Homoscedasticity

Profiles in the Presence of Homoscedastic or Heteroscedastic Noise, Anal. Chem. 66, 1994, 43-51. 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

The errors are only or essentially in the measured values y as the dependent variable (bsx sy) and in addition, the errors sy are constant in the several calibration points (Homoscedasticity) ... 

The several variants deriving from the items 1 to 4 are represented in the flow sheet given in Fig. 6.6. Common calibration by Gaussian least squares estimation (OLS) can only be applied if the measured values are independent and normal-distributed, free from outliers and leverage points and are characterized by homoscedastic errors. Additionally, the error of the values in the analytical quantity x (measurand) must be negligible compared with the errors of the measured values y. 

Commercial software packages are usually able to represent graphically the residual errors (deviations) of a given calibration model which can be examined visually. Typical plots as shown in Fig. 6.8 may give information on the character of the residuals and therefore on the tests that have to be carried out, such as randomness, normality, linearity, homoscedasticity, etc. 

Homoscedasticity. Unequal variances are recognizable from residual plots as in Fig. 6.8c where frequently ey is a function of x in the given trumpet-like form. In such a case, the test of homoscedasticity can be carried out in a simple way by means of the Hartley test (Fmax test), Fmax = smax/5min> see Sect. 4.3.4 (1). 

As noted above, the variations in the data representing the error must meet the usual conditions for statistical validity they must be random and statistically independent, and it is highly desirable that they be homoscedastic and Normally distributed. The data should be a representative sampling of the populations that the experiment is supposed... 

A detailed treatment of linearity evaluation is beyond the scope of this present book but a few general points are made below. It is important to establish the homogeneity of the variance ( homoscedasticity ) of the method across the working range. This can be done by carrying out ten replicate measurements at the extreme ends of the range. The variance of each set is calculated and a statistical test (F test) carried out to check if these two variances are statistically significantly different [9]. 

Bartlett s test does not test for normality, but rather homogeneity of variance (also called equality of variances or homoscedasticity). 

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

The / -test (covered in the next section) is actually a test for the two sample (that is, control and one test group) case of homoscedasticity. Bartlett s is designed for three or more samples. 

Figure 2 Example of graphical presentation of a % dissolved vs. time simulated data set obtained by using Eq. (2) (W0 = 100, 6 = 1, c = 3), assuming a specific sampling scheme (indicated in the text) and perturbing the data with homoscedastic error with a mean of 0 and SD = 4 (dotted line) and the corresponding fitted line obtained by fitting Eq. (2) to the specific data set (continuous line).

Both assumptions are mainly needed for constructing confidence intervals and tests for the regression parameters, as well as for prediction intervals for new observations in x. The assumption of normal distribution additionally helps avoid skewness and outliers, mean 0 guarantees a linear relationship. The constant variance, also called homoscedasticity, is also needed for inference (confidence intervals and tests). This assumption would be violated if the variance of y (which is equal to the residual variance a2, see below) is dependent on the value of x, a situation called heteroscedasticity, see Figure 4.8. 

It should be emphasized that if the concentration range is greater than one order of magnitude (long-range calibration), violations of the constant variance assumption of OLS are frequent. Especially, homoscedasticity is rarely found hence, a more general model needs to be applied. 

To model the relationship between PLA and PLR, we used each of these in ordinary least squares (OLS) multiple regression to explore the relationship between the dependent variables Mean PLR or Mean PLA and the independent variables (Berry and Feldman, 1985).OLS regression was used because data satisfied OLS assumptions for the model as the best linear unbiased estimator (BLUE). Distribution of errors (residuals) is normal, they are uncorrelated with each other, and homoscedastic (constant variance among residuals), with the mean of 0. We also analyzed predicted values plotted against residuals, as they are a better indicator of non-normality in aggregated data, and found them also to be homoscedastic and independent of one other. 

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 ... 

We will also assume that we have a prior estimate of for the system under investigation and that the variance is homoscedastic (see Section 3.3). Our reason for assuming the availability of an estimate of is to obviate the need for replication in the experimental design so that the effect of the location of the experiments in factor space can be discussed by itself. 

A similar effect is observed when non-interacting factors are not controlled. However, uncontrolled non-interacting factors usually produce homoscedastic noise (see Figure 3.6) uncontrolled interacting factors often produce heteroscedastic noise (see Figure 3.7), as they do in the present example. 

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 ... 

In Eq. 13.15, the squared standard deviations (variances) act as weights of the squared residuals. The standard deviations of the measurements are usually not known, and therefore an arbitrary choice is necessary. It should be stressed that this choice may have a large influence of the final best set of parameters. The scheme for appropriate weighting and, if appropriate, transformation of data (for example logarithmic transformation to fulfil the requirement of homoscedastic variance) should be based on reasonable assumptions with respect to the error distribution in the data, for example as obtained during validation of the plasma concentration assay. The choice should be checked afterwards, according to the procedures for the evaluation of goodness-of-fit (Section 13.2.8.5). 

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-... 

A good example of the effect of transformations on the variation of data around some given true value is found in viscosity measurements Here the variability is definitely related to the level of viscosity However, the logarithm of the viscosity is homoscedastic, as can be seen below. 

If we keep our designs orthogonal and our data homoscedastic, our decisions will be uniform no matter which way we look at them ... 

Homoscedastic having a variance which i ipdependent of the magnitude of the observations. ... 

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