Computational algorithms for the factor model

Algorithms for computing the maximum likelihood estimators and the estimated covariance matrix of the estimators of the factor model are derived. The algorithms are particularly suitable for large matrices and for samples that give zero estimates of some error variances. A method of constructing estimators for reduced models is presented. The algorithms can also be used for the multivariate errors-in-variables model with known error covariance matrix.     

Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

Prediction Error Estimation Under Bregman Divergence for Non-Parametric Regression and Classification

Abstract.  Prediction error is critical to assess model fit and evaluate model prediction. We propose the cross-validation (CV) and approximated CV methods for estimating prediction error under the Bregman divergence (BD), which embeds nearly all of the commonly used loss functions in the regression, classification procedures and machine learning literature. The approximated CV formulas are analytically derived, which facilitate fast estimation of prediction error under BD. We then study a data-driven optimal bandwidth selector for local-likelihood estimation that minimizes the overall prediction error or equivalently the covariance penalty. It is shown that the covariance penalty and CV methods converge to the same mean-prediction-error-criterion. We also propose a lower-bound scheme for computing the local logistic regression estimates and demonstrate that the algorithm monotonically enhances the target local likelihood and converges. The idea and methods are extended to the generalized varying-coefficient models and additive models.     

Multivariate survival mixed models for genetic analysis of longevity traits

A class of multivariate mixed survival models for continuous and discrete time with a complex covariance structure is introduced in a context of quantitative genetic applications. The methods introduced can be used in many applications in quantitative genetics although the discussion presented concentrates on longevity studies. The framework presented allows to combine models based on continuous time with models based on discrete time in a joint analysis. The continuous time models are approximations of the frailty model in which the baseline hazard function will be assumed to be piece-wise constant. The discrete time models used are multivariate variants of the discrete relative risk models. These models allow for regular parametric likelihood-based inference by exploring a coincidence of their likelihood functions and the likelihood functions of suitably defined multivariate generalized linear mixed models. The models include a dispersion parameter, which is essential for obtaining a decomposition of the variance of the trait of interest as a sum of parcels representing the additive genetic effects, environmental effects and unspecified sources of variability; as required in quantitative genetic applications. The methods presented are implemented in such a way that large and complex quantitative genetic data can be analyzed. Some key model control techniques are discussed in a supplementary online material.     

Multilevel maximum likelihood estimation with application to covariance matrices

The asymptotic variance of the maximum likelihood estimate is proved to decrease when the maximization is restricted to a subspace that contains the true parameter value. Maximum likelihood estimation allows a systematic fitting of covariance models to the sample, which is important in data assimilation. The hierarchical maximum likelihood approach is applied to the spectral diagonal covariance model with different parameterizations of eigenvalue decay, and to the sparse inverse covariance model with specified parameter values on different sets of nonzero entries. It is shown computationally that using smaller sets of parameters can decrease the sampling noise in high dimension substantially.     

Introduction of a new measure for detecting poor fit due to omitted nonlinear terms in SEM

The model chi-square that is used in linear structural equation modeling compares the fitted covariance matrix of a target model to an unstructured covariance matrix to assess global fit. For models with nonlinear terms, i.e., interaction or quadratic terms, this comparison is very problematic because these models are not nested within the saturated model that is represented by the unstructured covariance matrix. We propose a novel measure that quantifies the heteroscedasticity of residuals in structural equation models. It is based on a comparison of the likelihood for the residuals under the assumption of heteroscedasticity with the likelihood under the assumption of homoscedasticity. The measure is designed to respond to omitted nonlinear terms in the structural part of the model that result in heteroscedastic residual scores. In a small Monte Carlo study, we demonstrate that the measure appears to detect omitted nonlinear terms reliably when falsely a linear model is analyzed and the omitted nonlinear terms account for substantial nonlinear effects. The results also indicate that the measure did not respond when the correct model or an overparameterized model were used.     

Maximum likelihood estimates in the multivariate normal with patterned mean and covariance via the em algorithm

The maximum likelihood equations for a multivariate normal model with structured mean and structured covariance matrix may not have an explicit solution. In some cases the model's error term may be decomposed as the sum of two independent error terms, each having a patterned covariance matrix, such that if one of the unobservable error terms is artificially treated as "missing data", the EM algorithm can be used to compute the maximum likelihood estimates for the original problem. Some decompositions produce likelihood equations which do not have an explicit solution at each iteration of the EM algorithm, but within-iteration explicit solutions are shown for two general classes of models including covariance component models used for analysis of longitudinal data.     

Multivariate Poisson regression with covariance structure

In recent years the applications of multivariate Poisson models have increased, mainly because of the gradual increase in computer performance. The multivariate Poisson model used in practice is based on a common covariance term for all the pairs of variables. This is rather restrictive and does not allow for modelling the covariance structure of the data in a flexible way. In this paper we propose inference for a multivariate Poisson model with larger structure, i.e. different covariance for each pair of variables. Maximum likelihood estimation, as well as Bayesian estimation methods are proposed. Both are based on a data augmentation scheme that reflects the multivariate reduction derivation of the joint probability function. In order to enlarge the applicability of the model we allow for covariates in the specification of both the mean and the covariance parameters. Extension to models with complete structure with many multi-way covariance terms is discussed. The method is demonstrated by analyzing a real life data set.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic efficiency of a competing risks model: A two-sample case

A generalized Cox regression model is studied for the covariance analysis of competing risks data subject to independent random censoring. The information of the maximum partial likelihood estimates is compared with that of maximum likelihood estimates assuming a log linear hazard function.The method of generalized variance is used to define the efficiency of estimation between the two models. This is then applied to two-sample problems with two exponentially censoring rates. Numerical results are summarized ane presented graphically.The detailed results indicate that the semi-parametric model wrks well for a higher rate of censoring. A method of generalizing the result to type 1 censoring and the efficiency of estimating the coefficient of the covariate are discussecd. A brief account of using the results to help design experiments is also given.     

