Testing for heteroscedasticity

Another quick and dirty test for heteroscedasticity suggested by Carroll and Ruppert (1988) is to compute the Spearman rank correlation coefficient between absolute studentized residuals and predicted values. If Spearman s correlation coefficient is statistically significant, this is indicative of increasing variance, but in no manner should the degree of correlation be taken as a measure of the degree of heteroscedasticity. They also suggest that a further refinement to any residual plot would be to overlay a nonparametric smoothed curve, such as a LOESS or kernel fit. 

As a more formal test for heteroscedasticity for linear models, Goldfeld and Quandt (1965) proposed the following method  

Omit c observations where c is specified a priori and divide the remaining n-c observations into two groups of (n-c)/2  

Fit separate OLS fits to the two datasets and compute the residual sum of squares for each group and 

1 Caution should be used when plotting the log of the squared residuals because small squared residuals near zero will result in inflated log squared residuals. For this reason it may be useful to trim the smallest squared residuals prior to plotting. 

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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. 

In summary, the Type I error rate from using the LRT to test for the inclusion of a covariate in a model was inflated when the data were heteroscedastic and an inappropriate estimation method was used. Type I error rates with FOCE-I were in general near nominal values under most conditions studied and suggest that in most cases FOCE-I should be the estimation method of choice. In contrast, Type I error rates with FO-approximation and FOCE were very dependent on and sensitive to many factors, including number of samples per subject, number of subjects, and how the residual error was defined. The combination of high residual variability with sparse sampling was a particularly disastrous combination using... 

The procedure for testing goodness of fit for heteroscedastic data also requires modification of that used in the homoscedastic case (Equations [8.33-8.38]). For heteroscedastic data the following equations apply (Analytical Methods Committee 1994) ... 

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 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. 

Finally, we compute Glesjer s test statistics for the three models discussed in Section 14.3.5. We regress e2, e, and log e on 1, Xh and X2. We use the White estimator for the covariance matrix of the parameter estimates in these regressions as there is ample evidence now that the disturbances are heteroscedastic related to X2. To compute the Wald statistic, we need the two slope coefficients, which we denote q, and the 2x2 submatrix of the 3x3 covariance matrix of the coefficients, which we denote Vq. The statistic is W - q Vq 1q. For the three regressions, the values are 4.13, 6.51, and 6.60, respectively. The critical value from the chi-squared distribution with 2 degrees of freedom is 5.99, so the second and third are statistically significant while the first is not. 

Agterdenbos stated that in some circumstanees (he studied ICP data) the standard deviation of the measurements will probably be independent of the concentration values (homoscedasticity) as long as the latter are lower than 22 LD (LD = traditional limit of detection) and that, on the contrary, concentrations higher than 50 LD will lead to heteroscedasticity (an almost constant relative standard deviation). " " This may serve as a rough rule of thumb, but should be tested experimentally for particular instances and requirements. 

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