Basic Concepts
The Ryan, Einot, Gabriel, Welsh Studentized Range Q (REGWQ) test uses what is known as a step-down approach to control familywise error. In this test, no confidence intervals are calculated.
To conduct the REGWQ test, first, we arrange the sample means in descending order x̄1 ≥ x̄2 ≥ … ≥ x̄k. We next test whether there is a statistically significant difference between x̄1 and x̄k. If not, then we conclude that all the group means are equal and terminate the testing. If, instead, we find that there is a significant difference between x̄1 and x̄k, then we next test (1) whether there is a significant difference between x̄1 and x̄k-1 and (2) whether there is a significant difference between x̄2 and x̄k.
The process is recursive. At each stage, when we compare x̄i and x̄j (with x̄i ≥ x̄j), we define
where p = j – i + 1. The null hypothesis that x̄i = x̄j (and therefore all the means x̄i, x̄i+1, … x̄j are equal) is rejected if
This process is the same as for Tukey’s HSD test except that p and αp are used in calculating qcrit instead of k and α.
If the test shows that there is no significant difference between x̄i and x̄j (or if j = i + 1), then this part of the process terminates; otherwise, we test (1) whether there is a significant difference between x̄i and x̄j-1 and (2) whether there is a significant difference between x̄i+1 and x̄j.
Example
Example 1: Apply the REGWQ to the scenario described in Example 1 of Tukey-Kramer Test, but based on the data on the left side of Figure 1.
Figure 1 – Comparing five teaching methods
Based on the one-way ANOVA shown on the right side of the figure, we see there is a significant difference between the five teaching methods. We now use the REGWQ post-hoc test, as shown in Figure 2, to pinpoint which pairs of methods are significantly different.
Figure 2 – REGWQ test
The table on the left-hand side of the figure consists of the groups sorted from highest to lowest mean. We will refer to the index (i.e. rank) of these groups, where i = 1 refers to the group with the highest mean (i.e. Method 2) and i = 5 refers to the group with the lowest mean (i.e. Method 5).
Column U refers to the initial test comparing the groups with the highest and lowest means. Since p = 5 – 1 + 1 = 5 (cell U6) and p = 5 > 4 = k – 1, it follows that αp = α = .05 (cell U8). Thus qcrit =QCRIT(U6,U9,U8,2) = QCRIT(p,df,αp,2) = QCRIT(5,35,.05,2) = 4.066. QCRIT is based on the Studentized Range q table of critical values (using interpolation where necessary). We could have used the formula =QINV(U8,U6,U9) instead to get the value 4.0659.
Significance results
The standard error (cell U14) is calculated by the formula =SQRT($J$14/$H$5) and the q-stat (cell U15) by the formula =SQRT($J$14/$H$5). Since q = 4.7556 > 4.066 = qcrit, we conclude that 22.375 (cell U13) is a significant difference between the means of Method 2 and 5.
We calculate the p-value using the formula =QDIST(U15,U6,U9). Since p-value = .01533 < .05 = αp, we again conclude that there is a significant difference between Method 2 and 5. This time we could have used the formula =QPROB(U15,U6,U9) to calculate the p-value obtaining a similar value of .015025.
Since we obtained a significant result, we now proceed on to testing group 1 with group 4 and group 2 with group 5. These comparisons are shown in columns V and W, respectively. Since both of these are significant results we proceed to further testing. Based on the significant result in column V we next test group 1 with group 3 (column X) and group 2 with group 4 (column Y), and based on the significant result in column W we test group 2 with group 4 (column Z) and group 3 with group 5 (column AA).
Of these four tests, only the test in column Y is significant (groups 2 and 4). Thus we continue by testing group 2 with 3 (column AA again) and group 3 with 4 (column AB). Since neither of these yields a significant result, the testing stops.
We conclude there are four significant results: Method 2 with Method 5, Method 2 with Method 1, Method 3 with Method 5, and Method 3 with Method 1.
Data Analysis Tool
We can perform the same test using the Single Factor ANOVA tool. When the dialog box shown in Figure 1 of ANOVA Analysis Tool appears, fill in the Input Range with A3:E11, make sure that the Column headings included with data is checked, and choose the REGWQ option. After clicking on the OK button, the output shown in Figure 3 is displayed.
Figure 3 – REGWQ data analysis
We see from the figure that the same four comparisons are significant. We also see that there are no alpha (i.e. αp), q-crit, and p-value entries for the first and last comparisons. This is because these comparisons represent subsets of comparisons that have already been found to be non-significant.
Worksheet Function
Real Statistics Function: The Real Statistics Resource Pack uses the following array function to create the output for the REGWQ option of the Single Factor ANOVA data analysis tool.
REGWQ(R1, lab, alpha): outputs the results of the REGWQ test on the data in R1, including column headings; if lab = TRUE (default FALSE), then column headings are added to the output; alpha is the significance level (default .05).
For Example 1, =REGWQ(A4:E11,TRUE) yields the results shown in range G12:P22 of Figure 3.
Reference
Howell, D. C. (2010) Statistical methods for psychology (7th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf