Response heteroscedastic

It should be stressed that, depending on the size of the errors we are willing to tolerate in the predictions made from the regression equation, it might be that neither lack of fit nor response heteroscedasticity have any practical importance. In any case, it is well to be prepared to treat... 

Wavelength accuracy and reproducibility reproducibility of sensitivity/response factors Heteroscedasticity... 

Replication is often included in central composite designs. If the response surface is thought to be reasonably homoscedastic, only one of the factor combinations (commonly the center point) need be replicated, usually three or four times to provide sufficient degrees of freedom for s. If the response surface is thought to be heteroscedastic, the replicates can be spread over the response surface to obtain an average purely experimental uncertainty. 

Is the response surface shown in Figure 12.6 homoscedastic or heteroscedastic ... 

Results for which the mean values of the samples (treatments) are different, but which have the same variance, is said to be homoscedastic, as opposed to having different variance, which is said to be heteroscedastic. Thus, in the case of homoscedastic variation, the variance is constant with increasing mean response, whereas with heteroscedastic variation the variance increases with the mean response. ANOVA is quite sensitive to... 

These and most other equations developed by statisticians assume that the experimental error is the same over the entire response surface there is no satisfactory agreement for how to incorporate heteroscedastic errors. Note that there are several different equations in the literature according to the specific aims of the confidence interval calculations, but for brevity we introduce only two which can be generally applied to most situations. 

Finally it is often useful to be able estimate the experimental error (as discussed in Section 2.2.2), and one method is to perform extra replicates (typically five) in the centre. Obviously other approaches to replication are possible, but it is usual to replicate in the centre and assume that the error is the same throughout the response surface. If there are any overriding reasons to assume that heteroscedasticity of errors has an important role, replication could be performed at the star or factorial points. However, much of experimental design is based on classical statistics where there is no real detailed information about error distributions over an experimental domain, or at least obtaining such information would be unnecessarily laborious. 

The response error (A/ ) is not constant (heteroscedastic) so that the highest precision does not necessarily coincide with the highest sensitivity (Section 15.4). 

A second problem is that sometimes a nonlinear function cannot be found that adequately fits the response data. In this case, it may be possible to transform the independent variable such that the model becomes linear and a suitable regression function can easily be found. Another often used approach is to transform the dependent variable such that the model becomes linear. This approach cannot be advocated any longer since often times the transformation, while creating a linear regression model, often leads to heteroscedasticity and non-normality in the residual. In a sense, the analyst is robbing Peter to pay Paul, so to speak. 

If, however, the system is heteroscedastic and the standard deviation does indeed vary within the domain so that certain experimental conditions are known to give less precise results, this must be taken into account when calculating the model coefficients. One means of doing this is by weighting. Each experiment i is assigned a weight w inversely proportional to the variance of the response at that point. Equation 4.5, for least squares estimation of the model coefficients, may thus be rewritten as ... 

Assumes a heteroscedastic error structure (variance changes widi the response). The random error is assumed to be some function of the observed data (i.e., if Wi = 1 /, the variance is... 

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