Robust alternatives to the F-Test in mixed linear models based on MM-estimates

Mixed linear models are commonly used to analyze data in many settings. These models are generally fitted by means of (restricted) maximum likelihood techniques relying heavily on normality. The sensitivity of the resulting estimators and related tests to this underlying assumption has been identified as a weakness that can even lead to wrong interpretations. Very recently a highly robust estimator based on a scale estimate, that is, an S-estimator, has been proposed for general mixed linear models. It has the advantage of being easy to compute and allows the computation of a robust score test. However, this proposal cannot be used to define a likelihood ratio type test that is certainly the most direct route to robustify an F-test. As the latter is usually a key tool of hypothesis testing in mixed linear models, we propose two new robust estimators that allow the desired extension. They also lead to resistant Wald-type tests useful for testing contrasts and covariate effects. We study their properties theoretically and by means of simulations. The analysis of a real data set illustrates the advantage of the new approach in the presence of outlying observations.     

On the finite sample distribution of the likelihood ratio statistic for testing heterogeneity in meta-analysis

In meta-analysis, hypothesis testing is one of the commonly used approaches for assessing whether heterogeneity exists in effects between studies. The literature concluded that the Q-statistic is clearly the best choice and criticized the performance of the likelihood ratio test in terms of the type I error control and power. However, all the criticism for the likelihood ratio test is based on the use of a mixture of two chi-square distributions with 0 and 1 degrees of freedom, which is justified only asymptotically. In this study, we develop a novel method to derive the finite sample distribution of the likelihood ratio test and restricted likelihood ratio test statistics for testing the zero variance component in the random effects model for meta-analysis. We also extend this result to the heterogeneity test when metaregression is applied. A numerical study shows that the proposed statistics have superior performance to the Q-statistic, especially when the number of studies collected for meta-analysis is small to moderate.     

Testing random effects in linear mixed‐effects models with serially correlated errors

In linear mixed‐effects models, random effects are used to capture the heterogeneity and variability between individuals due to unmeasured covariates or unknown biological differences. Testing for the need of random effects is a nonstandard problem because it requires testing on the boundary of parameter space where the asymptotic chi‐squared distribution of the classical tests such as likelihood ratio and score tests is incorrect. In the literature several tests have been proposed to overcome this difficulty, however all of these tests rely on the restrictive assumption of i.i.d. measurement errors. The presence of correlated errors, which often happens in practice, makes testing random effects much more difficult. In this paper, we propose a permutation test for random effects in the presence of serially correlated errors. The proposed test not only avoids issues with the boundary of parameter space, but also can be used for testing multiple random effects and any subset of them. Our permutation procedure includes the permutation procedure in Drikvandi, Verbeke, Khodadadi, and Partovi Nia (2013) as a special case when errors are i.i.d., though the test statistics are different. We use simulations and a real data analysis to evaluate the performance of the proposed permutation test. We have found that random slopes for linear and quadratic time effects may not be significant when measurement errors are serially correlated.     

Retrospective versus prospective score tests for genetic association with case‐control data

Since the seminal work of Prentice and Pyke, the prospective logistic likelihood has become the standard method of analysis for retrospectively collected case‐control data, in particular for testing the association between a single genetic marker and a disease outcome in genetic case‐control studies. In the study of multiple genetic markers with relatively small effects, especially those with rare variants, various aggregated approaches based on the same prospective likelihood have been developed to integrate subtle association evidence among all the markers considered. Many of the commonly used tests are derived from the prospective likelihood under a common‐random‐effect assumption, which assumes a common random effect for all subjects. We develop the locally most powerful aggregation test based on the retrospective likelihood under an independent‐random‐effect assumption, which allows the genetic effect to vary among subjects. In contrast to the fact that disease prevalence information cannot be used to improve efficiency for the estimation of odds ratio parameters in logistic regression models, we show that it can be utilized to enhance the testing power in genetic association studies. Extensive simulations demonstrate the advantages of the proposed method over the existing ones. A real genome‐wide association study is analyzed for illustration.     

