On Testing an Unspecified Function Through a Linear Mixed Effects Model with Multiple Variance Components

Summary We examine a generalized F ‐test of a nonparametric function through penalized splines and a linear mixed effects model representation. With a mixed effects model representation of penalized splines, we imbed the test of an unspecified function into a test of some fixed effects and a variance component in a linear mixed effects model with nuisance variance components under the null. The procedure can be used to test a nonparametric function or varying‐coefficient with clustered data, compare two spline functions, test the significance of an unspecified function in an additive model with multiple components, and test a row or a column effect in a two‐way analysis of variance model. Through a spectral decomposition of the residual sum of squares, we provide a fast algorithm for computing the null distribution of the test, which significantly improves the computational efficiency over bootstrap. The spectral representation reveals a connection between the likelihood ratio test (LRT) in a multiple variance components model and a single component model. We examine our methods through simulations, where we show that the power of the generalized F ‐test may be higher than the LRT, depending on the hypothesis of interest and the true model under the alternative. We apply these methods to compute the genome‐wide critical value and p ‐value of a genetic association test in a genome‐wide association study (GWAS), where the usual bootstrap is computationally intensive (up to 10⁸ simulations) and asymptotic approximation may be unreliable and conservative.     

Fixed and Random Effects Selection in Mixed Effects Models

Summary We consider selecting both fixed and random effects in a general class of mixed effects models using maximum penalized likelihood (MPL) estimation along with the smoothly clipped absolute deviation (SCAD) and adaptive least absolute shrinkage and selection operator (ALASSO) penalty functions. The MPL estimates are shown to possess consistency and sparsity properties and asymptotic normality. A model selection criterion, called the IC_Q statistic, is proposed for selecting the penalty parameters ( Ibrahim, Zhu, and Tang, 2008 , Journal of the American Statistical Association 103, 1648–1658). The variable selection procedure based on IC_Q is shown to consistently select important fixed and random effects. The methodology is very general and can be applied to numerous situations involving random effects, including generalized linear mixed models. Simulation studies and a real data set from a Yale infant growth study are used to illustrate the proposed methodology.     

Association Models for Clustered Data with Binary and Continuous Responses

Summary .  We consider analysis of clustered data with mixed bivariate responses, i.e., where each member of the cluster has a binary and a continuous outcome. We propose a new bivariate random effects model that induces associations among the binary outcomes within a cluster, among the continuous outcomes within a cluster, between a binary outcome and a continuous outcome from different subjects within a cluster, as well as the direct association between the binary and continuous outcomes within the same subject. For the ease of interpretations of the regression effects, the marginal model of the binary response probability integrated over the random effects preserves the logistic form and the marginal expectation of the continuous response preserves the linear form. We implement maximum likelihood estimation of our model parameters using standard software such as PROC NLMIXED of SAS . Our simulation study demonstrates the robustness of our method with respect to the misspecification of the regression model as well as the random effects model. We illustrate our methodology by analyzing a developmental toxicity study of ethylene glycol in mice.     

On the Effects of Departure from Normality on the F-Ratios in a Nested Random Effect Model by Mixture of Normals

In this paper we consider the following nested random effect model where the α_i's, the β_ij's and the e_ijk are independent random variables. By assuming that these variables follow a mixture of two normal densities, we study the effects of departure from normality on the classical JT-tests for variance components. It is shown that the departure from normality has little effects on the type 1 error and the power function; this indicates that the classical F-tests are quite robust with respect to departure from normality.     

Robust linear mixed models using the skew t distribution with application to schizophrenia data

We consider an extension of linear mixed models by assuming a multivariate skew t distribution for the random effects and a multivariate t distribution for the error terms. The proposed model provides flexibility in capturing the effects of skewness and heavy tails simultaneously among continuous longitudinal data. We present an efficient alternating expectation‐conditional maximization (AECM) algorithm for the computation of maximum likelihood estimates of parameters on the basis of two convenient hierarchical formulations. The techniques for the prediction of random effects and intermittent missing values under this model are also investigated. Our methodologies are illustrated through an application to schizophrenia data.     

