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Function Reference: anova2

statistics: p = anova2 (x, reps)
statistics: p = anova2 (x, reps, displayopt)
statistics: p = anova2 (x, reps, displayopt, model)
statistics: [p, atab] = anova2 (…)
statistics: [p, atab, stats] = anova2 (…)

Performs two-way factorial (crossed) or a nested analysis of variance (ANOVA) for balanced designs. For unbalanced factorial designs, diagnostic plots and/or planned contrasts, use anovan instead.

anova2 requires two input arguments with an optional third and fourth:

  • x contains the data and it must be a matrix of at least two columns and two rows.
  • reps is the number of replicates for each combination of factor groups.
  • displayopt is an optional parameter for displaying the ANOVA table, when it is ’on’ (default) and suppressing the display when it is ’off’.
  • model is an optional parameter to specify the model type as either:
    • "interaction" or "full" (default): compute both main effects and their interaction
    • "linear": compute both main effects without an interaction. When reps > 1 the test is suitable for a balanced randomized block design. When reps == 1, the test becomes a One-way Repeated Measures (RM)-ANOVA with Greenhouse-Geisser correction to the column factor degrees of freedom to make the test robust to violations of sphericity
    • "nested": treat the row factor as nested within columns. Note that the row factor is considered a random factor in the calculation of the statistics.

anova2 returns up to three output arguments:

  • p is the p-value of the null hypothesis that all group means are equal.
  • atab is a cell array containing the results in a standard ANOVA table.
  • stats is a structure containing statistics useful for performing a multiple comparison of means with the MULTCOMPARE function.

If anova2 is called without any output arguments, then it prints the results in a one-way ANOVA table to the standard output as if displayopt is ’on’.

Examples:

 
 load popcorn;
 anova2 (popcorn, 3);
 
 [p, anovatab, stats] = anova2 (popcorn, 3, "off");
 disp (p);

See also: anova1, anovan, multcompare

Source Code: anova2

Example: 1

  


 # Factorial (Crossed) Two-way ANOVA with Interaction

 popcorn = [5.5, 4.5, 3.5; 5.5, 4.5, 4.0; 6.0, 4.0, 3.0; ...
            6.5, 5.0, 4.0; 7.0, 5.5, 5.0; 7.0, 5.0, 4.5];

 [p, atab, stats] = anova2(popcorn, 3, "on");


                      ANOVA Table

Source             SS      df        MS       F      Prob>F
-----------------------------------------------------------
Columns         15.7500     2     7.8750    56.70    0.0000
Rows             4.5000     1     4.5000    32.40    0.0001
Interaction      0.0833     2     0.0417     0.30    0.7462
Error            1.6667    12     0.1389
Total           22.0000    17

Example: 2

  


 # One-way Repeated Measures ANOVA (Rows are a crossed random factor)

 data = [54, 43, 78, 111;
         23, 34, 37, 41;
         45, 65, 99, 78;
         31, 33, 36, 35;
         15, 25, 30, 26];

 [p, atab, stats] = anova2 (data, 1, "on", "linear");


                      ANOVA Table

Source             SS      df        MS       F      Prob>F
-----------------------------------------------------------
Columns       2174.9500     3   724.9833     3.62    0.0873
Rows          8371.7000     4  2092.9250    10.45    0.0007
Error         2404.3000    12   200.3583
Total        12950.9500    19

Note: Greenhouse-Geisser's correction was applied to the
degrees of freedom for the Column factor: F(1.74,6.95)

Example: 3

  


 # Balanced Nested One-way ANOVA (Rows are a nested random factor)

 data = [4.5924 7.3809 21.322; -0.5488 9.2085 25.0426; ...
         6.1605 13.1147 22.66; 2.3374 15.2654 24.1283; ...
         5.1873 12.4188 16.5927; 3.3579 14.3951 10.2129; ...
         6.3092 8.5986 9.8934; 3.2831 3.4945 10.0203];

 [p, atab, stats] = anova2 (data, 4, "on", "nested");


                      ANOVA Table

Source             SS      df        MS       F      Prob>F
-----------------------------------------------------------
Columns        745.3603     2   372.6802     4.02    0.1416
Rows           278.0185     3    92.6728     9.26    0.0006
Error          180.1804    18    10.0100
Total         1203.5592    23

Note: Rows are a random factor nested within the columns.
The Column F statistic uses the Row MS instead of the MSE.