multcompare

Multiple comparison test

Syntax

c = multcompare(stats)

c = multcompare(stats,Name,Value)

[c,m] = multcompare(___)

[c,m,h] = multcompare(___)

[c,m,h,gnames] = multcompare(___)

Description

c = multcompare(stats) returns a matrix c of the pairwise comparison results from a multiple comparison test using the information contained in the stats structure. multcompare also displays an interactive graph of the estimates and comparison intervals. Each group mean is represented by a symbol, and the interval is represented by a line extending out from the symbol. Two group means are significantly different if their intervals are disjoint; they are not significantly different if their intervals overlap. If you use your mouse to select any group, then the graph will highlight all other groups that are significantly different, if any.

c = multcompare(stats,Name,Value) returns a matrix of pairwise comparison results, c, using additional options specified by one or more Name,Value pair arguments. For example, you can specify the confidence interval, or the type of critical value to use in the multiple comparison.

[c,m] = multcompare(___) also returns a matrix, m, which contains estimated values of the means (or whatever statistics are being compared) for each group and the corresponding standard errors. You can use any of the previous syntaxes.

[c,m,h] = multcompare(___) also returns a handle, h, to the comparison graph.

[c,m,h,gnames] = multcompare(___) also returns a cell array, gnames, which contains the names of the groups.

Examples

Multiple Comparison of Group Means

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]}

Multiple Comparisons for Two-Way ANOVA

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

• The mean squared error (sigmasq)

• The estimates of the mean yield for each popcorn brand (colmeans)

• The number of observations for each popcorn brand (coln)

• The estimate of the mean yield for each popper type (rowmeans)

• The number of observations for each popper type (rown)

• The number of interactions (inter)

• The p-value that shows the significance level of the interaction term (pval)

• The error degrees of freedom (df).

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.

Multiple Comparisons for Three-Way ANOVA

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    0.0272
    1.0000    3.0000    4.4896    6.9500    9.4104    0.0170
    1.0000    4.0000    6.1396    8.6000   11.0604    0.0136
    2.0000    3.0000    8.8896   11.3500   13.8104    0.0101
    2.0000    4.0000   10.5396   13.0000   15.4604    0.0087
    3.0000    4.0000   -0.8104    1.6500    4.1104    0.0737

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.

Input Arguments

stats — Test data

structure

Test data, specified as a structure. You can create a structure using one of the following functions:

• anova1 — One-way analysis of variance.

• anova2 — Two-way analysis of variance.

• anovan — N-way analysis of variance.

• aoctool — Interactive analysis of covariance tool.

• friedman — Friedman’s test.

• kruskalwallis — Kruskal-Wallis test.

multcompare does not support multiple comparisons using anovan output for a model that includes random or nested effects. The calculations for a random effects model produce a warning that all effects are treated as fixed. Nested models are not accepted.

Data Types: struct

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Alpha',0.01,'CType','bonferroni','Display','off' computes the Bonferroni critical values, conducts the hypothesis tests at the 1% significance level, and omits the interactive display.

Alpha — Significance level

0.05 (default) | scalar value in the range (0,1)

Significance level of the multiple comparison test, specified as the comma-separated pair consisting of 'Alpha' and a scalar value in the range (0,1). The value specified for 'Alpha' determines the 100 × (1 – α) confidence levels of the intervals returned in the matrix c and in the figure.

Example: 'Alpha',0.01

Data Types: single | double

CType — Type of critical value

'tukey-kramer' (default) | 'hsd' | 'lsd' | 'bonferroni' | 'dunn-sidak' | 'scheffe'

Type of critical value to use for the multiple comparison, specified as the comma-separated pair consisting of 'CType' and one of the following.

Value	Description
'tukey-kramer' or 'hsd'	Tukey's honest significant difference criterion
'bonferroni'	Bonferroni method
'dunn-sidak'	Dunn and Sidák’s approach
'lsd'	Fisher's least significant difference procedure
'scheffe'	Scheffé's S procedure

Example: 'CType','bonferroni'

Display — Display toggle

'on' (default) | 'off'

Display toggle, specified as the comma-separated pair consisting of 'Display' and either 'on' or 'off'. If you specify 'on', then multcompare displays a graph of the estimates and their comparison intervals. If you specify 'off', then multcompare omits the graph.

Example: 'Display','off'

Dimension — Dimension over which to calculate marginal means

1 (default) | positive integer value | vector of positive integer values

A vector specifying the dimension or dimensions over which to calculate the population marginal means, specified as a positive integer value, or a vector of such values. Use the 'Dimension' name-value pair only if you create the input structure stats using the function anovan.

For example, if you specify 'Dimension' as 1, then multcompare compares the means for each value of the first grouping variable, adjusted by removing effects of the other grouping variables as if the design were balanced. If you specify 'Dimension'as [1,3], then multcompare computes the population marginal means for each combination of the first and third grouping variables, removing effects of the second grouping variable. If you fit a singular model, some cell means may not be estimable and any population marginal means that depend on those cell means will have the value NaN.

