Likelihood-based Confidence Intervals Research Articles

Likelihood-based inferences for active-arm trial with counterfactual incidence based on recency assay

Abstract Objectives The approach of using HIV recency assay to estimate the counterfactual incidence rate is being used as the primary efficacy method in a few ongoing large-scale HIV pre-exposure prophylaxis (PrEP) trials, and the current available approach for the inference is based on the Wald method that leverages the asymptotic distribution of the estimators. One issue with the Wald test is that it does not work well when the number of HIV infections are small in the active arm, and it fails to work when there are zero HIV infections. As future long-acting PrEP products are becoming more efficacious, it is very likely that a small or zero number of infections will be observed in HIV prevention trials, especially for subgroup analyses or interim analyses, hence there is a pressing need to develop inference methods that work under such scenarios. Methods It is well known that when the sample size is small to moderate, likelihood ratio tests are more reliable than Wald tests in terms of actual error probabilities coming close to matching nominal levels. In this manuscript we derive the likelihood ratio test and the likelihood-based confidence intervals for HIV prevention trials based on recency assays. Results Compared with the Wald test, the proposed method works when there are zero infections. Additionally, unlike the Wald test, the p-value from the likelihood ratio test is an increasing function with respect to the number of infections, which is a desirable property as otherwise it will cause confusions. Conclusions For HIV PrEP trials based on recency assay, the likelihood-based p-value and confidence interval can be preferable to the Wald based inference methods when the number of HIV infections is expected to be small.

Correcting for Sampling Error in between-Cluster Effects: An Empirical Bayes Cluster-Mean Approach with Finite Population Corrections

With clustered data, such as where students are nested within schools or employees are nested within organizations, it is often of interest to estimate and compare associations among variables separately for each level. While researchers routinely estimate between-cluster effects using the sample cluster means of a predictor, previous research has shown that such practice leads to biased estimates of coefficients at the between level, and recent research has recommended the use of latent cluster means with the multilevel structural equation modeling framework. However, the latent cluster mean approach may not always be the best choice as it (a) relies on the assumption that the population cluster sizes are close to infinite, (b) requires a relatively large number of clusters, and (c) is currently only implemented in specialized software such as Mplus. In this paper, we show how using empirical Bayes estimates of the cluster means can also lead to consistent estimates of between-level coefficients, and illustrate how the empirical Bayes estimate can incorporate finite population corrections when information on population cluster sizes is available. Through a series of Monte Carlo simulation studies, we show that the empirical Bayes cluster-mean approach performs similarly to the latent cluster mean approach for estimating the between-cluster coefficients in most conditions when the infinite-population assumption holds, and applying the finite population correction provides reasonable point and interval estimates when the population is finite. The performance of EBM can be further improved with restricted maximum likelihood estimation and likelihood-based confidence intervals. We also provide an R function that implements the empirical Bayes cluster-mean approach, and illustrate it using data from the classic High School and Beyond Study.

Confidence intervals for a proportion using a fixed-inverse double sampling scheme when the data are subject to false-positive misclassification

Of interest in this paper is the development of a model that uses fixed, then inverse sampling of binary data that is subject to false-positive misclassification in an effort to estimate a proportion. From this model, both the proportion of success and false-positive misclassification rate may be estimated. Also, three first-order likelihood-based confidence intervals for the proportion of success are mathematically derived and studied via a Monte Carlo simulation. The simulation results indicate that the likelihood ratio interval is generally preferable over the Wald and score interval. Lastly, the model is applied to two different real-world medical data sets.

A double-Cox model for non-proportional hazards survival analysis with frailty.

The Cox regression, a semi-parametric method of survival analysis, is extremely popular in biomedical applications. The proportional hazards assumption is a key requirement in the Cox model. To accommodate non-proportional hazards, we propose to parameterize the shape parameter of the baseline hazard function using the additional, separate Cox-regression term which depends on the vector of the covariates. This parametrization retains the general form of the hazard function over the strata and is similar to one in Devarajan and Ebrahimi (Comput Stat Data Anal. 2011;55:667-676) in the case of the Weibull distribution, but differs for other hazard functions. We call this model the double-Cox model. We formally introduce the double-Cox model with shared frailty and investigate, by simulation, the estimation bias and the coverage of the proposed point and interval estimation methods for the Gompertz and the Weibull baseline hazards. For real-life applications with low frailty variance and a large number of clusters, the marginal likelihood estimation is almost unbiased and the profile likelihood-based confidence intervals provide good coverage for all model parameters. We also compare the results from the over-parametrized double-Cox model to those from the standard Cox model with frailty in the case of the scale-only proportional hazards. The model is illustrated on an example of the survival after a diagnosis of type 2 diabetes mellitus. The R programs for fitting the double-Cox model are available on Github.

