Analysis of covariance in parallel-group clinical trials with pretreatment baselines

Analysis of covariance (ANCOVA) techniques are often employed in the analysis of clinical trials to try to account for the effects of varying pretreatment baseline values of an outcome variable on posttreatment measurements of the same variable. Baseline measurements of outcome variables are typically random variables, which violates the usual ANCOVA assumption that covariate values are fixed. Therefore, the usual ANCOVA hypothesis tests of treatment effects may be invalid, and the ANCOVA slope parameter estimator biased, for this application. We show, however, that if the pretreatment - posttreatment measurements have a bivariate normal distribution, then (i) the ANCOVA model with residual error independent of the covariate is a valid expression of the relationship between pretreatment and posttreatment measurements; (ii) the usual (fixed-covariate analysis) ANCOVA estimates of the slope parameter and treatment effect contrasts are unbiased; and (iii) the usual ANCOVA treatment effect contrast t-tests are valid significance tests for treatment effects. Moreover, as long as the magnitudes of the treatment effects do not depend on the "true" pretreatment value of the outcome variable, the true slope parameter must lie in the interval (0, 1) and the ANCOVA model has a clear interpretation as an adjustment (based on between- and within-subject variability) to an analysis of variance model applied to the posttreatment-pretreatment differences.     

Analysis of Homeostatic Mechanisms in Biochemical Networks

Cell metabolism is an extremely complicated dynamical system that maintains important cellular functions despite large changes in inputs. This “homeostasis” does not mean that the dynamical system is rigid and fixed. Typically, large changes in external variables cause large changes in some internal variables so that, through various regulatory mechanisms, certain other internal variables (concentrations or velocities) remain approximately constant over a finite range of inputs. Outside that range, the mechanisms cease to function and concentrations change rapidly with changes in inputs. In this paper we analyze four different common biochemical homeostatic mechanisms: feedforward excitation, feedback inhibition, kinetic homeostasis, and parallel inhibition. We show that all four mechanisms can occur in a single biological network, using folate and methionine metabolism as an example. Golubitsky and Stewart have proposed a method to find homeostatic nodes in networks. We show that their method works for two of these mechanisms but not the other two. We discuss the many interesting mathematical and biological questions that emerge from this analysis, and we explain why understanding homeostatic control is crucial for precision medicine.     

Age correction in dementia--matching to a healthy brain

In recent research, many univariate and multivariate approaches have been proposed to improve automatic classification of various dementia syndromes using imaging data. Some of these methods do not provide the possibility to integrate possible confounding variables like age into the statistical evaluation. A similar problem sometimes exists in clinical studies, as it is not always possible to match different clinical groups to each other in all confounding variables, like for example, early-onset (age<65 years) and late-onset (age≥65) patients with Alzheimer's disease (AD). Here, we propose a simple method to control for possible effects of confounding variables such as age prior to statistical evaluation of magnetic resonance imaging (MRI) data using support vector machine classification (SVM) or voxel-based morphometry (VBM). We compare SVM results for the classification of 80 AD patients and 79 healthy control subjects based on MRI data with and without prior age correction. Additionally, we compare VBM results for the comparison of three different groups of AD patients differing in age with the same group of control subjects obtained without including age as covariate, with age as covariate or with prior age correction using the proposed method. SVM classification using the proposed method resulted in higher between-group classification accuracy compared to uncorrected data. Further, applying the proposed age correction substantially improved univariate detection of disease-related grey matter atrophy using VBM in AD patients differing in age from control subjects. The results suggest that the approach proposed in this work is generally suited to control for confounding variables such as age in SVM or VBM analyses. Accordingly, the approach might improve and extend the application of these methods in clinical neurosciences.     

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%.     

