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Fit repeated measures model

`rm = fitrm(t,modelspec)`

`rm = fitrm(t,modelspec,Name,Value)`

returns
a repeated measures model, with additional options specified by one
or more `rm`

= fitrm(`t`

,`modelspec`

,`Name,Value`

)`Name,Value`

pair arguments.

For example, you can specify the hypothesis for the within-subject factors.

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 = table([1 2 3 4]','VariableNames',{'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)

rm = RepeatedMeasuresModel with properties: Between Subjects: BetweenDesign: [150x5 table] ResponseNames: {'meas1' 'meas2' 'meas3' 'meas4'} BetweenFactorNames: {'species'} BetweenModel: '1 + species' Within Subjects: WithinDesign: [4x1 table] WithinFactorNames: {'Measurements'} WithinModel: 'separatemeans' Estimates: Coefficients: [3x4 table] Covariance: [4x4 table]

Display the coefficients.

rm.Coefficients

`ans=`*3×4 table*
meas1 meas2 meas3 meas4
________ ________ ______ ________
(Intercept) 5.8433 3.0573 3.758 1.1993
species_setosa -0.83733 0.37067 -2.296 -0.95333
species_versicolor 0.092667 -0.28733 0.502 0.12667

`fitrm`

uses the `'effects'`

contrasts which means that the coefficients sum to 0. The `rm.DesignMatrix`

has one column of 1s for the intercept, and two other columns `species_setosa`

and `species_versicolor`

, which are as follows:

$$\begin{array}{c}species\_setosa=\{\begin{array}{ll}1& if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\\ 0& if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1& if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\end{array}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}and\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}species\_versicolor=\{\begin{array}{ll}0& if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\\ 1& if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1& if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\end{array}\end{array}$$

Display the covariance matrix.

rm.Covariance

`ans=`*4×4 table*
meas1 meas2 meas3 meas4
________ ________ ________ ________
meas1 0.26501 0.092721 0.16751 0.038401
meas2 0.092721 0.11539 0.055244 0.03271
meas3 0.16751 0.055244 0.18519 0.042665
meas4 0.038401 0.03271 0.042665 0.041882

Load the sample data.

load(fullfile(matlabroot,'examples','stats','longitudinalData.mat'));

The matrix `Y`

contains response data for 16 individuals. The response is the blood level of a drug measured at five time points (time = 0, 2, 4, 6, and 8). Each row of `Y`

corresponds to an individual, and each column corresponds to a time point. The first eight subjects are female, and the second eight subjects are male. This is simulated data.

Define a variable that stores gender information.

Gender = ['F' 'F' 'F' 'F' 'F' 'F' 'F' 'F' 'M' 'M' 'M' 'M' 'M' 'M' 'M' 'M']';

Store the data in a proper table array format to conduct repeated measures analysis.

t = table(Gender,Y(:,1),Y(:,2),Y(:,3),Y(:,4),Y(:,5),... 'VariableNames',{'Gender','t0','t2','t4','t6','t8'});

Define the within-subjects variable.

Time = [0 2 4 6 8]';

Fit a repeated measures model, where blood levels are the responses and gender is the predictor variable. Also define the hypothesis for within-subject factors.

rm = fitrm(t,'t0-t8 ~ Gender','WithinDesign',Time,'WithinModel','orthogonalcontrasts')

rm = RepeatedMeasuresModel with properties: Between Subjects: BetweenDesign: [16x6 table] ResponseNames: {'t0' 't2' 't4' 't6' 't8'} BetweenFactorNames: {'Gender'} BetweenModel: '1 + Gender' Within Subjects: WithinDesign: [5x1 table] WithinFactorNames: {'Time'} WithinModel: 'orthogonalcontrasts' Estimates: Coefficients: [2x5 table] Covariance: [5x5 table]

Load the sample data.

`load repeatedmeas`

The table `between`

includes the eight repeated measurements, `y1`

through `y8`

, as responses and the between-subject factors `Group`

, `Gender`

, `IQ`

, and `Age`

. `IQ`

and `Age`

as continuous variables. The table `within`

includes the within-subject factors `w1`

and `w2`

.

Fit a repeated measures model, where age, IQ, and group, gender are the predictor variables, and the model includes the interaction effect of group and gender. Also define the within-subject factors.

rm = fitrm(between,'y1-y8 ~ Group*Gender+Age+IQ','WithinDesign',within)

rm = RepeatedMeasuresModel with properties: Between Subjects: BetweenDesign: [30x12 table] ResponseNames: {'y1' 'y2' 'y3' 'y4' 'y5' 'y6' 'y7' 'y8'} BetweenFactorNames: {'Age' 'IQ' 'Group' 'Gender'} BetweenModel: '1 + Age + IQ + Group*Gender' Within Subjects: WithinDesign: [8x2 table] WithinFactorNames: {'w1' 'w2'} WithinModel: 'separatemeans' Estimates: Coefficients: [8x8 table] Covariance: [8x8 table]

Display the coefficients.