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.

Statistical inference of the rates of cell proliferation and phenotypic switching in cancer

Recent evidence suggests that nongenetic (epigenetic) mechanisms play an important role at all stages of cancer evolution. In many cancers, these mechanisms have been observed to induce dynamic switching between two or more cell states, which commonly show differential responses to drug treatments. To understand how these cancers evolve over time, and how they respond to treatment, we need to understand the state-dependent rates of cell proliferation and phenotypic switching. In this work, we propose a rigorous statistical framework for estimating these parameters, using data from commonly performed cell line experiments, where phenotypes are sorted and expanded in culture. The framework explicitly models the stochastic dynamics of cell division, cell death and phenotypic switching, and it provides likelihood-based confidence intervals for the model parameters. The input data can be either the fraction of cells or the number of cells in each state at one or more time points. Through a combination of theoretical analysis and numerical simulations, we show that when cell fraction data is used, the rates of switching may be the only parameters that can be estimated accurately. On the other hand, using cell number data enables accurate estimation of the net division rate for each phenotype, and it can even enable estimation of the state-dependent rates of cell division and cell death. We conclude by applying our framework to a publicly available dataset.

An in-silico analysis of experimental designs to study ventricular function: A focus on the right ventricle.

In-vivo studies of pulmonary vascular disease and pulmonary hypertension (PH) have provided key insight into the progression of right ventricular (RV) dysfunction. Additional in-silico experiments using multiscale computational models have provided further details into biventricular mechanics and hemodynamic function in the presence of PH, yet few have assessed whether model parameters are practically identifiable prior to data collection. Moreover, none have used modeling to devise synergistic experimental designs. To address this knowledge gap, we conduct a practical identifiability analysis of a multiscale cardiovascular model across four simulated experimental designs. We determine a set of parameters using a combination of Morris screening and local sensitivity analysis, and test for practical identifiability using profile likelihood-based confidence intervals. We employ Markov chain Monte Carlo (MCMC) techniques to quantify parameter and model forecast uncertainty in the presence of noise corrupted data. Our results show that model calibration to only RV pressure suffers from practical identifiability issues and suffers from large forecast uncertainty in output space. In contrast, parameter and model forecast uncertainty is substantially reduced once additional left ventricular (LV) pressure and volume data is included. A comparison between single point systolic and diastolic LV data and continuous, time-dependent LV pressure-volume data reveals that at least some quantitative data from both ventricles should be included for future experimental studies.

Improved inference on risk measures for univariate extremes

We discuss the use of likelihood asymptotics for inference on risk measures in univariate extreme value problems, focusing on estimation of high quantiles and similar summaries of risk for uncertainty quantification. We study whether higher-order approximation, based on the tangent exponential model, can provide improved inferences. We conclude that inference based on maxima is generally robust to mild model misspecification and that profile likelihood-based confidence intervals will often be adequate, whereas inferences based on threshold exceedances can be badly biased but may be improved by higher-order methods, at least for moderate sample sizes. We use the methods to shed light on catastrophic rainfall in Venezuela, flooding in Venice, and the lifetimes of Italian semisupercentenarians.

Partial linear additive distortion measurement errors models

We consider partial linear regression models when all the variables are measured with additive distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional mean calibration to obtain calibrated variables. A profile least squares estimator for the parameter is obtained, associated with its normal approximation based and empirical likelihood based confidence intervals. For the hypothesis testing on parameters, a restricted estimator under the null hypothesis and a test statistic are proposed. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Lastly, a score-type test statistic is then proposed for checking the validity of partial linear models. Simulation studies demonstrate the performance of our proposed procedure and a real example is analyzed as illustrate its practical usage.

Additive distortion measurement errors regression models with exponential calibration

In this paper, we used the newly proposed exponential calibration for the additive distortion measurement errors models, where neither the response variable nor the covariates can be directly observed but are distorted in additive fashions by an observed confounding variable. By using the exponential calibrated variables, three estimators of parameters and empirical likelihood-based confidence intervals are proposed, and we studied the asymptotic properties of the proposed estimators. For the hypothesis testing of model checking, an adaptive Neyman test statistic restricted is proposed. Simulation studies demonstrate the performance of the proposed estimators and the test statistic. A real example is analysed to illustrate its practical usage.

