Some comments on a class of simultaneous inference procedures in ancova

Methods for constructing simultaneous confidence intervals for contrasts of treatment effects in analysis of covariance (ANCOVA) models are discussed and compared. A simple procedure is given by which, under a normality assumption made on the covariates, any method appropriate in an analysis of variance (ANOVA) may be applied to a corresponding ANCOVA by means of a simple adjustment of the required critical values. Some properties of this procedure are noted.     

Comparison of Statistical Methods for Pretest–Posttest Designs in Terms of Type I Error Probability and Statistical Power

The pretest–posttest design is widely used to investigate the effect of an experimental treatment in biomedical research. The treatment effect may be assessed using analysis of variance (ANOVA) or analysis of covariance (ANCOVA). The normality assumption for parametric ANOVA and ANCOVA may be violated due to outliers and skewness of data. Nonparametric methods, robust statistics, and data transformation may be used to address the nonnormality issue. However, there is no simultaneous comparison for the four statistical approaches in terms of empirical type I error probability and statistical power. We studied 13 ANOVA and ANCOVA models based on parametric approach, rank and normal score-based nonparametric approach, Huber M-estimation, and Box–Cox transformation using normal data with and without outliers and lognormal data. We found that ANCOVA models preserve the nominal significance level better and are more powerful than their ANOVA counterparts when the dependent variable and covariate are correlated. Huber M-estimation is the most liberal method. Nonparametric ANCOVA, especially ANCOVA based on normal score transformation, preserves the nominal significance level, has good statistical power, and is robust for data distribution.     

Properties of the rank transformation in factorial analysis of covariance

Real world data often fail to meet the underlying assumption of population normality. The Rank Transformation (RT) procedure has been recommended as an alternative to the parametric factorial analysis of covariance (ANCOVA). The purpose of this study was to compare the Type I error and power properties of the RT ANCOVA to the parametric procedure in the context of a completely randomized balanced 3 × 4 factorial layout with one covariate. This study was concerned with tests of homogeneity of regression coefficients and interaction under conditional (non)normality. Both procedures displayed erratic Type I error rates for the test of homogeneity of regression coefficients under conditional nonnormality. With all parametric assumptions valid, the simulation results demonstrated that the RT ANCOVA failed as a test for either homogeneity of regression coefficients or interaction due to severe Type I error inflation. The error inflation was most severe when departures from conditional normality were extreme. Also associated with the RT procedure was a loss of power. It is recommended that the RT procedure not be used as an alternative to factorial ANCOVA despite its encouragement from SAS, IMSL, and other respected sources.     

Bayesian analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable from the Bayesian point of view. We develop a fully Bayesian model for ANCOVA based on the conjugate prior distributions for parameters contained in the model. The Bayes estimator of parameters, ANCOVA model and adjusted effects for both treatments and covariates along with predictive distribution of future observations are developed. We also provide the essentials for comparing adjusted treatments effects and adjusted factor effects. A simulation study and a real world application are also performed to illustrate and evaluate the proposed Bayesian model.     

Gaussian mixture analysis of covariance

ABSTRACT

In many real-world applications, the traditional theory of analysis of covariance (ANCOVA) leads to inadequate and unreliable results because of violation of the response variable observations from the essential Gaussian assumption that may be due to the heterogeneity of population, the presence of outlier or both of them. In this paper, we develop a Gaussian mixture ANCOVA model for modelling heterogeneous populations with a finite number of subpopulation. We provide the maximum likelihood estimates of the model parameters via an EM algorithm. We also drive the adjusted effects estimators for treatments and covariates. The Fisher information matrix of the model and asymptotic confidence intervals for the parameter are also discussed. We performed a simulation study to assess the performance of the proposed model. A real-world example is also worked out to explained the methodology.     

Analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable. We develop the essential methodology for estimating the model parameters via maximum likelihood method. The general form of the maximum likelihood estimator is obtained in color closed form. Adjusted treatment effects and adjusted covariate effects are given, too. We also provide the asymptotic distribution of the proposed estimators. A simulation study and a real world application are also performed to illustrate and evaluate the proposed methodology.     

