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Analysis of variance (ANOVA) may be required by researchers as an inferential statistical test when more than two means require comparison. This paper explains how to perform ANOVA in Microsoft Excel., (Copyright © 2018 Elsevier B.V. All rights reserved.)

"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%., (© 2019, The International Biometric Society.)

In this study, we investigate how Wilks' lambda, Pillai's trace, Hotelling's trace, and Roy's largest root test statistics can be affected when the normal and homogeneous variance assumptions of the MANOVA method are violated. In other words, in these cases, the robustness of the tests is examined. For this purpose, a simulation study is conducted in different scenarios. In different variable numbers and different sample sizes, considering the group variances are homogeneous ( σ 12 = σ 22 = ⋯ = σ g 2 ) and heterogeneous (increasing) ( σ 12 < σ 22 < ⋯< σ g 2 ), random numbers are generated from Gamma(4-4-4; 0.5), Gamma(4-9-36; 0.5), Student's t (2), and Normal(0; 1) distributions. Furthermore, the number of observations in the groups being balanced and unbalanced is also taken into account. After 10000 repetitions, type-I error values are calculated for each test for α  = 0.05. In the Gamma distribution, Pillai's trace test statistic gives more robust results in the case of homogeneous and heterogeneous variances for 2 variables, and in the case of 3 variables, Roy's largest root test statistic gives more robust results in balanced samples and Pillai's trace test statistic in unbalanced samples. In Student's t distribution, Pillai's trace test statistic gives more robust results in the case of homogeneous variance and Wilks' lambda test statistic in the case of heterogeneous variance. In the normal distribution, in the case of homogeneous variance for 2 variables, Roy's largest root test statistic gives relatively more robust results and Wilks' lambda test statistic for 3 variables. Also in the case of heterogeneous variance for 2 and 3 variables, Roy's largest root test statistic gives robust results in the normal distribution. The test statistics used with MANOVA are affected by the violation of homogeneity of covariance matrices and normality assumptions particularly from unbalanced number of observations., Competing Interests: The authors declare that there are no conflicts of interest regarding the publication of this paper.

Randomized clinical trials are considered the ideal source for generation of robust evidence for clinical and public health decision making. Estimation of treatment effect in observational studies is always subject to varying degrees of bias due to lack of random allocation, blindness, precise definition of intervention, as well as the existence of potential unknown and unmeasured confounding variables. Unlike other conventional methods, instrumental variable analysis (IVA), as a method for controlling confounding bias in non-randomized studies, attempts to estimate the treatment effect with the least bias even without knowing and measuring the potential confounders in the causal pathway. In this paper, after understanding the main concepts of this approach, it has been attempted to provide a method for analyzing and reporting the IVA for clinical researchers through a simplified example. The data used in this paper is derived from the clinical data of the follow-up of multiple sclerosis (MS) patients treated with two class of interferon., (© 2018 The Author(s). This is an open-access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.)

Inconsistencies in the research findings on F-test robustness to variance heterogeneity could be related to the lack of a standard criterion to assess robustness or to the different measures used to quantify heterogeneity. In the present paper we use Monte Carlo simulation to systematically examine the Type I error rate of F-test under heterogeneity. One-way, balanced, and unbalanced designs with monotonic patterns of variance were considered. Variance ratio (VR) was used as a measure of heterogeneity (1.5, 1.6, 1.7, 1.8, 2, 3, 5, and 9), the coefficient of sample size variation as a measure of inequality between group sizes (0.16, 0.33, and 0.50), and the correlation between variance and group size as an indicator of the pairing between them (1, .50, 0, -.50, and -1). Overall, the results suggest that in terms of Type I error a VR above 1.5 may be established as a rule of thumb for considering a potential threat to F-test robustness under heterogeneity with unequal sample sizes.

The genomics era has led to an increase in the dimensionality of data collected in the investigation of biological questions. In this context, dimension-reduction techniques can be used to summarise high-dimensional signals into low-dimensional ones, to further test for association with one or more covariates of interest. This paper revisits one such approach, previously known as principal component of heritability and renamed here as principal component of explained variance (PCEV). As its name suggests, the PCEV seeks a linear combination of outcomes in an optimal manner, by maximising the proportion of variance explained by one or several covariates of interest. By construction, this method optimises power; however, due to its computational complexity, it has unfortunately received little attention in the past. Here, we propose a general analytical PCEV framework that builds on the assets of the original method, i.e. conceptually simple and free of tuning parameters. Moreover, our framework extends the range of applications of the original procedure by providing a computationally simple strategy for high-dimensional outcomes, along with exact and asymptotic testing procedures that drastically reduce its computational cost. We investigate the merits of the PCEV using an extensive set of simulations. Furthermore, the use of the PCEV approach is illustrated using three examples taken from the fields of epigenetics and brain imaging.

This paper describes the estimation and analysis of variance distribution of surface electromyogram (EMG) signals based on a stochastic EMG model. With the assumption that EMG signals at a certain time follow Gaussian distribution, their variance is handled as a random variable that follows inverse gamma distribution, and noise superimposed onto this variance can be expressed accordingly. The paper proposes variance distribution estimation based on marginal likelihood maximization of EMG signals. A simulation experiment using artificially generated signals to verify its accuracy indicated that the method can estimate variance distribution with high accuracy for a wide range of variance distribution shaping. Analysis of variance distribution using measured EMG signals revealed the relationship between muscle force and variance distribution involving signal-dependent noise.