Living with observational data in biological anthropology

The establishment of cause and effect relationships is a fundamental objective of scientific research. Many lines of evidence can be used to make cause–effect inferences. When statistical data are involved, alternative explanations for the statistical relationship need to be ruled out. These include chance (apparent patterns due to random factors), confounding effects (a relationship between two variables because they are each associated with an unmeasured third variable), and sampling bias (effects due to preexisting properties of compared groups). The gold standard for managing these issues is a controlled randomized experiment. In disciplines such as biological anthropology, where controlled experiments are not possible for many research questions, causal inferences are made from observational data. Methods that statisticians recommend for this difficult objective have not been widely adopted in the biological anthropology literature. Issues involved in using statistics to make valid causal inferences from observational data are discussed.     

AC-PCoA: Adjustment for confounding factors using principal coordinate analysis

Confounding factors exist widely in various biological data owing to technical variations, population structures and experimental conditions. Such factors may mask the true signals and lead to spurious associations in the respective biological data, making it necessary to adjust confounding factors accordingly. However, existing confounder correction methods were mainly developed based on the original data or the pairwise Euclidean distance, either one of which is inadequate for analyzing different types of data, such as sequencing data.In this work, we proposed a method called Adjustment for Confounding factors using Principal Coordinate Analysis, or AC-PCoA, which reduces data dimension and extracts the information from different distance measures using principal coordinate analysis, and adjusts confounding factors across multiple datasets by minimizing the associations between lower-dimensional representations and confounding variables. Application of the proposed method was further extended to classification and prediction. We demonstrated the efficacy of AC-PCoA on three simulated datasets and five real datasets. Compared to the existing methods, AC-PCoA shows better results in visualization, statistical testing, clustering, and classification.     

Nonparametric analysis of covariance for comparing change in randomized studies with baseline values subject to error

Change from baseline to a follow-up examination can be compared among two or more randomly assigned treatment groups by using analysis of variance on the change scores. However, a generally more sensitive (powerful) test can be performed using analysis of covariance (ANOVA) on the follow-up data with the baseline data as a covariate. This approach is not without potential problems, though. The assumption of ordinary ANCOVA of normally distributed errors is speculative for many variables employed in biomedical research. Furthermore, the baseline values are inevitably random variables and often are measured with error. This report investigates, in this situation, the validity and relative power of the ordinary ANCOVA test and two asymptotically distribution-free alternative tests, one based on the rank transformation and the other based on the normal scores transformation. The procedures are illustrated with data from a clinical trial. Normal and several nonnormal distributions, as well as varying degree of variable error, are studied by Monte Carlo methods. The normal scores test is generally recommended for statistical practice.     

Variable complementary network: a novel approach for identifying biomarkers and their mutual associations

Biological variables involved in a disease process often correlate with each other through for example shared metabolic pathways. In addition to their correlation, these variables contain complementary information that is particularly useful for disease classification and prediction. However, complementary information between variables is rarely explored. Therefore, establishing methods for the investigation of variable??s complementary information is very necessary. We propose a model population analysis approach that aggregates information of a number of classification models obtained with the help of Monte Carlo sampling in variable space for quantitatively calculating the complementary information between variables. We then assemble these complementary information to construct a variable complementary network (VCN) to give an overall visualization of how biological variables complement each other. Using a simulated dataset and two metabolomics datasets, we show that the complementary information is effective in biomarker discovery and that mutual associations of metabolites revealed by this method can provide information for exploring altered metabolic pathways. (The source codes for implementing VCN in MATLAB are freely available at: http://code.google.com/p/vcn2011/.)     

Analysis of covariance with pre-treatment measurements in randomized trials under the cases that covariances and post-treatment variances differ between groups

