Analysis of Covariance

Analysis of covariance is a technique for estimating and testing the effects of certain variables when so-called covariables are present the technique has, unfortunately not yet attracted the interest it merits. 

In environmental studies in particular, very often one is not able to adjust these concomitant variables but only to measure them. Typical situations occur during sampling. In principle all sampling circumstances are concomitant variables, for example pH, temperature, salinity in water sampling, or humidity in air sampling, redox potential, and particle size distribution in soil sampling. 

If the undesigned effect of these covariables is not taken into account, the results of analysis of variance may be biased and serious misinterpretation is possible. 

In analysis of covariance the influence of the covariable(s) is basically corrected for by means of a regression model with the covariable(s) as the independent variable(s). Hence analysis of covariance appears as a combination of both regression analysis and analysis of variance. 

It is beyond the scope of this introduction to give further mathematical details see, e.g., [HOCHSTADTER and KAISER, 1988]. Let us rely on the developers of dedicated 

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. 

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. 

The baseline counts tend to differ in various regions of the country—and hence, hospitals—an aspect that can potentially reduce the study s ability to compare the results and, therefore, different infection rates. 

Quantitative variables in the covariance model are termed concomitant variables or covariates. The covariate relationship is intended to provide reduction in error. If it does not, a covariance model should be replaced by an ANOVA, because one is losing degrees of freedom using covariates. 

The best way to assure that the covariate is related to the dependent variable y is to have familiarity with the intended covariates before the study begins. 

Another technique which can often be used to advantage by food research workers is that known as analysis of covariance. Essentially, this technique is an extension of analysis of variance methods to include cases where measurements are obtained on two or more characteristics from each experimental unit. A moment s thought should convince us that this is, then, only a scheme for combining regression and analysis of variance into a single technique. Clearly, this device, where applicable, should permit a more discriminating analysis of the sample data. 

A covariance analysis may be performed on appropriate data for any of the standard designs that is, completely randomized, randomized complete block, Latin square, split plot, and so on. The technique is the 

An example of such an analysis is provided by Peterson, Tucker et al. (1951), who studied moisture content of turnip greens with a view to examining the variation due to leaf-to-leaf differences, plant-to-plant differences, and time held prior to making the moisture determinations. A Latin square design was used. The data on moisture are shown in Tables LV and LVI. The data on leaf weights are not given, except as 

Moisture Content of Turnip Greens, as Affected by Drsring and Size Increasing leaf size 

Moisture Content of Turnip Greens as Affected by Holding Time and Leaf Size  

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There are instances where it is important to know if a given regression line is linear. For example, simple competitive antagonism should yield a linear Schild regression (see Chapter 6). A statistical method used to assess whether or not a regression is linear utilizes analysis of covariance. A prerequisite to this approach is that there... 

Polynomial regression with indicator variables is another recommended statistical method for analysis of fish-mercury data. This procedure, described by Tremblay et al. (1998), allows rigorous statistical comparison of mercury-to-length relations among years and is considered superior to simple hnear regression and analysis of covariance for analysis of data on mercury-length relations in fish. 

It should be noted that the analysis of covariance assumes a linear relationship between the standard value and the variable value, for a given treatment, and assumes independence between an effect of a given treatment and the value of the standard. The relationship between these may be more complex in many drift studies. Therefore, the standard treatment should be near the median of the variables being investigated. The only alternative involves a great deal of patience and time to obtain very similar... 

The above analysis establishes that there was no significant sex difference, as indicated by the tail probabilities for sex (p = 0.2667) and sexxtreatment interaction (p = 0.9784). There was also some indication that there may have been some treatment effect across the treatment groups in both sexes (p = 0.0559). Examination of the variate means indicated that both sexes seemed to have lower means than their respective controls. The picture was clouded by the fact that there was a similar slightly lower tendency, though not very consistent, in the covariate means as well. Under this circumstance, it is more appropriate to take both the covariate and the variate into any optimal analysis. Table 16.19 shows an analysis of covariance for the factorial model. 

Homoscedasticity is an important assumption for Student s /-test, analysis of variance, and analysis of covariance. 

Spence, S., Soper, K., Hoe, C.M. and Coleman, J. (1998) The heart rate-corrected QT interval of conscious beagle dogs a formula based on analysis of covariance. Toxicological Sciences, 45, 247-258. 

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. 

Fig. 8.10 Illustration of analysis of covariance using data from Rikkers et al., 1978.

Any statistical procedure that defines a set of experimental data in terms of a mathematical function. The most common curve-fitting protocol is the least squares method (also known as analysis of covariance). 

As we shall see later the data type to a large extent determines the class of statistical tests that we undertake. Commonly for continuous data we use the t-tests and their extensions analysis of variance and analysis of covariance. For binary, categorical and ordinal data we use the class of chi-square tests (Pearson chi-square for categorical data and the Mantel-Haenszel chi-square for ordinal data) and their extension, logistic regression. 

We will say quite a bit later, in the chapter on adjusted analysis and analysis of covariance (Chapter 6) about additional improvements to this kind of analysis that increase sensitivity further and also avoid the so-called potential problem of regression towards the mean. For the moment though, it is test 3 that is the best way to compare the treatments. 

As the number of baseline factors increases, however, this approach becomes a little unwieldy. The method of analysis of covariance is a more general methodology that can deal with this increase in complexity and which can also give improved ways of exploring interactions. We will develop this methodology later in the chapter. 

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. 

Analysis of covariance offers a number of advantages over simple two treatment group comparisons ... 

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