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# fitrm

Fit repeated measures model

## Syntax

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

## Description

example

````rm = fitrm(t,modelspec)` returns a repeated measures model, specified by `modelspec`, fitted to the variables in the table or dataset array `t`.```

example

````rm = fitrm(t,modelspec,Name,Value)` returns a repeated measures model, with additional options specified by one or more `Name,Value` pair arguments.For example, you can specify the hypothesis for the within-subject factors.```

## Examples

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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:

`$\begin{array}{c}species_setosa=\left\{\begin{array}{ll}1& if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\\ 0& if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1& if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\end{array}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}and\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}species_versicolor=\left\{\begin{array}{ll}0& if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\\ 1& if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1& if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\end{array}\end{array}$`

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.

## Input Arguments

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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`

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'`

### Name-Value Pair 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: `'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'`.

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, w1, w2, ..., wk.

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`

Model 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`

## Output Arguments

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Repeated measures model, returned as a `RepeatedMeasuresModel` object.

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

## More About

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### Model Specification Using Wilkinson Notation

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 NotationMeaning
`Y1,Y2,Y3`Specific list of variables
`Y1-Y5`All table variables from `Y1` through `Y5`

The following rules specify terms in `modelspec`.

Wilkinson notationFactors in Standard Notation
`1`Constant (intercept) term
`X^k`, where `k` is a positive integer`X`, `X2`, ..., `Xk`
`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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