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

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 specification

character 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'. Specify the terms 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 a cell array of character vectors.

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

Data Types: char | string

Properties

BetweenDesign — Design for between-subject factors

table

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

Data Types: table

BetweenModel — Model for between-subjects factors

character vector

Model for between-subjects factors, stored as a character vector. This character vector 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

BetweenFactorNames — Names of variables used as between-subject factors

cell array of character vectors

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

Data Types: cell

ResponseNames — Names of variables used as response variables

cell array of character vectors

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

Data Types: cell

WithinDesign — Values of within-subject factors

table

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

Data Types: table

WithinModel — Model for within-subjects factors

character vector

Model for within-subjects factors, stored as a character vector.

You can specify WithinModel as a character vector or a string scalar using dot notation: Mdl.WithinModel = newWithinModelValue.

WithinFactorNames — Names of within-subject factors

cell array of character vectors

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

Data Types: cell

Coefficients — Values of estimated coefficients

table

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

Covariance — Estimated response covariances

table

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

DFE — Error degrees of freedom

scalar 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

anova	Analysis of variance for between-subject effects
epsilon	Epsilon adjustment for repeated measures anova
grpstats	Compute descriptive statistics of repeated measures data by group
manova	Multivariate analysis of variance
margmean	Estimate marginal means
mauchly	Mauchly’s test for sphericity
multcompare	Multiple comparison of estimated marginal means
plot	Plot data with optional grouping
plotprofile	Plot expected marginal means with optional grouping
predict	Compute predicted values given predictor values
random	Generate new random response values given predictor values
ranova	Repeated measures analysis of variance

Examples

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=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:

$$species\_setosa=\{\begin{array}{c}1,ifsetosa\\ 0,ifversicolor\\ -1,ifvirginica\end{array}$$

and

$$species\_versicolor=\{\begin{array}{c}0,ifsetosa\\ 1,ifversicolor\\ -1,ifvirginica\end{array}.$$

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

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.

More About

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 Notation	Description
Y1,Y2,Y3	Specific list of variables
Y1-Y5	All table variables from Y1 through Y5

Use these rules to specify terms in modelspec.

Wilkinson Notation	Factors 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.