Likelihood-based Confidence Interval Research Articles

Semlbci: An R package for Forming Likelihood-Based Confidence Intervals for Parameter Estimates, Correlations, Indirect Effects, and Other Derived Parameters

There are three common types of confidence interval (CI) in structural equation modeling (SEM): Wald-type CI, bootstrapping CI, and likelihood-based CI (LBCI). LBCI has the following advantages: (1) it has better coverage probabilities and Type I error rate compared to Wald-type CI when the sample size is finite; (2) it correctly tests the null hypothesis of a parameter based on likelihood ratio chi-square difference test; (3) it is less computationally intensive than bootstrapping CI; and (4) it is invariant to transformations. However, LBCI is not available in many popular SEM software packages. We developed an R package, semlbci, for forming LBCI for parameters in models fitted by lavaan, a popular open-source SEM package, such that researchers have more options in forming CIs for parameters in SEM. The package supports both unstandardized and standardized estimates, derived parameters such as indirect effect, multisample models, and the robust LBCI proposed by Falk.

Novel empirical likelihood inference for the mean difference with right-censored data.

This paper focuses on comparing two means and finding a confidence interval for the difference of two means with right-censored data using the empirical likelihood method combined with the independent and identically distributed random functions representation. In the literature, some early researchers proposed empirical link-based confidence intervals for the mean difference based on right-censored data using the synthetic data approach. However, their empirical log-likelihood ratio statistic has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose an empirical likelihood method based on the independent and identically distributed representation of Kaplan-Meier weights involved in the empirical likelihood ratio. We obtain the standard chi-squared distribution. We also apply the adjusted empirical likelihood to improve coverage accuracy for small samples. In addition, we investigate a new empirical likelihood method, the mean empirical likelihood, within the framework of our study. The performances of all the empirical likelihood methods are compared via extensive simulations. The proposed empirical likelihood-based confidence interval has better coverage accuracy than those from existing methods. Finally, our findings are illustrated with a real data set.

SEM-Based Methods to Form Confidence Intervals for Indirect Effect: Still Applicable Given Nonnormality, Under Certain Conditions.

A SEM-based approach using likelihood-based confidence interval (LBCI) has been proposed to form confidence intervals for unstandardized and standardized indirect effect in mediation models. However, when used with the maximum likelihood estimation, this approach requires that the variables are multivariate normally distributed. This can affect the LBCIs of unstandardized and standardized effect differently. In the present study, the robustness of this approach when the predictor is not normally distributed but the error terms are conditionally normal, which does not violate the distributional assumption of ordinary least squares (OLS) estimation, is compared to four other approaches: nonparametric bootstrapping, two variants of LBCI, LBCI assuming the predictor is fixed (LBCI-Fixed-X) and LBCI based on ADF estimation (LBCI-ADF), and Monte Carlo. A simulation study was conducted using a simple mediation model and a serial mediation model, manipulating the distribution of the predictor. The Monte Carlo method performed worst among the methods. LBCI and LBCI-Fixed-X had suboptimal performance when the distributions had high kurtosis and the population indirect effects were medium to large. In some conditions, the problem was severe even when the sample size was large. LBCI-ADF and nonparametric bootstrapping had coverage probabilities close to the nominal value in nearly all conditions, although the coverage probabilities were still suboptimal for the serial mediation model when the sample size was small with respect to the model. Implications of these findings in the context of this special case of nonnormal data were discussed.

A statistical measure for the skewness of X chromosome inactivation based on family trios

BackgroundX chromosome inactivation (XCI) is an important gene regulation mechanism in females to equalize the expression levels of X chromosome between two sexes. Generally, one of two X chromosomes in females is randomly chosen to be inactivated. Nonrandom XCI (XCI skewing) is also observed in females, which has been reported to play an important role in many X-linked diseases. However, there is no statistical measure available for the degree of the XCI skewing based on family data in population genetics.ResultsIn this article, we propose a statistical approach to measure the degree of the XCI skewing based on family trios, which is represented by a ratio of two genotypic relative risks in females. The point estimate of the ratio is obtained from the maximum likelihood estimates of two genotypic relative risks. When parental genotypes are missing in some family trios, the expectation-conditional-maximization algorithm is adopted to obtain the corresponding maximum likelihood estimates. Further, the confidence interval of the ratio is derived based on the likelihood ratio test. Simulation results show that the likelihood-based confidence interval has an accurate coverage probability under the situations considered. Also, we apply our proposed method to the rheumatoid arthritis data from USA for its practical use, and find out that a locus, rs2238907, may undergo the XCI skewing against the at-risk allele. But this needs to be further confirmed by molecular genetics.ConclusionsThe proposed statistical measure for the skewness of XCI is applicable to complete family trio data or family trio data with some paternal genotypes missing. The likelihood-based confidence interval has an accurate coverage probability under the situations considered. Therefore, our proposed statistical measure is generally recommended in practice for discovering the potential loci which undergo the XCI skewing.

