Load the sample data.

load carsmall

Perform a one-way analysis of variance (ANOVA) to see if there is any difference between the mileage of the cars by origin.

[p,t,stats] = anova1(MPG,Origin,'off');

Perform a multiple comparison of the group means.

[c,m,h,nms] = multcompare(stats);

multcompare displays the estimates with comparison intervals around them. You can click the graphs of each country to compare its mean to those of other countries.

Now display the mean estimates and the standard errors with the corresponding group names.

[nms num2cell(m)]

ans=6×3 cell array
    {'USA'    }    {[21.1328]}    {[0.8814]}
    {'Japan'  }    {[31.8000]}    {[1.8206]}
    {'Germany'}    {[28.4444]}    {[2.3504]}
    {'France' }    {[23.6667]}    {[4.0711]}
    {'Sweden' }    {[22.5000]}    {[4.9860]}
    {'Italy'  }    {[     28]}    {[7.0513]}

Load the sample data.

load popcorn
popcorn

popcorn = 6×3

    5.5000    4.5000    3.5000
    5.5000    4.5000    4.0000
    6.0000    4.0000    3.0000
    6.5000    5.0000    4.0000
    7.0000    5.5000    5.0000
    7.0000    5.0000    4.5000

The data is from a study of popcorn brands and popper types (Hogg 1987). The columns of the matrix popcorn are brands (Gourmet, National, and Generic). The rows are popper types oil and air. In the study, researchers popped a batch of each brand three times with each popper. The values are the yield in cups of popped popcorn.

Perform a two-way ANOVA. Also compute the statistics that you need to perform a multiple comparison test on the main effects.

[~,~,stats] = anova2(popcorn,3,'off')

stats = struct with fields:
      source: 'anova2'
     sigmasq: 0.1389
    colmeans: [6.2500 4.7500 4]
        coln: 6
    rowmeans: [4.5000 5.5000]
        rown: 9
       inter: 1
        pval: 0.7462
          df: 12

The stats structure includes

Perform a multiple comparison test to see if the popcorn yield differs between pairs of popcorn brands (columns).

c = multcompare(stats)

Note: Your model includes an interaction term.  A test of main effects can be 
difficult to interpret when the model includes interactions.

c = 3×6

    1.0000    2.0000    0.9260    1.5000    2.0740    0.0000
    1.0000    3.0000    1.6760    2.2500    2.8240    0.0000
    2.0000    3.0000    0.1760    0.7500    1.3240    0.0116

The first two columns of c show the groups that are compared. The fourth column shows the difference between the estimated group means. The third and fifth columns show the lower and upper limits for 95% confidence intervals for the true mean difference. The sixth column contains the p-value for a hypothesis test that the corresponding mean difference is equal to zero. All p-values (0, 0, and 0.0116) are very small, which indicates that the popcorn yield differs across all three brands.

The figure shows the multiple comparison of the means. By default, the group 1 mean is highlighted and the comparison interval is in blue. Because the comparison intervals for the other two groups do not intersect with the intervals for the group 1 mean, they are highlighted in red. This lack of intersection indicates that both means are different than group 1 mean. Select other group means to confirm that all group means are significantly different from each other.

Perform a multiple comparison test to see the popcorn yield differs between the two popper types (rows).

c = multcompare(stats,'Estimate','row')

Note: Your model includes an interaction term.  A test of main effects can be 
difficult to interpret when the model includes interactions.

c = 1×6

    1.0000    2.0000   -1.3828   -1.0000   -0.6172    0.0001

The small p-value of 0.0001 indicates that the popcorn yield differs between the two popper types (air and oil). The figure shows the same results. The disjoint comparison intervals indicate that the group means are significantly different from each other.

Load the sample data.

y = [52.7 57.5 45.9 44.5 53.0 57.0 45.9 44.0]';
g1 = [1 2 1 2 1 2 1 2];
g2 = {'hi';'hi';'lo';'lo';'hi';'hi';'lo';'lo'};
g3 = {'may';'may';'may';'may';'june';'june';'june';'june'};

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.

[~,~,stats] = anovan(y,{g1 g2 g3},'model','interaction',...
    'varnames',{'g1','g2','g3'});

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 = multcompare(stats,'Dimension',[1 2])

results = 6×6

    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.