Load the sample data.

`y`

is the response vector and `g1`

, `g2`

, and `g3`

are the grouping variables (factors). Each factor has two levels, and every observation in `y`

is identified by a combination of factor levels. For example, observation `y(1)`

is associated with level 1 of factor `g1`

, level `'hi'`

of factor `g2`

, and level `'may'`

of factor `g3`

. Similarly, observation `y(6)`

is associated with level 2 of factor `g1`

, level `'hi'`

of factor `g2`

, and level `'june'`

of factor `g3`

.

Test if the response is the same for all factor levels. Also compute the statistics required for multiple comparison tests.

The *p*-value of 0.2578 indicates that the mean responses for levels `'may'`

and `'june'`

of factor `g3`

are not significantly different. The *p*-value of 0.0347 indicates that the mean responses for levels `1`

and `2`

of factor `g1`

are significantly different. Similarly, the *p*-value of 0.0048 indicates that the mean responses for levels `'hi'`

and `'lo'`

of factor `g2`

are significantly different.

Perform multiple comparison tests to find out which groups of the factors `g1`

and `g2`

are significantly different.

results =
1.0000 2.0000 -6.8604 -4.4000 -1.9396 0.0280
1.0000 3.0000 4.4896 6.9500 9.4104 0.0177
1.0000 4.0000 6.1396 8.6000 11.0604 0.0143
2.0000 3.0000 8.8896 11.3500 13.8104 0.0108
2.0000 4.0000 10.5396 13.0000 15.4604 0.0095
3.0000 4.0000 -0.8104 1.6500 4.1104 0.0745

`multcompare`

compares the combinations of groups (levels) of the two grouping variables, `g1`

and `g2`

. In the `results`

matrix, the number 1 corresponds to the combination of level `1`

of `g1`

and level `hi`

of `g2`

, the number 2 corresponds to the combination of level `2`

of `g1`

and level `hi`

of `g2`

. Similarly, the number 3 corresponds to the combination of level `1`

of `g1`

and level `lo`

of `g2`

, and the number 4 corresponds to the combination of level `2`

of `g1`

and level `lo`

of `g2`

. The last column of the matrix contains the *p*-values.

For example, the first row of the matrix shows that the combination of level `1`

of `g1`

and level `hi`

of `g2`

has the same mean response values as the combination of level `2`

of `g1`

and level `hi`

of `g2`

. The *p*-value corresponding to this test is 0.0280, which indicates that the mean responses are significantly different. You can also see this result in the figure. The blue bar shows the comparison interval for the mean response for the combination of level `1`

of `g1`

and level `hi`

of `g2`

. The red bars are the comparison intervals for the mean response for other group combinations. None of the red bars overlap with the blue bar, which means the mean response for the combination of level `1`

of `g1`

and level `hi`

of `g2`

is significantly different from the mean response for other group combinations.

You can test the other groups by clicking on the corresponding comparison interval for the group. The bar you click on turns to blue. The bars for the groups that are significantly different are red. The bars for the groups that are not significantly different are gray. For example, if you click on the comparison interval for the combination of level `1`

of `g1`

and level `lo`

of `g2`

, the comparison interval for the combination of level `2`

of `g1`

and level `lo`

of `g2`

overlaps, and is therefore gray. Conversely, the other comparison intervals are red, indicating significant difference.