Homoscedastic data

After outliers have been purged from the data and a model has been evaluated visually and/or by, e.g. residual plots, the model fit should also be tested by appropriate statistical methods [2, 6, 9, 10, 14], The fit of unweighted regression models (homoscedastic data) can be tested by the ANOVA lack-of-fit test [6, 9]. A detailed discussion of alternative statistical tests for both unweighted and weighted calibration models can be found in Ref. [16]. The widespread practice to evaluate a calibration model via its coefficients of correlation or determination is not acceptable from a statistical point of view [9]. 

Figure 2 A regression line through mean values of homoscedastic data otd" = SSjyKn - 2) = liy, - ydVin - 2)...

Half-width, 15 Hammering distance, 140 Heaviside function, 144 Hidden layers, in ANN, 151 Homoscedastic data, 159... 

Heteroscedastic data The variance of data in a calibration is not independent of their magnitude. Usually this is seen as an increase in variance with increasing concentration (e.g., when the relative standard deviation is constant for a calibration). (Section 5.3.1) Homoscedastic data The variance of data in a calibration is independent of their magnitude (i.e., the standard deviation is constant). (Section 5.3.1)... 

Figure 6.2 Regression line through mean values of homoscedastic data...

Figure 2 Examples of plots of residuals of calibration graphs. Top normally distributed residuals with constant variance (homoscedastic data). Middle data showing an increasing variance with increasing concentration (heteroscedastic data). Bottom residuai piot of a curved calibration relation.

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... 

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... 

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).

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 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). 

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. 

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 ... 

A fully parametric model/estimator provides consistent, efficient, and comparatively precise results. The semiparametric model/estimator, by comparison, is relatively less precise in general terms. But, the payoff to this imprecision is that the semiparametric formulation is more likely to be robust to failures of the assumptions of the parametric model. Consider, for example, the binary probit model of Chapter 21, which makes a strong assumption of normality and homoscedasticity. If the assumptions are coirect, the probit estimator is the most efficient use of the data. However, if the normality assumption or the homoscedasticity assumption are incorrect, then the probit estimator becomes inconsistent in an unknown fashion. Lewbel s semiparametric estimator for the binary choice model, in contrast, is not very precise in comparison to the probit model. But, it will remain consistent if the normality assumption is violated, and it is even robust to certain kinds of heteroscedasticity. 

Homoscedastic noise. This is the simplest to envisage. The features of the noise, normally file mean and standard deviation, remain constant over the entire data series. The most common type of noise is given by a normal distribution, with mean zero, and standard deviation dependent on the instrument used. In most real world situations, there are several sources of instrumental noise, but a combination of different symmetric noise distributions often tends towards a normal distribution. Hence this is a good approximation in the absence of more detailed knowledge of a system. 

The calibration lines are usually calculated by ordinary least squares (OLS) regression. A precondition for the application of OLS regression is that the variance of the signal should be independent from the signal itself. This property is also called homoscedasticity. When this is not the case, one is dealing with a heteroscedastic situation. The heteroscedastic property of data can be observed by reviewing a graph that displays residuals. 

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. 

The assumption of homoscedasticity means that the residual variability should be constant over all available data dimensions (predictions, covariates, time, etc). If we observe heteroscedasticity, then we need to change the residual error model to account for this. In practice, this means that we should weight the data differently by using a different model for the residual variability. 

Use ANOYA if you want to know if there is significant difference among a number of instances of a factor. Always use ANOVA for more than one factor. ANOVA data must be normally distributed and homoscedastic. Use a Mest for testing pairs of instances. The data must be normally distributed but need not be homoscedastic. (Sections 3.8, 4.2)... 

The data are known as homoscedastic, which means that the errors in y are independent of the concentration. Data for which the uncertainty, for example, grows with the concentration are heteroscedastic data. 

Sometimes it is useful to transform a nonlinear model into a linear one when the distribution of error terms is approximately normal and homoscedastic. Such a case might be when a suitable nonlinear function cannot be found to model the data. One might then try to change the relationship between x and Y so that a model can be found. One way to do this is to change the model... 

At first glance, the data suggests an Emax model might best describe the increase in percent inhibition with increasing concentration, eventually plateauing at a maximal value of 70% inhibition. See Mager, Wyska, and Jusko (2003) for a useful review of pharmacodynamic models. The first model examined was an Emax model with additive (homoscedastic) residual error... 

When the variance model is heteroscedastic, the algorithm for bootstrapping the residuals will not be valid because the bootstrapped data set might not have the same variance model as the original data. In fact, more than likely, bootstrapping heteroscedastic residuals will lead to a homoscedastic model. Heteroscedas-ticity is not a problem for the random case because heteroscedasticity will be preserved after bootstrapping. In the heteroscedastic case, the modified residuals need to be corrected for their variance so that Eq. (A. 102) becomes... 

In determining a mathematical model, whether by linear combinations or by multilinear regression, we have assumed the standard deviation of random experimental error to be (approximately) constant (homoscedastic) over the experimental region. Mathematical models were fitted to the data and their statistical significance or that of their coefficients was calculated on the basis of this constant experimental variance. Now the standard deviation is often approximately constant. All experiments may then be assumed equally reliable and so their usefulness depends solely on their positions within the domain. 

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