Confidence Intervals for the Risk Ratio Using Double Sampling with Misclassified Binomial Data

We derive three likelihood-based confidence intervals for the risk ratio of two proportion parameters using a double sampling scheme for mis classified binomial data. The risk ratio is also known as the relative risk. We obtain closed-form maximum likelihood estimators of the model parameters by maximizing the full-likelihood function. Moreover, we develop three confidence intervals: a naive Wald interval, a modified Wald interval, and a Fieller-type interval. We apply the three confidence intervals to cervical cancer data. Finally, we perform two Monte Carlo simulation studies to assess and compare the coverage probabilities and average lengths of the three interval estimators. Unlike the other two interval estimators, the modified Wald interval always produces close-to-nominal confidence intervals for the various simulation scenarios examined here. Hence, the modified Wald confidence interval is preferred in practice.

Confidence Intervals for the Risk Ratio Using Double Sampling with Misclassified Binomial Data

We derive three likelihood-based confidence intervals for the risk ratio of two proportion parameters using a double sampling scheme for misclassified binomial data. The risk ratio is also known as the relative risk. We obtain closed-form maximum likelihood estimators of the model parameters by maximizing the full-likelihood function. Moreover, we develop three confidence intervals: a naive Wald interval, a modified Wald interval, and a Fieller-type interval. We apply the three confidence intervals to cervical cancer data. Finally, we perform two Monte Carlo simulation studies to assess and compare the coverage probabilities and average lengths of the three interval estimators. Unlike the other two interval estimators, the modified Wald interval always produces close-to-nominal confidence intervals for the various simulation scenarios examined here. Hence, the modified Wald confidence interval is preferred in practice.

Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors

The estimators of the slope and the intercept of simple linear regression model with normal errors are normally distributed and their exact confidence intervals are constructed using the t-distribution. However, when the normality assumption is not fulfilled, it is not possible to obtain exact confidence intervals. The Wald method of interval estimation is commonly used to provide approximate confidence intervals in such cases, and since it is derived from the central limit theorem it requires large samples in order to provide reliable approximate confidence intervals. This paper considers an alternative method of constructing approximate confidence intervals for the parameters of a simple linear regression model with Cauchy errors which is based the normal approximation to the Cauchy likelihood. The normal approximation to the Cauchy likelihood is obtained by a Tailor series expansion of the Cauchy log-likelihood function about the maximum likelihood estimate of the parameters and ignoring terms of order greater two. The maximized relative log-likelihood function for each parameter is then derived from the normal Cauchy relative log-likelihood function. The approximate confidence intervals for the parameters are constructed from their respective maximized relative log-likelihood functions. These confidence intervals have closed form confidence limits, are short and have coverage probabilities close to the nominal value 0.95.

Influence of the spatial and temporal monitoring design on the identification of an instantaneous pollutant release in a river

In recent years, several approaches in the field of pollution source identification have been developed to identify the unknown pollution source parameters (PSP) of a river pollution incident. These approaches rely on monitoring data which have to be collected in the case of the pollution incident. Up until now, the influence of the often arbitrarily selected monitoring data on the identification results is rarely evaluated. Nevertheless, when considering an inverse problem, monitoring data can be of major importance regarding parameter identifiability and the reliable estimation of unknown parameters. Accordingly, in this work, the influence of the spatial and temporal monitoring design on the identification of an instantaneous pollutant release is analyzed. Assuming steady flow conditions, an analytical solution of the two-dimensional advection-dispersion-reaction-equation is used to model the pollutant transport in the river. The mirror-image technique is applied to capture no-flux conditions across the lateral boundaries. Parameter identifiability is analyzed using the profile likelihood approach. The approach additionally enables the user to determine causes in the case of poor or non-identifiability and to derive likelihood-based confidence intervals. The results imply that the spatial and temporal monitoring design has a significant influence on the identifiability and reliable estimation of PSP. This work provides an overview of first recommendations to guide the setup of a monitoring campaign in the case of a pollution incident.

Meta-analysis of published excess relative risk estimates.

