Model ANCOVA

The b in this model is as previously, but now a = and c = 2 fli- We refer to this in mathematics as a re-parameterisation, don t be put off by it The hypothesis Hq Ui = U2 is now replaced by the hypothesis Hj c = 0. None of this changes the analysis in any sense, it is just a more convenient way to write down the model and will be useful later when we bring together the ideas of ANOVA and 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. 

Equally important is the use of suitable quantitative statistical analyses, including more complicated models, because they can hold certain variables constant, control for artifacts, and provide supplementary information. Whatever statistics are used, they should be explicitly described in sufficient detail so the reader knows exactly what was done and can make a judgment about their appropriateness. For example, there are many different types of analyses of variance (ANOVA), analyses of covariance (ANCOVA), or multivariate analyses of variance (MANOVA), and some are not appropriate to the task at hand. If only the results of an ANOVA with a p < 0.001 are provided, the reader should be justifiably dubious, because this model may not be proper ( p is an estimate of the probability that the results occurred by chance). Sufficient details are required to clarify which model was used, because the p value may be invalid with an inappropriate model. 

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. 

The best approach is to choose a limited number of covariates that are all related to the outcome variable. The question then becomes How does one decide on this limited group of related covariates One strategy would be to look at all baseline values at the end of the trial and to include those that seem very different between the groups in the ANCOVA model. However, this strategy is not ideal. The safest approach is to select prognostic covariates before the trial and identify... 

ANCOVA, in theory, is fairly straightforward. The statistical model includes qualitative independent factors as in ANOVA, say, three product formulations, A, B, and C, with corresponding quantitative response variables (Table 11.1). This is the ANOVA portion. 

Note that we are using I as the symbol in place of z. The ANCOVA model can be computed in ANOVA terms or as a regression. We look at both the approaches. 

Fc is the calculated ANCOVA value for treatments. If done by regression, both full and reduced models are computed. 

We perform a single-factor ANCOVA in two ways. This is because different computer software packages do it differently. Note, though, that one can also perform ANCOVA by using two regression analyses one for the full model, and the other for a reduced model. 

Letus begin with a very simple, yet often encountered evaluation comparing effects of three different topical antimicrobial products on two distinct groups— males and females—and how an ANCOVA model ultimately was used. 

Recall that, in ANCOVA, the model has an ANOVA portion and a regression portiOTi. The covariant is the regression portion. Hence, we have a b or slope for the covariate, which is b = 0.9733 (Table 11.8). This, in itself, can be used to determine if the covariate is significant in reducing overall error. If the b value is zero, then the use of a covariate is not of value in reducing error, and ANOVA would probably be a better application. A 95% confidence interval for the (3 value can be determined. 

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. 

One type of test that incorporates all of these (t-test, ANOVA, ANCOVA, MANOVA, and MANCOVA and also ordinary linear regression) is the general linear model. This method is included in several statistical software packages and is a convenient tool for analyzing many different types of data. 

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. 

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