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.     

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%.     

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 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.     

RESPONSE OF A LENTIC PERIPHYTON COMMUNITY TO NUTRIENT ENRICHMENT AT LOW N:P RATIOS1

We examined the effect of nitrogen:phosphorus (N:P) ratios and nutrient concentrations on periphyton when nutrients (N and P) are provided in excess. A gradient of seven N:P ratios ranging from 7.5:1 to 1:7.5 and each at three absolute concentrations, was established using nutrient‐releasing substrata placed in a meso‐oligotrophic lake. Differences in total algal biovolume among nutrient ratios were significant (analysis of covariance [ANCOVA]) when P concentration was entered as the co‐variate. In addition, total algal biovolume was significantly correlated with N concentration but not P. To further evaluate the relationship between nutrient ratios and biovolume, we analyzed (using four 1‐way analysis of variances [ANOVAs]) four subsets of data defined as a series of treatments where one nutrient concentration remained relatively constant as the other changed creating different N:P ratios. Ratios of data subsets ranged from 1:1 to 7.5:1 and 1:1 to 1:7.5 with low and high concentrations of both series. Only diatom biovolume varied with ratio but these differences are most likely related to increased green algal abundance. Species richness and diversity differed among N:P ratios (ANCOVA) when P concentration was used as the co‐variate. Stigeoclonium tenue (Ag.) Gomont, which generally accounted for the increase in green algal abundance, varied with nutrient ratio (ANCOVA) when P was the co‐variate. Based on the ANCOVAs, correlations, and one‐way ANOVAs, periphyton in this system appears to be affected by N concentration but not by N:P treatment ratios under nutrient‐rich conditions. When compared with previous studies, these data also suggest that the response of periphyton to in situ treatments constructed with nutrient‐releasing substrata vary between years.     

A half a century of measuring ungulate body condition using indices: is it time for a change?

From a literature review of five wildlife ecology journals since 1937, we document how using indices to monitor ungulate body condition is common practice, with the kidney fat index (KFI = weight of fat around the kidneys/weight of kidneys without fat × 100) as the favoured tool (82% of studies). In this context, we highlight the problems of using indices when underlying statistical assumptions are not met (isometry, parallel slopes between treatments). We show, with real and simulated data for two cervids with contrasting fat storage strategies, how results from analysis of variance of KFI values differ from analysis of covariance (ANCOVA) of raw data. We conclude that the KFI is affected by the restrictions typically associated with derived index values, and as a consequence, statistical analysis of the KFI could generate spurious results leading to erroneous interpretations concerning variation in body condition of ungulate populations. Thus, we recommend analysing fat weight as an untransformed variable in ANCOVA (kidney weight as covariate) to describe body condition variation in ungulates. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Detection of abnormal fermentations in wine process by multivariate statistics and pattern recognition techniques

Three multivariate statistical techniques (Multiway Principal Component Analysis, Multiway Partial Least Squares, and Stepwise Linear Discriminant Analysis) and one artificial intelligence method (Artificial Neural Networks) were evaluated to detect and predict early abnormal behaviors of wine fermentations. The techniques were tested with data of thirty-two variables at different stages of fermentation from industrial wine fermentations of Cabernet Sauvignon. All the techniques studied considered a pre-treatment to obtain a homogeneous space and reduce the overfitting. The results were encouraging; it was possible to classify at 72h 100% of the fermentation correctly with three variables using Multiway Partial Least Squares and Artificial Neural Networks. Additional and complementary results were obtained with Stepwise Linear Discriminant Analysis, which found that ethanol, sugars and density measurements are able to discriminate abnormal behavior.     

