Nonnormality in Linear Regression

III.I.4 Nonnormality in Linear Regression

As we have discussed before, the MLE approach is only valid under the assumption that the error component is distributed according to a prespecified probability distribution function. Most of the time the normal distribution is implicitly assumed.

In order to check the error component for normality there are some diagnostics which may be valuable assistants: the histograms, and measures for kurtosis and skewness.

The histogram is a graph consisting of a set of rectangles. Each rectangle has its base on the x axis and the length of each base is equal to the class interval sizes. The area of each rectangle is proportional to the class frequency. With some practice it should not be hard to see at one glance if the histogram "fits" the normal curve.

The kurtosis of a distribution, which is said to be a measure for peakedness, can for instance be measured by

(III.I.4-1)

which is the fourth moment about the mean divided by the square of the second moment about the mean. It can be proved that a normally distributed variable has a kurtosis of 3 (according to eq. (III.I.4-1)). Therefore a distribution can be called "leptokurtic" if the measure is larger than 3 and "platykurtic" if it is smaller than 3.

The coefficient of kurtosis should only be used with care ! In the statistics literature there have been given many examples where this measure for kurtosis fails (for small samples). Also, it is possible for a variable to have a kurtosis of 3, without being normally distributed.

The skewness of a distribution, which is a measure for asymmetry, can for example be defined by

(III.I.4-2)

Pearson's second coefficient of skewness is defined by

(III.I.4-3)

The skewness of a distribution can also be defined by means of moments as in

(III.I.4-4)

The distribution of a stochastic variable can also be tested with respect to any given theoretical distribution. The main idea behind these tests is that the difference between histogram-frequencies of the variable and theoretical frequencies are computed. It is then tested whether these differences are "large enough" in order to reject the null-hypothesis that the variable is distributed according to the prespecified theoretical distribution. Most of these tests are based on the Chi-square distribution. An excellent review (with illustrations) of these tests (including graphical methods such as the "suspended root-display") can be found in Mills 1990.

The suspended rootogram is based on the same principle, except for the fact that the deviations between the normal and actual frequencies is computed on using transformations.