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**Class: **RepeatedMeasuresModel

Multiple comparison of estimated marginal means

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 model`RepeatedMeasuresModel`

objectRepeated measures model, returned as a `RepeatedMeasuresModel`

object.

For properties and methods of this object, see `RepeatedMeasuresModel`

.

`var`

— Variables for which to compute marginal meanscharacter 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`

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`

.

`'Alpha'`

— Significance level0.05 (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 bycharacter 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 $$\left|t\right|=\frac{\left|{\overline{y}}_{i}-{\overline{y}}_{j}\right|}{\sqrt{MSE\left(\frac{1}{{n}_{i}}+\frac{1}{{n}_{j}}\right)}}>{t}_{1-\eta /2,v,}$$ where $$\eta =1-{\left(1-\alpha \right)}^{{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(\begin{array}{l}k\\ 2\end{array}\right)$}\right.}}$$ and |

`'bonferroni'` | Use critical values from the $${t}_{\alpha /2\left(\begin{array}{c}ng\\ 2\end{array}\right),v},$$ where |

`'scheffe'` | Use critical values from Scheffé's $$\sqrt{\left(ng-1\right){F}_{\alpha ,ng-1,v}},$$ where |

`'lsd'` |
Least significant difference. This option uses plain
$${t}_{\alpha /2,v},$$ where |

**Example: **`'ComparisonType','dunn-sidak'`

`tbl`

— Results of multiple comparisontable

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 |

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 $$p$$-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=`*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 $$p$$-value of 0.0021493 indicates that there is significant difference between the marginal means of groups A and C. The $$p$$-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 $$p$$-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=`*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 $$p$$-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.

`fitrm`

| `margmean`

| `plotprofile`

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