Analysis of covariance with pre-treatment measurements in randomized trials: comparison of equal and unequal slopes

In randomized trials, an analysis of covariance (ANCOVA) is often used to analyze post-treatment measurements with pre-treatment measurements as a covariate to compare two treatment groups. Random allocation guarantees only equal variances of pre-treatment measurements. We hence consider data with unequal covariances and variances of post-treatment measurements without assuming normality. Recently, we showed that the actual type I error rate of the usual ANCOVA assuming equal slopes and equal residual variances is asymptotically at a nominal level under equal sample sizes, and that of the ANCOVA with unequal variances is asymptotically at a nominal level, even under unequal sample sizes. In this paper, we investigated the asymptotic properties of the ANCOVA with unequal slopes for such data. The estimators of the treatment effect at the observed mean are identical between equal and unequal variance assumptions, and these are asymptotically normal estimators for the treatment effect at the true mean. However, the variances of these estimators based on standard formulas are biased, and the actual type I error rates are not at a nominal level, irrespective of variance assumptions. In equal sample sizes, the efficiency of the usual ANCOVA assuming equal slopes and equal variances is asymptotically the same as those of the ANCOVA with unequal slopes and higher than that of the ANCOVA with equal slopes and unequal variances. Therefore, the use of the usual ANCOVA is appropriate in equal sample sizes.     

Longitudinal analysis of pre- and post-treatment measurements with equal baseline assumptions in randomized trials

For continuous variables of randomized controlled trials, recently, longitudinal analysis of pre- and posttreatment measurements as bivariate responses is one of analytical methods to compare two treatment groups. Under random allocation, means and variances of pretreatment measurements are expected to be equal between groups, but covariances and posttreatment variances are not. Under random allocation with unequal covariances and posttreatment variances, we compared asymptotic variances of the treatment effect estimators in three longitudinal models. The data-generating model has equal baseline means and variances, and unequal covariances and posttreatment variances. The model with equal baseline means and unequal variance–covariance matrices has a redundant parameter. In large sample sizes, these two models keep a nominal type I error rate and have high efficiency. The model with equal baseline means and equal variance–covariance matrices wrongly assumes equal covariances and posttreatment variances. Only under equal sample sizes, this model keeps a nominal type I error rate. This model has the same high efficiency with the data-generating model under equal sample sizes. In conclusion, longitudinal analysis with equal baseline means performed well in large sample sizes. We also compared asymptotic properties of longitudinal models with those of the analysis of covariance (ANCOVA) and t-test.     

Robustness of ANCOVA in randomized trials with unequal randomization

Randomized trials with continuous outcomes are often analyzed using analysis of covariance (ANCOVA), with adjustment for prognostic baseline covariates. The ANCOVA estimator of the treatment effect is consistent under arbitrary model misspecification. In an article recently published in the journal, Wang et al proved the model-based variance estimator for the treatment effect is also consistent under outcome model misspecification, assuming the probability of randomization to each treatment is 1/2. In this reader reaction, we derive explicit expressions which show that when randomization is unequal, the model-based variance estimator can be biased upwards or downwards. In contrast, robust sandwich variance estimators can provide asymptotically valid inferences under arbitrary misspecification, even when randomization probabilities are not equal.     

Estimation of genetic covariance from joint offspring-parent and sib-sib statistics

Formulae for calculating the variances and covariances of estimates of genetic variance or covariance from offspring-parent and sib covariance data are presented. The model consists of a one-way classification of families with unequal numbers of offspring, where normally distributed traits x (a parental measure), y and z (offspring measures) are recorded. Procedures for pooling offspring-parent and sib covariance estimators are discussed. An application of the results to estimating genetic variance is presented, and the offspring-parent and sib covariance estimators and a pooled statistic are compared in terms of exact and asymptotic formulae. Implications for experimental design and allocation of experimental resources are also reported.     