Repeated-measure analysis of variance is a general term that can imply a number of different statistical models used to analyze data from studies in which measurements are taken from each subject on more than one occasion. Repeated-measure analyses encompass univariate models (with or without sphericity adjustment), multivariate models, mixed models, analysis of covariance, multilevel models, latent growth models, and hybrids of these models. These models are based on different assumptions, especially regarding correlations (sphericity) between within-subject factors, which comprise the variance-covariance matrix. Violation of this assumption may lead to misleading and erroneous conclusions. Because many papers do not provide enough information about what analysis was really conducted, and about why it was done, the reader is unable to evaluate the validity of the analysis. Here a brief overview of several of the most commonly used models for analyzing data from repeated-measure designs is provided, and guidance is suggested for describing the statistical approach employed. The goals of this paper are (1) to give authors an overview of the diversity of commonly used models and associated assumptions, and (2) to facilitate reporting sufficient information about the tests to allow the reader to evaluate the validity of the tests and the credibility of the inferences made by the authors. Among the available approaches to repeated-measure analyses, the mixed model is recommended for its flexibility in handling different covariance structures and its insensitivity to missing data. Whether or not it is used, the overall guiding principles in reporting should always be Accuracy, Completeness, and Transparency (ACT principles): tell the reader precisely all what you did and why., (Copyright © 2016 Elsevier Inc. All rights reserved.)

Measurement invariance (MI) is a property of measurement that is often implicitly assumed, but in many cases, not tested. When the assumption of MI is tested, it generally involves determining if the measurement holds longitudinally or cross-culturally. A growing literature shows that other groupings can, and should, be considered as well. Additionally, it is noted that the standard techniques for investigating MI have been focused almost exclusively on the case of 2 groups, with very little work on the case of more than 2 groups, even though the need for such techniques is apparent in many fields of research. This paper introduces and illustrates a model building technique to investigating MI for more than 2 groups. This technique is an extension of the already-existing hierarchy for testing MI introduced by Meredith (1993). An example using data on father involvement in 5 different groups of families of children with and without developmental disabilities from the Early Childhood Longitudinal Study-Birth Cohort dataset will be given. We show that without considering the possible differential functioning of the measurements on multiple developmental groups, the differences present between the groups in terms of the measurements may be obscured. This could lead to incorrect conclusions., ((c) 2016 APA, all rights reserved).)

Multistage sampling is a frequently-used method in random sampling survey in public health. Clustering or independence between observations often exists in the sampling, often called complex sample, generated by multistage sampling. Sampling error may be underestimated and the probability of type I error may be increased if the multistage sample design was not taken into consideration in analysis. As variance (error) estimator in complex sample is often complicated, statistical software usually adopt ultimate cluster variance estimate (UCVE) to approximate the estimation, which simply assume that the sample comes from one-stage sampling. However, with increased sampling fraction of primary sampling unit, contribution from subsequent sampling stages is no more trivial, and the ultimate cluster variance estimate may, therefore, lead to invalid variance estimation. This paper summarize a method of variance estimation considering multistage sampling design. The performances are compared with UCVE and the method considering multistage sampling design by simulating random sampling under different sampling schemes using real world data. Simulation showed that as primary sampling unit (PSU) sampling fraction increased, UCVE tended to generate increasingly biased estimation, whereas accurate estimates were obtained by using the method considering multistage sampling design.

Background: Identifying genes that are differentially expressed (DE) between two or more conditions with multiple patterns of expression is one of the primary objectives of gene expression data analysis. Several statistical approaches, including one-way analysis of variance (ANOVA), are used to identify DE genes. However, most of these methods provide misleading results for two or more conditions with multiple patterns of expression in the presence of outlying genes. In this paper, an attempt is made to develop a hybrid one-way ANOVA approach that unifies the robustness and efficiency of estimation using the minimum β-divergence method to overcome some problems that arise in the existing robust methods for both small- and large-sample cases with multiple patterns of expression., Results: The proposed method relies on a β-weight function, which produces values between 0 and 1. The β-weight function with β = 0.2 is used as a measure of outlier detection. It assigns smaller weights (≥ 0) to outlying expressions and larger weights (≤ 1) to typical expressions. The distribution of the β-weights is used to calculate the cut-off point, which is compared to the observed β-weight of an expression to determine whether that gene expression is an outlier. This weight function plays a key role in unifying the robustness and efficiency of estimation in one-way ANOVA., Conclusion: Analyses of simulated gene expression profiles revealed that all eight methods (ANOVA, SAM, LIMMA, EBarrays, eLNN, KW, robust BetaEB and proposed) perform almost identically for m = 2 conditions in the absence of outliers. However, the robust BetaEB method and the proposed method exhibited considerably better performance than the other six methods in the presence of outliers. In this case, the BetaEB method exhibited slightly better performance than the proposed method for the small-sample cases, but the the proposed method exhibited much better performance than the BetaEB method for both the small- and large-sample cases in the presence of more than 50% outlying genes. The proposed method also exhibited better performance than the other methods for m > 2 conditions with multiple patterns of expression, where the BetaEB was not extended for this condition. Therefore, the proposed approach would be more suitable and reliable on average for the identification of DE genes between two or more conditions with multiple patterns of expression.

Instrumental variable (IV) analysis has been widely used in economics, epidemiology, and other fields to estimate the causal effects of covariates on outcomes, in the presence of unobserved confounders and/or measurement errors in covariates. However, IV methods for time-to-event outcome with censored data remain underdeveloped. This paper proposes a Bayesian approach for IV analysis with censored time-to-event outcome by using a two-stage linear model. A Markov chain Monte Carlo sampling method is developed for parameter estimation for both normal and non-normal linear models with elliptically contoured error distributions. The performance of our method is examined by simulation studies. Our method largely reduces bias and greatly improves coverage probability of the estimated causal effect, compared with the method that ignores the unobserved confounders and measurement errors. We illustrate our method on the Women's Health Initiative Observational Study and the Atherosclerosis Risk in Communities Study., (Copyright © 2014 John Wiley & Sons, Ltd.)