Analysis of Repeated Measurements with Ante-Dependence Covariance Models

This paper considers the use of ante-dependence models in problems with repeated measures through time. These are conditional regression models which reflect the dependence of a measure on some of the previous observations from the same subject. We present maximum likelihood estimators of the covariance matrix and procedures for selecting the order of ante-degendence based on penalized like-lihoods. Extensions to missing data situations are discussed. We propose Wald-type test statistics and apply them in two situations common in experiments with repeated measures: one with pre-study observations and another one with small sample size relative to the number of time periods. In these examples, tests assuming ante-dependence find effects which are not detected using competing procedures.     

Nonlinear mixed effects models for repeated measures data

We propose a general, nonlinear mixed effects model for repeated measures data and define estimators for its parameters. The proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood (or restricted maximum likelihood) estimators for linear mixed effects models. We implement Newton-Raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models. Two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed.     

An exact trend test for correlated binary data

The problem of testing a dose-response relationship in the presence of exchangeably correlated binary data has been addressed using a variety of models. Most commonly used approaches are derived from likelihood or generalized estimating equations and rely on large-sample theory to justify their inferences. However, while earlier work has determined that these methods may perform poorly for small or sparse samples, there are few alternatives available to those faced with such data. We propose an exact trend test for exchangeably correlated binary data when groups of correlated observations are ordered. This exact approach is based on an exponential model derived by Molenberghs and Ryan (1999) and Ryan and Molenberghs (1999) and provides natural analogues to Fisher's exact test and the binomial trend test when the data are correlated. We use a graphical method with which one can efficiently compute the exact tail distribution and apply the test to two examples.     

On Mendelian randomization analysis of case-control study

Mendelian randomization (MR) analysis uses genotypes as instruments to estimate the causal effect of an exposure in the presence of unobserved confounders. The existing MR methods focus on the data generated from prospective cohort studies. We develop a procedure for studying binary outcomes under a case-control design. The proposed procedure is built upon two working models commonly used for MR analyses and adopts a quasi-empirical likelihood framework to address the ascertainment bias from case-control sampling. We derive various approaches for estimating the causal effect and hypothesis testing under the empirical likelihood framework. We conduct extensive simulation studies to evaluate the proposed methods. We find that the proposed empirical likelihood estimate is less biased than the existing estimates. Among all the approaches considered, the Lagrange multiplier (LM) test has the highest power, and the confidence intervals derived from the LM test have the most accurate coverage. We illustrate the use of our method in MR analysis of prostate cancer case-control data with vitamin D level as exposure and three single nucleotide polymorphisms as instruments.     

A Note on Maximum Likelihood Ratio Test of No Outliers in Regression Models

A derivation of the maximum likelihood ratio test for testing no outliers in regression models is given using the method of WETHERILL (1981, pp. 106–107) for estimating the regression parameters. This method is essentially similar to the one outlined in BARNETT and LEWIS (1978, p. 263), although by our detailed derivation it is easier to see that the maximum likelihood estimate of θ of model (3) under the hypothesis that the i^th observation in an outlier is the same as that obtained from model (1) when the i^th observation is removed.     

Regression Analysis with a Misclassified Covariate from a Current Status Observation Scheme

Summary Naive use of misclassified covariates leads to inconsistent estimators of covariate effects in regression models. A variety of methods have been proposed to address this problem including likelihood, pseudo‐likelihood, estimating equation methods, and Bayesian methods, with all of these methods typically requiring either internal or external validation samples or replication studies. We consider a problem arising from a series of orthopedic studies in which interest lies in examining the effect of a short‐term serological response and other covariates on the risk of developing a longer term thrombotic condition called deep vein thrombosis. The serological response is an indicator of whether the patient developed antibodies following exposure to an antithrombotic drug, but the seroconversion status of patients is only available at the time of a blood sample taken upon the discharge from hospital. The seroconversion time is therefore subject to a current status observation scheme, or Case I interval censoring, and subjects tested before seroconversion are misclassified as nonseroconverters. We develop a likelihood‐based approach for fitting regression models that accounts for misclassification of the seroconversion status due to early testing using parametric and nonparametric estimates of the seroconversion time distribution. The method is shown to reduce the bias resulting from naive analyses in simulation studies and an application to the data from the orthopedic studies provides further illustration.     

