# One-Way ANOVA Post Hoc Tests

Once you have determined that differences exist among the means, post hoc range tests and pairwise multiple comparisons can determine which means differ. Range tests identify homogeneous subsets of means that are not different from each other. Pairwise multiple comparisons test the difference between each pair of means and yield a matrix where asterisks indicate significantly different group means at an alpha level of 0.05.

Equal Variances Assumed

Tukey's honestly significant difference test,
Hochberg's GT2, Gabriel, and Scheffé are multiple comparison
tests and range tests. Other available range tests are Tukey's *b*, S-N-K (Student-Newman-Keuls), Duncan,
R-E-G-W *F* (Ryan-Einot-Gabriel-Welsch *F* test), R-E-G-W *Q* (Ryan-Einot-Gabriel-Welsch range test), and Waller-Duncan.
Available multiple comparison tests are Bonferroni, Tukey's honestly
significant difference test, Sidak, Gabriel, Hochberg, Dunnett, Scheffé,
and LSD (least significant difference).

- LSD. Uses t tests to perform all pairwise comparisons between group means. No adjustment is made to the error rate for multiple comparisons.
- Bonferroni. Uses t tests to perform pairwise comparisons between group means, but controls overall error rate by setting the error rate for each test to the experimentwise error rate divided by the total number of tests. Hence, the observed significance level is adjusted for the fact that multiple comparisons are being made.
- Sidak. Pairwise multiple comparison test based on a t statistic. Sidak adjusts the significance level for multiple comparisons and provides tighter bounds than Bonferroni.
- Scheffe. Performs simultaneous joint pairwise comparisons for all possible pairwise combinations of means. Uses the F sampling distribution. Can be used to examine all possible linear combinations of group means, not just pairwise comparisons.
- R-E-G-W F. Ryan-Einot-Gabriel-Welsch multiple stepdown procedure based on an F test.
- R-E-G-W Q. Ryan-Einot-Gabriel-Welsch multiple stepdown procedure based on the Studentized range.
- S-N-K. Makes all pairwise comparisons between means using the Studentized range distribution. With equal sample sizes, it also compares pairs of means within homogeneous subsets, using a stepwise procedure. Means are ordered from highest to lowest, and extreme differences are tested first.
- Tukey. Uses the Studentized range statistic to make all of the pairwise comparisons between groups. Sets the experimentwise error rate at the error rate for the collection for all pairwise comparisons.
- Tukey's b. Uses the Studentized range distribution to make pairwise comparisons between groups. The critical value is the average of the corresponding value for the Tukey's honestly significant difference test and the Student-Newman-Keuls.
- Duncan. Makes pairwise comparisons using a stepwise order of comparisons identical to the order used by the Student-Newman-Keuls test, but sets a protection level for the error rate for the collection of tests, rather than an error rate for individual tests. Uses the Studentized range statistic.
- Hochberg's GT2. Multiple comparison and range test that uses the Studentized maximum modulus. Similar to Tukey's honestly significant difference test.
- Gabriel. Pairwise comparison test that used the Studentized maximum modulus and is generally more powerful than Hochberg's GT2 when the cell sizes are unequal. Gabriel's test may become liberal when the cell sizes vary greatly.
- Waller-Duncan. Multiple comparison test based on a t statistic; uses a Bayesian approach.
- Dunnett. Pairwise multiple comparison t test that compares a set of treatments against a single control mean. The last category is the default control category. Alternatively, you can choose the first category. 2-sided tests that the mean at any level (except the control category) of the factor is not equal to that of the control category. < Control tests if the mean at any level of the factor is smaller than that of the control category. > Control tests if the mean at any level of the factor is greater than that of the control category.

Equal Variances Not Assumed

Multiple comparison tests that do not assume equal
variances are Tamhane's T2, Dunnett's T3, Games-Howell, and Dunnett's *C*.

- Tamhane's T2. Conservative pairwise comparisons test based on a t test. This test is appropriate when the variances are unequal.
- Dunnett's T3. Pairwise comparison test based on the Studentized maximum modulus. This test is appropriate when the variances are unequal.
- Games-Howell. Pairwise comparison test that is sometimes liberal. This test is appropriate when the variances are unequal.
- Dunnett's C. Pairwise comparison test based on the Studentized range. This test is appropriate when the variances are unequal.

*Note*:
You may find it easier to interpret the output from post hoc tests
if you deselect Hide empty rows and columns in the Table Properties dialog box (in an activated pivot table,
choose Table Properties from
the Format menu).

Obtaining Post Hoc Tests for One-Way ANOVA

This feature requires the Statistics Base option.

- From the menus choose:
- In the One-Way ANOVA dialog box, click Post Hoc.

*Note*: You may find it easier to interpret
the output from post hoc tests if you deselect Hide empty rows and columns in the Table Properties dialog
box (in an activated pivot table, choose Table Properties from the Format menu).