Lilliefors Test for Normality

The normality assumption is at the core of a majority of standard statistical procedures, and it is important to be able to test this assumption. In addition, showing that a sample does not come from a normally distributed population is sometimes of importance per se. Among the many procedures used to test this assumption, one of the most well known is a modification of the Kolmogorov-Smirnov test of goodness of fit, generally referred to as the Lilliefors test for normality (or Lilliefors test, for short). This test was developed independently by Lilliefors and by Van Soest. The null hypothesis for this test is that the error is normally distributed (i.e., there is no difference between the observed distribution of the error and a normal distribution). The alternative hypothesis is that the error is not normally distributed.

Like most statistical tests, this test of normality defines a criterion and gives its sampling distribution. When the probability associated with the criterion is smaller than a given α level, the alternative hypothesis is accepted (i.e., we conclude that the sample does not come from a normal distribution). An interesting peculiarity of the Lilliefors test is the technique used to derive the sampling distribution of the criterion. In general, mathematical statisticians derive the sampling distribution of the criterion using analytical techniques. However, in this case, this approach fails, and consequently, Lilliefors decided to calculate an approximation of the sampling distribution by using the Monte Carlo technique. Essentially, the procedure consists of extracting a large number of samples from a normal population and computing the value of the criterion for each of these samples. The empirical distribution of the values of the criterion gives an approximation of the sampling distribution of the criterion under the null hypothesis.

Specifically, both Lilliefors and Van Soest used, for each sample size chosen, 1,000 random samples derived from a standardized normal distribution to approximate the sampling distribution of a Kolmogorov-Smirnov criterion of goodness of fit. The critical values given by Lilliefors and Van Soest are quite similar, the relative error being of the order of 10−2.

According to Lilliefors, this test of normality is more powerful than other procedures for a wide range of nonnormal conditions. Dagnelie indicated, in addition, that the critical values reported by Lilliefors can be approximated by an analytical formula. Such a formula facilitates writing computer routines because it eliminates the risk of creating errors when keying in the values of the table. Recently, Molin and Abdi refined the approximation given by Dagnelie and computed new tables using a larger number of runs (i.e., K = 100,000) in their simulations.

Notation

The sample for the test is made of N scores, each of them denoted Xi. The sample mean is denoted MX and is computed as

the sample variance is denoted

[Page 541] and the standard deviation of the sample, denoted SX, is equal to the square root of the sample variance.

The first step of the test is to transform each of the Xi scores into z scores as follows:

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