Multilevel Analysis with Few Clusters: Improving Likelihood-Based Methods to Provide Unbiased Estimates and Accurate Inference

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

This work describes a Gaussian Markov random field model that includes several previously proposed models, and studies properties of its maximum likelihood (ML) and restricted maximum likelihood (REML) estimators in a special case. Specifically, for models where a particular relation holds between the regression and precision matrices of the model, we provide sufficient conditions for existence and uniqueness of ML and REML estimators of the covariance parameters, and provide a straightforward way to compute them. It is found that the ML estimator always exists while the REML estimator may not exist with positive probability. A numerical comparison suggests that for this model ML estimators of covariance parameters have, overall, better frequentist properties than REML estimators.

Genetic correlations (rho ( g )) are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of rho ( g ) are not known, and accurate methods have not been described for estimating the precision of estimates of rho ( g ). Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of rho ( g ) by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of rho ( g ), and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for rho ( g ). MANOVA estimates of rho ( g ) were normally distributed. REML and ML estimates were normally distributed for rho ( g ) = 0.1, but skewed for rho ( g ) = 0.5 and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of rho ( g ) close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, rho ( g ), n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, rho ( g ), n, and r.

The EM algorithm is a widely used technique for finding maximum likelihood (ML) estimates when the data are not fully observed. Despite its popularity for computing ML estimates in unrestricted problems and the need for simplified computations for problems with equality and inequality restrictions, there have been few applications of the algorithm to restricted ML estimation. The EM algorithm is presented for restricted ML estimation and provides its applications to the probit model under equality and inequality restrictions using two small data sets.

On the Information Content of Partial Observations: A Bayesian Approach

Approximate ML and REML estimation for regression models with spatial or time series AR(1) noise

A one-stage individual participant data (IPD) meta-analysis synthesizes IPD from multiple studies using a general or generalized linear mixed model. This produces summary results (eg, about treatment effect) in a single step, whilst accounting for clustering of participants within studies (via a stratified study intercept, or random study intercepts) and between-study heterogeneity (via random treatment effects). We use simulation to evaluate the performance of restricted maximum likelihood (REML) and maximum likelihood (ML) estimation of one-stage IPD meta-analysis models for synthesizing randomized trials with continuous or binary outcomes. Three key findings are identified. First, for ML or REML estimation of stratified intercept or random intercepts models, a t-distribution based approach generally improves coverage of confidence intervals for the summary treatment effect, compared with a z-based approach. Second, when using ML estimation of a one-stage model with a stratified intercept, the treatment variable should be coded using "study-specific centering" (ie, 1/0 minus the study-specific proportion of participants in the treatment group), as this reduces the bias in the between-study variance estimate (compared with 1/0 and other coding options). Third, REML estimation reduces downward bias in between-study variance estimates compared with ML estimation, and does not depend on the treatment variable coding; for binary outcomes, this requires REML estimation of the pseudo-likelihood, although this may not be stable in some situations (eg, when data are sparse). Two applied examples are used to illustrate the findings.

The restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations. In this paper, we show the REML estimates are consistent if the model is asymptotically identifiable and infinitely informative under the (location) invariant class, and are asymptotically normal (A.N.) if in addition the model is asymptotically nondegenerate. The result does not require normality or boundedness of the rank p of design matrix of fixed effects. Moreover, we give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates (MLE) in non-normal cases. As an application, we show for all unconfounded balanced mixed models of the analysis of variance the REML (ANOVA) estimates are consistent; and are also A.N. provided the models are nondegenerate; the MLE are consistent (A.N.) if and only if certain constraints on p are satisfied.

A data matrix is said to be ipsative when the sum of the scores obtained over the variables for each subject is a constant. In this article, a general type of ipsative data known as partially additive ipsative data (PAID) is defined. Ordinary additive ipsative data (All311 is a special case. Due to the specific nature of the research design or measurement process, the observed vector is X PAID with an underlying nonipsative vector y. It is shown that if the underlying distribution of y is multivariate normal with structured covariance matrix Σ = Σ(Θ), the observed X will have a degenerate normal distribution. As a result, ordinary maximum likelihood estimation of Θ cannot be carried out directly. A transformation of X is suggested so that the transformed vector X* = BX will have a nonsingular density and restricted maximum likelihood (REML) estimation can be applied. A simulation study is conducted to investigate the effect of sample size and other model characteristics on the performance of the ML estimators and the sampling behavior of the goodness of fit statistic. It is found that REML estimates are in general close to the true parameter values, but they have larger dard errors as compared with the ordinary MLE based on y. The test statistic is well behaved when sample size is large enough. Moreover, the likelihood of obtaining a convergent solution depends on a number of factors such as sample size, number of indicators per latent factor, and degree of ipsativity. Finally, statistical decisions (reject or not reject the hypothesized model) based on X* are in general consistent with that based on y.