When primary endpoints of randomized trials are continuous variables, the analysis of covariance (ANCOVA) with pre-treatment measurements as a covariate is often used to compare two treatment groups. In the ANCOVA, equal slopes (coefficients of pre-treatment measurements) and equal residual variances are commonly assumed. However, random allocation guarantees only equal variances of pre-treatment measurements. Unequal covariances and variances of post-treatment measurements indicate unequal slopes and, usually, unequal residual variances. For non-normal data with unequal covariances and variances of post-treatment measurements, it is known that the ANCOVA with equal slopes and equal variances using an ordinary least-squares method provides an asymptotically normal estimator for the treatment effect. However, the asymptotic variance of the estimator differs from the variance estimated from a standard formula, and its property is unclear. Furthermore, the asymptotic properties of the ANCOVA with equal slopes and unequal variances using a generalized least-squares method are unclear. In this paper, we consider non-normal data with unequal covariances and variances of post-treatment measurements, and examine the asymptotic properties of the ANCOVA with equal slopes using the variance estimated from a standard formula. Analytically, we show that the actual type I error rate, thus the coverage, of the ANCOVA with equal variances is asymptotically at a nominal level under equal sample sizes. That of the ANCOVA with unequal variances using a generalized least-squares method is asymptotically at a nominal level, even under unequal sample sizes. In conclusion, the ANCOVA with equal slopes can be asymptotically justified under random allocation.     

Correcting for measurement error in individual ancestry estimates in structured association tests

We present theoretical explanations and show through simulation that the individual admixture proportion estimates obtained by using ancestry informative markers should be seen as an error-contaminated measurement of the underlying individual ancestry proportion. These estimates can be used in structured association tests as a control variable to limit type I error inflation or reduce loss of power due to population stratification observed in studies of admixed populations. However, the inclusion of such error-containing variables as covariates in regression models can bias parameter estimates and reduce ability to control for the confounding effect of admixture in genetic association tests. Measurement error correction methods offer a way to overcome this problem but require an a priori estimate of the measurement error variance. We show how an upper bound of this variance can be obtained, present four measurement error correction methods that are applicable to this problem, and conduct a simulation study to compare their utility in the case where the admixed population results from the intermating between two ancestral populations. Our results show that the quadratic measurement error correction (QMEC) method performs better than the other methods and maintains the type I error to its nominal level.     

Enhanced identification and biological validation of differential gene expression via Illumina whole-genome expression arrays through the use of the model-based background correction methodology

Despite the tremendous growth of microarray usage in scientific studies, there is a lack of standards for background correction methodologies, especially in single-color microarray platforms. Traditional background subtraction methods often generate negative signals and thus cause large amounts of data loss. Hence, some researchers prefer to avoid background corrections, which typically result in the underestimation of differential expression. Here, by utilizing nonspecific negative control features integrated into Illumina whole genome expression arrays, we have developed a method of model-based background correction for BeadArrays (MBCB). We compared the MBCB with a method adapted from the Affymetrix robust multi-array analysis algorithm and with no background subtraction, using a mouse acute myeloid leukemia (AML) dataset. We demonstrated that differential expression ratios obtained by using the MBCB had the best correlation with quantitative RT–PCR. MBCB also achieved better sensitivity in detecting differentially expressed genes with biological significance. For example, we demonstrated that the differential regulation of Tnfr2, Ikk and NF-kappaB, the death receptor pathway, in the AML samples, could only be detected by using data after MBCB implementation. We conclude that MBCB is a robust background correction method that will lead to more precise determination of gene expression and better biological interpretation of Illumina BeadArray data.     

Bayesian Dynamical Systems Modelling in the Social Sciences

Data arising from social systems is often highly complex, involving non-linear relationships between the macro-level variables that characterize these systems. We present a method for analyzing this type of longitudinal or panel data using differential equations. We identify the best non-linear functions that capture interactions between variables, employing Bayes factor to decide how many interaction terms should be included in the model. This method punishes overly complicated models and identifies models with the most explanatory power. We illustrate our approach on the classic example of relating democracy and economic growth, identifying non-linear relationships between these two variables. We show how multiple variables and variable lags can be accounted for and provide a toolbox in R to implement our approach.     

