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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 = 3x4 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:

Display the covariance matrix.

rm.Covariance

ans = 4x4 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 = 8x8 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 of the form

`'y1-yk ~ terms'`

Formula for model specification, specified as a character vector
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, char arrays, or a 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 single 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 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 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`

`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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