rm.Coefficients

`ans=`*8×8 table*
y1 y2 y3 y4 y5 y6 y7 y8
________ _______ _______ _______ _________ ________ _______ ________
(Intercept) 141.38 195.25 9.8663 -49.154 157.77 0.23762 -42.462 76.111
Age 0.32042 -4.7672 -1.2748 0.6216 -1.0621 0.89927 1.2569 -0.38328
IQ -1.2671 -1.1653 0.05862 0.4288 -1.4518 -0.25501 0.22867 -0.72548
Group_A -1.2195 -9.6186 22.532 15.303 12.602 12.886 10.911 11.487
Group_B 2.5186 1.417 -2.2501 0.50181 8.0907 3.1957 11.591 9.9188
Gender_Female 5.3957 -3.9719 8.5225 9.3403 6.0909 1.642 -2.1212 4.8063
Group_A:Gender_Female 4.1046 10.064 -7.3053 -3.3085 4.6751 2.4907 -4.325 -4.6057
Group_B:Gender_Female -0.48486 -2.9202 1.1222 0.69715 -0.065945 0.079468 3.1832 6.5733

The display shows the coefficients for fitting the repeated measures as a function of the terms in the between-subjects model.

`t`

— Input data table

Input data, which includes the values of the response variables and the between-subject factors to use as predictors in the repeated measures model, specified as a table.

**Data Types: **`table`

`modelspec`

— Formula for model specificationcharacter vector or string scalar of the form

```
'y1-yk ~
terms'
```

Formula for model specification, specified as a character vector or string scalar of the form
`'y1-yk ~ terms'`

. The responses and terms are
specified using Wilkinson
notation. `fitrm`

treats the variables used in
model terms as categorical if they are categorical (nominal or ordinal),
logical, character arrays, string arrays, or cell arrays of character
vectors.

For example, if you have four repeated measures as responses
and the factors `x1`

, `x2`

, and `x3`

as
the predictor variables, then you can define a repeated measures model
as follows.

**Example: **`'y1-y4 ~ x1 + x2 * x3'`

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`

.

`'WithinDesign','W','WithinModel','w1+w2'`

specifies
the matrix `w`

as the design matrix for within-subject
factors, and the model for within-subject factors `w1`

and `w2`

is `'w1+w2'`

.`'WithinDesign'`

— Design for within-subject factorsnumeric vector of length

Design for within-subject factors, specified as the comma-separated
pair consisting of `'WithinDesign'`

and one of the
following:

Numeric vector of length

*r*, where*r*is the number of repeated measures.In this case,

`fitrm`

treats the values in the vector as continuous, and these are typically time values.*r*-by-*k*numeric matrix of the values of the*k*within-subject factors,*w*_{1},*w*_{2}, ...,*w*._{k}In this case,

`fitrm`

treats all*k*variables as continuous.*r*-by-*k*table that contains the values of the*k*within-subject factors.In this case,

`fitrm`

treats all numeric variables as continuous, and all categorical variables as categorical.

For example, if the table `weeks`

contains
the values of the within-subject factors, then you can define the
design table as follows.

**Example: **`'WithinDesign',weeks`

**Data Types: **`single`

| `double`

| `table`

`'WithinModel'`

— Model specifying within-subject hypothesis test`'separatemeans'`

(default) | `'orthogonalcontrasts'`

| character vector or string scalar that defines a modelModel specifying the within-subject hypothesis test, specified
as the comma-separated pair consisting of `'WithinModel'`

and
one of the following:

`'separatemeans'`

— Compute a separate mean for each group.`'orthogonalcontrasts'`

— This is valid only when the within-subject model has a single numeric factor*T*. Responses are the average, the slope of centered*T*, and, in general, all orthogonal contrasts for a polynomial up to*T*^(*p*– 1), where*p*is the number if rows in the within-subject model.A character vector or string scalar that defines a model specification in the within-subject factors. You can define the model based on the rules for the

`terms`

in`modelspec`

.

For example, if there are three within-subject factors `w1`

, `w2`

,
and `w3`

, then you can specify a model for the within-subject
factors as follows.

**Example: **`'WithinModel','w1+w2+w2*w3'`

**Data Types: **`char`

| `string`

`rm`

— Repeated measures model`RepeatedMeasuresModel`

objectRepeated measures model, returned as a `RepeatedMeasuresModel`

object.

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

.

Wilkinson notation describes the factors present in models. It does not describe the multipliers (coefficients) of those factors.

The following rules specify the responses in `modelspec`

.

Wilkinson Notation | Meaning |
---|---|

`Y1,Y2,Y3` | Specific list of variables |

`Y1-Y5` | All table variables from `Y1` through `Y5` |

The following rules specify terms in `modelspec`

.

Wilkinson notation | Factors in Standard Notation |
---|---|

`1` | Constant (intercept) term |

`X^k` , where `k` is a positive
integer | `X` , `X` ,
..., `X` |

`X1 + X2` | `X1` , `X2` |

`X1*X2` | `X1` , `X2` , `X1*X2` |

`X1:X2` | `X1*X2` only |

`-X2` | Do not include `X2` |

`X1*X2 + X3` | `X1` , `X2` , `X3` , `X1*X2` |

`X1 + X2 + X3 + X1:X2` | `X1` , `X2` , `X3` , `X1*X2` |

`X1*X2*X3 - X1:X2:X3` | `X1` , `X2` , `X3` , `X1*X2` , `X1*X3` , `X2*X3` |

`X1*(X2 + X3)` | `X1` , `X2` , `X3` , `X1*X2` , `X1*X3` |

Statistics and Machine
Learning Toolbox™ notation always includes a constant term
unless you explicitly remove the term using `-1`

.

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