Asymptotically robust variance estimation for person‐time incidence rates

Person‐time incidence rates are frequently used in medical research. However, standard estimation theory for this measure of event occurrence is based on the assumption of independent and identically distributed (iid) exponential event times, which implies that the hazard function remains constant over time. Under this assumption and assuming independent censoring, observed person‐time incidence rate is the maximum‐likelihood estimator of the constant hazard, and asymptotic variance of the log rate can be estimated consistently by the inverse of the number of events. However, in many practical applications, the assumption of constant hazard is not very plausible. In the present paper, an average rate parameter is defined as the ratio of expected event count to the expected total time at risk. This rate parameter is equal to the hazard function under constant hazard. For inference about the average rate parameter, an asymptotically robust variance estimator of the log rate is proposed. Given some very general conditions, the robust variance estimator is consistent under arbitrary iid event times, and is also consistent or asymptotically conservative when event times are independent but nonidentically distributed. In contrast, the standard maximum‐likelihood estimator may become anticonservative under nonconstant hazard, producing confidence intervals with less‐than‐nominal asymptotic coverage. These results are derived analytically and illustrated with simulations. The two estimators are also compared in five datasets from oncology studies.     

Some unexamined aspects of analysis of covariance in pretest-posttest studies

The use of an analysis of covariance (ANCOVA) model in a pretest-posttest setting deserves to be studied separately from its use in other (non-pretest-posttest) settings. For pretest-posttest studies, the following points are made in this article: (a) If the familiar change from baseline model accurately describes the data-generating mechanism for a randomized study then it is impossible for unequal slopes to exist. Conversely, if unequal slopes exist, then it implies that the change from baseline model as a data-generating mechanism is inappropriate. An alternative data-generating model should be identified and the validity of the ANCOVA model should be demonstrated. (b) Under the usual assumptions of equal pretest and posttest within-subject error variances, the ratio of the standard error of a treatment contrast from a change from baseline analysis to that from ANCOVA is less than 2(1)/(2). (c) For an observational study it is possible for unequal slopes to exist even if the change from baseline model describes the data-generating mechanism. (d) Adjusting for the pretest variable in observational studies may actually introduce bias where none previously existed.     

Conditional Maximum Likelihood Estimate and Exact Test of the Common Relative Difference in Combination of 2 × 2 Tables under Inverse Sampling

On the basis of the conditional distribution, given the marginal totals of non-cases fixed for each of independent 2 × 2 tables under inverse sampling, this paper develops the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance. This paper further provides for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level and for investigating whether the underlying common RD equals any specified value an exact test procedure, of which Type I error is always less than or equal to the nominal α-level. These exact interval estimation and exact hypothesis testing procedures are especially useful for the situation in which the number of index subjects in a study is small and the asymptotically approximate methods may not be appropriate for use. This paper also notes the condition under which the CMLE of RD uniquely exists and includes a simple example to illustrate use of these techniques.     

Analysis of covariance in randomized trials: More precision and valid confidence intervals,without model assumptions

“Covariate adjustment” in the randomized trial context refers to an estimator of the average treatment effect that adjusts for chance imbalances between study arms in baseline variables (called “covariates”). The baseline variables could include, for example, age, sex, disease severity, and biomarkers. According to two surveys of clinical trial reports, there is confusion about the statistical properties of covariate adjustment. We focus on the analysis of covariance (ANCOVA) estimator, which involves fitting a linear model for the outcome given the treatment arm and baseline variables, and trials that use simple randomization with equal probability of assignment to treatment and control. We prove the following new (to the best of our knowledge) robustness property of ANCOVA to arbitrary model misspecification: Not only is the ANCOVA point estimate consistent (as proved by Yang and Tsiatis, 2001) but so is its standard error. This implies that confidence intervals and hypothesis tests conducted as if the linear model were correct are still asymptotically valid even when the linear model is arbitrarily misspecified, for example, when the baseline variables are nonlinearly related to the outcome or there is treatment effect heterogeneity. We also give a simple, robust formula for the variance reduction (equivalently, sample size reduction) from using ANCOVA. By reanalyzing completed randomized trials for mild cognitive impairment, schizophrenia, and depression, we demonstrate how ANCOVA can achieve variance reductions of 4 to 32%.     

On the Effect of Unbalancedness and Heteroscedasticity on the ANOVA Estimator of Group Variance Component in One-Way Random Model

The expression for rth cumulant of ANOVA estimator of group variance component is derived in the One-way unbalanced random model under heteroscedasticity. The expression is used to study the effect of unbalancedness and heteroscedasticity on the mean and variance of the estimator, numerically. The computed results reveal that the unbalancedness and heteroscedasticity have a combined effect on the mean and variance of the estimator. For certain situations of unequal group sizes and error variances, the mean and variance of the estimator are increased and for certain other situations the values are decreased.     

