multcompare
Multiple comparison of estimated marginal means
Description
returns
multiple comparisons of the estimated marginal means with additional
options specified by one or more tbl
= multcompare(rm
,var
,Name,Value
)Name,Value
pair
arguments.
For example, you can specify the comparison type or which variable to group by.
Examples
Multiple Comparison of Estimated Marginal Means
Load the sample data.
load fisheriris
The column vector species
consists of iris flowers of three different species: setosa, versicolor, and virginica. The double matrix meas
consists of four types of measurements on the flowers: the length and width of sepals and petals in centimeters, respectively.
Store the data in a table array.
t = table(species,meas(:,1),meas(:,2),meas(:,3),meas(:,4),... 'VariableNames',{'species','meas1','meas2','meas3','meas4'}); Meas = dataset([1 2 3 4]','VarNames',{'Measurements'});
Fit a repeated measures model, where the measurements are the responses and the species is the predictor variable.
rm = fitrm(t,'meas1-meas4~species','WithinDesign',Meas);
Perform a multiple comparison of the estimated marginal means of species.
tbl = multcompare(rm,'species')
tbl=6×7 table
species_1 species_2 Difference StdErr pValue Lower Upper
______________ ______________ __________ ________ __________ ________ ________
{'setosa' } {'versicolor'} -1.0375 0.060539 9.5606e-10 -1.1794 -0.89562
{'setosa' } {'virginica' } -1.7495 0.060539 9.5606e-10 -1.8914 -1.6076
{'versicolor'} {'setosa' } 1.0375 0.060539 9.5606e-10 0.89562 1.1794
{'versicolor'} {'virginica' } -0.712 0.060539 9.5606e-10 -0.85388 -0.57012
{'virginica' } {'setosa' } 1.7495 0.060539 9.5606e-10 1.6076 1.8914
{'virginica' } {'versicolor'} 0.712 0.060539 9.5606e-10 0.57012 0.85388
The small -values (in the pValue
field) indicate that the estimated marginal means for the three species significantly differ from each other.
Perform Multiple Comparisons with Specified Options
Load the sample data.
load repeatedmeas
The table between
includes the between-subject variables age, IQ, group, gender, and eight repeated measures y1
through y8
as responses. The table within
includes the within-subject variables w1
and w2
. This is simulated data.
Fit a repeated measures model, where the repeated measures y1
through y8
are the responses, and age, IQ, group, gender, and the group-gender interaction are the predictor variables. Also specify the within-subject design matrix.
R = fitrm(between,'y1-y8 ~ Group*Gender + Age + IQ','WithinDesign',within);
Perform a multiple comparison of the estimated marginal means based on the variable Group
.
T = multcompare(R,'Group')
T=6×7 table
Group_1 Group_2 Difference StdErr pValue Lower Upper
_______ _______ __________ ______ _________ _______ _______
A B 4.9875 5.6271 0.65436 -9.1482 19.123
A C 23.094 5.9261 0.0021493 8.2074 37.981
B A -4.9875 5.6271 0.65436 -19.123 9.1482
B C 18.107 5.8223 0.013588 3.4805 32.732
C A -23.094 5.9261 0.0021493 -37.981 -8.2074
C B -18.107 5.8223 0.013588 -32.732 -3.4805
The small -value of 0.0021493 indicates that there is significant difference between the marginal means of groups A and C. The -value of 0.65436 indicates that the difference between the marginal means for groups A and B is not significantly different from 0.
multcompare
uses the Tukey-Kramer test statistic by default. Change the comparison type to the Scheffe procedure.
T = multcompare(R,'Group','ComparisonType','Scheffe')
T=6×7 table
Group_1 Group_2 Difference StdErr pValue Lower Upper
_______ _______ __________ ______ _________ _______ _______
A B 4.9875 5.6271 0.67981 -9.7795 19.755
A C 23.094 5.9261 0.0031072 7.5426 38.646
B A -4.9875 5.6271 0.67981 -19.755 9.7795
B C 18.107 5.8223 0.018169 2.8273 33.386
C A -23.094 5.9261 0.0031072 -38.646 -7.5426
C B -18.107 5.8223 0.018169 -33.386 -2.8273
The Scheffe test produces larger -values, but similar conclusions.
Perform multiple comparisons of estimated marginal means based on the variable Group
for each gender separately.