Life-history invariants with bounded variables cannot be distinguish from data generated by random processes using standard analyses

A dimensionless approach to the study of life-history evolution has been applied to a wide variety of variables in the search for life-history invariants. This approach usually employs ordinary least squares (OLS) regressions of log-transformed data. In several well-studied combinations of variables the range of values of one parameter is bounded or limited by the value of the other. In this situation, the null hypothesis normally applied to regression analysis is not appropriate. We generate the null expectations and confidence intervals (CI) for OLS and reduced major axis (RMA) regressions using random variables that are bounded in this way. Comparisons of these CI show that, for log-transformed data, the patterns generated by random data and those predicted by life history invariant theory often could not be distinguished because both predict a slope of 1. We recommend that tests based on the putative invariant ratios and not the correlations between the two variables be used in the exploration of life-history invariants using bounded data. Because empirical data are often not normally distributed randomization test may be more appropriate than standard statistical tests.     

An improved method for studying the condition of fish, with an example using Pacific sardine Sardinops sagax (Jenyns)

The statistical treatment of condition used in many previous studies has been to determine a predicted weight for a fish, and to calculate a ratio of actual to predicted weight. The ratios obtained, known as condition factor K , are then further analysed using analysis of variance or other techniques to relate changes in K to other independent variables. It is suggested that condition can be studied better by using a single model to analyse the response of fish weight to a number of factors simultaneously. This approach affords benefit from the simplicity of using an integrated analysis and avoids problems with skewed distributions of ratios. An example using data on the Pacific sardine, Sardinops sagax (Jenyns), is given. Changes in condition of this fish due to monthly, yearly and temperature effects are calculated.     

Ratios as a size adjustment in morphometrics

Simple ratios in which a measurement variable is divided by a size variable are commonly used but known to be inadequate for eliminating size correlations from morphometric data. Deficiencies in the simple ratio can be alleviated by incorporating regression coefficients describing the bivariate relationship between the measurement and size variables. Recommendations have included: 1) subtracting the regression intercept to force the bivariate relationship through the origin (intercept-adjusted ratios); 2) exponentiating either the measurement or the size variable using an allometry coefficient to achieve linearity (allometrically adjusted ratios); or 3) both subtracting the intercept and exponentiating (fully adjusted ratios). These three strategies for deriving size-adjusted ratios imply different data models for describing the bivariate relationship between the measurement and size variables (i.e., the linear, simple allometric, and full allometric models, respectively). Algebraic rearrangement of the equation associated with each data model leads to a correctly formulated adjusted ratio whose expected value is constant (i.e., size correlation is eliminated). Alternatively, simple algebra can be used to derive an expected value function for assessing whether any proposed ratio formula is effective in eliminating size correlations. Some published ratio adjustments were incorrectly formulated as indicated by expected values that remain a function of size after ratio transformation. Regression coefficients incorporated into adjusted ratios must be estimated using least-squares regression of the measurement variable on the size variable. Use of parameters estimated by any other regression technique (e.g., major axis or reduced major axis) results in residual correlations between size and the adjusted measurement variable. Correctly formulated adjusted ratios, whose parameters are estimated by least-squares methods, do control for size correlations. The size-adjusted results are similar to those based on analysis of least-squares residuals from the regression of the measurement on the size variable. However, adjusted ratios introduce size-related changes in distributional characteristics (variances) that differentially alter relationships among animals in different size classes. © 1993 Wiley-Liss, Inc.     

VEGAN,a package of R functions for community ecology

Abstract. VEGAN adds vegetation analysis functions to the general‐purpose statistical program R. Both R and VEGAN can be downloaded for free. VEGAN implements several ordination methods, including Canonical Correspondence Analysis and Non‐metric Multidimensional Scaling, vector fitting of environmental variables, randomization tests, and various other analyses of vegetation data. It can be used for large data. Graphical output can be customized using the R language's extensive graphics capabilities. VEGAN is appropriate for routine and research use, if you are willing to learn some R.     