Extending Tests of Random Effects to Assess for Measurement Invariance in Factor Models

Factor analysis models are widely used in health research to summarize hard-to-measure predictor or outcome variable constructs. For example, in the ELEMENT study, factor models are used to summarize lead exposure biomarkers which are thought to indirectly measure prenatal exposure to lead. Classic latent factor models are fitted assuming that factor loadings are constant across all covariate levels (e.g., maternal age in ELEMENT); that is, measurement invariance (MI) is assumed. When the MI is not met, measurement bias is introduced. Traditionally, MI is examined by defining subgroups of the data based on covariates, fitting multi-group factor analysis, and testing differences in factor loadings across covariate groups. In this paper, we develop novel tests of measurement invariance by modeling the factor loadings as varying coefficients, i.e., letting the factor loading vary across continuous covariate values instead of groups. These varying coefficients are estimated using penalized splines, where spline coefficients are penalized by treating them as random coefficients. The test of MI is then carried out by conducting a likelihood ratio test for the null hypothesis that the variance of the random spline coefficients equals zero. We use a Monte Carlo EM algorithm for estimation, and obtain the likelihood using Monte Carlo integration. Using simulations, we compare the Type I error and power of our testing approach and the multi-group testing method. We apply the proposed methods to summarize data on prenatal biomarkers of lead exposure from the ELEMENT study and find violations of MI due to maternal age.     

Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods

We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise.     

FLCRM: Functional linear cox regression model

Summary

We consider a functional linear Cox regression model for characterizing the association between time‐to‐event data and a set of functional and scalar predictors. The functional linear Cox regression model incorporates a functional principal component analysis for modeling the functional predictors and a high‐dimensional Cox regression model to characterize the joint effects of both functional and scalar predictors on the time‐to‐event data. We develop an algorithm to calculate the maximum approximate partial likelihood estimates of unknown finite and infinite dimensional parameters. We also systematically investigate the rate of convergence of the maximum approximate partial likelihood estimates and a score test statistic for testing the nullity of the slope function associated with the functional predictors. We demonstrate our estimation and testing procedures by using simulations and the analysis of the Alzheimer's Disease Neuroimaging Initiative (ADNI) data. Our real data analyses show that high‐dimensional hippocampus surface data may be an important marker for predicting time to conversion to Alzheimer's disease. Data used in the preparation of this article were obtained from the ADNI database ( adni.loni.usc.edu ).     

Quantile Function Models for Quantal Response Analysis

The usual analysis of quantal response data occurring in diverse fields such as economics, medicine, psychology and toxicology use probit and logit models or their extensions with generalized least squares or the principle of likelihood as the method of statistical inference. The symmetric alternative models lead to practically comparable results and the choice of model or method is determined by considerations of familiarity and computational convenience. Recent attempts at improvement involve larger parametric families of tolerance distributions and employ the method of maximum likelihood in analysis. In this paper we consider models with the tolerance distributions based upon the Tukey-lambda distributions which are described in terms of their quantile functions. The likelihood methods for fitting the models and testing their adequacies are developed and illustrated using classical data due to BLISS (1935) and ASHFORD and SMITH (1964).     

Likelihood inference for exchangeable binary data with varying cluster sizes

This article investigates maximum likelihood estimation with saturated and unsaturated models for correlated exchangeable binary data, when a sample of independent clusters of varying sizes is available. We discuss various parameterizations of these models, and propose using the EM algorithm to obtain maximum likelihood estimates. The methodology is illustrated by applications to a study of familial disease aggregation and to the design of a proposed group randomized cancer prevention trial.     

Analyzing Incomplete Data Subject to a Threshold using Empirical Likelihood Methods: An Application to a Pneumonia Risk Study in an ICU Setting

Summary .  The initial detection of ventilator-associated pneumonia (VAP) for inpatients at an intensive care unit needs composite symptom evaluation using clinical criteria such as the clinical pulmonary infection score (CPIS). When CPIS is above a threshold value, bronchoalveolar lavage (BAL) is performed to confirm the diagnosis by counting actual bacterial pathogens. Thus, CPIS and BAL results are closely related and both are important indicators of pneumonia whereas BAL data are incomplete. To compare the pneumonia risks among treatment groups for such incomplete data, we derive a method that combines nonparametric empirical likelihood ratio techniques with classical testing for parametric models. This technique augments the study power by enabling us to use any observed data. The asymptotic property of the proposed method is investigated theoretically. Monte Carlo simulations confirm both the asymptotic results and good power properties of the proposed method. The method is applied to the actual data obtained in clinical practice settings and compares VAP risks among treatment groups.     