A meta-analytic summary effect estimate often is calculated as an inverse-variance-weighted average of study-specific estimates of association. The variances of published estimates of association often are derived from their associated confidence intervals under assumptions typical of Wald-type statistics, such as normality of the parameter. However, in some research areas, such as radiation epidemiology, epidemiological results typically are obtained by fitting linear relative risk models, and associated likelihood-based confidence intervals are often asymmetric; consequently, reasonable estimates of variances associated with study-specific estimates of association may be difficult to infer from the standard approach based on the assumption of a Wald-type interval. Here, a novel method is described for meta-analysis of published results from linear relative risk models that uses a parametric transformation of published results to improve on the normal approximation used to assess confidence intervals. Using simulations, it is illustrated that the meta-analytic summary obtained using the proposed approach yields less biased summary estimates, with better confidence interval coverage, than the summary obtained using the more classical approach to meta-analysis. The proposed approach is illustrated using a previously published example of meta-analysis of epidemiological findings regarding circulatory disease following exposure to low-level ionizing radiation.

Measuring symmetry and asymmetry of multiplicative distortion measurement errors data

This paper studies the measure of symmetry or asymmetry of a continuous variable under the multiplicative distortion measurement errors setting. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. First, two direct plug-in estimation procedures are proposed, and the empirical likelihood based confidence intervals are constructed to measure the symmetry or asymmetry of the unobserved variable. Next, we propose four test statistics for testing whether the unobserved variable is symmetric or not. The asymptotic properties of the proposed estimators and test statistics are examined. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.

Modified empirical likelihood-based confidence intervals for data containing many zero observations

Data containing many zeroes is popular in statistical applications, such as survey data. A confidence interval based on the traditional normal approximation may lead to poor coverage probabilities, especially when the nonzero values are highly skewed and the sample size is small or moderately large. The empirical likelihood (EL), a powerful nonparametric method, was proposed to construct confidence intervals under such a scenario. However, the traditional empirical likelihood experiences the issue of under-coverage problem which causes the coverage probability of the EL-based confidence intervals to be lower than the nominal level, especially in small sample sizes. In this paper, we investigate the numerical performance of three modified versions of the EL: the adjusted empirical likelihood, the transformed empirical likelihood, and the transformed adjusted empirical likelihood for data with various sample sizes and various proportions of zero values. Asymptotic distributions of the likelihood-type statistics have been established as the standard chi-square distribution. Simulations are conducted to compare coverage probabilities with other existing methods under different distributions. Real data has been given to illustrate the procedure of constructing confidence intervals.

Logarithmic calibration for multiplicative distortion measurement errors regression models

In this article, we propose a new identifiability condition by using the logarithmic calibration for the distortion measurement error models, where neither the response variable nor the covariates can be directly observed but are measured with multiplicative measurement errors. Under the logarithmic calibration, the direct‐plug‐in estimators of parameters and empirical likelihood based confidence intervals are proposed, and we studied the asymptotic properties of the proposed estimators. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.

Statistical inference for Markov chains with applications to credit risk

The focus of this paper is on the derivation of confidence and credibility intervals for Markov chains when discrete-time, continuous-time or discretely observed continuous-time data are available. Thereby, our contribution is threefold: First, we discuss and compare multinomial confidence regions for the rows of discrete-time Markov transition matrices in the light of empirical characteristics of credit rating migrations. Second, we derive an analytical expression for the expected Fisher information matrix of a continuous-time Markov chain which is used to construct credibility intervals using a non-informative Jeffreys prior distribution and a Metropolis-Hastings Algorithm. Third, we concretize profile and estimated/pseudo likelihood based confidence intervals in the continuous-time data settings, which in contrast to asymptotic normality based intervals explicitly consider non-negativity constraints for the parameters. Furthermore, we illustrate the described methods by Moody’s corporate ratings data with exact continuous-time transitions.

Correlation coefficient-based measure for checking symmetry or asymmetry of a continuous variable with additive distortion

This paper studies how to estimate and test the symmetry of a continuous variable under the additive distortion measurement errors setting. The unobservable variable is distorted in a additive fashion by an observed confounding variable. In this paper, a direct plug-in estimation procedure and two-step direct plug-in estimation procedure for correlation coefficient are proposed to measure the symmetry or asymmetry of a continuous variable, and empirical likelihood based confidence intervals are constructed to test the symmetry of the unobserved variable. The asymptotic properties of the proposed estimators and test statistics are investigated. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.