The adjustment of random baseline measurements in treatment effect estimation

The analysis of covariance (ANCOVA) is often used in analyzing clinical trials that make use of “baseline” response. Unlike Crager [1987. Analysis of covariance in parallel-group clinical trials with pretreatment baseline. Biometrics 43, 895–901.], we show that for random baseline covariate, the ordinary least squares (OLS)-based ANCOVA method provides invalid unconditional inference for the test of treatment effect when heterogeneous regression exists for the baseline covariate across different treatments. To correctly address the random feature of baseline response, we propose to directly model the pre- and post-treatment measurements as repeated outcome values of a subject. This bivariate modeling method is evaluated and compared with the ANCOVA method by a simulation study under a wide variety of settings. We find that the bivariate modeling method, applying the Kenward–Roger approximation and assuming distinct general variance–covariance matrix for different treatments, performs the best in analyzing a clinical trial that makes use of random baseline measurements.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

Rank repeated measures analysis of covariance

When analyzing a response variable at the presence of both factors and covariates, with potentially correlated responses and violated assumptions of the normal residual or the linear relationship between the response and the covariates, rank-based tests can be an option for inferential procedures instead of the parametric repeated measures analysis of covariance (ANCOVA) models. This article derives a rank-based method for multi-way ANCOVA models with correlated responses. The generalized estimating equations (GEE) technique is employed to construct the proposed rank tests. Asymptotic properties of the proposed tests are derived. Simulation studies confirmed the performance of the proposed tests.     

A Comparison of Analysis of Covariate-Adjusted Residuals and Analysis of Covariance

Various methods to control the influence of a covariate on a response variable are compared. These methods are ANOVA with or without homogeneity of variances (HOV) of errors and Kruskal–Wallis (K–W) tests on (covariate-adjusted) residuals and analysis of covariance (ANCOVA). Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels or factors. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and covariate means are equal for all treatments. Empirical size and power performance of the methods are compared by extensive Monte Carlo simulations. We manipulated the conditions such as assumptions of normality and HOV, sample size, and clustering of the covariates. The parametric methods on residuals and ANCOVA exhibited similar size and power when error terms have symmetric distributions with variances having the same functional form for each treatment, and covariates have uniform distributions within the same interval for each treatment. In such cases, parametric tests have higher power compared to the K–W test on residuals. When error terms have asymmetric distributions or have variances that are heterogeneous with different functional forms for each treatment, the tests are liberal with K–W test having higher power than others. The methods on covariate-adjusted residuals are severely affected by the clustering of the covariates relative to the treatment factors when covariate means are very different for treatments. For data clusters, ANCOVA method exhibits the appropriate level. However, such a clustering might suggest dependence between the covariates and the treatment factors, so makes ANCOVA less reliable as well.     

Nearly optimal covariate designs—Part I

We propose to discuss at length several examples from standard text books. All of these examples deal with analysis of covariance (ANCOVA) models and related analyses of data. We intend to capitalize on our understanding of optimal covariate designs (OCDs) in different ANCOVA models and re-visit these examples with a view to suggest optimal/nearly optimal designs for estimation of the covariate parameter(s). As we will see, for some examples our task is very much routine but for others, it is indeed a highly non trivial exercise.

?We intent to cover a total of six examples—divided in two parts. This is Part I—dealing with two examples.     

A note reconciling ANOVA tests under unequal error variances

The purpose of this note is to derive simple testing procedures for ANOVA under heteroscedasticity by a single approach that are equivalent to the prior art in the literature obtained by the Parametric Bootstrap and the Generalized Fiducial approach. By similar approach, researchers are encouraged to derive generalized tests in other applications, as alternative to parametric bootstrap tests and fiducial tests, including ANCOVA and MANOVA under heteroscedasticity, especially in Mixed Model applications, where the bootstrap approach fails.     

