The latest version, JMP 19, adds an option to perform a Games-Howell test for mean comparisons in the One-Way Analysis report (with Y as continuous and X as nominal or ordinal) in the Bivariate Platform.
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Games-Howell test(From JMP hover help)
This test compares the means of all pairs and deals with situations where it cannot be assumed that the error variances of each group are all equal.
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The Tukey's HSD test is a well-known multiple comparison test that compares the means of all pairs, but this test is appropriate when it can be assumed that the error variances of each group are all equal.
On the other hand, there is a method called "Welch's analysis of variance (ANOVA)" that does not assume equal variances and examines the difference in means between groups, and the Games-Howell test is widely used as one of its post-hoc tests.
Assuming normality of each group, whether or not there is an assumption of equal variance × "Test of mean difference・Post-hoc testing is as follows:
| |
Overall test of the difference in means |
Testing the mean differences of all pairs (multiple comparisons/post-hoc tests) |
| Assume equal variances |
Analysis of variance (ANOVA) |
Tukey-Kramer HSD test |
| Do not assume equal variances |
Welch's analysis of variance (ANOVA) |
Games-Howell test |
*Post-hoc test: A multiple comparison method that identifies which pairs are significantly different after an overall difference has been shown by ANOVA or other methods.
Example: Comparison of pain caused by different painkillers
Consider the following example:
(example)Forty-seven patients with pain were given one of three analgesics (A, B, or C) with different ingredients. The degree of pain after administration was measured using a visual analog scale (VAS, 0-100).
In this case, we would like to find out whether the degree of pain differs depending on the type of analgesic used, and whether there are any significant differences between groups.
Enter the collected data into JMP, specify "VAS" (continuous scale) for Y, and "Group" (nominal scale) for X in the "Bivariate" platform, and a "One-Way Analysis" report will be displayed.
(This data table is available for download from this blog.)
Additionally, we will show a report with several options implemented, such as displaying the mean and standard deviation for each group.
The graph shows that Group C has a larger variance than Groups A and B. The average value for Group C is also higher.
Since it seems unlikely that equality of variances can be assumed, we run the option Test for equality of variances and check the results of Welch's analysis of variance.
If the significance level is set to 0.05, the p-value is 0.0360, which means there is a significant difference. In other words, if equal variances are not assumed, there is a difference in the means of each group.
Now that we have determined the difference in average values between groups, we will use the Games-Howell test to determine which groups specifically have differences (post-hoc test).
To perform this test, select Compare Means > All Pairs, Games-Howell from the red triangle in the upper left.
This will add a report called "Comparison of all pairs using the Games-Howell test." The "Rank of Differences Report" within it displays the difference between each group (C and B, C and A, A and 
, the standard error of the difference, the degrees of freedom, and so on, and calculates the p-value based on these.
In this example, only the comparison between Group C and Group B is significant.
Reference: Test formula
The Games-Howell test assumes unequal variances, so standard errors are calculated from the group variances and the degrees of freedom are calculated using the Welch-Satterthwaite formula.
Below is the formula in JSL for comparing Group C and Group B.
/* Games-Howell検定の計算例 */
nGroup = 3;
alpha = 0.05;
n_C = 15; n_B = 20;
mean_C = 62.22; std_C = 12.3493;
mean_B = 52.62; std_B = 5.8470;
/* dif_CB:差、dif_SE_CB:差の標準誤差、df_CB:自由度 */
dif_CB = mean_C - mean_B;
/* 注意：JMPでは、分母に1/2を入れている */
difSE_CB = Sqrt((std_C^2/n_C + std_B^2/n_B)*0.5);
dfnum_CB = (std_C^2/n_C + std_B^2/n_B)^2;
dfden_CB = (std_C^2/n_C)^2/(n_C - 1) + (std_B^2/n_B)^2/(n_B - 1);
df_CB = dfnum_CB / dfden_CB;
Show(dif_CB, difSE_CB,df_CB);
/* dast_CB: 臨界値q*、pval_CB:p値 */
q_CB = dif_CB / Sqrt(std_C^2/n_C + std_B^2/n_B);
qast_CB = Tukey HSD Quantile(1 - alpha,nGroup,df_CB);
pval_CB = Tukey HSD P value(q_CB,nGroup,df_CB);
Show(qast_CB,pval_CB);
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