A covariance estimator for GEE with improved small-sample properties

In this paper, we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations (GEE). Hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small. Resampling methods, such as the jackknife and bootstrap, have been suggested for covariance estimation when the number of clusters is small. A drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near-zero cell counts caused by resampling. We propose a bias-corrected covariance estimator that avoids this problem. In a small simulation study, we compare the bias-corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10-40 subjects with equal and unequal cluster sizes of 16-64 observations. The bias-corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal, whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2-3 times the nominal level. The methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease.     

How Informative Is Wright's Estimator of the Number of Genes Affecting a Quantitative Character?

S. Wright suggested an estimator, m, of the number of loci, m, contributing to the difference in a quantitative character between two differentiated populations, which is calculated from the phenotypic means and variances in the two parental populations and their F1 and F2 hybrids. The same method can also be used to estimate m contributing to the genetic variance within a single population, by using divergent selection to create differentiated lines from the base population. In this paper we systematically examine the utility and problems of this technique under the influences of unequal allelic effects and initial allele frequencies, and linkage, which are known to lead m to underestimate m. In addition, we examine the effects of population size and selection intensity during the generations of selection. During selection, the estimator m rapidly approaches its expected value at the selection limit. With reasonable assumptions about unequal allelic effects and initial allele frequencies, the expected value of m without linkage is likely to be on the order of one-third of the number of genes. The estimates suffer most seriously from linkage. The practical maximum expectation of m is just about the number of chromosomes, considerably less than the "recombination index" which has been assumed to be the upper limit. The estimates are also associated with large sampling variances. An estimator of the variance of m derived by R. Lande substantially underestimates the actual variance. Modifications to the method can ameliorate some of the problems. These include using F3 or later generation variances or the genetic variance in the base population, and replicating the experiments and estimation procedure. However, even in the best of circumstances, information from m is very limited and can be misleading.     

Closed Form Estimators of Latent Cell Frequencies

One-stage and two-stage closed form estimators of latent cell frequencies in multidimensional contingency tables are derived from the weighted least squares criterion. The first stage estimator is asymptotically equivalent to the conditional maximum likelihood estimator and does not necessarily have minimum asymptotic variance. The second stage estimator does have minimum asymptotic variance relative to any other existing estimator. The closed form estimators are defined for any number of latent cells in contingency tables of any order under exact general linear constraints on the logarithms of the nonlatent and latent cell frequencies.     

MIVQUE and REML Estimators of Variance Components under Proportionality Condition

In this study, a one-way random effect model with unequal cell variances is considered, and the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and Restricted Maximum Likelihood (REML) estimator of the variance components are studied. The algebraic inversion of the variance matrix of the observation vector is obtained to achieve some computational convenience. Using the proportionality condition described by Talukder (1992) that the cell sizes are proportional to the cell variances, MIVQUE and REML estimators are shown to be the same as the ANOVA estimators.     

Comparison of asymptotic population growth rates with heterogeneous variances

This report explores how the heterogeneity of variances affects randomization tests used to evaluate differences in the asymptotic population growth rate, λ. The probability of Type I error was calculated in four scenarios for populations with identical λ but different variance of λ: (1) Populations have different projection matrices: the same λ may be obtained from different sets of vital rates, which gives room for different variances of λ. (2) Populations have identical projection matrices but reproductive schemes differ and fecundity in one of the populations has a larger associated variance. The two other scenarios evaluate a sampling artifact as responsible for heterogeneity of variances. The same population is sampled twice, (3) with the same sampling design, or (4) with different sampling effort for different stages. Randomization tests were done with increasing differences in sample size between the two populations. This implies additional differences in the variance of λ. The probability of Type I error keeps at the nominal significance level (α = .05) in Scenario 3 and with identical sample sizes in the others. Tests were too liberal, or conservative, under a combination of variance heterogeneity and different sample sizes. Increased differences in sample size exacerbated the difference between observed Type I error and the nominal significance level. Type I error increases or decreases depending on which population has a larger sample size, the population with the smallest or the largest variance. However, by their own, sample size is not responsible for changes in Type I errors.     

