Homoscedastic analysis

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

Most of the statistical tests we use (t test, F test, analysis of variance, multiple regression analysis) are predicated on the assumption that the variation being studied is the same, regardless of whether the property averages 10 or 50 or 75,000. For example, a homoscedastic variable might show variation as follows (several measurements on the same sample )... 

In all types of data analysis there are assumptions made. In a parametric approach, like the one in NONMEM, many assumptions concern the handling of the residual error (9,12) and, in a sense, the validity of the whole analysis rests on the degree to which we have accounted for the residual variability appropriately. The two most important assumptions in this respect are (a) that the residual variability is homoscedastic and (b) that the residuals are symmetrically distributed. 

If the variance model is not homoscedastic, a suitable transformation needs to be found prior to performing the algorithm. Carroll et al. (1995) stress that the extrapolation step should be approached as any other modeling problem, with attention paid to the adequacy of the extrapolant based on theoretical considerations, residual analysis, and possible use of linearizing transformations and that extrapolation is risky in general even when model diagnostics fail to indicate problems. ... 

The residuals may be distributed in various different ways. First of all they may be scattered more or less symmetrically about zero. This dispersion can be described by a standard deviation of random experimental error. If this is (approximately) constant over the experimental region the system is homoscedastic, as has been assumed up to now. However the analysis of residuals may show that the standard deviation varies within the domain, and the system is heteroscedastic. On the other hand it may reveal systematic errors where the residuals are not distributed symmetrically about zero, but show trends which indicate model inadequacy. 

Of course the analogy is not perfect, since in a meta-analysis the weights depend on the observed variances from trial to trial and the weights are, therefore, random variables. In a linear, model, however, the assumption of homoscedasticity is imposed. This means that the weights are fixed. This point is taken up in the next section. 

Changing the regression method by using either weighted, least-squares analysis, if the variance is not homoscedastic, or nonlinear, least-squares analysis to determine the parameter values. 

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