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.
rm
— Repeated measures modelRepeatedMeasuresModel
objectRepeated measures model, returned as a RepeatedMeasuresModel
object.
For properties and methods of this object, see RepeatedMeasuresModel
.
var
— Variables for which to compute marginal meansstringVariables for which to compute the marginal means, specified
as a string 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
| cell
Specify optional comma-separated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
'Alpha'
— Confidence level0.05 (default) | scalar value in the range of 0 through 1Confidence 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 bystringFactor to do the comparisons by, specified as the comma-separated
pair consisting of 'By'
and a string. The comparison
between levels of var
occurs separately for each
value of the factor you specify.
If you have more then 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
'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 $$\left|t\right|=\frac{\left|{\overline{y}}_{i}-{\overline{y}}_{j}\right|}{\sqrt{MSE\left(\frac{1}{{n}_{i}}+\frac{1}{{n}_{j}}\right)}}>{t}_{1-\eta /2,v,}$$ $$\eta =1-{\left(1-\alpha \right)}^{{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(\begin{array}{l}k\\ 2\end{array}\right)$}\right.}}$$ |
'bonferroni' | Use critical values from the t distribution,
after a Bonferroni adjustment to compensate for multiple comparisons.
The critical value is $${t}_{\alpha /2\left(\begin{array}{c}ng\\ 2\end{array}\right),v},$$ |
'scheffe' | Use critical values from Scheffé's S procedure,
derived from the Fdistribution. The critical value
is $$\sqrt{\left(ng-1\right){F}_{\alpha ,ng-1,v}},$$ |
'lsd' | Least significant difference. This option uses plain t-tests.
The critical value is $${t}_{\alpha /2,v},$$ |
Example: 'ComparisonType','dunn-sidak'
tbl
— Results of multiple comparisontableResults 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 |
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 = 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 p-values (in the pValue
field)
indicate that the estimated marginal means for the three species significantly
differ from each other.
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 = 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 p-value of 0.0021493 indicates that there is significant difference between the marginal means of groups A and C. The p-values 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 = 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 p-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 = 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 p-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 p-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 (p-value is 0.049957), it is not significantly different from 0 for males (p-value is 0.14414).
[1] G. A. Milliken, and Johnson, D. E. Analysis of Messy Data. Volume I: Designed Experiments. New York, NY: Chapman & Hall, 1992.
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