ANCOVA

Real fancy (compared to univariate ANOVA/ANCOVA techniques). 

As the ANCOVA table indicates, there was definite significant treatment effect (p = 0.0104), but this effect was not sex specific because there was no significant... 

In other words, we have gained about 2.3-fold precision by ANCOVA over ANOVA in resolving treatment effect. 

The underlying assumptions for ANCOVA are fairly rigid and restrictive. 

This calculation is also equivalent to r = sample covariance/ (SxSy), as was seen earlier under 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. 

To illustrate ANCOVA we consider the data in Table 8.7 taken from a study reported by Rikkers et al7 that compared the effect of two types... 

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. 

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. 

Adjusted analyses presented earlier in this chapter also share some of these advantages and provide improvements in efficiency, can also account for baseline imbalances and allow the evaluation of the homogeneity of the treatment effect. On this final point, however, adjusted analyses are less able to identify the nature of those interactions. With ANCOVA it is possible to say which particular covariates are causing such interactions. A further point to note here and, as mentioned... 

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. 

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. 

It probably comes as no surprise to learn that there are mathematical connections between ANOVA and ANCOVA for continuous data. 

Suppose that we have just two centres (and two treatment groups). If we define binary indicators, say z and x, to denote treatment group and centre, respectively, then including z and x in an ANCOVA is identical mathematically to the corresponding two-way ANOVA. This connection is true more generally. If we were now to add more centres (say four in total) to the ANOVA then defining binary indicators to uniquely define these centres Xj = 1 for a patient in centre 1, Xj = 1 for a patient in centre 2, X3 = 1 for a patient in centre 3 with 0 values otherwise, then ANCOVA with terms z, Xj, Xj and X3 would be mathematically the same as ANOVA. We would obtain the same p-values, (adjusted) estimates of treatment effect, confidence intervals etc. 

We can see, therefore, that ANCOVA is a very powerful technique that mathematically incorporates ANOVA. 

This link applies also to the p-value from the unpaired t-test and the confidence interval for p, the mean difference between the treatments, and in addition extends to adjusted analyses including ANOVA and ANCOVA and similarly for regression. For example, if the test for the slope b of the regression line gives a significant p-value (at the 5 per cent level) then the 95 per cent confidence interval for the slope will not contain zero and vice versa. 

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 t-tests and their extensions ANOVA, ANCOVA and regression all make assumptions about the distribution of the data in the background populations. If these assumptions are not appropriate then strictly speaking the p-values coming out of those tests together with the associated confidence intervals are not valid. 

B observations in each of several centres) and also with more complex structures which form the basis of ANCOVA and regression. For example, in regression the assumption of normality applies to the vertical differences between each patient s observation y and the value of y on the underlying straight line that describes the relationship between x andy. We therefore look for the normality of the residuals the vertical differences between each observation and the corresponding value on the fitted line. 

In the paired t-test setting it is the normality of the differences (response on A — response on B) that is required for the validity of the test. The log transformation on the original data can sometimes be effective in this case in recovering normality for these differences. In other settings, such as ANOVA, ANCOVA and regression, log transforming the outcome variable is always worth trying, where this is a strictly positive quantity, as an initial attempt to recover normality. 

Extending non-parametric tests to more complex settings, such as regression, ANOVA and ANCOVA is not straightforward and this is one aspect of these methods that limits their usefulness. 

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. 

It is this specific feature that has led to the development of special methods to deal with data of this kind. If censoring were not present then we would probably just takes logs of the patient survival times and undertake the unpaired t-test or its extension ANCOVA to compare our treatments. Note that the survival times, by definition, are always positive and frequently the distribution is positively skewed so taking logs would often be successful in recovering normality. 

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. 

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