The arithmetic mean of ratio-normalized experiments overestimates the true E/C ratio

Biological assays often suffer from large systematic variation between sets of experiments. This variation is sometimes countered by normalizing the results of an "exposed" (E) experiment to that of a simultaneously performed "control" (C). We demonstrate that the arithmetic mean of such ratios overestimates the "true" E/C ratio. Fortunately, the overestimation may be calculated from experimentally accessible information, and it is generally possible to correct for this factor using formulas presented in this paper. We have studied the impact of this effect on a set of studies in the bioelectromagnetics literature and find that, although most results are weakened by the correction, few are significantly altered. Some of the papers used for our literature study are controversial; we believe that the present study may strengthen the quoted results by removing doubts about the statistical treatment of E/C ratios. Both false positives and negatives are possible if the proper correction is not made to the arithmetic mean of a set of E/C data. Realistic examples of erroneous statistical conclusions demonstrate that this is a real concern for E/C data which are marginal in both magnitude (mean < 2) and variance (standard deviation > 0.5).     

Mass-length relationship of Mediterranean barbel as an indicator of environmental status in South-west European stream ecosystems

Analysis of mass-length relationships of 10 populations of Mediterranean barbel from three basins with different environmental conditions showed that there were significant differences (ANCOVA, P <0∣05) in relationships between sampling sites, which could imply differences in environmental conditions. Oxygen concentration and riparian cover were found to be the environmental variables that accounted for most variation (83%) in parameter a of the mass-length relationship between populations. Thus, mass-length relationships of barbels can be used to assess environmental conditions in southwestern European streams.     

Characterization of the interplay between the main factors contributing to lipoplex-mediated transfection in cell cultures

Transfection efficiency of lipoplex-mediated gene delivery is multifactorial. However, the mode of interaction between the factors which affect transfection is not fully understood. To help fill this deficiency we evaluated the effect of the interplay between several variables that affect transfection efficiency in cell cultures. For this, we applied the Analysis of Variance Model with Fixed Effects and Repeated Measures to assess the data. The variables studied include: two different genes, Luc, and human growth hormone (hGH), in three different plasmids (two of which contain the luciferase (Luc) gene, but different promoter-enhancer regions (CMV and H19) and one plasmid coding hGH with a S16 promoter); three topoisoforms of pDNA (supercoiled (SC), open circular (OC), and closed circular (CC)); three cationic lipid compositions, all based on the monocationic lipid DOTAP (100% DOTAP, DOTAP/DOPE 1 : 1, and DOTAP/cholesterol 1 : 1, all ratios are mole ratios); two DNA-/L+ charge ratios (0.2 and 0.5); and two cell lines (NIH 3T3 and MBT-2). Our statistical analysis confirmed that the cell type, the gene used for transfection, the promoter type, the type of helper lipid, and DNA-/DOTAP+ charge ratio, all affect transfection efficiency in a statistically significant manner. The most efficient lipoplex formulation in both cell lines was that based on DOTAP (without helper lipid), having CC plasmid DNA. We suggest that for obtaining the most transfection-efficient lipoplex one should select the best topoisoform of pDNA for each particular cell type, and complex it with cationic liposomes having optimal lipid composition.     

Predicting patterns of vascular plant species richness with composite variables: a meso-scale study in Finnish Lapland

This paper is an attempt, using statistical modelling techniques, to understand the patterns of vascular plant species richness at the poorly studied meso-scale within a relatively unexplored subarctic zone. Species richness is related to floristic-environmental composite variables, using occurrence data of vascular plants and environmental and spatial predictor variables in 362 1 km² grid squares in the Kevo Nature Reserve. Species richness is modelled in two different way. First, by detecting the major floristic-environmental gradients with the ordination procedure of canonical correspondence analysis, and subsequently relating these ordination axes to species richness by generalized linear modelling. Second, species richness is directly related to the composite environmental factors of explanatory variables, using partial least squares regression. The most important explanatory variables, as suggested by both approaches, are relatively similar, and largely reflect the influence of altitude or altitudinally related variables in the models. The most prominent floristic gradient in the data runs from alpine habitats to river valleys, and this gradient is the main source of variation in species richness. Some local environmental variables are also relatively important predictors; the grid squares rich in vascular plant taxa are mainly located in the lowlands of the reserve and are characterised by rivers and brooks, as well as by abundant cliff walls. The two statistical models account for approximately the same amount of variation in the species richness, with more than half of the variation unexplained. Potential reasons for the relatively modest fit are discussed, and the results are compared to the characteristics of the diversity-environment relationships at both broader- and finer-scales.     

