Unequal Error Variances

A simple transformation procedure can often remove the unequal scatter in e. But this is not the only procedure available weighted least-squares regression can also be useful. 

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There is a great body of literature on transformations of the dependent variable since these type of transformations are often used to account for unequal error variances and/or nonnormality of the error distribution. Useful references for this type of transformation are Carroll and Ruppert (1988, 1984). 

We have discussed transforming y and x values to linearize them, as well as removing effects of serial correlation. But transformations can also be valuable in eliminating nonconstant error variances. Unequal error variances are often easily determined by a residual plot. For a simple linear regression, y = ho + hi JCi e, the residual plot will appear similar to Figure 8.8, if a constant variance is present. 

With unequal variances we cannot speak of an "overall standard error". In that case s2 computed by (3.11) yields an unbiased estimate of the constant a2 in the weighting coefficients. Therefore, s 2 = s2/wi is an unbiased estimate of the error variance. If we have a different independent estimate of the same variance, for example computed from the replicates at the value x of the independent variable, then our assumptions can be checked by an F-test, involving the ratio of the two estimates, see e.g. Himmelblau (ref. 5). Though this is the best way to measure the goodness-of-fit, it requires additional information (i.e., replicates), not always available. 

MI Ml E S>. ° i= - U )2 del S, where s = su = - fik)(yu - Hfl) experimental errors interdependent with known covariance matrix are elements of the inverted matrix 53 1 experimental errors interdependent with unequal known variances experimental errors independent with equal (known or unknown) variances experimental errors interdependent with unknown covariance matrix no missing observations... 

The variables X , and in Table 3.10 are the sample mean, sample variance, and number of data points of the (th sample, respectively. In the calculation of v in the case of unequal population variances, the result should be rounded to the nearest integer. The t distribution can approximate the difference of sample means for populations with unequal population. The t variable is obtained via Table 3.3. Table 3.10 is also important one should not assume that the population variances are identical without evidence. When in doubt, assume that the populations have differing variances, as this may provide a wider confidence interval (and a larger margin of error in drawing conclusions). 

DeShon, R. P. and Alexander, R. A. (1996], "Alternative procedures for testing regression slope homogeneity when group error variances are unequal," Psychological Methods, 1 (3], 261-77. 

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