Analysis of covariance ANCOVA

In studies in which there are important prognostic factors accounting for them as part of the analysis can be important in increasing the precision with which treatment effects can be estimated. Such analyses generally involve the use of an analysis of covariance (ANCOVA) type of approach. 

This technique is called analysis of covariance (ANCOVA) and size of the primary tumour is termed the covariate. Taking account of the covariate here has led to a much more powerful analysis than that provided by the simple unpaired t-test. Of course the main reason why we are seeing such an improvement in sensitivity is that the covariate is such a strong predictor of outcome. These improvements will not be quite so great with weaker predictors. 

A11 F values, except total cerebral volume, are for analysis of covariance (ANCOVA) using total cerebral volume as a covariate. Adapted from Kumra et al. (2000)... 

Analysis of Covariance (ANCOVA) for Testing Similarity of Slopes The first step is determining whether or not the degradation rates for all batches behave in a similar fashion. The following hypothesis will be tested ... 

Analysis of covariance (ANCOVA) reverses drug-induced depression of the CNS. Statistical method to determine if two or more related dependent variables exposed to two or more related variables differ significantly from chance. 

In any study, it is important that researchers first establish whether or not their data demonstrate a relationship between POP tissue concentration and tissue lipid levels. This is seldom done, as it is typically assumed that such a relationship must exist for lipophilic contaminants. As is compellingly demonstrated by Hebert and Keenley-side47 in their paper To normalize or not to normalize Fat is the question , such assumptions can lead to lipid normalized POP concentrations that are completely at odds with measured wet weight POP values. Further, since factors other than total lipid (such as differences in lipid class, for example) can affect POP levels in organisms, simple ratios (e.g. ng POP/ng lipid) are often inadequate and may actually increase data variability. In many cases, analysis of covariance (ANCOVA) may prove to be a more appropriate method for lipid normalization of POP concentrations47. 

Analysis of covariance (ANCOVA) employs both analysis of variance (ANOVA) and regression analyses in its procedures. In the present author s previous book Applied Statistical Designs for the Researcher), ANCOVA was not reported mainly because it presented statistical analysis that did not require the use of a computer. For this book, a computer with statistical software is a requirement hence, ANCOVA is discussed here, particularly because many statisticians refer to it as a special t5q>e of regression. 

In regression models, the independent variables are usually quantitative or continuous variables. When the independent variables consist of all qualitative (grouped or categorical) variables, the model is the ANOVA model. When the independent variables consist of both qualitative variables and quantitative variables, the model is the analysis of covariance (ANCOVA) model. The next section illustrates the ANOVA model. 

In order to assess the impact of contextual control variables on the endogenous constructs, an analysis of covariance (ANCOVA) was conducted for each dependent variable (customer orientation, domain-specific innovativeness, opinion leadership) and each model (with and without empathy as predictor variable), which results in six different analyses. All assumptions for conducting ANCOVAs were met (Keselman et al. 1998 Owen and Froman 1998). The models included the original, hypothesized relationships (covarlates) and control variables for firm and department affiliation (fixed factors 3 firms, 5 department groups (product management, sales, marketing, R D, other)). Only direct effects were modeled. The results of the analyses are shown in Table 20. 

By analysis of covariance (ANCOVA), we evaluated the effects of sociodemographic variables on psychometric tests. 

Non-parametric procedures tend to be simple two group comparisons. In particular, a general non-parametric version of analysis of covariance does not exist. So the advantages of ANCOVA, correcting for baseline imbalances, increasing precision, looking for treatment-by-covariate interactions, are essentially lost within a non-parametric framework. 

In Chapter 6 we covered methods for adjusted analyses and analysis of covariance in relation to continuous (ANOVA and ANCOVA) and binary and ordinal data (CMH tests and logistic regression). Similar methods exist for survival data. As with these earlier methods, particularly in relation to binary and ordinal data, there are numerous advantages in accounting for such factors in the analysis. If the randomisation has been stratified, then such factors should be incorporated into the analysis in order to preserve the properties of the resultant p-values. 

Abbreviations ANCOVA, analysis of covariance CR, conditioned response. 

Should treatment-by-covariate interactions be found, either through a test of homogeneity in an adjusted analysis or through ANCOVA, then analysis usually proceeds by looking at treatment differences within subgroups. Plots of treatment effects with associated confidence intervals within these subgroups are useful in this regard. 

The name ANCOVA indicates that covariates are taken into account in the analysis. A covariate is a variable other than the main variable of interest. In the case of our ongoing example of examining decreases in SBP, subjects baseline SBP is likely to be of considerable interest. This technique of ANCOVA can be used to control statistically for baseline differences and to prevent them from skewing the results for the treatment effect. 

Our data were submitted to descriptive analysis in terms of mean values, range and frequency distributions. Since the distributions of lead exposure parameters were skewed, a log transformation of values was applied. Pearson s correlation coefficients were calculated to evaluate the association between neuropsychological impairments and lead exposure. The level of significance assumed was p < 0.05 (one-tailed). To evaluate the influence of potential confounding variables on measured parameters, analysis of variance (ANOVA) was performed. In addition, we estimated the variance of psychometric tests explained by covariates (ALA-D, PbB, PbH, PbT) after regressing out the effects of demographic variables (ANCOVA). 

Allows assessment of prognostic factors. Fitting the ANCOVA model provides coefficients for the covariates and although this is not the primary focus of the analysis, these coefficients and associated confidence intervals provide information on the effect of the baseline covariates on outcome. 

One disadvantage of ANCOVA is that the modelling does involve a number of assumptions and if those assumptions are not valid then the approach could mislead. For example, it is assumed (usually) that the covariates affect outcome in a linear way there is invariably too little information in the data to be able to assess this assumption in any effective way. In contrast, with an adjusted analysis, assumptions about the way in which covariates affect outcome are not made and in that sense it can be seen as a more robust approach. In some regulatory circles adjusted analyses are preferred to ANCOVA for these reasons. 

Remember however that variables used to stratify the randomisation should be included. It is also not usually appropriate to select covariates within ANCOVA models using stepwise (or indeed any other) techniques. The main purpose of the analysis is to compare the treatment groups not to select covariates. 

Using several different statistical methods, for example, an unpaired t-test, an analysis adjusted for centre effects, ANCOVA adjusting for centre and including baseline risk as a covariate, etc., and choosing that method which produces the smallest p-value is another form of multiplicity and is inappropriate. 

The technique of ANCOVA allows more than one covariate to be added to the analysis model. This means that a wide range of variables measured at baseline can potentially be used. While this possibility can initially appear advantageous, it raises a potential concern. Using all of the possible variables is neither practical nor desirable, and so a decision has to be made concerning which ones to include in the ANCOVA model. Importantly, if the covariate is not related to the primary variable of interest, including it in the ANCOVA model is of no benefit. 

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