# fitrm

Fit repeated measures model

## Description

returns a repeated measures model, specified by `rm`

= fitrm(`t`

,`modelspec`

)`modelspec`

, fitted
to the variables in the table or dataset array `t`

. The model
`rm`

is returned as a `RepeatedMeasuresModel`

object. For more information about the properties of this model object, see `RepeatedMeasuresModel`

.

returns a repeated measures model with additional options specified by one or more
name-value arguments. For example, you can specify the hypothesis for the
within-subject factors.`rm`

= fitrm(`t`

,`modelspec`

,`Name=Value`

)

## Examples

### Fit a Repeated Measures Model

Load the `fisheriris`

data set.

`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 for the flowers: the length and width of sepals and petals in centimeters.

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`

with these values:

$$\begin{array}{c}species\_setosa=\{\begin{array}{ll}1& if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\\ 0& if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1& if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\end{array}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}and\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}species\_versicolor=\{\begin{array}{ll}0& if\phantom{\rule{0.2777777777777778em}{0ex}}setosa\\ 1& if\phantom{\rule{0.2777777777777778em}{0ex}}versicolor\\ -1& if\phantom{\rule{0.2777777777777778em}{0ex}}virginica\end{array}\end{array}$$

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

### Specify Within-Subject Hypothesis

Load the sample data.

`load('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 data is simulated.

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 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-subject 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]

### Fit Model with Covariates

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`

are continuous variables. The table `within`

includes the within-subject factors `w1`

and `w2`

.

Fit a repeated measures model, where age, IQ, group, and 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

`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.

The variable names in `t`

must be valid MATLAB^{®} identifiers. You can verify the variable names by using the
`isvarname`

function. If
the variable names are not valid, then you can convert them by using the `matlab.lang.makeValidName`

function.

**Data Types: **`table`

`modelspec`

— Formula for model specification

character vector or string scalar of the form ```
'y1-yk ~
terms'
```

Formula for the 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 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: **`WithinDesign="W",WithinModel="w1+w2"`

specifies the matrix
`w`

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

and `w2`

as `"w1+w2"`

.

`WithinDesign`

— Design for within-subject factors

numeric vector of length *r* (default) | *r*-by-*k* numeric matrix | *r*-by-*k* table

Design for the within-subject factors, specified as 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. These values 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

Model specifying the within-subject hypothesis test, specified as one of the following:

`'separatemeans'`

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

— This value 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 of 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 you have 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`

## More About

### 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 Notation | Description |
---|---|

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

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

The following rules specify the 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 includes a constant term unless you explicitly remove the
term using `-1`

.

## Version History

**Introduced in R2014a**

## See Also

`RepeatedMeasuresModel`

| `manova`

| `manova1`

| `ranova`

| `anova`

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