A semiparametric estimation procedure of dependence parameters in multivariate families of distributions

This paper investigates the properties of a semiparametric methodfor estimating the dependence parameters in a family of multivariatedistributions. The proposed estimator, obtained as a solutionof a pseudo-likelihood equation, is shown to be consistent,asymptotically normal and fully efficient at independence. Anatural estimator of its asymptotic variance is proved to beconsistent. Comparisons are made with alternative semiparametricestimators in the special case of Clayton's model for associationin bivariate data.     

Nonlinear selection and the evolution of variances and covariances for continuous characters in an anole

The pattern of genetic variances and covariances among characters, summarized in the additive genetic variance‐covariance matrix, G , determines how a population will respond to linear natural selection. However, G itself also evolves in response to selection. In particular, we expect that, over time, G will evolve correspondence with the pattern of multivariate nonlinear natural selection. In this study, we substitute the phenotypic variance‐covariance matrix ( P ) for G to determine if the pattern of multivariate nonlinear selection in a natural population of Anolis cristatellus, an arboreal lizard from Puerto Rico, has influenced the evolution of genetic variances and covariances in this species. Although results varied among our estimates of P and fitness, and among our analytic techniques, we find significant evidence for congruence between nonlinear selection and P , suggesting that natural selection may have influenced the evolution of genetic constraint in this species.     

Multiple Comparisons of Biodiversity

A common goal in statistical ecology is to compare several communities and or time points with respect to taxonomic diversity (usually species diversity). For this purpose, the current literature recommends the application of traditional ANOVA techniques to “replicates” of diversity indices. This approach is not even asymptotically correct because diversity index estimates have unequal variances, even when sample sizes are equal and even when the hypothesis of equality of diversity indices is true. It is shown that transformations of the data can not be used to remedy this situation. We construct an asymptotically correct method and illustrate its implementation using dinosaur extinction data.     

Comparison of recent estimators of interclass correlation from familial data

The asymptotic and finite-sample properties of several recent estimators of interclass correlation are compared to more traditional estimators in the case of a variable number of siblings per family. It is shown that Karlin's family-weighted pairwise estimator (Karlin, Cameron, and Williams, 1981, Proceedings of the National Academy of Science 78, 2664-2668) is virtually equivalent to the ensemble estimator (Rosner, Donner, and Hennekens, 1977, Applied Statistics 26, 179-187), thus suggesting an estimator of the former's asymptotic variance. Further, an estimator proposed by Srivastava (1984, Biometrika 71, 177-185) is shown to be identical to the modified sib-mean estimator (Konishi, 1982, Annals of the Institute of Statistical Mathematics 34, 505-515) when the sib-sib correlation is estimated by the method of unweighted group means. Although the estimator due to Srivastava has smaller asymptotic variance than the other two, the gain in efficiency is slight, for familial data, both asymptotically and in finite samples.     

The Nonparametric Behrens‐Fisher Problem: Asymptotic Theory and a Small‐Sample Approximation

A generalization of the Behrens‐Fisher problem for two samples is examined in a nonparametric model. It is not assumed that the underlying distribution functions are continuous so that data with arbitrary ties can be handled. A rank test is considered where the asymptotic variance is estimated consistently by using the ranks over all observations as well as the ranks within each sample. The consistency of the estimator is derived in the appendix. For small samples (n₁, n₂ ≥ 10), a simple approximation by a central t‐distribution is suggested where the degrees of freedom are taken from the Satterthwaite‐Smith‐Welch approximation in the parametric Behrens‐Fisher problem. It is demonstrated by means of a simulation study that the Wilcoxon‐Mann‐Whitney‐test may be conservative or liberal depending on the ratio of the sample sizes and the variances of the underlying distribution functions. For the suggested approximation, however, it turns out that the nominal level is maintained rather accurately. The suggested nonparametric procedure is applied to a data set from a clinical trial. Moreover, a confidence interval for the nonparametric treatment effect is given.     

Prediction of the merits of single crosses

Summary Best linear unbiased prediction of single crosses is described for a model that is somewhat more complete than those previously published. The method applies to unequal subclass numbers with unequal means of the observations. The lines are assumed to be a random sample from some population. Also, methods for unbiased estimation of the variances and covariances from such data are presented.