Sunday, March 2, 2008

Lecture 8 - Residual Analysis - Checking the Equal Variance Assumption

Homoscadasticity (Not that there's anything wrong with that.)
We now turn to checking the assumption of equal variance of errors, the "E" in our LINE mnemonic. This assumptions states that not only is the error at each x-value distributed normally, but the variance in the error is equal at each point.

Equal variance of errors is known as homoscadasticity. Unequal variance of errors is called heteroscadasticity.

For this analysis we turn again to the plot of residuals vs. the independent variable (x) that we used in when we validated the linearity assumption. For linearity, we were just looking to see if the residuals were evenly distributed above and below the x-axis. To check for equal variance of errors, we check to see if there's any pattern in the distribution of the residuals around the x-axis.

Running the residual plot versus x in Minitab:
1. Load up your data.
2. Select Stat-Regression-Regression from the menu bar.
3. Put Annual Sales in the Response box and Square Feet in the Predictor box.
4. Click the Graphs button. Put Square Feet in the Residuals versus the variables box.
5. Click OK in both the Graphs and Regression dialogs. The residual plot appears.

Review the graph and ask yourself: Is there any pattern in the residuals? Do they get increasing larger or small as x changes? If so, then you have a case of heteroscadasticity. But if the residuals are distributed evenly and consistently around the x-axis, then you can conclude that the variances are consistent and the assumption of equality of variances is valid.

In our example, I'd be somewhat concerned with the fact that the residuals are closer to the x-axis for small values of x, but broaden out for larger values. The variance does seem to taper off as x gets very large, which is an indication that the variances are equal for x>2 or so. (Click the graph for a larger view of the plot.)

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