Tests for regression coefficients such as global, local, and partial F-tests are common in applied research. In the framework of multiple imputation, there are several papers addressing tests for regression coefficients. However, for simultaneous hypothesis testing, the existing methods are computationally intensive because they involve calculation with vectors and (inversion of) matrices. In this paper, we propose a simple method based on the scalar entity, coefficient of determination, to perform (global, local, and partial) F-tests with multiply imputed data. The proposed method is evaluated using simulated data and applied to suicide prevention data., (Copyright © 2014 John Wiley & Sons, Ltd.)

This paper demonstrates the usefulness and flexibility of the general structural equation modelling (SEM) approach to fitting direct covariance patterns or structures (as opposed to fitting implied covariance structures from functional relationships among variables). In particular, the MSTRUCT modelling language (or syntax) of the CALIS procedure (SAS/STAT version 9.22 or later: SAS Institute, 2010) is used to illustrate the SEM approach. The MSTRUCT modelling language supports a direct covariance pattern specification of each covariance element. It also supports the input of additional independent and dependent parameters. Model tests, fit statistics, estimates, and their standard errors are then produced under the general SEM framework. By using numerical and computational examples, the following tests of basic covariance patterns are illustrated: sphericity, compound symmetry, and multiple-group covariance patterns. Specification and testing of two complex correlation structures, the circumplex pattern and the composite direct product models with or without composite errors and scales, are also illustrated by the MSTRUCT syntax. It is concluded that the SEM approach offers a general and flexible modelling of direct covariance and correlation patterns. In conjunction with the use of SAS macros, the MSTRUCT syntax provides an easy-to-use interface for specifying and fitting complex covariance and correlation structures, even when the number of variables or parameters becomes large., (© 2014 The British Psychological Society.)

Methods for choosing an appropriate sample size in animal experiments have received much attention in the statistical and biological literature. Due to ethical constraints the number of animals used is always reduced where possible. However, as the number of animals decreases so the risk of obtaining inconclusive results increases. By using a more efficient experimental design we can, for a given number of animals, reduce this risk. In this paper two popular cases are considered, where planned comparisons are made to compare treatments back to control and when researchers plan to make all pairwise comparisons. By using theoretical and empirical techniques we show that for studies where all pairwise comparisons are made the traditional balanced design, as suggested in the literature, maximises sensitivity. For studies that involve planned comparisons of the treatment groups back to the control group, which are inherently more sensitive due to the reduced multiple testing burden, the sensitivity is maximised by increasing the number of animals in the control group while decreasing the number in the treated groups.

In this paper we propose a latent class distance association model for clustering in the predictor space of large contingency tables with a categorical response variable. The rows of such a table are characterized as profiles of a set of explanatory variables, while the columns represent a single outcome variable. In many cases such tables are sparse, with many zero entries, which makes traditional models problematic. By clustering the row profiles into a few specific classes and representing these together with the categories of the response variable in a low-dimensional Euclidean space using a distance association model, a parsimonious prediction model can be obtained. A generalized EM algorithm is proposed to estimate the model parameters and the adjusted Bayesian information criterion statistic is employed to test the number of mixture components and the dimensionality of the representation. An empirical example highlighting the advantages of the new approach and comparing it with traditional approaches is presented., (© 2014 The British Psychological Society.)

Background: Pairwise association between neurons is a key feature in understanding neural coding. Statistical neuroscience provides tools to estimate and assess these associations. In the mammalian brain, activating ascending pathways arise from neuronal nuclei located at the brainstem and at the basal forebrain that regulate the transition between sleep and awake neuronal firing modes in extensive regions of the cerebral cortex, including the primary visual cortex, where neurons are known to be selective for the orientation of a given stimulus. In this paper, the estimation of neural synchrony as a function of time is studied in data obtained from anesthetized cats. A functional data analysis of variance model is proposed. Bootstrap statistical tests are introduced in this context; they are useful tools for the study of differences in synchrony strength regarding 1) transition between different states (anesthesia and awake), and 2) affinity given by orientation selectivity., Results: An analysis of variance model for functional data is proposed for neural synchrony curves, estimated with a cross-correlation based method. Dependence arising from the experimental setting needs to be accounted for. Bootstrap tests allow the identification of differences between experimental conditions (modes of activity) and between pairs of neurons formed by cells with different affinities given by their preferred orientations. In our test case, interactions between experimental conditions and preferred orientations are not statistically significant., Conclusions: The results reflect the effect of different experimental conditions, as well as the affinity regarding orientation selectivity in neural synchrony and, therefore, in neural coding. A cross-correlation based method is proposed that works well under low firing activity. Functional data statistical tools produce results that are useful in this context. Dependence is shown to be necessary to account for, and bootstrap tests are an appropriate method with which to do so.

Methodologists have recently proposed robust variance estimation as one way to handle dependent effect sizes in meta-analysis. Software macros for robust variance estimation in meta-analysis are currently available for Stata (StataCorp LP, College Station, TX, USA) and spss (IBM, Armonk, NY, USA), yet there is little guidance for authors regarding the practical application and implementation of those macros. This paper provides a brief tutorial on the implementation of the Stata and spss macros and discusses practical issues meta-analysts should consider when estimating meta-regression models with robust variance estimates. Two example databases are used in the tutorial to illustrate the use of meta-analysis with robust variance estimates., (Copyright © 2013 John Wiley & Sons, Ltd.)

