Response homoscedastic

In view of Problem 3.17 and what ever insight you might have on the shapes of response surfaces in general, what fraction of response surfaces do you think might be homoscedastic Why Is it possible to consider any local region of the response surface shown in Figure 3.7 to be approximately homoscedastic ... 

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 ... 

A suitable response variable is selected. This variable should be chosen such that it has a homoscedastical error and results in simple models. For reasons stated below is chosen (see Section 6.2.10). 

In the previous sections it has been stipulated that there are several response variables which can be modeled. The success of the optimization procedure depends on the selection of the response variable(s). There are several criteria which can be used to select a response variable [12,17]. The response variable should have a homoscedastical error structure and have to change continuously and smoothly. Both experimental data and chromatographic theory can be used to check these properties. 

From chromatographic theory [2] it is clear that the R value should result in simple models. For this reason it is preferred over, the k or the Rj. These latter response values can be calculated from predicted R values. It is more difficult to determine the error structure of the R . It is believed however that logarithmic transformation of the k values should result in homoscedastical error structures [3]. 

If there is no theory available to determine a suitable transformation, statistical methods can be used to determine a transformation. The Box-Cox transformation [18] is a common approach to determine if a transformation of a response is needed. With the Box-Cox transformation the response, y, is taken to different powers A, (e.g. -2transformed response can be fitted by a predefined (simple) model. Both an optimal value and a confidence interval for A can be estimated. The transformation which results in the lowest value for the residual variance is the optimal value and should give a combination of a homoscedastical error structure and be suitable for the predefined model. When A=0 the trans-... 

The most widely used type of simultaneous assay is the one in which the response has a homoscedastic linear regression on a logarithmic scale. Homoscedasticity means that the variance of all experimental groups is the same. For such an assay, the condition of similarity requires that the straight lines of the standard and the samples should be parallel. Otherwise, the condition of similarity between the sample and standard is not established, that is, it would not be valid to assume that dilutions of one behave the same as dilutions of the other, which is the assay s underlying principle. 

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... 

Assumes a homoscedastic error structure (common or homogeneous variance regardless of response). The random error is the same for all observations. 

The error variance is constant for all the investigated range of X, and is equal to a certain value This hypothesis is often made by stating that the observed responses are homoscedastic. 

Equations [8.25-8.29] are the statistical quantities relevant to a simple linear regression (Equation [8.19a]), e.g., as in a calibration experiment for an analytical method that determines instrument response (Y) as a function of analyte concentration or amount (x) note again that these equations are appropriate for homoscedastic data for... 

---