Week 18: The Siegel-Tukey Test for Equal Variability


Today we’re going to talk about another nonparametric test: the Siegel-Tukey test for equal variability!

When Would You Use It?
The Siegel-Tukey test for equal variability is a nonparametric test used to determine if two independent samples represent two populations with different variances.

What Type of Data?
The Siegel-Tukey test for equal variability requires ordinal data.

Test Assumptions

  • Each sample is a simple random sample from the population it represents.
  • The two samples are independent.
  • The underlying distributions of the samples have equal medians.

Test Process
Step 1: Formulate the null and alternative hypotheses. The null hypothesis claims that the two population variances are equal. The alternative hypothesis claims otherwise (one variance is greater than the other, or that they are simply not equal).

[Note that from here on out, the calculations are exactly the same as for the Mann-Whitney U test. The only thing that differs is how the data are ranked.]

Step 2: Compute the test statistics: U1 and U2. Since this is best done with data, please see the example shown below to see how this is done.

Step 3: Obtain the critical value. Unlike most of the tests we’ve done so far, you don’t get a precise p-value when computing the results here. Rather, you calculate your U values and then compare them to a specific value. This is done using a table (such as the one here). Find the number at the intersection of your sample sizes for both samples at the specified alpha-level. Compare this value with the smaller of your U1 and U2 values.

Step 4: Determine the conclusion. If your test statistic is equal to or less than the table value, reject the null hypothesis. If your test statistic is greater than the table value, fail to reject the null (that is, claim that the variances are equal in the population).

Example
Today’s data come from my 2012 music selection. I wanted to see if the median play counts for two genres—pop and electronic—were the same. I chose these two because I think most of my favorite songs are of one of the two genres. To keep things relatively simple for the example, I sampled n = 8 electronic songs and n = 8 pop songs. Set α = 0.05.

H0: σ2pop = σ2electronic
Ha: σ2pop ≠ σ2electronic

The following table shows several different columns of information. I will explain the columns below.

test17a

Column 1 is the genre of each song.
Column 2 is the play count for each song, ranked from least to greatest
Column 3 is the rank of each play count. In order to obtain the ranks for this test, start by giving a rank of “1” to the lowest play count value. Then a rank of “2” to the highest play count value, a rank of “3” to the second highest play count value, a rank of “4” to the second lowest play count value, etc. (that is, assign ranks by alternating from one extreme to the other).

To compute U1 and U2, use the following equations:

test15b

So here,

test17c

The test statistic itself is the smaller of the above values; in this case, they’re both the same, so we get U = 32. In the table, the critical value for n1 = 8 and n2 = 8 and α = 0.05 for a two-tailed test is 13. Since U > 13, we fail to reject the null and retain the claim that the population variances are equal.

Example in R
No R example this week; most of this is easy enough to do by hand for a small-ish sample.

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