Least squares estimation of regression parameters in mixed effects models with unmeasured covariates

We consider mixed effects models for longitudinal, repeated measures or clustered data. Unmeasured or omitted covariates in such models may be correlated with the included covanates, and create model violations when not taken into account. Previous research and experience with longitudinal data sets suggest a general form of model which should be considered when omitted covariates are likely, such as in observational studies. We derive the marginal model between the response variable and included covariates, and consider model fitting using the ordinary and weighted least squares methods, which require simple non-iterative computation and no assumptions on the distribution of random covariates or error terms, Asymptotic properties of the least squares estimators are also discussed. The results shed light on the structure of least squares estimators in mixed effects models, and provide large sample procedures for statistical inference and prediction based on the marginal model. We present an example of the relationship between fluid intake and output in very low birth weight infants, where the model is found to have the assumed structure.     

Longitudinal models for polytomous responses

Recent work by Miller and Landis (1991) discusses generalized variance component models for polytomous responses. This work is adapted to longitudinal models for repeated measures of individuals having polytomous responses. In this setting, individuals are considered to be “clusters”. The resulting simplifications are discussed. First, each response has a multinomial distribution with N=l. Second, observed cluster proportions in the variance component estimates must be replaced by their expectations. This technique accommodates patients with missing data in a sequence of repeated observations.     

A monte carlo comparison of tests of parallelism of the ponses over time of two populations vith non-uniform observation times

Repeated measures data collected at random observation times are quite common in clinical studies and are often difficult to analyze. A Monte Carlo comparison of four analysis procedures with respect to significance level and power is presented. The basic procedures compared are successive difference analyses and three procedures using the data as summarized in the estimated quadratic polynomial regression coefficients for each subject. These three procedures are (1) Hotelling's T-square, (2) Multivariate Multisample Rank Sum Test (MMRST) and (3) Multivariate Multisample Median Test (MMMT).

For the variety of dispersion structures, sample sizes and treatement groups simulated the MMRST and successive difference analysis were the most satisfactory.     

Variable order ante-dependence models

Ante-dependence models can be used to model the covariance structure in problems involving repeated measures through time. They are conditional regression models which generalize Gabriel’s constant-order ante-dependence model. Likelihood-based procedures are presented, together with simple expressions for likelihood ratio test statistics in terms of sum of squares from appropriate analysis of covariance. The estimation of the orders is approached as a model selection problem, and penalized likelihood criteria are suggested. Extensions of all procedures discussed here to situations with a monotone pattern of missing data are presented.     

Bias from the use of generalized estimating equations to analyze incomplete longitudinal binary data

Patient dropout is a common problem in studies that collect repeated binary measurements. Generalized estimating equations (GEE) are often used to analyze such data. The dropout mechanism may be plausibly missing at random (MAR), i.e. unrelated to future measurements given covariates and past measurements. In this case, various authors have recommended weighted GEE with weights based on an assumed dropout model, or an imputation approach, or a doubly robust approach based on weighting and imputation. These approaches provide asymptotically unbiased inference, provided the dropout or imputation model (as appropriate) is correctly specified. Other authors have suggested that, provided the working correlation structure is correctly specified, GEE using an improved estimator of the correlation parameters (‘modified GEE’) show minimal bias. These modified GEE have not been thoroughly examined. In this paper, we study the asymptotic bias under MAR dropout of these modified GEE, the standard GEE, and also GEE using the true correlation. We demonstrate that all three methods are biased in general. The modified GEE may be preferred to the standard GEE and are subject to only minimal bias in many MAR scenarios but in others are substantially biased. Hence, we recommend the modified GEE be used with caution.     

Nonparametric analysis of treatment effects with missing observations

This paper develops a test for comparing treatment effects when observations are missing at random for repeated measures data on independent subjects. It is assumed that missingness at any occasion follows a Bernoulli distribution. It is shown that the distribution of the vector of linear rank statistics depends on the unknown parameters of the probability law that governs missingness, which is absent in the existing conditional methods employing rank statistics. This dependence is through the variance–covariance matrix of the vector of linear ranks. The test statistic is a quadratic form in the linear rank statistics when the variance–covariance matrix is estimated. The limiting distribution of the test statistic is derived under the null hypothesis. Several methods of estimating the unknown components of the variance–covariance matrix are considered. The estimate that produces stable empirical Type I error rate while maintaining the highest power among the competing tests is recommended for implementation in practice. Simulation studies are also presented to show the advantage of the proposed test over other rank-based tests that do not account for the randomness in the missing data pattern. Our method is shown to have the highest power while also maintaining near-nominal Type I error rates. Our results clearly illustrate that even for an ignorable missingness mechanism, the randomness in the pattern of missingness cannot be ignored. A real data example is presented to highlight the effectiveness of the proposed method.     

Testing the Equality of Covariance Operators in Functional Samples

Abstract. We propose a non‐parametric test for the equality of the covariance structures in two functional samples. The test statistic has a chi‐square asymptotic distribution with a known number of degrees of freedom, which depends on the level of dimension reduction needed to represent the data. Detailed analysis of the asymptotic properties is developed. Finite sample perfo‐rmance is examined by a simulation study and an application to egg‐laying curves of fruit flies.     

Leverage analysis for linear mixed models

We consider a generalized leverage matrix useful for the identification of influential units and observations in linear mixed models and show how a decomposition of this matrix may be employed to identify high leverage points for both the marginal fitted values and the random effect component of the conditional fitted values. We illustrate the different uses of the two components of the decomposition with a simulated example as well as with a real data set.     

Berechnung der verteilungsfunktion quadratischer formen von normalverteilen zufallsgröben and deren anwendung in der statistik

Zusammenfassung: Für die Bestimmung der Verteilungsfunktion F(x) von quadrastischen Formen normalverteilter Zufallsgröen werden Verfahren angegeben.Diese Verfahren werden mit den aus der Literatur bekannten Ergebnissen verglichen.Im Staz 2 wird eine quadratische Faltungsgleichung hergeleitet, aus der F(x) berechenbar ist. Weiterhin werden Abschätzungen nach oben und unten für F(x) hergeleitet.Die für endliche quadratischer Formen übertragen.Anwendungsbeispiele aus der Statistik zufälliger Prozesse (Identifikation und Konfidenzintervalle für die Mittelwerfunktion eines GAUSSPROZESSES) und aus der parameterfreien Statistik werden angegeben.Da zur Berechnung von F(x) eine quadratische Faltungsgleichung zu lösen ist, werden im letzten Teil der Arbeit Lösungs möglichkeiten fü diese Gleichung diskutiert und numerische Ergebnisse angegeben.     

On tests in variance components models 2

A procedure for finding exact tests for some hypotheses on variance components in unbalanced models is proposed. It is based on F-distributed statistics got by an orthogonal decomposition of the sample space.