Background: The ultra-large von Willebrand factor (vWF) multimers are very active and must be degraded by ADAMTS13 for optimal activity. A severe functional deficiency of ADAMTS13 has been associated with thrombotic thrombocytopenic purpura. The correct interpretation of patient vWF and ADAMTS13 plasma levels requires an understanding of the biological variation associated with these analytes. In the present paper, we aimed to determine the biological variation of ADAMTS13 and vWF in human adults., Materials and Methods: Blood samples were collected weekly from 19 healthy subjects for 5 consecutive weeks. vWF activity and antigenicity were determined using aggregometric and immunoturbidimetric methods. ADAMTS13 antigenicity and activity were determined by ELISA., Results: The within-subject biological variations for vWF activity and antigenicity were 8.06% and 14.37%, respectively, while the between-subject biological variations were 18.5% and 22.59%, respectively. The index of individuality for vWF activity was 0.44, while vWF antigenicity was 0.64. Similarly, ADAMTS13 activity and antigenicity within-subject biological variations were 12.73% and 9.75%, respectively, while between-subject biological variations were 9.63% and 6.28%, respectively. The ADAMTS13 indexes of individuality were 1.32 and 1.55, respectively., Conclusion: We report high biological variation and individuality in vWF antigenicity and activity levels. However, ADAMTS13 antigenicity and activity displayed high biological variation, but low individuality. Thus, population-based reference intervals may be useful for monitoring ADAMTS13 antigenicity and activity, but not for vWF, which displays high individuality. These findings should be considered when determining the reference interval and other clinical variables associated with ADAMTS13 and vWF levels.

For one-way fixed effects ANOVA, it is well known that the conventional F test of the equality of means is not robust to unequal variances, and numerous methods have been proposed for dealing with heteroscedasticity. On the basis of extensive empirical evidence of Type I error control and power performance, Welch's procedure is frequently recommended as the major alternative to the ANOVA F test under variance heterogeneity. To enhance its practical usefulness, this paper considers an important aspect of Welch's method in determining the sample size necessary to achieve a given power. Simulation studies are conducted to compare two approximate power functions of Welch's test for their accuracy in sample size calculations over a wide variety of model configurations with heteroscedastic structures. The numerical investigations show that Levy's (1978a) approach is clearly more accurate than the formula of Luh and Guo (2011) for the range of model specifications considered here. Accordingly, computer programs are provided to implement the technique recommended by Levy for power calculation and sample size determination within the context of the one-way heteroscedastic ANOVA model., (© 2013 The British Psychological Society.)

The correlated-error ANOVA method proposed by Obuchowski and Rockette (OR) has been a useful procedure for analyzing reader-performance outcomes, such as the area under the receiver-operating-characteristic curve, resulting from multireader multicase radiological imaging data. This approach, however, has only been formally derived for the test-by-reader-by-case factorial study design. In this paper, I show that the OR model can be viewed as a marginal-mean ANOVA model. Viewing the OR model within this marginal-mean ANOVA framework is the basis for the marginal-mean ANOVA approach, the topic of this paper. This approach (1) provides an intuitive motivation for the OR model, including its covariance-parameter constraints; (2) provides easy derivations of OR test statistics and parameter estimates, as well as their distributions and confidence intervals; and (3) allows for easy generalization of the OR procedure to other study designs. In particular, I show how one can easily derive OR-type analysis formulas for any balanced study design by following an algorithm that only requires an understanding of conventional ANOVA methods., (Copyright © 2013 John Wiley & Sons, Ltd.)

This paper considers five test statistics for comparing the recovery of a rapid growth-based enumeration test with respect to the compendial microbiological method using a specific nonserial dilution experiment. The finite sample distributions of these test statistics are unknown, because they are functions of correlated count data. A simulation study is conducted to investigate the type I and type II error rates. For a balanced experimental design, the likelihood ratio test and the main effects analysis of variance (ANOVA) test for microbiological methods demonstrated nominal values for the type I error rate and provided the highest power compared with a test on weighted averages and two other ANOVA tests. The likelihood ratio test is preferred because it can also be used for unbalanced designs. It is demonstrated that an increase in power can only be achieved by an increase in the spiked number of organisms used in the experiment. The power is surprisingly not affected by the number of dilutions or the number of test samples. A real case study is provided to illustrate the theory., (Copyright © 2013 John Wiley & Sons, Ltd.)

In this journal, Zimmerman (2004, 2011) has discussed preliminary tests that researchers often use to choose an appropriate method for comparing locations when the assumption of normality is doubtful. The conceptual problem with this approach is that such a two-stage process makes both the power and the significance of the entire procedure uncertain, as type I and type II errors are possible at both stages. A type I error at the first stage, for example, will obviously increase the probability of a type II error at the second stage. Based on the idea of Schmider et al. (2010), which proposes that simulated sets of sample data be ranked with respect to their degree of normality, this paper investigates the relationship between population non-normality and sample non-normality with respect to the performance of the ANOVA, Brown-Forsythe test, Welch test, and Kruskal-Wallis test when used with different distributions, sample sizes, and effect sizes. The overall conclusion is that the Kruskal-Wallis test is considerably less sensitive to the degree of sample normality when populations are distinctly non-normal and should therefore be the primary tool used to compare locations when it is known that populations are not at least approximately normal., (© 2012 The British Psychological Society.)

