Heteroscedasticity

This is a heteroscedastic regression model in which the matrix X is a column of ones. The efficient 

For the model in the previous exercise, what is the probability limit of s1 = (l/( -l))Z (y - y )2 Note that this is the least squares estimate of the residual variance. It is also n times the conventional estimator of the variance of the OLS estimator, Est.Var[ y ]=. vTX X) 1 = s1 In. How does this compare to the true value you found in part (b) of Exercise 1 Does the conventional estimator produce the coirect estimate of the true asymptotic variance of the least squares estimator  

The appropriate variance of the least squares estimator is Var[ y ]= (o1/n1 fLlx2. which is, of course, precisely what we have been analyzing above. It follows that the conventional estimator of the variance of the OLS estimator in this model is an appropriate estimator of the true variance of the least squares estimator. This follows from the fact that the regressor in the model, i, is unrelated to the source of heteroscedasticity, as discussed in the text. 

Two samples of 50 observations each produce the following moment matrices (In each case, X is a constant and one variable.) 

The parameter estimates are computed directly using the results of Chapter 6. 

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

This method will be fully exploited later but cannot be used here. It is to be noted that when it is attempted to estimate the standard deviation of the different available values at the same temperature the resulting standard deviations are not constant and are strongly correlated to the average of the values. In other words, the higher the average, the higher the standard deviations. This is the phenomenon of heteroscedasticity. It does not provide any possibility of estimating uncertainty of measurement. 

The diagram below shows in graphical form the same property for alcohols. The two lines joining the four extreme values show the funnel-shaped configuration of values. It is the standard way of spotting the values of heteroscedasticity. 

In these conditions it is impossible to estimate the uncertainty of measurement that would enable unrealistic experimental points to be noted. It would be interesting to analyse the origin of this heteroscedasticity.,... 

When analysing the standard deviation value, which measures the dispersion of measurements, the effect of heteroscedasticity, already discussed in connection with the measurement of vapour pressure, is noted ie the dependency between standard deviation and average (the higher the average, the greater the dispersion of measurements). One way to make this unfortunate property obvious when it comes to analysing data is to calculate the coefficient of variation for each distribution (C 0- If it is more or less constant, there is heteroscedasticity. 

In order to apply RBL or GRAFA successfully some attention has to be paid to the quality of the data. Like any other multivariate technique, the results obtained by RBL and GRAFA are affected by non-linearity of the data and heteroscedast-icity of the noise. By both phenomena the rank of the data matrix is higher than the number of species present in the sample. This has been demonstrated on the PCA results obtained for an anthracene standard solution eluted and detected by three different brands of diode array detectors [37]. In all three cases significant second eigenvalues were obtained and structure is seen in the second principal component. 

In case that the measuring errors sy vary and therefore heteroscedasticity must be assumed, the original LS criterion (Eq. 6.14) must be applied and the model of weighted least squares (WLS) results from this criterion as will be shown in Sect. 6.2.3. 

Fig. 6.8. Typical plots of residual deviations random scattering (a), systematic deviations indicating nonlinearity (b), and trumpet-like form of heteroscedasticity (c)...

Bartlett s is very sensitive to departures from normality. As a result, a finding of a significant chi square value in Bartlett s may indicate nonnormality rather than heteroscedasticity. Such a finding can be brought about by outliers, and the sensitivity to such erroneous findings is extreme with small sample sizes. 

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. 

FIGURE 4.8 Examples of residual plots from linear regression. In the upper left plot, the residuals are randomly scattered around 0 (eventually normally distributed) and fulfill a requirement of OLS. The upper right plot shows heteroscedasticity because the residuals increase with y (and thus they also depend on x). The lower plot indicates a nonlinear relationship between x and y. 

Decision diamond Are the classical assumptions for fitting regression lines met N0 Clearly the measurements at the different x-levels differ in their variability. This can be shown by using the F-test. Another method is outlined in another chapter of this text (10). In this case weighted least squares will resolve the problem of heteroscedasticity or unequal variance across the graph. I have chosen weights of 1, 1, 0.1, 0.01 and 0.01 for the resolution of this problem. 

The solution to the problem of non-constant variance (or heteroscedasticity) rests in several suggestions. The simplest is to limit the range of the graph ( 1 ). The range, however, would be so small that it would be ineffective to use it practically. 

The methods used were those of Mitchell ( 1 ), Kurtz, Rosenberger, and Tamayo ( 2 ), and Wegscheider T ) Mitchell accounted for heteroscedastic error variance by using weighted least squares regression. Mitchell fitted a curve either to all or part of the calibration range, using either a linear or a quadratic model. Kurtz, et al., achieved constant variance by a... 

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

The ANOVA test, which is also recommended by the Analytical Methods Committee of The Royal Society of Chemistry (UK), can be generalized to other regression models, and it can be extended to handle heteroscedasticity. For a more detailed prescription and the extension of the test see further reading. 

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 ... 

A graphical display of the residuals tells us a lot about our data. They should be normally distributed (top left). If the variances increase with the concentration, we have inhomogeneous variances, called heteroscedasticity (bottom left). The consequences are discussed in the next slide. If we have a linear trend in the residuals, we probably used the wrong approach or we have a calculation error in our procedure (top right). Non-linearity of data deliver the situation described on bottom right, if we nevertheless use the linear function. 

Heteroscedastic having a variance which changes with the magnitude of the observations. 

Homoscedasticity can be of real concern in fields other than abnormal psychology. Some deviates are known to be homoscedastic, and others are definitely heteroscedastic. Statisticians, in general, much prefer homoscedastic deviates to those other kind/... 

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