On a convergent stochastic estimation algorithm for frailty models

A maximum likelihood estimation procedure is presented for the frailty model. The procedure is based on a stochastic Expectation Maximization algorithm which converges quickly to the maximum likelihood estimate. The usual expectation step is replaced by a stochastic approximation of the complete log-likelihood using simulated values of unobserved frailties whereas the maximization step follows the same lines as those of the Expectation Maximization algorithm. The procedure allows to obtain at the same time estimations of the marginal likelihood and of the observed Fisher information matrix. Moreover, this stochastic Expectation Maximization algorithm requires less computation time. A wide variety of multivariate frailty models without any assumption on the covariance structure can be studied. To illustrate this procedure, a Gaussian frailty model with two frailty terms is introduced. The numerical results based on simulated data and on real bladder cancer data are more accurate than those obtained by using the Expectation Maximization Laplace algorithm and the Monte-Carlo Expectation Maximization one. Finally, since frailty models are used in many fields such as ecology, biology, economy, …, the proposed algorithm has a wide spectrum of applications.     

FITTING SPATIAL CORRELATION MODELS: APPROXIMATING A LIKELIHOOD APPROXIMATION

The paper considers a class of spatial correlation models (stationary Gaussian processes) which includes (spatial) conditional autoregressive, simultaneous autoregressive, moving average and direct covariance models. Given observations on a finite rectangular lattice, a likelihood approximation for estimating the parameters in the spectral density of the model is discussed. The approximation consists of applying the trapezoidal rule, with a her grid of frequencies than the usual Fourier frequencies, to compute the integral in an appraximation due to Whittle (1954) and later modified by Guyon (1984). With this approximation, a Fisher scoring type algorithm has a simple form and in some casea reduces to iteratively reweighted least squares. Methods for computing the unbiased two-dimensional periodogram required by the method are presented and the accuracy of the approximation is discussed. The asymptotic distribution of the parameter estimates computed from the likelihood approximation is also given.     

Mixed effects smoothing spline analysis of variance

We propose a general family of nonparametric mixed effects models. Smoothing splines are used to model the fixed effects and are estimated by maximizing the penalized likelihood function. The random effects are generic and are modelled parametrically by assuming that the covariance function depends on a parsimonious set of parameters. These parameters and the smoothing parameter are estimated simultaneously by the generalized maximum likelihood method. We derive a connection between a nonparametric mixed effects model and a linear mixed effects model. This connection suggests a way of fitting a nonparametric mixed effects model by using existing programs. The classical two-way mixed models and growth curve models are used as examples to demonstrate how to use smoothing spline analysis-of-variance decompositions to build nonparametric mixed effects models. Similarly to the classical analysis of variance, components of these nonparametric mixed effects models can be interpreted as main effects and interactions. The penalized likelihood estimates of the fixed effects in a two-way mixed model are extensions of James–Stein shrinkage estimates to correlated observations. In an example three nested nonparametric mixed effects models are fitted to a longitudinal data set.     

Random-effect models with singular precision

We show that smoothing spline, intrinsic autoregression (IAR) and state-space model can be formulated as partially specified random-effect model with singular precision (SP). Various fitting methods have been suggested for the aforementioned models and this paper investigates the relationships among them, once the models have been placed under a single framework. Some methods have been previously shown to give the best linear unbiased predictors (BLUPs) under some random-effect models and here we show that they are in fact uniformly BLUPs (UBLUPs) under a class of models that are generated by the SP of random effects. We offer some new interpretations of the UBLUPs under models of SP and define BLUE and BLUP in these partially specified models without having to specify the covariance. We also show how the full likelihood inferences for random-effect models can be made for these models, so that the maximum likelihood (ML) and restricted maximum likelihood (REML) estimators can be used for the smoothing parameters in splines, etc.     

Response transformations in repeated measures and growth curve models

This article describes estimation and inference procedures for the parameters of the Box-Cox and foided-power transformations in repeated measures and growth curve models. Procedures for computing maximum likelihood estimates of the transformation and covariance parameters under several covanance structures (omnibus sphericity, local sphericity, and unstructured) are described. Lack of fit statistics and hypothesis tests for comparing these structures also are described. The procedures are illustrated on three data sets. Software for performing the analyses in the SAS System is described and is available from the authors.     

A cautionary note on generalized linear models for covariance of unbalanced longitudinal data

Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations.     

Fitting a non-linear model with errors in both variables and its application *

Estimation of the parameters of a non-linear model is considered when both measured variables have random errors. The maximum likelihood estimates with the asymptotic variance and covariance matrix are presented. Real data are used to illustrate the procedure discussed.     

Longitudinal conditional models with intermittent missingness: SAS code and applications

This work provides a set of macros performed with SAS (Statistical Analysis System) for Windows, which can be used to fit conditional models under intermittent missingness in longitudinal data. A formalized transition model, including random effects for individuals and measurement error, is presented. Model fitting is based on the missing completely at random or missing at random assumptions, and the separability condition. The problem translates to maximization of the marginal observed data density only, which for Gaussian data is again Gaussian, meaning that the likelihood can be expressed in terms of the mean and covariance matrix of the observed data vector. A simulation study is presented and misspecification issues are considered. A practical application is also given, where conditional models are fitted to the data from a clinical trial that assessed the effect of a Cuban medicine on a disease of the respiratory system.