The importance of providing structural validity evidence for test score(s) derived from psychometric test instruments is highlighted by several institutions; for example, the American Psychological Association (2014) demands that evidence for the validity of an instruments' internal structure and its underlying measurement model must be provided before it is applied in psychological assessment. The knowledge about the latent structure of data obtained with tests addressing the major question "What is/are the construct[s] being measured" by psychological tests under investigation (Ziegler, 2014 (Ziegler, , 2020)) . The study of structural validity is typically addressed with factor analyses when the test scores reflect continuous latent traits. As most submissions to Psychological Test Adaptation and Development (PTAD) deal with the adaptation and further development of existing measures, authors typically test a measurement model that is based on theoretical considerations and prior findings on original versions (or adaptations) of the test under investigation. Our literature review of PTAD's publications showed that more than 90% of the articles contain at least one confirmatory factor analysis (CFA). As editor and reviewers of PTAD, we appreciate that authors are rigorous in providing evidence on the structural validity of their tests' data. However, since PTAD's inception in 2019, we experience that one comment is frequently communicated to authors during the review process, namely, the request to adjust the analytic approach in CFA from maximum likelihood (ML) estimation toward using the mean-and variance-adjusted weighted least squares (WLSMV; Muthén et al., 1997) estimator to account for the ordinal nature of the data that psychological instruments typically generate on the item level. In this editorial, we discuss the rationale behind choosing the WLSMV estimator when analyzing test adaptations and developments that are based on ordinal categorical data and concisely illustrate the problems associated with using the ML estimator (potentially in combination with robust tests of model fit) for such data.

A restricted maximum likelihood estimator for truncated height samples

Maximum likelihood (ML) estimation of spatial autoregressive models for large spatial data sets is well established by making use of the commonly sparse nature of the contiguity matrix on which spatial dependence is built. Adding a measurement error that naturally separates the spatial process from the measurement error process are not well established in the literature, however, and ML estimation of such models to large data sets is challenging. Recently a reduced rank approach was suggested which re-expresses and approximates such a model as a spatial random effects model (SRE) in order to achieve fast fitting of large data sets by fitting the corresponding SRE. In this paper we propose a fast and exact method to accomplish ML estimation and restricted ML estimation of complexity of $$O(n^{3/2})$$ operations when the contiguity matrix is based on a local neighbourhood. The methods are illustrated using the well known data set on house prices in Lucas County in Ohio.

In their paper on maximum likelihood will) Incomplete data. Dempster. Laird, and Rubin (1977) noted that both maximum likelihood (ML) and restricted ML (REML) estimators of variance components in the mixed model analysis oi variance can be computed via the LM algorithm. Thi-follows from treating the random effects as missing data and using the incomplete data framework outlined in Dempster, et al. (1977). We elaborate on this idea, introducing a class of generalized ML estimates, indexed by a parameter τ, which contain REML and ordinary ML estimates as special limiting cases. This device enables us to derive a single set of iterative EM equations which yields either ML or REML estimates of the variance components, depending upon the value specified for τ.

The maximum likelihood (ML) procedure of Hartley aud Rao [2] is modified by adapting a transformation from Pattersou and Thompson [7] which partitions the likelihood render normality into two parts, one being free of the fixed effects. Maximizing this part yields what are called restricted maximum likelihood (REML) estimators. As well as retaining the property of invariance under translation that ML estimators have, the REML estimators have the additional property of reducing to the analysis variance (ANOVA) estimators for many, if not all, cases of balanced data (equal subclass numbers). A computing algorithm is developed, adapting a transformation from Hemmerle and Hartley [6], which reduces computing requirements to dealing with matrices having order equal to the dimension of the parameter space rather than that of the sample space. These same matrices also occur in the asymptotic sampling variances of the estimators.

Meta‐regression can be used to examine the association between effect size estimates and the characteristics of the studies included in a meta‐analysis using regression‐type methods. By searching for those characteristics (i.e., moderators) that are related to the effect sizes, we seek to identify a model that represents the best approximation to the underlying data generating mechanism. Model selection via testing, either through a series of univariate models or a model including all moderators, is the most commonly used approach for this purpose. Here, we describe alternative model selection methods based on information criteria, multimodel inference, and relative variable importance. We demonstrate their application using an illustrative example and present results from a simulation study to compare the performance of the various model selection methods for identifying the true model across a wide variety of conditions. Whether information‐theoretic approaches can also be used not only in combination with maximum likelihood (ML) but also restricted maximum likelihood (REML) estimation was also examined. The results indicate that the conventional methods for model selection may be outperformed by information‐theoretic approaches. The latter are more often among the set of best methods across all of the conditions simulated and can have higher probabilities for identifying the true model under particular scenarios. Moreover, their performance based on REML estimation was either very similar to that from ML estimation or at times even better depending on how exactly the REML likelihood was computed. These results suggest that alternative model selection methods should be more widely applied in meta‐regression.