite Sample Performance of Inference Methods for Propensity Score Matching and Weighting Estimators

ABSTRACT

This article investigates the finite sample properties of a range of inference methods for propensity score-based matching and weighting estimators frequently applied to evaluate the average treatment effect on the treated. We analyze both asymptotic approximations and bootstrap methods for computing variances and confidence intervals in our simulation designs, which are based on German register data and U.S. survey data. We vary the design w.r.t. treatment selectivity, effect heterogeneity, share of treated, and sample size. The results suggest that in general, theoretically justified bootstrap procedures (i.e., wild bootstrapping for pair matching and standard bootstrapping for “smoother” treatment effect estimators) dominate the asymptotic approximations in terms of coverage rates for both matching and weighting estimators. Most findings are robust across simulation designs and estimators.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Estimation of Main Effect When Covariates Have Non-Proportional Hazards

The Cox proportional hazards (PH) regression model has been widely used to analyze survival data in clinical trials and observational studies. In addition to estimating the main treatment or exposure group effect, it is common to adjust for additional covariates using the Cox model. It is well known that violation of the PH assumption can lead to estimates that are biased and difficult to interpret, and model checking has become a routine procedure. However, such checking might focus on the primary group comparisons, and the assumption can still be violated when adjusting for many of the potential covariates. We study the effect of violation of the PH assumption of the covariates on the estimation of the main group effect in the Cox model. The results are summarized in terms of the bias and the coverage properties of the confidence intervals. Overall in randomized clinical trials, the bias caused by misspecifying the PH assumption on the covariates is no more than 15% in absolute value regardless of sample size. In observational studies where the covariates are likely correlated with the group variable, however, the bias can be very severe. The coverage properties largely depend on sample size, as expected, as bias becomes dominating with increasing sample size. These findings should serve as cautionary notes when adjusting for potential confounders in observational studies, as the violation of PH assumption on the confounders can lead to erroneous results.     

Nonparametric confidence and tolerance intervals from record values data

In a number of situations only observations that exceed or only those that fall below the current extreme value are recorded. Examples include meteorology, hydrology, athletic events and mining. Industrial stress testing is also an example in which only items that are weaker than all the observed items are destroyed. In this paper, it is shown that, how record values can be used to provide distribution-free confidence intervals for population quantiles and tolerance intervals. We provide some tables that help one choose the appropriate record values and present a numerical example. Also universal upper bounds for the expectation of the length of the confidence intervals are derived. The results may be of interest in situation where only record values are stored.     

Generalized confidence intervals for proportions of total variance in mixed linear models

Exact confidence intervals for a proportion of total variance, based on pivotal quantities, only exist for mixed linear models having two variance components. Generalized confidence intervals (GCIs) introduced by Weerahandi [1993. Generalized confidence intervals (Corr: 94V89 p726). J. Am. Statist. Assoc. 88, 899–905] are based on generalized pivotal quantities (GPQs) and can be constructed for a much wider range of models. In this paper, the author investigates the coverage probabilities, as well as the utility of GCIs, for a proportion of total variance in mixed linear models having more than two variance components. Particular attention is given to the formation of GPQs and GCIs in mixed linear models having three variance components in situations where the data exhibit complete balance, partial balance, and partial imbalance. The GCI procedure is quite general and provides a useful method to construct confidence intervals in a variety of applications.     

Nonparametric asymptotic confidence intervals for extreme quantiles

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.     

Robust confidence regions for the semi-parametric regression model with responses missing at random

In this paper, a regression semi-parametric model is considered where responses are assumed to be missing at random. From the empirical likelihood function defined based on the rank-based estimating equation, robust confidence intervals/regions of the true regression coefficient are derived. Monte Carlo simulation experiments show that the proposed approach provides more accurate confidence intervals/regions compared to its normal approximation counterpart under different model error structure. The approach is also compared with the least squares approach, and its superiority is shown whenever the error distribution in the simulation study is heavy tailed or contaminated. Finally, a real data example is given to illustrate our proposed method.     

Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data

Abstract. In this article, a naive empirical likelihood ratio is constructed for a non‐parametric regression model with clustered data, by combining the empirical likelihood method and local polynomial fitting. The maximum empirical likelihood estimates for the regression functions and their derivatives are obtained. The asymptotic distributions for the proposed ratio and estimators are established. A bias‐corrected empirical likelihood approach to inference for the parameters of interest is developed, and the residual‐adjusted empirical log‐likelihood ratio is shown to be asymptotically chi‐squared. These results can be used to construct a class of approximate pointwise confidence intervals and simultaneous bands for the regression functions and their derivatives. Owing to our bias correction for the empirical likelihood ratio, the accuracy of the obtained confidence region is not only improved, but also a data‐driven algorithm can be used for selecting an optimal bandwidth to estimate the regression functions and their derivatives. A simulation study is conducted to compare the empirical likelihood method with the normal approximation‐based method in terms of coverage accuracies and average widths of the confidence intervals/bands. An application of this method is illustrated using a real data set.     

Confidence intervals for the difference in paired Youden indices

Comparison of accuracy between two diagnostic tests can be implemented by investigating the difference in paired Youden indices. However, few literature articles have discussed the inferences for the difference in paired Youden indices. In this paper, we propose an exact confidence interval for the difference in paired Youden indices based on the generalized pivotal quantities. For comparison, the maximum likelihood estimate‐based interval and a bootstrap‐based interval are also included in the study for the difference in paired Youden indices. Abundant simulation studies are conducted to compare the relative performance of these intervals by evaluating the coverage probability and average interval length. Our simulation results demonstrate that the exact confidence interval outperforms the other two intervals even with small sample size when the underlying distributions are normal. A real application is also used to illustrate the proposed intervals. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of average causal effect using the restricted mean residual lifetime as effect measure

Although mean residual lifetime is often of interest in biomedical studies, restricted mean residual lifetime must be considered in order to accommodate censoring. Differences in the restricted mean residual lifetime can be used as an appropriate quantity for comparing different treatment groups with respect to their survival times. In observational studies where the factor of interest is not randomized, covariate adjustment is needed to take into account imbalances in confounding factors. In this article, we develop an estimator for the average causal treatment difference using the restricted mean residual lifetime as target parameter. We account for confounding factors using the Aalen additive hazards model. Large sample property of the proposed estimator is established and simulation studies are conducted in order to assess small sample performance of the resulting estimator. The method is also applied to an observational data set of patients after an acute myocardial infarction event.     

Confidence intervals for ratios of AUCs in the case of serial sampling: a comparison of seven methods

Pharmacokinetic studies are commonly performed using the two-stage approach. The first stage involves estimation of pharmacokinetic parameters such as the area under the concentration versus time curve (AUC) for each analysis subject separately, and the second stage uses the individual parameter estimates for statistical inference. This two-stage approach is not applicable in sparse sampling situations where only one sample is available per analysis subject similar to that in non-clinical in vivo studies. In a serial sampling design, only one sample is taken from each analysis subject. A simulation study was carried out to assess coverage, power and type I error of seven methods to construct two-sided 90% confidence intervals for ratios of two AUCs assessed in a serial sampling design, which can be used to assess bioequivalence in this parameter.     

Empirical likelihood ratio confidence intervals in terms of cumulative hazard function for left-truncated and right-censored data

Abstract

Based on the approach of Pan and Zhou, we demonstrate that empirical likelihood ratios in terms of cumulative hazard function for left-truncated and right-censored (LTRC) data can be used to form confidence intervals for the parameters that are linear functionals of the cumulative hazard function. Simulation studies indicate that the empirical likelihood ratio based confidence intervals work well in finite samples.