Testing homogeneity of dispersions may be of its own scientific interest as well as an important auxiliary step verifying assumptions of a main analysis. The problem is that many biological and ecological data are highly skewed and zero-inflated. Also the number of variables often exceeds the sample size. Thus data analysts often do not rely on parametric assumptions, but use a particular dissimilarity measure to calculate a matrix of pairwise differences. This matrix is then the basis for further statistical inference. Anderson (2006) proposed a distance-based test of homogeneity of multivariate dispersions for a one-way ANOVA design, for which a matrix of pairwise dissimilarities can be taken as an input. The key idea, like in Levene's test, is to replace each observation with its distance to an estimated group center. In this paper we suggest an alternative approach that is based on the means of within-group distances and does not require group centre calculations to obtain the test statistic. We show that this approach can have theoretical as well as practical advantages. A permutation procedure that gives type I error close to the prescribed value even in small samples is described., (Copyright © 2013, The International Biometric Society.)

In clinical trials with time-to-event outcomes, it is common to estimate the marginal hazard ratio from the proportional hazards model, even when the proportional hazards assumption is not valid. This is unavoidable from the perspective that the estimator must be specified a priori if probability statements about treatment effect estimates are desired. Marginal hazard ratio estimates under non-proportional hazards are still useful, as they can be considered to be average treatment effect estimates over the support of the data. However, as many have shown, under non-proportional hazard, the 'usual' unweighted marginal hazard ratio estimate is a function of the censoring distribution, which is not normally considered to be scientifically relevant when describing the treatment effect. In addition, in many practical settings, the censoring distribution is only conditionally independent (e.g., differing across treatment arms), which further complicates the interpretation. In this paper, we investigate an estimator of the hazard ratio that removes the influence of censoring and propose a consistent robust variance estimator. We compare the coverage probability of the estimator to both the usual Cox model estimator and an estimator proposed by Xu and O'Quigley (2000) when censoring is independent of the covariate. The new estimator should be used for inference that does not depend on the censoring distribution. It is particularly relevant to adaptive clinical trials where, by design, censoring distributions differ across treatment arms., (Copyright © 2012 John Wiley & Sons, Ltd.)

The paper assesses biased-coin designs for sequential treatment allocation in clinical trials. Comparisons emphasise the importance of considering randomness, as well as treatment balance, which are calculated as bias and loss. In the numerical examples, the responses are assumed normally distributed, perhaps after transformation, and balance is required over a set of covariates. The effect of covariate distribution on the properties of five allocation rules is investigated, with an emphasis on methods of comparison, which also apply to other forms of response. The concept of admissibility shows that the widely used minimisation rule is outperformed by Atkinson's rule derived from the theory of optimum experimental design. We present a simplified form of this rule. For this rule, the ability to guess the next treatment allocation decreases with study size. For the other rules, it is constant., (Copyright © 2012 John Wiley & Sons, Ltd.)

Although the concentration index (CI) and the health achievement index (HAI) have been extensively used, previous studies have relied on bootstrapping to compute the variance of the HAI, whereas competing variance estimators exist for the CI. This paper provides methods of statistical inference for the HAI and compares the available variance estimators for both the CI and the HAI using Monte Carlo simulation. Results for both the CI and the HAI suggest that analytical methods and bootstrapping are well behaved. The convenient regression method gives standard errors close to the other methods, provided the CI is not too large (< 0.2), but otherwise tends to understate the standard errors. In our simulation setting, the improvement from the Newey-West correction over the convenient regression method has mixed evidence when the CI ≤ 0.1 and is modest when the CI > 0.1. Published 2011. This article is a US Government work and is in the public domain in the USA., (Published 2011. This article is a US Government work and is in the public domain in the USA.)

In epidemiology and in clinical research, risk factors often have special distributions. A common situation is that a proportion of individuals have exposure zero, and among those exposed, we have some continuous distribution. We call this a 'spike at zero'. Examples for this are smoking, duration of breastfeeding, or alcohol consumption. Furthermore, the empirical distribution of laboratory values and other measurements may have a semi-continuous distribution as a result of the lower detection limit of the measurement. To model the dose-response function, an extension of the fractional polynomial approach was recently proposed. In this paper, we suggest a modification of the previously suggested FP procedure. We first give the theoretical justification of this modified procedure by investigating relevant distribution classes. Here, we systematically derive the theoretical shapes of dose-response curves under given distributional assumptions (normal, log normal, gamma) in the framework of a logistic regression model. Further, we check the performance of the procedure in a simulation study and compare it to the previously suggested method, and finally we illustrate the procedures with data from a case-control study on breast cancer., (© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

This paper deals with bias-reduction techniques for observational studies in evidence-based laboratory medicine (EBLM). In the field of laboratory medicine, many observational studies have been performed since it is difficult to design randomized experimental studies. The results of these observational studies have usually been affected by various types of biases in observational data that could not be controlled by the researchers. In randomized experiments, random assignment provides unbiased estimations of the treatment effect. In contrast, in observational studies, incorrect (biased) estimations arise from the imbalance between the covariates for the treatment/exposure group and the control group; therefore, information regarding confounding factors that affect both an outcome variable and assignment should be used to construct a multivariate model for minimizing bias. Covariate adjustment helps to reduce bias by correcting the imbalance in covariates. Analysis of covariance (ANCOVA) is an important method for covariate adjustment. The ANCOVA model is an extension of multiple regression models that can statistically control the effects of covariates. The propensity score method has recently been used as a covariate adjustment method in applied research. Because propensity scores concentrate the information on covariates, conditional expectations can be easily computed. In this paper, both methods were exemplified in a study on sex-based differences in HDL cholesterol levels. Similar unbiased estimates of sex-based differences were obtained using both methods, as opposed to an incorrect estimate obtained using univariate analysis. The results emphasize that covariate adjustment should be used to obtain credible evidence in observational studies.