Robust Joint Modeling of Longitudinal Measurements and Competing Risks Failure Time Data

Existing methods for joint modeling of longitudinal measurements and survival data can be highly influenced by outliers in the longitudinal outcome. We propose a joint model for analysis of longitudinal measurements and competing risks failure time data which is robust in the presence of outlying longitudinal observations during follow‐up. Our model consists of a linear mixed effects sub‐model for the longitudinal outcome and a proportional cause‐specific hazards frailty sub‐model for the competing risks data, linked together by latent random effects. Instead of the usual normality assumption for measurement errors in the linear mixed effects sub‐model, we adopt a t ‐distribution which has a longer tail and thus is more robust to outliers. We derive an EM algorithm for the maximum likelihood estimates of the parameters and estimate their standard errors using a profile likelihood method. The proposed method is evaluated by simulation studies and is applied to a scleroderma lung study (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient estimation for patient-specific rates of disease progression using nonnormal linear mixed models

Summary .  This article presents a new class of nonnormal linear mixed models that provide an efficient estimation of subject-specific disease progression in the analysis of longitudinal data from the Modification of Diet in Renal Disease (MDRD) trial. This new analysis addresses the previously reported finding that the distribution of the random effect characterizing disease progression is negatively skewed. We assume a log-gamma distribution for the random effects and provide the maximum likelihood inference for the proposed nonnormal linear mixed model. We derive the predictive distribution of patient-specific disease progression rates, which demonstrates rather different individual progression profiles from those obtained from the normal linear mixed model analysis. To validate the adequacy of the log-gamma assumption versus the usual normality assumption for the random effects, we propose a lack-of-fit test that clearly indicates a better fit for the log-gamma modeling in the analysis of the MDRD data. The full maximum likelihood inference is also advantageous in dealing with the missing at random (MAR) type of dropouts encountered in the MDRD data.     

Construction and Analysis of an Augmented Lattice Square Design

Augmented designs are useful for screening experiments involving large numbers of new and untried treatments. Since resolvable row‐column designs are useful for controlling extraneous variation, it is desirable to use such designs for the check or standard treatments to construct augmented lattice square experiment designs. A simple procedure is described for constructing such designs using c = 2k and c = 3k check treatments and n = rk(k ‐— 2) and n = rk(k — 3) new treatments, respectively, r being the number of complete blocks. A trend analysis for these designs, which allows for solutions of fixed effects, is presented. The random effects case is also discussed. A SAS computer code and the output from this code illustrated with a small numerical example are available from the author.     

Validity of f-test in a randomized block design with missing data

In this paper the first two moments of the test criterion (Treat. S.S.)/(Treat.S.S. + Error S.S.) have been derived assuming that one observation corresponding to the first plot of the first block which is under treatment ‘m’ (say), is missing in a randomized block design with ‘V’ treatments and ‘r’ blocks. To keep the analysis simple the case of one missing observation has been considered. It is concluded that in general the design is not unbiased in YATES (1951) sense and the usual F-test is satisfactory iff /(vr - r - 1)=(S - S₁)/(v - 1) (r-1), the block errors are homogeneous and (vr - r - 1) is large. The analysis of three unifornity trial data indicates that the first condition is the most important for the F-test to be satisfactory. However, if one observation is missing at random from some plot of some block, the F-test is unbiased. If the block errors are homogeneous and (vr - r - 1) GT96, the F-test also provides a good approximation to the corresponding randomization test in this case.     

Mixed model analysis of censored longitudinal data with flexible random-effects density

Mixed models are commonly used to represent longitudinal or repeated measures data. An additional complication arises when the response is censored, for example, due to limits of quantification of the assay used. While Gaussian random effects are routinely assumed, little work has characterized the consequences of misspecifying the random-effects distribution nor has a more flexible distribution been studied for censored longitudinal data. We show that, in general, maximum likelihood estimators will not be consistent when the random-effects density is misspecified, and the effect of misspecification is likely to be greatest when the true random-effects density deviates substantially from normality and the number of noncensored observations on each subject is small. We develop a mixed model framework for censored longitudinal data in which the random effects are represented by the flexible seminonparametric density and show how to obtain estimates in SAS procedure NLMIXED. Simulations show that this approach can lead to reduction in bias and increase in efficiency relative to assuming Gaussian random effects. The methods are demonstrated on data from a study of hepatitis C virus.     

