anova2

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, respectively. The rows are popper types, oil and air. In the study, researchers popped a batch of each brand three times with each popper, that is, the number of replications is 3. The first three rows correspond to the oil popper, and the last three rows correspond to the air popper. The response values are the yield in cups of popped popcorn.

Perform a two-way ANOVA. Save the ANOVA table in the cell array tbl for easy access to results.

[p,tbl] = anova2(popcorn,3);

The column Prob>F shows the p-values for the three brands of popcorn (0.0000), the two popper types (0.0001), and the interaction between brand and popper type (0.7462). These values indicate that popcorn brand and popper type affect the yield of popcorn, but there is no evidence of an interaction effect of the two.

Display the cell array containing the ANOVA table.

tbl

tbl=6×6 cell array
    {'Source'     }    {'SS'     }    {'df'}    {'MS'      }    {'F'       }    {'Prob>F'    }
    {'Columns'    }    {[15.7500]}    {[ 2]}    {[  7.8750]}    {[ 56.7000]}    {[7.6790e-07]}
    {'Rows'       }    {[ 4.5000]}    {[ 1]}    {[  4.5000]}    {[ 32.4000]}    {[1.0037e-04]}
    {'Interaction'}    {[ 0.0833]}    {[ 2]}    {[  0.0417]}    {[  0.3000]}    {[    0.7462]}
    {'Error'      }    {[ 1.6667]}    {[12]}    {[  0.1389]}    {0x0 double}    {0x0 double  }
    {'Total'      }    {[     22]}    {[17]}    {0x0 double}    {0x0 double}    {0x0 double  }

Store the F-statistic for the factors and factor interaction in separate variables.

Fbrands = tbl{2,5}

Fbrands = 
56.7000

Fpoppertype = tbl{3,5}

Fpoppertype = 
32.4000

Finteraction = tbl{4,5}

Finteraction = 
0.3000

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. The first three rows correspond to the oil popper, and the last three rows correspond to the air popper. 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).

c1 = multcompare(stats);

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

The figure shows the multiple comparisons 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.

Display the multiple comparison results in a table.

tbl1 = array2table(c1,"VariableNames", ...
    ["Group A","Group B","Lower Limit","A-B","Upper Limit","P-value"])

tbl1=3×6 table
    Group A    Group B    Lower Limit    A-B     Upper Limit     P-value  
    _______    _______    ___________    ____    ___________    __________

       1          2         0.92597       1.5       2.074       4.1188e-05
       1          3           1.676      2.25       2.824       6.1588e-07
       2          3         0.17597      0.75       1.324         0.011591

The first two columns of c1 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 are very small, which indicates that the popcorn yield differs across all three brands.

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

c2 = 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.

tbl2 = array2table(c2,"VariableNames", ...
    ["Group A","Group B","Lower Limit","A-B","Upper Limit","P-value"])

tbl2=1×6 table
    Group A    Group B    Lower Limit    A-B    Upper Limit     P-value  
    _______    _______    ___________    ___    ___________    __________

       1          2         -1.3828      -1      -0.61722      0.00010037

The small p-value 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.