Likelihood-Based Confidence Intervals for a Parameter With an Upper or Lower Bound

The precision of estimates in many statistical models can be expressed by a confidence interval (CI). CIs based on standard errors (SEs) are common in practice, but likelihood-based CIs are worth consideration. In comparison to SEs, likelihood-based CIs are typically more difficult to estimate, but are more robust to model (re)parameterization. In latent variable models, some parameters might take on values outside of their interpretable range. Therefore, it is desirable to place a bound to keep the parameter interpretable. For likelihood-based CI, a correction is needed when a parameter is bounded. The correction is known (Wu & Neale, 2012), but is difficult to implement in practice. A novel automatic implementation that is simple for an applied researcher to use is introduced. A simulation study demonstrates the accuracy of the correction using a latent growth curve model and the method is illustrated with a multilevel confirmatory factor analysis.

Formula omitted]-statistic with side information

In this paper we study U-statistics with side information incorporated using the method of empirical likelihood. Some basic properties of the proposed statistics are investigated. We find that by implementing the side information properly, the proposed U-statistics can have smaller asymptotic variance than the existing U-statistics in the literature. The proposed U-statistics can achieve asymptotic efficiency in a formal sense and their weak limits admit a convolution result. We also find that the corresponding U-likelihood ratio procedure, as well as the U-empirical likelihood based confidence interval construction, do not benefit from incorporating side information, a result that is consistent with the result under the standard empirical likelihood ratio procedure. The impact of incorrect side information implementation in the proposed U-statistics is also explored. Simulation studies are conducted to assess the finite sample performance of the proposed method. The numerical results show that with side information implemented, the deduction of asymptotic variance can be substantial in some cases, and the coverage probability of the confidence interval using the U-empirical likelihood ratio based method outperforms that of the normal approximation based method, in particular in the cases when the underlying distribution is skewed.

TOWARD A UNIFIED INTERVAL ESTIMATION OF AUTOREGRESSIONS

An empirical likelihood–based confidence interval is proposed for interval estimations of the autoregressive coefficient of a first-order autoregressive model via weighted score equations. Although the proposed weighted estimate is less efficient than the usual least squares estimate, its asymptotic limit is always normal without assuming stationarity of the process. Unlike the bootstrap method or the least squares procedure, the proposed empirical likelihood–based confidence interval is applicable regardless of whether the underlying autoregressive process is stationary, unit root, near-integrated, or even explosive, thereby providing a unified approach for interval estimation of an AR(1) model to encompass all situations. Finite-sample simulation studies confirm the effectiveness of the proposed method.

Likelihood-based confidence intervals for the risk difference of two-sample binary data with a fallible classifier and a gold standard

We develop likelihood-based confidence intervals for risk difference in two-sample misclassified binary data. Such data consist of two studies. The first study is the main study where individuals are classified by an inexpensive fallible classifier which may misclassify. The second study is a validation substudy where individuals are classified by using both the fallible classifier and an expensive gold standard which classifies perfectly. We propose and examine three likelihood-based confidence interval methods and conclude that the modified Wald method applied to small-number adjusted new data performs well and has nominal coverage probabilities.

Empirical Likelihood-Based Confidence Interval of ROC Curves

In this article we propose an empirical likelihood-based confidence interval for receiver operating characteristic curves which are based on a continuous-scale test. The approach is easily understood, simply implemented, and computationally efficient. The results from our simulation studies indicate that the finite-sample numerical performance slightly outperforms the most promising methods published recently. Two real datasets are analyzed by using the proposed method and the existing bootstrap-based method.

Confidence intervals for a common mean with missing data with applications in an AIDS study

In practical data analysis, nonresponse phenomenon frequently occurs. In this paper, we propose an empirical likelihood based confidence interval for a common mean by combining the imputed data, assuming that data are missing completely at random. Simulation studies show that such confidence intervals perform well, even when the missing proportion is high. Our method is applied to an analysis of a real data set from an AIDS clinic trial study.

Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.

Rate of Foot‐and‐Mouth Disease Virus Transmission by Carriers Quantified from Experimental Data

Upon infection with foot-and-mouth disease virus (FMDV) a considerable number of animals become carriers of the virus. These carriers are considered to be a risk for new outbreaks, but the rate at which these animals can transmit the infection has not been quantified. An analysis was carried out using data from previously published experiments in order to quantify the transmission rate parameter beta of FMDV infection from carriers to susceptible animals. The parameter beta was estimated at 0.0256 (likelihood-based confidence interval: 0.008-0.059) infections per carrier per month. Moreover, analysis of published experimental data indicates that the proportion of FMDV carriers decreases at a rate of 0.115 per month. Both parameters obtained from this study are useful for quantitative risk analyses of the trade of animals from FMDV-infected areas or the lifting of vaccination programs.