Identification of Relevant Phytochemical Constituents for Characterization and Authentication of Tomatoes by General Linear Model Linked to Automatic Interaction Detection (GLM-AID) and Artificial Neural Network Models (ANNs)

There are a large number of tomato cultivars with a wide range of morphological, chemical, nutritional and sensorial characteristics. Many factors are known to affect the nutrient content of tomato cultivars. A complete understanding of the effect of these factors would require an exhaustive experimental design, multidisciplinary scientific approach and a suitable statistical method. Some multivariate analytical techniques such as Principal Component Analysis (PCA) or Factor Analysis (FA) have been widely applied in order to search for patterns in the behaviour and reduce the dimensionality of a data set by a new set of uncorrelated latent variables. However, in some cases it is not useful to replace the original variables with these latent variables. In this study, Automatic Interaction Detection (AID) algorithm and Artificial Neural Network (ANN) models were applied as alternative to the PCA, AF and other multivariate analytical techniques in order to identify the relevant phytochemical constituents for characterization and authentication of tomatoes. To prove the feasibility of AID algorithm and ANN models to achieve the purpose of this study, both methods were applied on a data set with twenty five chemical parameters analysed on 167 tomato samples from Tenerife (Spain). Each tomato sample was defined by three factors: cultivar, agricultural practice and harvest date. General Linear Model linked to AID (GLM-AID) tree-structured was organized into 3 levels according to the number of factors. p-Coumaric acid was the compound the allowed to distinguish the tomato samples according to the day of harvest. More than one chemical parameter was necessary to distinguish among different agricultural practices and among the tomato cultivars. Several ANN models, with 25 and 10 input variables, for the prediction of cultivar, agricultural practice and harvest date, were developed. Finally, the models with 10 input variables were chosen with fit’s goodness between 44 and 100%. The lowest fits were for the cultivar classification, this low percentage suggests that other kind of chemical parameter should be used to identify tomato cultivars.     

Analysis of Covariance Models for Data From Observational Field Studies

Abstract We outline the features of a general class of statistical models (i.e., analysis of covariance [ANCOVA] models) that has proven to be effective for the analysis of data from observational studies. In observational studies, treatments are assigned by Nature in a decidedly nonrandom manner; consequently, many of the crucial assumptions and safeguards of the classic experimental design either fail or are absent. Hence, inferences (causal or associative) are more difficult to justify. Typically, investigators can expect the primary factors of interest, which are usually called environmental exposures rather than treatments, to be involved in complex interactions with each other and with other factors, and these factors will be confounded with still other factors. We provide examples illustrating the application of ANCOVA models to adjust for confounding factors and complex interactions, thereby providing relatively clean estimates of association between exposure and response. We summarize information on available software and supporting literature for implementing ANCOVA models for the analysis of cross-sectional and longitudinal observational field data. We conclude with a brief discussion of critical model fitting issues, including proper specification of the functional form of continuous covariates and problems associated with overfitted models and misspecified models that lack important covariates.     

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.     

A guide to analysis of mouse energy metabolism

We present a consolidated view of the complexity and challenges of designing studies for measurement of energy metabolism in mouse models, including a practical guide to the assessment of energy expenditure, energy intake and body composition and statistical analysis thereof. We hope this guide will facilitate comparisons across studies and minimize spurious interpretations of data. We recommend that division of energy expenditure data by either body weight or lean body weight and that presentation of group effects as histograms should be replaced by plotting individual data and analyzing both group and body-composition effects using analysis of covariance (ANCOVA).