It has been well known that ignoring measurement error may result in substantially biased estimates in many contexts including linear and nonlinear regressions. For survival data with measurement error in covariates, there has been extensive discussion in the literature with the focus on proportional hazards (PH) models. Recently, research interest has extended to accelerated failure time (AFT) and additive hazards (AH) models. However, the impact of measurement error on other models, such as the proportional odds model, has received relatively little attention, although these models are important alternatives when PH, AFT, or AH models are not appropriate to fit data. In this paper, we investigate this important problem and study the bias induced by the naive approach of ignoring covariate measurement error. To adjust for the induced bias, we describe the simulation-extrapolation method. The proposed method enjoys a number of appealing features. Its implementation is straightforward and can be accomplished with minor modifications of existing software. More importantly, the proposed method does not require modeling the covariate process, which is quite attractive in practice. As the precise values of error-prone covariates are often not observable, any modeling assumption on such covariates has the risk of model misspecification, hence yielding invalid inferences if this happens. The proposed method is carefully assessed both theoretically and empirically. Theoretically, we establish the asymptotic normality for resulting estimators. Numerically, simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error, along with an application to a data set arising from the Busselton Health Study. Sensitivity of the proposed method to misspecification of the error model is studied as well., (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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., (Copyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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., (Copyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

Background: Meta-analysis typically involves combining the estimates from independent studies in order to estimate a parameter of interest across a population of studies. However, outliers often occur even under the random effects model. The presence of such outliers could substantially alter the conclusions in a meta-analysis. This paper proposes a methodology for identifying and, if desired, downweighting studies that do not appear representative of the population they are thought to represent under the random effects model., Methods: An outlier is taken as an observation (study result) with an inflated random effect variance. We used the likelihood ratio test statistic as an objective measure for determining whether observations have inflated variance and are therefore considered outliers. A parametric bootstrap procedure was used to obtain the sampling distribution of the likelihood ratio test statistics and to account for multiple testing. Our methods were applied to three illustrative and contrasting meta-analytic data sets., Results: For the three meta-analytic data sets our methods gave robust inferences when the identified outliers were downweighted., Conclusions: The proposed methodology provides a means to identify and, if desired, downweight outliers in meta-analysis. It does not eliminate them from the analysis however and we consider the proposed approach preferable to simply removing any or all apparently outlying results. We do not however propose that our methods in any way replace or diminish the standard random effects methodology that has proved so useful, rather they are helpful when used in conjunction with the random effects model.

A great deal of educational and social data arises from cluster sampling designs where clusters involve schools, classrooms, or communities. A mistake that is sometimes encountered in the analysis of such data is to ignore the effect of clustering and analyse the data as if it were based on a simple random sample. This typically leads to an overstatement of the precision of results and too liberal conclusions about precision and statistical significance of mean differences. This paper gives simple corrections to the test statistics that would be computed in an analysis of variance if clustering were (incorrectly) ignored. The corrections are multiplicative factors depending on the total sample size, the cluster size, and the intraclass correlation structure. For example, the corrected F statistic has Fisher's F distribution with reduced degrees of freedom. The corrected statistic reduces to the F statistic computed by ignoring clustering when the intraclass correlations are zero. It reduces to the F statistic computed using cluster means when the intraclass correlations are unity, and it is in between otherwise. A similar adjustment to the usual statistic for testing a linear contrast among group means is described.

Longitudinally collected outcomes are increasingly common in cell biology and gene therapy research. In this article, we review the current practice of statistical analysis of longitudinal data in these fields, and recommend the "best performing" statistical method among those available in most statistical packages. A survey of papers published in Molecular Therapy indicates that longitudinal data are only properly analyzed in a small fraction of articles, and the most popular approach was analyzing each measurement time point data separately using an analysis of variance (ANOVA) model with Tukey's post hoc tests. We show that first, such cross-sectional ANOVA approach does not utilize all the power that the longitudinal design of a study provides, and second, Tukey's post hoc tests applied at each measurement time separately could result in a false positivity rate as high as 30% using a simulation study. We recommend mixed effects model analysis instead. We also discuss the complexities of multiple comparison adjustment in the post hoc testing that result from within experimental unit correlation existing in longitudinal data. We recommend resampling as a method that readily adjusts the post hoc testing to be limited to only interesting comparisons and thereby avoids unduly sacrificing the power.

This paper presents some considerations about the use of adequate statistical techniques in the framework of the neuroelectromagnetic brain mapping. With the use of advanced EEG/MEG recording setup involving hundred of sensors, the issue of the protection against the type I errors that could occur during the execution of hundred of univariate statistical tests, has gained interest. In the present experiment, we investigated the EEG signals from a mannequin acting as an experimental subject. Data have been collected while performing a neuromarketing experiment and analyzed with state of the art computational tools adopted in specialized literature. Results showed that electric data from the mannequin's head presents statistical significant differences in power spectra during the visualization of a commercial advertising when compared to the power spectra gathered during a documentary, when no adjustments were made on the alpha level of the multiple univariate tests performed. The use of the Bonferroni or Bonferroni-Holm adjustments returned correctly no differences between the signals gathered from the mannequin in the two experimental conditions. An partial sample of recently published literature on different neuroscience journals suggested that at least the 30% of the papers do not use statistical protection for the type I errors. While the occurrence of type I errors could be easily managed with appropriate statistical techniques, the use of such techniques is still not so largely adopted in the literature., (Copyright (c) 2010 Elsevier B.V. All rights reserved.)