Testing for cubic smoothing splines under dependent data

Summary In most research on smoothing splines the focus has been on estimation, while inference, especially hypothesis testing, has received less attention. By defining design matrices for fixed and random effects and the structure of the covariance matrices of random errors in an appropriate way, the cubic smoothing spline admits a mixed model formulation, which places this nonparametric smoother firmly in a parametric setting. Thus nonlinear curves can be included with random effects and random coefficients. The smoothing parameter is the ratio of the random‐coefficient and error variances and tests for linear regression reduce to tests for zero random‐coefficient variances. We propose an exact F‐test for the situation and investigate its performance in a real pine stem data set and by simulation experiments. Under certain conditions the suggested methods can also be applied when the data are dependent.     

On the Impact of Parametric Assumptions and Robust Alternatives for Longitudinal Data Analysis

Models for longitudinal data are employed in a wide range of behavioral, biomedical, psychosocial, and health‐care‐related research. One popular model for continuous response is the linear mixed‐effects model (LMM). Although simulations by recent studies show that LMM provides reliable estimates under departures from the normality assumption for complete data, the invariable occurrence of missing data in practical studies renders such robustness results less useful when applied to real study data. In this paper, we show by simulated studies that in the presence of missing data estimates of the fixed effect of LMM are biased under departures from normality. We discuss two robust alternatives, the weighted generalized estimating equations (WGEE) and the augmented WGEE (AWGEE), and compare their performances with LMM using real as well as simulated data. Our simulation results show that both WGEE and AWGEE provide valid inference for skewed non‐normal data when missing data follows the missing at random, the most popular missing data mechanism for real study data.     

On Analysis of Covariance Under Heteroscedasticity and Robustness with Respect to Departure from Normality

In this paper we consider a covariance model under heteroscedastic variances and propose a procedure for testing treatment effects. A simultaneous procedure for comparing the treatment effects is also developed. Some Monte Carlo studies indicate that the effects of departure from normality may not be very serious although some effects do exist due to high level of heteroscedasticity.     

A scaled linear mixed model for multiple outcomes

We propose a scaled linear mixed model to assess the effects of exposure and other covariates on multiple continuous outcomes. The most general form of the model allows a different exposure effect for each outcome. An important special case is a model that represents the exposure effects using a common global measure that can be characterized in terms of effect sizes. Correlations among different outcomes within the same subject are accommodated using random effects. We develop two approaches to model fitting, including the maximum likelihood method and the working parameter method. A key feature of both methods is that they can be easily implemented by repeatedly calling software for fitting standard linear mixed models, e.g., SAS PROC MIXED. Compared to the maximum likelihood method, the working parameter method is easier to implement and yields fully efficient estimators of the parameters of interest. We illustrate the proposed methods by analyzing data from a study of the effects of occupational pesticide exposure on semen quality in a cohort of Chinese men.     

基于线性混合效应的红松人工林枝条生物量模型

基于黑龙江省孟家岗林场60株红松解析木3643个枝条生物量的实测数据,利用全部子回归技术建立了枝条生物量模型(枝、叶和枝总生物量模型),最终选择lnw=k₁+k₂lnL_b+k₃lnD_b为枝条生物量最优基础模型.利用SAS 9.3统计软件的PROC MIXED模块建立枝条生物量混合模型,并采用AIC、BIC、对数似然值和似然比等统计指标评价不同模型的拟合效果.结果表明: 红松解析木的叶和枝总生物量混合模型以k₁、k₂、k₃作为随机效应参数的拟合效果最好,而枝生物量混合模型以k₁、k₂作为随机效应参数的拟合效果最好.最后将枝条生物量最优基础模型与最优混合模型进行模型检验.混合模型各项指标优于基础模型,能有效地提高模型的预估精度,并且通过方差协方差结构校正随机参数来反映树木之间的差异.     