T = multcompare(R,'Group','By','Gender')
T=12×8 table
Gender Group_1 Group_2 Difference StdErr pValue Lower Upper
______ _______ _______ __________ ______ ________ _________ __________
Female A B 4.1883 8.0177 0.86128 -15.953 24.329
Female A C 24.565 8.2083 0.017697 3.9449 45.184
Female B A -4.1883 8.0177 0.86128 -24.329 15.953
Female B C 20.376 8.1101 0.049957 0.0033459 40.749
Female C A -24.565 8.2083 0.017697 -45.184 -3.9449
Female C B -20.376 8.1101 0.049957 -40.749 -0.0033459
Male A B 5.7868 7.9498 0.74977 -14.183 25.757
Male A C 21.624 8.1829 0.038022 1.0676 42.179
Male B A -5.7868 7.9498 0.74977 -25.757 14.183
Male B C 15.837 8.0511 0.14414 -4.3881 36.062
Male C A -21.624 8.1829 0.038022 -42.179 -1.0676
Male C B -15.837 8.0511 0.14414 -36.062 4.3881
The results indicate that the difference between marginal means for groups A and B is not significant from 0 for either gender (corresponding -values are 0.86128 for females and 0.74977 for males). The difference between marginal means for groups A and C is significant for both genders (corresponding -values are 0.017697 for females and 0.038022 for males). While the difference between marginal means for groups B and C is significantly different from 0 for females (-value is 0.049957), it is not significantly different from 0 for males (-value is 0.14414).
Input Arguments
rm
— Repeated measures model
RepeatedMeasuresModel
object
Repeated measures model, returned as a RepeatedMeasuresModel
object.
For properties and methods of this object, see RepeatedMeasuresModel
.
var
— Variables for which to compute marginal means
character vector | string scalar
Variables for which to compute the marginal means, specified as a character vector or string
scalar representing the name of a between- or within-subjects factor in
rm
. If var
is a between-subjects factor, it
must be categorical.
Data Types: char
| string
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: tbl = multcompare(rm,var,'By',C)
Alpha
— Significance level
0.05 (default) | scalar value in the range of 0 through 1
Significance level of the confidence intervals for population marginal means,
specified as the comma-separated pair consisting of 'alpha'
and a
scalar value in the range of 0 through 1. The confidence level is
100*(1–alpha
)%.
Example: 'alpha',0.01
Data Types: double
| single
By
— Factor to perform comparisons by
character vector | string scalar
Factor to do the comparisons by, specified as the comma-separated pair consisting of
'By'
and a character vector or string scalar. The comparison
between levels of var
occurs separately for each value of the
factor you specify.
If you have more than one between-subjects factors, A,
B, and C, and if you want to do the comparisons
of A levels separately for each level of C, then
specify A as the var
argument and specify
C using the 'By'
argument as follows.
Example: 'By',C
Data Types: char
| string
ComparisonType
— Type of critical value to use
'tukey-kramer'
(default) | 'dunn-sidak'
| 'bonferroni'
| 'scheffe'
| 'lsd'
Type of critical value to use, specified as the comma-separated
pair consisting of 'ComparisonType'
and one of
the following.
Comparison Type | Definition |
---|---|
'tukey-kramer' | Default. Also called Tukey’s Honest Significant Difference procedure. It is based on the Studentized range distribution. According to the unproven Tukey-Kramer conjecture, it is also accurate for problems where the quantities being compared are correlated, as in analysis of covariance with unbalanced covariate values. |
'dunn-sidak' |
Use critical values from the t distribution, after an adjustment for multiple comparisons that was proposed by Dunn and proved accurate by Sidák. The critical value is where and ng is the number of groups (marginal means). This procedure is similar to, but less conservative than, the Bonferroni procedure. |
'bonferroni' | Use critical values from the t distribution, after a Bonferroni adjustment to compensate for multiple comparisons. The critical value is where ng is the number of groups (marginal means), and v is the error degrees of freedom. This procedure is conservative, but usually less so than the Scheffé procedure. |
'scheffe' | Use critical values from Scheffé's S procedure, derived from the F distribution. The critical value is where ng is the number of groups (marginal means), and v is the error degrees of freedom. This procedure provides a simultaneous confidence level for comparisons of all linear combinations of the means, and it is conservative for comparisons of simple differences of pairs. |
'lsd' |
Least significant difference. This option uses plain t-tests. The critical value is where v is the error degrees of freedom. It provides no protection against the multiple comparison problem. |
Example: 'ComparisonType','dunn-sidak'
Output Arguments
tbl
— Results of multiple comparison
table
Results of multiple comparisons of estimated marginal means,
returned as a table. tbl
has the following columns.
Column Name | Description |
---|---|
Difference | Estimated difference between the corresponding two marginal means |
StdErr | Standard error of the estimated difference between the corresponding two marginal means |
pValue | p-value for a test that the difference between the corresponding two marginal means is 0 |
Lower | Lower limit of simultaneous 95% confidence intervals for the true difference |
Upper | Upper limit of simultaneous 95% confidence intervals for the true difference |
References
[1] G. A. Milliken, and Johnson, D. E. Analysis of Messy Data. Volume I: Designed Experiments. New York, NY: Chapman & Hall, 1992.
Version History
Introduced in R2014a
See Also
margmean
| plotprofile
| fitrm
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