Population marginal means are described by Milliken and Johnson (1992) and by Searle, Speed, and Milliken (1980). The idea behind population marginal means is to remove any effect of an unbalanced design by fixing the values of the factors specified by 'Dimension', and averaging out the effects of other factors as if each factor combination occurred the same number of times. The definition of population marginal means does not depend on the number of observations at each factor combination. For designed experiments where the number of observations at each factor combination has no meaning, population marginal means can be easier to interpret than simple means ignoring other factors. For surveys and other studies where the number of observations at each combination does have meaning, population marginal means may be harder to interpret.

Example: 'Dimension',[1,3]

Data Types: single | double

Estimate — Estimates to be compared

'column' (default) | 'row' | 'slope' | 'intercept' | 'pmm'

Estimates to be compared, specified as the comma-separated pair consisting of 'Estimate' and an allowable value. The allowable values for 'Estimate' depend on the function used to generate the input structure stats, according to the following table.

Source	Values
anova1	None. This name-value pair is ignored, and multcompare always compares the group means.
anova2	Either 'column' to compare column means, or 'row' to compare row means.
anovan	None. This name-value pair is ignored, and multcompare always compares the population marginal means as specified by the 'Dimension' name-value pair argument.
aoctool	Either 'slope', 'intercept', or 'pmm' to compare slopes, intercepts, or population marginal means, respectively. If the analysis of covariance model did not include separate slopes, then 'slope' is not allowed. If it did not include separate intercepts, then no comparisons are possible.
friedman	None. This name-value pair is ignored, and multcompare always compares the average column ranks.
kruskalwallis	None. This name-value pair is ignored, and multcompare always compares the average group ranks.

Example: 'Estimate','row'

Output Arguments

c — Matrix of multiple comparison results

matrix of scalar values

Matrix of multiple comparison results, returned as an p-by-6 matrix of scalar values, where p is the number of pairs of groups. Each row of the matrix contains the result of one paired comparison test. Columns 1 and 2 contain the indices of the two samples being compared. Column 3 contains the lower confidence interval, column 4 contains the estimate, and column 5 contains the upper confidence interval. Column 6 contains the p-value for the hypothesis test that the corresponding mean difference is not equal to 0.

For example, suppose one row contains the following entries.

2.0000  5.0000  1.9442  8.2206  14.4971 0.0432

These numbers indicate that the mean of group 2 minus the mean of group 5 is estimated to be 8.2206, and a 95% confidence interval for the true difference of the means is [1.9442, 14.4971]. The p-value for the corresponding hypothesis test that the difference of the means of groups 2 and 5 is significantly different from zero is 0.0432.

In this example the confidence interval does not contain 0, so the difference is significant at the 5% significance level. If the confidence interval did contain 0, the difference would not be significant. The p-value of 0.0432 also indicates that the difference of the means of groups 2 and 5 is significantly different from 0.

m — Matrix of estimates

matrix of scalar values

Matrix of the estimates, returned as a matrix of scalar values. The first column of m contains the estimated values of the means (or whatever statistics are being compared) for each group, and the second column contains their standard errors.

h — Handle to the figure

handle

Handle to the figure containing the interactive graph, returned as a handle. The title of this graph contains instructions for interacting with the graph, and the x-axis label contains information about which means are significantly different from the selected mean. If you plan to use this graph for presentation, you may want to omit the title and the x-axis label. You can remove them using interactive features of the graph window, or you can use the following commands.

title('')
xlabel('')

gnames — Group names

cell array of character vectors

Group names, returned as a cell array of character vectors. Each row of gnames contains the name of a group.

More About

Multiple Comparison Tests

Analysis of variance compares the means of several groups to test the hypothesis that they are all equal, against the general alternative that they are not all equal. Sometimes this alternative may be too general. You may need information about which pairs of means are significantly different, and which are not. A multiple comparison test can provide this information.

When you perform a simple t-test of one group mean against another, you specify a significance level that determines the cutoff value of the t-statistic. For example, you can specify the value alpha = 0.05 to insure that when there is no real difference, you will incorrectly find a significant difference no more than 5% of the time. When there are many group means, there are also many pairs to compare. If you applied an ordinary t-test in this situation, the alpha value would apply to each comparison, so the chance of incorrectly finding a significant difference would increase with the number of comparisons. Multiple comparison procedures are designed to provide an upper bound on the probability that any comparison will be incorrectly found significant.

References

[1] Hochberg, Y., and A. C. Tamhane. Multiple Comparison Procedures. Hoboken, NJ: John Wiley & Sons, 1987.

[2] Milliken, G. A., and D. E. Johnson. Analysis of Messy Data, Volume I: Designed Experiments. Boca Raton, FL: Chapman & Hall/CRC Press, 1992.

[3] Searle, S. R., F. M. Speed, and G. A. Milliken. “Population marginal means in the linear model: an alternative to least-squares means.” American Statistician. 1980, pp. 216–221.