Harmonic Mean Approach to Regression Analysis Under Heteroscedasticity and Nonnormality

In this paper, by combining the harmonic mean approach with the Welch and the James procedure (see WELCH 1951, JAMES, 1951), we develop some robust procedures for testing parallelism in several straight lines under heteroscedasticity and nonnormality. Through Monte Carlo simulations it is shown that these new tests are quite robust with respect to departure from normality. For small sample sizes, however, the TAN-TABATABAI (1984) F_β and F*_β tests appear to be more powerful than the new tests, although when sample sizes are not small, there are hardly any differences between the Tan-Tabatabai F_β and F*_β tests and the new tests.     

A gene's eye view of epistasis,selection and speciation

In this mini‐review, I discuss the effects of gene interaction or epistasis from a `gene's eye view.' By a `gene's eye view' of epistasis, I mean that I will consider a single, bi‐allelic locus, A , whose effects on fitness result only from its interactions with alleles of another, unknown locus, X . I will show how changes in the frequencies of alleles at the background locus affect the relationship of alleles at the A ‐locus to fitness. Changing the genetic background changes the fundamental characteristics of the A ‐locus, such as the magnitude and sign of allelic effects on fitness, and, consequently, it changes the strength and pattern of selection. I consider each of the four kinds of pair–wise interactions between two loci and show that some kinds of epistasis are more sensitive than others to population genetic subdivision. Lastly, I show that some kinds of epistasis are more likely than others to affect the process of speciation and contribute to or be responsible for general genetic features of interspecific hybrids, such as Haldane's rule.     

Estimation of Fixed and Random Effects in the Functional Nonlinear Errors-in-Variable Model

Consider an experiment where a nonlinear continuous functional relationship exists between y and X. Assume that this relationship has been measured at n replicated points of X from each of t treatments or populations. Assume further that the X are fixed unknown vectors and that the location parameter v is either a fixed unknown vector or a vector of random variables. In the first case various linear hypotheses are to be tested about v, such as tests for main effects and interaction; in the second case, the mean and variance of the random variable v are to be estimated. A two-step procedure based on asymptotic theory is presented to test hypotheses or develop estimates for the fixed effects or random effects functional errors-in-variable model. An example of a one-way random effects model is given.     

Conformational change of poly( -lysine) induced by lipid vesicles of dilauroylphosphatidic acid

The effect of negatively charged dilauroylphosphatidic acid (DLPA) vesicles on the conformation of poly( -lysine) was investigated by circular dichroism measurements. DLPA vesicles induced a confomiational change Of poly( -lysine) from the random coil to β-structure in 5 mM Tes, pH 7.0. The fraction of induced β-structure (F_β) was determined via a procedure of curve fit the observed spectra to the reference spectra. F_β increased linearly with the molar ratio, r, of DLPA to lysine residues up to r 0.7, and reached a saturation value of 1 at r > 1. Within the range 0.7 r 1, precipitation occurred. The effect of dilution of the negative charge on vesicle membranes was examined by mixing DLPA with dilauroylphosphatidylcholine (DLPC). Although the β-structure Of poly -lysine) was also induced by mixed vesicles, the saturation value of F_β decreased with decreasing DLPA content in mixed vesicles. The variation in saturation value of F_β with the composition of mixed vesicles was interpreted in terms of the change in average distance between DLPA head groups in mixed vesicles.     

ESTIMATING HIERARCHICAL F-STATISTICS

This paper presents an analysis of variance (ANOVA) approach by which estimation of F-statistics can be made from data with an arbitrary s-level hierarchical population structure. Assuming a complete random-effect model, a general ANOVA procedure is developed to estimate F-statistics as ratios of different variance components for all levels of population subdivision in the hierarchy. A generalized relationship among F-statistics is also derived to extend the well-known relationship originally found by Sewall Wright. Although not entirely free from the bias particular to small number of subdivisions at each hierarchy and extreme gene frequencies, the ANOVA estimators of F-statistics consider sampling effects at each level of hierarchy, thus removing the bias incurred in the other estimators that are commonly based on direct substitution of unknown gene frequencies by their sample estimates. Therefore, the ANOVA estimation procedure presented here may become increasingly useful in analyzing complex population structure because of increasing use of the estimated hierarchical F-statistics to infer genetic and demographic structures of natural populations within and among species.