Profile Likelihood for Estimation and Confidence Intervals

Normal-based confidence intervals for a parameter of interest are inaccurate when the sampling distribution of the estimate is nonnormal. The technique known as profile likelihood can produce confidence intervals with better coverage. It may be used when the model includes only the variable of interest or several other variables in addition. Profile-likelihood confidence intervals are particularly useful in nonlinear models. The command pllf computes and plots the maximum likelihood estimate and profile likelihood–based confidence interval for one parameter in a wide variety of regression models.

Likelihood-based confidence interval for the ratio of scale parameters of two independent Weibull distributions

The Weibull distribution is widely used in lifetime data analysis. For example, in studies on the time to the occurrence of tumors in human populations or in laboratory animals, the time of occurrence of tumors is generally assumed to be distributed as a Weibull distribution. Moreover, in engineering, the voltage levels at which failure occurred in electrical cable insulation has been shown to be distributed as a Weibull distribution. When comparing two independent Weibull distributions, it is often assumed that only the scale parameter is altered. In this paper, we propose a simple and accurate procedure to obtain inference concerning the ratio of the two scale parameters of two independent distributions. The performance of the proposed method is assessed through Monte Carlo simulation studies. The numerical results show that the proposed method is extremely accurate even for very small samples. The method is applied to a set of real-life data.

Bootstrap confidence intervals

This article surveys bootstrap methods for producing good approximate confidence intervals. The goal is to improve by an order of magnitude upon the accuracy of the standard intervals $\hat{\theta} \pm z^{(\alpha)} \hat{\sigma}$, in a way that allows routine application even to very complicated problems. Both theory and examples are used to show how this is done. The first seven sections provide a heuristic overview of four bootstrap confidence interval procedures: $BC_a$, bootstrap-t , ABC and calibration. Sections 8 and 9 describe the theory behind these methods, and their close connection with the likelihood-based confidence interval theory developed by Barndorff-Nielsen, Cox and Reid and others.

A Bayesian Analysis of Extreme Rainfall Data

SUMMARY Understanding and quantifying the behaviour of a rainfall process at extreme levels has important applications for design in civil engineering. As in the extremal analysis of any environmental process, estimates often are required of the probability of events that are rarer than those already recorded. As data on extremes are scarce, all available sources of information should be used in inference. Consequently, research has focused on the development of techniques that make optimal use of available data. I n this paper a daily rainfall series is analysed within a Bayesian framework, illustrating how the careful elicitation of prior expert information can supplement data and lead to improved estimates of extremal behaviour. For example, using the prior knowledge of an expert hydrologist, a Bayesian 95% interval estimate of the 1 00-year return level for daily rainfall is found to be approximately half of the width of the corresponding likelihoodbased confidence interval,

Modeling Incidence Rate Ratio and Rate Difference: Additivity or Multiplicativity of Human Immunodeficiency Virus Parenteral and Sexual Transmission among Intravenous Drug Users

The analysis concerns data from the Northern Italian Seronegative Drug Addicts Study, a multicenter longitudinal study about the incidence of human immunodeficiency virus infection in intravenous drug users from Milan and other areas of northern Italy between 1987 and 1991. Different measures of parenteral and heterosexual exposure effects were estimated by fitting multiplicative models for rate ratio and additive models for both rate ratio and rate difference into a Poisson regression model for grouped cohort data. In areas of high human immunodeficiency virus prevalence among intravenous drug users, the adjusted rate ratio under a multiplicative structure was 6.2 (95% likelihood-based confidence interval (LCI) 2.9-14.4) for parenteral and 2.9 (95% LCI 1.3-6.1) for sexual transmission. Under the additive model, the rate ratio was 7.8 (95% LCI 3.4-20.2) for parenteral and 9.2 (95% LCI 2.2-29.7) for sexual transmission, and the rate difference per 100 person-years was 9.8 (95% LCI 5.3-15.6) for parenteral and 10.5 (95% LCI 1.8-24.2) for sexual transmission (controlled for each other). Because of the small sample size, a clear discrimination between models could not be reached. However, in spite of the greater risk associated with parenteral transmission under a multiplicative model, the additive model suggests that the relative impact of measures aimed at inducing condom use is similar to that which would be obtained by measures aimed at stopping syringe sharing.

LC 50 estimates and their confidence intervals derived for tests with only one concentration with partial effect

Linear approximation fails to determine the likelihood based confidence interval for the LC 50 parameter in a toxicity experiment with only one concentration causing partial mortality. An equation for this confidence interval is derived given a log-logistic concentration response relation. A table of the 95 and 99% confidence intervals is given for the case that partial survival occurs only at one concentration, under the condition that a fixed number of animals is used at each test concentration and a constant ratio is used between the test concentrations.