Chapter Twenty-Four - Additional Resources

The k-sample slippage test (Conover, 1971)

An alternative to the Kruskal-Wallis one-way analysis of variance by ranks is suggested by Conover (1991). The test is very quick and easy to compute without recourse to calculators, and has the added advantage that ties caused by several observations equalling one another present little difficulty when using the procedure.

The term ‘slippage’ in the title of the test refers to its central purpose, which is to ascertain whether one or more population distribution functions have ‘slipped to the right’ of others, that is to say, whether they contain larger values than others.

The slippage test relies heavily on the largest values in each of the samples. It is powerful, Conover says, as long as it remains the case that the populations having the larger means also have larger variances. In this event, the extreme values of a sample are sensitive indicators of differences in means.

Where this does not hold true, as for example, when the variances of populations remain equal despite some means being larger than others, or where variances decrease as means become larger, then the slippage test is somewhat insensitive to differences in means and the Kruskal-Wallis test is better employed. One further limitation is that the k-sample slippage test may only be employed when all samples contain the same number of observations.

The slippage test makes the following assumptions:

• that each sample is a random sample from some population;
• that the k samples are independent of one another;
• that the random variables are continuous;
• that the data are at least at the ordinal level of measurement;
• that either the k population distribution functions are identical or that some of the populations contain larger observations than other populations.

By way of example, suppose that a Department of Education and Skills directive requires standardized tests of mathematical achievement to be administered in secondary schools in six counties close to London in order to publicize the performances of the respective students. Twelve students are randomly selected from the Year 10 classes (age 15) of the six secondary schools. Their scores are set out thus:

Mathematical achievement scores in six secondary schools

The null hypothesis is that the mathematical achievement of the Year 10 classes in the six secondary schools are identically distributed.

Procedures

• Place an asterisk next to the greatest score in each of the six samples.
• Underline the smallest asterisked score once and the greatest asterisked score twice.
• Add up the number of scores in the sample with the greatest asterisked score (that is, Sample B) that exceed the smallest asterisked score (that is, 166 in Sample E). The obtained value is 7.
• Enter Tables 14 in Appendix A where k = the number of samples, that is 6, and n = the number of observations, that is 12.
• We see that a critical value of 6 is statistically significant at r=.05. Our obtained value exceeds to critical value and is therefore statistically significant at less than r=.05.

We do not support the null hypothesis, therefore, and conclude that at least one of the six secondary schools is associated with higher mathematical achievement scores than at least one of the other five.

Conover points to the way in which further multiple comparisons can be undertaken when the k-slippage test has not supported the null hypothesis as in out example above. The ‘highest’ samples (that is Sample B) is simply discarded and the analysis is repeated on the remaining k-1 samples, interpolating the Table 15 in Appendix A at k=5, n=12. Where the null hypothesis is, again, not supported, the procedure is repeated on the remaining k-2 samples, and so on, until the null hypothesis is supported.

The reader may wish to verify the following results from further multiple comparisons of the data set out in the table above.

The comparison of Sample A (233 *) with Sample E (166 *) results in an obtained value of 6, statistically significant at less than r=.05 where k=4 and n=12.

The comparison of Sample F (219 *) with Sample E (166 *) results in an obtained value of 5, statistically significant at less than r=.10 but greater than r=.05 where k=3 and n=12. The comparison of Sample C (191 *) with Sample E (166 *) results in an obtained value of 2, which is not statistically significant, where k=2 and n=12.