Choice of Baselines in Clinical Trials: A Simulation Study from Statistical Power Perspective

Multiple assessments of an efficacy variable are often conducted prior to the initiation of randomized treatments in clinical trials as baseline information. Two goals are investigated in this article, where the first goal is to investigate the choice of these baselines in the analysis of covariance (ANCOVA) to increase the statistical power, and the second to investigate the magnitude of power loss when a continuous efficacy variable is dichotomized to categorical variable as commonly reported the biomedical literature. A statistical power analysis is developed with extensive simulations based on data from clinical trials in study participants with end stage renal disease (ESRD). It is found that the baseline choices primarily depend on the correlations among the baselines and the efficacy variable, with substantial gains for correlations greater than 0.6 and negligible for less than 0.2. Continuous efficacy variables always give higher statistical power in the ANCOVA modeling and dichotomizing the efficacy variable generally decreases the statistical power by 25%, which is an important practicum in designing clinical trials for study sample size and realistically budget. These findings can be easily applied in and extended to other clinical trials with similar design.     

Improving precision and power in randomized trials with a two-stage study design: Stratification using clustering method

In a randomized controlled trial (RCT), it is possible to improve precision and power and reduce sample size by appropriately adjusting for baseline covariates. There are multiple statistical methods to adjust for prognostic baseline covariates, such as an ANCOVA method. In this paper, we propose a clustering-based stratification method for adjusting for the prognostic baseline covariates. Clusters (strata) are formed only based on prognostic baseline covariates, not outcome data nor treatment assignment. Therefore, the clustering procedure can be completed prior to the availability of outcome data. The treatment effect is estimated in each cluster, and the overall treatment effect is derived by combining all cluster-specific treatment effect estimates. The proposed implementation of the procedure is described. Simulations studies and an example are presented.     

Nearly optimal covariate designs—Part II

This is a continuation to Part I toward our efforts for providing illustrative examples in the context of analysis of covariance (ANCOVA) models and related analyses of data. We discuss four more examples here, and these are derived from standard textbooks. We re-visit these examples with a view to suggest optimal/nearly optimal designs for estimation of the covariate parameter(s).     

Statistical power to detect violation of the proportional hazards assumption when using the Cox regression model

The use of the Cox proportional hazards regression model is wide-spread. A key assumption of the model is that of proportional hazards. Analysts frequently test the validity of this assumption using statistical significance testing. However, the statistical power of such assessments is frequently unknown. We used Monte Carlo simulations to estimate the statistical power of two different methods for detecting violations of this assumption. When the covariate was binary, we found that a model-based method had greater power than a method based on cumulative sums of martingale residuals. Furthermore, the parametric nature of the distribution of event times had an impact on power when the covariate was binary. Statistical power to detect a strong violation of the proportional hazards assumption was low to moderate even when the number of observed events was high. In many data sets, power to detect a violation of this assumption is likely to be low to modest.     

Comparison of the estimators of the intra-cluster correlation for the nested error regression model

The intra-cluster correlation is insisted on nested error regression model that, in practice, is rarely known. This article demonstrates the size in generalized least squares (GLS) F-test using Fuller–Battese transformation and modification F-test. For the balanced case, the former using strictly positive, analysis of covariance (ANCOVA) and analysis of variance (ANOVA) estimators of intra-cluster correlation can control the size for moderate intra-cluster correlations. For small intra-cluster correlation, they perform well when the numbers of cluster are large. The latter using the ANOVA estimator performs well except for small numbers of cluster. When intra-cluster correlation is large, it cannot control the size. For the unbalanced case, the GLS F-test using the Fuller–Battese transformation and the modification F-test using the strictly positive, the ANCOVA and the ANOVA estimators maintain the significance level for small total sample size and small intra-cluster correlations when there is a large variation in cluster sizes, but they perform well in controlling the size for large total sample size and small different variation in cluster sizes. Besides, Henderson’s method 3 estimator maintains the significance level for a few situations.     

The performance of multistage group screening designs

This article deals with multistage group screening in which group-factors contain the same number of factors. A usual assumption of this procedure is that the directions of possible effects are known. In practice, however, this assumption i s often unreasonable. This paper examines, in the case of no errors in observations, the performance of multistage group screening when this assumption is false . This enails consideration of cancellation effects within group-factors.     

Robust Estimation and Hypothesis Testing of Linear Contrasts in Analysis of Covariance with Stochastic Covariates

Estimators of parameters are derived by using the method of modified maximum likelihood (MML) estimation when the distribution of covariate X and the error e are both non-normal in a simple analysis of covariance (ANCOVA) model. We show that our estimators are efficient. We also develop a test statistic for testing a linear contrast and show that it is robust. We give a real life example.     

Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures