RepeatedMeasuresModel class

Repeated measures model class

Description

A RepeatedMeasuresModel object represents a model fitted to data with multiple measurements per subject. The object comprises data, fitted coefficients, covariance parameters, design matrix, error degrees of freedom, and between- and within-subjects factor names for a repeated measures model. You can predict model responses using the predict method and generate random data at new design points using the random method.

Construction

You can fit a repeated measures model using fitrm(t,modelspec).

Input Arguments

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t — Input datatable

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 specificationstring of the form 'y1-yk ~ terms'

Formula for model specification, specified as a string of the form 'y1-yk ~ terms'. Specify the terms 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 array of strings.

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

Properties

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BetweenDesign Design for between-subject factorstable

Design for between-subject factors and values of repeated measures, stored as a table.

Data Types: table

BetweenModelModel for between-subjects factorsstring

Model for between-subjects factors, stored as a string. This string is the text representation to the right of the tilde in the model specification you provide when fitting the repeated measures model using fitrm.

Data Types: char

BetweenFactorNamesNames of variables used as between-subject factorscell array of strings

Names of variables used as between-subject factors in the repeated measures model, rm, stored as a cell array of strings.

Data Types: cell

ResponseNamesNames of variables used as response variablescell array of strings

Names of variables used as response variables in the repeated measures model, rm, stored as a cell array of strings.

Data Types: cell

WithinDesignValues of within-subject factorstable

Values of the within-subject factors, stored as a table.

Data Types: table

WithinModelModel for within-subjects factorsstring

Model for within-subjects factors, stored as a string.

Data Types: char

WithinFactorNamesNames of within-subject factorscell array of strings

Names of the within-subject factors, stored as a cell array of strings.

Data Types: cell

CoefficientsValues of estimated coefficientstable

Values of the estimated coefficients for fitting the repeated measures as a function of the terms in the between-subjects model, stored as a table.

fitrm' defines the coefficients for a categorical term using 'effects' coding, which means coefficients sum to 0. There is one coefficient for each level except the first. The implied coefficient for the first level is the sum of the other coefficients for the term.

You can display the coefficient values as a matrix rather than a table using coef = r.Coefficients{:,:}.

You can display marginal means for all levels using the margmean method.

Data Types: table

CovarianceEstimated response covariancestable

Estimated response covariances, that is, covariance of the repeated measures, stored as a table. fitrm computes the covariances around the mean returned by the fitted repeated measures model rm.

You can display the covariance values as a matrix rather than a table using coef = r.Covariance{:,:}.

Data Types: table

DFEError degrees of freedomscalar value

Error degrees of freedom, stored as a scalar value. DFE is the number of observations minus the number of estimated coefficients in the between-subjects model.

Data Types: double

Methods

anovaAnalysis of variance for between-subject effects
epsilonEpsilon adjustment for repeated measures anova
grpstatsCompute descriptive statistics of repeated measures data by group
manovaMultivariate analysis of variance
margmean Estimate marginal means
mauchlyMauchly's test for sphericity
multcompareMultiple comparison of estimated marginal means
plotPlot data with optional grouping
plotprofile Plot expected marginal means with optional grouping
predictCompute predicted values given predictor values
random Generate new random response values given predictor values
ranovaRepeated measures analysis of variance

Definitions

Wilkinson Notation

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

Use these rules to specify the responses in modelspec.

Wilkinson NotationDescription
Y1,Y2,Y3Specific list of variables
Y1-Y5All table variables from Y1 through Y5

Use these rules to specify terms in modelspec.

Wilkinson NotationFactors in Standard Notation
1Constant (intercept) term
X^k, where k is a positive integerX, X2, ..., Xk
X1 + X2X1, X2
X1*X2X1, X2, X1*X2
X1:X2X1*X2 only
-X2Do not include X2
X1*X2 + X3X1, X2, X3, X1*X2
X1 + X2 + X3 + X1:X2X1, X2, X3, X1*X2
X1*X2*X3 - X1:X2:X3X1, X2, X3, X1*X2, X1*X3, X2*X3
X1*(X2 + X3)X1, X2, X3, X1*X2, X1*X3

Statistics Toolbox™ notation always includes a constant term unless you explicitly remove the term using -1.

Examples

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Fit a Repeated Measures Model

Load the sample data.

load fisheriris

The column vector, species, consists of iris flowers of three different species: setosa, versicolor, 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 = 

                           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:

species_setosa={1,ifsetosa0,ifversicolor1,ifvirginicaandspecies_versicolor={0,ifsetosa1,ifversicolor1,ifvirginica

Display the covariance matrix.

rm.Covariance
ans = 

              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

Display the error degrees of freedom.

rm.DFE
ans =

   147

The error degrees of freedom is the number of observations minus the number of estimated coefficients in the between-subjects model, e.g. 150 – 3 = 147.

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