Background: Sensitivity analysis is an indispensable tool for the analysis of complex systems. In a recent paper, we have introduced a thermodynamically consistent variance-based sensitivity analysis approach for studying the robustness and fragility properties of biochemical reaction systems under uncertainty in the standard chemical potentials of the activated complexes of the reactions and the standard chemical potentials of the molecular species. In that approach, key sensitivity indices were estimated by Monte Carlo sampling, which is computationally very demanding and impractical for large biochemical reaction systems. Computationally efficient algorithms are needed to make variance-based sensitivity analysis applicable to realistic cellular networks, modeled by biochemical reaction systems that consist of a large number of reactions and molecular species., Results: We present four techniques, derivative approximation (DA), polynomial approximation (PA), Gauss-Hermite integration (GHI), and orthonormal Hermite approximation (OHA), for analytically approximating the variance-based sensitivity indices associated with a biochemical reaction system. By using a well-known model of the mitogen-activated protein kinase signaling cascade as a case study, we numerically compare the approximation quality of these techniques against traditional Monte Carlo sampling. Our results indicate that, although DA is computationally the most attractive technique, special care should be exercised when using it for sensitivity analysis, since it may only be accurate at low levels of uncertainty. On the other hand, PA, GHI, and OHA are computationally more demanding than DA but can work well at high levels of uncertainty. GHI results in a slightly better accuracy than PA, but it is more difficult to implement. OHA produces the most accurate approximation results and can be implemented in a straightforward manner. It turns out that the computational cost of the four approximation techniques considered in this paper is orders of magnitude smaller than traditional Monte Carlo estimation. Software, coded in MATLAB, which implements all sensitivity analysis techniques discussed in this paper, is available free of charge., Conclusions: Estimating variance-based sensitivity indices of a large biochemical reaction system is a computationally challenging task that can only be addressed via approximations. Among the methods presented in this paper, a technique based on orthonormal Hermite polynomials seems to be an acceptable candidate for the job, producing very good approximation results for a wide range of uncertainty levels in a fraction of the time required by traditional Monte Carlo sampling.

Analysis of motor performance variability in tasks with redundancy affords insight about synergies underlying central nervous system (CNS) control. Preferential distribution of variability in ways that minimally affect task performance suggests sophisticated neural control. Unfortunately, in the analysis of variability the choice of coordinates used to represent multi-dimensional data may profoundly affect analysis, introducing an arbitrariness which compromises its conclusions. This paper assesses the influence of coordinates. Methods based on analyzing a covariance matrix are fundamentally dependent on an investigator's choices. Two reasons are identified: using anisotropy of a covariance matrix as evidence of preferential distribution of variability; and using orthogonality to quantify relevance of variability to task performance. Both are exquisitely sensitive to coordinates. Unless coordinates are known a priori, these methods do not support unambiguous inferences about CNS control. An alternative method uses a two-level approach where variability in task execution (expressed in one coordinate frame) is mapped by a function to its result (expressed in another coordinate frame). An analysis of variability in execution using this function to quantify performance at the level of results offers substantially less sensitivity to coordinates than analysis of a covariance matrix of execution variables. This is an initial step towards developing coordinate-invariant analysis methods for movement neuroscience.

In 2007, Rhiel presented a technique to estimate the coefficient of variation from the range when sampling from skewed distributions. To provide an unbiased estimate, a correction factor (a(n)) for the mean was included. Numerical correction factors for a number of skewed distributions were provided. In a follow-up paper, he provided a proof he claimed showed the correction factor was independent of the mean and standard deviation, making the factors useful as these parameters vary; however, that proof did not establish independence. Herein is a proof which establishes the independence.

We wish to suggest the categorical analysis of variance as a means of quantifying the proportion of total genetic variation attributed to different sources of variation. This method potentially challenges researchers to rethink conclusions derived from a well-known method known as the analysis of molecular variance (AMOVA). The CATANOVA framework allows explicit definition, and estimation, of two measures of genetic differentiation. These parameters form the subject of interest in many research programmes, but are often confused with the correlation measures defined in AMOVA, which cannot be interpreted as relative contributions of particular sources of variation. Through a simulation approach, we show that under certain conditions, researchers who use AMOVA to estimate these measures of genetic differentiation may attribute more than justified amounts of total variation to population labels. Moreover, the two measures can also lead to incongruent conclusions regarding the genetic structure of the populations of interest. Fortunately, one of the two measures seems robust to variations in relative sample sizes used. Its merits are illustrated in this paper using mitochondrial haplotype and amplified fragment length polymorphism (AFLP) data.

Sensitivity analysis is an indispensable tool for studying the robustness and fragility properties of biochemical reaction systems as well as for designing optimal approaches for selective perturbation and intervention. Deterministic sensitivity analysis techniques, using derivatives of the system response, have been extensively used in the literature. However, these techniques suffer from several drawbacks, which must be carefully considered before using them in problems of systems biology. We develop here a probabilistic approach to sensitivity analysis of biochemical reaction systems. The proposed technique employs a biophysically derived model for parameter fluctuations and, by using a recently suggested variance-based approach to sensitivity analysis [Saltelli et al., Chem. Rev. (Washington, D.C.) 105, 2811 (2005)], it leads to a powerful sensitivity analysis methodology for biochemical reaction systems. The approach presented in this paper addresses many problems associated with derivative-based sensitivity analysis techniques. Most importantly, it produces thermodynamically consistent sensitivity analysis results, can easily accommodate appreciable parameter variations, and allows for systematic investigation of high-order interaction effects. By employing a computational model of the mitogen-activated protein kinase signaling cascade, we demonstrate that our approach is well suited for sensitivity analysis of biochemical reaction systems and can produce a wealth of information about the sensitivity properties of such systems. The price to be paid, however, is a substantial increase in computational complexity over derivative-based techniques, which must be effectively addressed in order to make the proposed approach to sensitivity analysis more practical.