A comparison of the multilevel MIMIC model to the multilevel regression and mixed ANOVA model for the estimation and testing of a cross-level interaction effect: A simulation study

When observing data on a patient-reported outcome measure in, for example, clinical trials, the variables observed are often correlated and intended to measure a latent variable. In addition, such data are also often characterized by a hierarchical structure, meaning that the outcome is repeatedly measured within patients. To analyze such data, it is important to use an appropriate statistical model, such as structural equation modeling (SEM). However, researchers may rely on simpler statistical models that are applied to an aggregated data structure. For example, correlated variables are combined into one sum score that approximates a latent variable. This may have implications when, for example, the sum score consists of indicators that relate differently to the latent variable being measured. This study compares three models that can be applied to analyze such data: the multilevel multiple indicators multiple causes (ML-MIMIC) model, a univariate multilevel model, and a mixed analysis of variance (ANOVA) model. The focus is on the estimation of a cross-level interaction effect that presents the difference over time on the patient-reported outcome between two treatment groups. The ML-MIMIC model is an SEM-type model that considers the relationship between the indicators and the latent variable in a multilevel setting, whereas the univariate multilevel and mixed ANOVA model rely on sum scores to approximate the latent variable. In addition, the mixed ANOVA model uses aggregated second-level means as outcome. This study showed that the ML-MIMIC model produced unbiased cross-level interaction effect estimates when the relationships between the indicators and the latent variable being measured varied across indicators. In contrast, under similar conditions, the univariate multilevel and mixed ANOVA model underestimated the cross-level interaction effect.     

Integrating biological knowledge into variable selection: an empirical Bayes approach with an application in cancer biology

ABSTRACT: BACKGROUND: An important question in the analysis of biochemical data is that of identifying subsets of molecular variables that may jointly influence a biological response. Statistical variable selection methods have been widely used for this purpose. In many settings, it may be important to incorporate ancillary biological information concerning the variables of interest. Pathway and network maps are one example of a source of such information. However, although ancillary information is increasingly available, it is not always clear how it should be used nor how it should be weighted in relation to primary data. RESULTS: We put forward an approach in which biological knowledge is incorporated using informative prior distributions over variable subsets, with prior information selected and weighted in an automated, objective manner using an empirical Bayes formulation. We employ continuous, linear models with interaction terms and exploit biochemically-motivated sparsity constraints to permit exact inference. We show an example of priors for pathway- and network-based information and illustrate our proposed method on both synthetic response data and by an application to cancer drug response data. Comparisons are also made to alternative Bayesian and frequentist penalised-likelihood methods for incorporating network-based information. CONCLUSIONS: The empirical Bayes method proposed here can aid prior elicitation for Bayesian variable selection studies and help to guard against mis-specification of priors. Empirical Bayes, together with the proposed pathway-based priors, results in an approach with a competitive variable selection performance. In addition, the overall procedure is fast, deterministic, and has very few user-set parameters, yet is capable of capturing interplay between molecular players. The approach presented is general and readily applicable in any setting with multiple sources of biological prior knowledge.     

The plasticity of human social organization and behavior: Contextual variables and proximal mechanisms

The purpose of this paper is to illustrate a behavioral-ecological, contextual approach to thinking about cultural variation as well as the problem of the gap between actual behavior and optimal behavior. This approach emphasizes sociobiologically predicted, genetically based, central tendencies in human behavior rather than genetic variation, but it is consistent with theories emphasizing genetic variation. The variable of economic production is introduced as a contextual variable associated, in sociobiologically predictable ways, with variation in sexual competition, family and social structure, and the socialization of children. Social controls on individual behavior as well as personal ideology are described as contextual variables that strongly affect individual fitness within societies, but do so in ways that are underdetermined by biological theory. For example, there is no way derived from biological theory to predict whether ideology or social controls in a society will be egalitarian or antiegalitarian. Individual behavior is also strongly affected by the interactions of these contextual variables with proximal mechanisms. Examples of maladaptive behavior emphasizing the interactions among the proposed contextual variables, the sociobiologically expected central tendencies in human behavior, and the proximal mechanisms proposed by psychologists are provided.     

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.