Bootstrap Restricted Likelihood Ratio Test for the Detection of Rare Variants

In this paper the detection of rare variants association with continuous phenotypes of interest is investigated via the likelihood-ratio based variance component test under the framework of linear mixed models. The hypothesis testing is challenging and nonstandard, since under the null the variance component is located on the boundary of its parameter space. In this situation the usual asymptotic chisquare distribution of the likelihood ratio statistic does not necessarily hold. To circumvent the derivation of the null distribution we resort to the bootstrap method due to its generic applicability and being easy to implement. Both parametric and nonparametric bootstrap likelihood ratio tests are studied. Numerical studies are implemented to evaluate the performance of the proposed bootstrap likelihood ratio test and compare to some existing methods for the identification of rare variants. To reduce the computational time of the bootstrap likelihood ratio test we propose an effective approximation mixture for the bootstrap null distribution. The GAW17 data is used to illustrate the proposed test.     

Estimating demographic models for the range dynamics of plant species

Aims To better understand how demographic processes shape the range dynamics of woody plants (in this case, Proteaceae), we introduce a likelihood framework for fitting process‐based models of range dynamics to spatial abundance data. Location The fire‐prone Fynbos biome (Cape Floristic Region, South Africa). Methods Our process‐based models have a spatially explicit demographic submodel (describing dispersal, reproduction, mortality and local extinction) as well as an observation submodel (describing imperfect detection of individuals), and are constrained by species‐specific predictions of habitat distribution models and process‐based models for seed dispersal by wind. Free model parameters were varied to find parameter sets with the highest likelihood. After testing this approach with simulated data, we applied it to eight Proteaceae species that differ in breeding system (monoecy versus dioecy) and adult fire survival. We assess the importance of Allee effects and negative density dependence for range dynamics, by using the Akaike information criterion to select between alternative models fitted for the same species. Results The best model for all dioecious study species included Allee effects, whereas this was true for only one of four monoecious species. As expected, sprouters (in which adults survive fire) were estimated to have lower rates of reproduction and catastrophic population extinction than related non‐sprouters. Overcompensatory population dynamics seem important for three of four non‐sprouters. We also found good quantitative agreement between independent data and most estimates of reproduction, carrying capacity and extinction probability. Main conclusions This study shows that process‐based models can quantitatively describe how large‐scale abundance distributions arise from the movement and interaction of individuals. It stresses links between the life history, demography and range dynamics of Proteaceae: dioecious species seem more susceptible to Allee effects which reduce migration ability and increase local extinction risk, and sprouters seem to have high persistence of established populations, but their low reproduction limits habitat colonization and migration.     

Estimating environmental effects on population dynamics: consequences of observation error

Within the paradigm of population dynamics a central task is to identify environmental factors affecting population change and to estimate the strength of these effects. We here investigate the impact of observation errors in measurements of population densities on estimates of environmental effects. Adding observation errors may change the autocorrelation of a population time series with potential consequences for estimates of effects of autocorrelated environmental covariates. Using Monte Carlo simulations, we compare the performance of maximum likelihood estimates from three stochastic versions of the Gompertz model (log–linear first order autoregressive model), assuming 1) process error only, 2) observation error only, and 3) both process and observation error (the linear state–space model on log‐scale). We also simulated population dynamics using the Ricker model, and evaluated the corresponding maximum likelihood estimates for process error models. When there is observation error in the data and the considered environmental variable is strongly autocorrelated, its estimated effect is likely to be biased when using process error models. The environmental effect is overestimated when the sign of the autocorrelations of the intrinsic dynamics and the environment are the same and underestimated when the signs differ. With non‐autocorrelated environmental covariates, process error models produce fairly exact point estimates as well as reliable confidence intervals for environmental effects. In all scenarios, observation error models produce unbiased estimates with reasonable precision, but confidence intervals derived from the likelihood profiles are far too optimistic if there is process error present. The safest approach is to use state–space models in presence of observation error. These are factors worthwhile to consider when interpreting earlier empirical results on population time series, and in future studies, we recommend choosing carefully the modelling approach with respect to intrinsic population dynamics and covariate autocorrelation.