In this paper errors in variables methods for fitting straight lines to data are reviewed. In these methods the x and y variables are both assumed to be subject to measurement error and not, as in simple least squares linear regression, just one of them. The methods are described in a unified context using the statistical principle of the method of moments. Guidance is given on the choice of an appropriate method of estimating the slope and intercept of the fitted line. Formulas for the approximate standard errors of the estimators are provided in a technical appendix. A numerical example from biochemical studies is included to illustrate the methodology.

In biomedical research, researchers frequently use statistical procedures such as the t-test, standard analysis of variance (ANOVA), or the repeated measures ANOVA to compare means between the groups of interest. There are frequently some misuses in applying these procedures since the conditions of the experiments or statistical assumptions necessary to apply these procedures are not fully taken into consideration. In this paper, we demonstrate the correct use of repeated measures ANOVA to prevent or minimize ethical or scientific problems due to its misuse. We also describe the appropriate use of multiple comparison tests for follow-up analysis in repeated measures ANOVA. Finally, we demonstrate the use of repeated measures ANOVA by using real data and the statistical software package SPSS (SPSS Inc., USA).

In randomised controlled trials, an overlap of confidence intervals is often cited as evidence of 'no statistically significant difference' between intervention groups. This paper illustrates the limitations of this strategy and compares different univariate linear regression models with baseline and follow-up response measures. The researchers also demonstrate that using 'change in response' or exit score as a function of the baseline response in clinical studies leads to the same results. Further, using a model that includes baseline response as covariate leads to more precise estimates. The implications for future trials are discussed.

Etiological research aims to investigate the causal relationship between putative risk factors (or determinants) and a given disease or other outcome. In contrast, prognostic research aims to predict the probability of a given clinical outcome and in this perspective the pathophysiology of the disease is not an issue. Multivariate modeling is a fundamental tool both to infer causality and to investigate prognostic factors in epidemiological research. The analytical approaches to etiological and prognostic studies are strictly dependent on the research question and imply knowledge of the main statistical procedures for model building and data interpretation. In this paper we describe the application of multivariate statistical modeling in etiological and prognostic research. We will mainly focus on the differences in model building and data interpretation between these two areas of epidemiologic research.

In his 1923 landmark article, Neyman introduced randomization-based inference to estimate average treatment effects from experiments under the completely randomized design. Under this framework, Neyman considered the statistical estimation of the sample average treatment effect and derived the variance of the standard estimator using the treatment assignment mechanism as the sole basis of inference. In this paper, I extend Neyman's analysis to randomized experiments under the matched-pair design where experimental units are paired based on their pre-treatment characteristics and the randomization of treatment is subsequently conducted within each matched pair. I study the variance identification for the standard estimator of average treatment effects and analyze the relative efficiency of the matched-pair design over the completely randomized design. I also show how to empirically evaluate the relative efficiency of the two designs using experimental data obtained under the matched-pair design. My randomization-based analysis differs from previous studies in that it avoids modeling and other assumptions as much as possible. Finally, the analytical results are illustrated with numerical and empirical examples., (Copyright 2008 John Wiley & Sons, Ltd.)

Objective: Typical methods of analyzing data from clinical trials have shortcomings, notably comparisons of group means, use of change scores from pre- and post-treatment assessments, ignoring intervening assessments, and focusing on direct effects of treatment. A comparison of group means disregards the likelihood that individuals have different trajectories of change. Moreover, change scores ignore intervening assessments that may provide useful information about change. This paper compares results from traditional regression-based methods for analyzing data from a clinical trial (e.g., regression with change scores) with those of latent growth curve modeling (LGM)., Methods: LGM is a method that uses structural equation modeling techniques to model individual change, assess treatment effects and the relationship among multiple outcomes simultaneously, and model measurement error. The consequence is more precise parameter estimates while using data from all available time points., Results: Results demonstrate that LGM can yield stronger parameter estimates than the traditional regression-based approach and explain more variance in the outcome. In trials where there is a true effect, but it is non-significant or marginally significant using the traditional methods, LGM may provide evidence of this effect., Conclusions: Analysts are encouraged to consider LGM as an additional and informative tool for analyzing clinical trial or other longitudinal data.

Nonlinear mixed effects models are now widely used in biometrical studies, especially in pharmacokinetic research or for the analysis of growth traits for agricultural and laboratory species. Most of these studies, however, are often based on ML estimation procedures, which are known to be biased downwards. A few REML extensions have been proposed, but only for approximated methods. The aim of this paper is to present a REML implementation for nonlinear mixed effects models within an exact estimation scheme, based on an integration of the fixed effects and a stochastic estimation procedure. This method was implemented via a stochastic EM, namely the SAEM algorithm. The simulation study showed that the proposed REML estimation procedure considerably reduced the bias observed with the ML estimation, as well as the residual mean squared error of the variance parameter estimations, especially in the unbalanced cases. ML and REML based estimators of fixed effects were also compared via simulation. Although the two kinds of estimates were very close in terms of bias and mean square error, predictions of individual profiles were clearly improved when using REML vs. ML. An application of this estimation procedure is presented for the modelling of growth in lines of chicken., ((c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)