Class: RepeatedMeasuresModel
Analysis of variance for between-subject effects
rm — Repeated measures modelRepeatedMeasuresModel objectRepeated measures model, returned as a RepeatedMeasuresModel object.
For properties and methods of this object, see RepeatedMeasuresModel.
WM — Within-subject model'separatemeans' (default) | 'orthogonalcontrasts' | character vector or string scalar defining a model specification | r-by-nc matrix specifying
nc contrastsWithin-subject model, specified as one of the following:
'separatemeans' — The response
is the average of the repeated measures (average across the within-subject
model).
'orthogonalcontrasts' —
This is valid 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. anova multiplies Y, the
response you use in the repeated measures model rm by
the orthogonal contrasts, and uses the columns of the resulting product
matrix as the responses.
anova computes the orthogonal contrasts for T using
the Q factor of a QR factorization of the Vandermonde matrix.
A character vector or string scalar that defines a model specification in the within-subject
factors. Responses are defined by the terms in that model.
anova multiplies Y, the
response matrix you use in the repeated measures model
rm by the terms of the model, and uses the
columns of the result as the responses.
For example, if there is a Time factor and 'Time' is
the model specification, then anova uses two terms,
the constant and the uncentered Time term. The default is '1' to
perform on the average response.
An r-by-nc matrix, C,
specifying nc contrasts among the r repeated
measures. If Y represents the matrix of repeated
measures you use in the repeated measures model rm,
then the output tbl contains a separate analysis
of variance for each column of Y*C.
The anova table contains a separate univariate
analysis of variance results for each response.
Example: 'WithinModel','Time'
Example: 'WithinModel','orthogonalcontrasts'
anovatbl — Results of analysis of varianceResults of analysis of variance for between-subject effects, returned as a table. This includes all terms on the between-subjects model and the following columns.
| Column Name | Definition |
|---|---|
Within | Within-subject factors |
Between | Between-subject factors |
SumSq | Sum of squares |
DF | Degrees of freedom |
MeanSq | Mean squared error |
F | F-statistic |
pValue | p-value corresponding to the F-statistic |
Load the sample data.
load fisheririsThe 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 = dataset([1 2 3 4]','VarNames',{'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);
Perform analysis of variance.
anova(rm)
ans=3×7 table
Within Between SumSq DF MeanSq F pValue
________ ________ ______ ___ _______ ______ ___________
Constant constant 7201.7 1 7201.7 19650 2.0735e-158
Constant species 309.61 2 154.8 422.39 1.1517e-61
Constant Error 53.875 147 0.36649
There are 150 observations and 3 species. The degrees of freedom for species is 3 - 1 = 2, and for error it is 150 - 3 = 147. The small -value of 1.1517e-61 indicates that the measurements differ significantly according to species.
Load the sample panel data.
load('panelData.mat');The dataset array, panelData, contains yearly observations on eight cities for 6 years. The first variable, Growth, measures economic growth (the response variable). The second and third variables are city and year indicators, respectively. The last variable, Employ, measures employment (the predictor variable). This is simulated data.
Store the data in a table array and define city as a nominal variable.
t = table(panelData.Growth,panelData.City,panelData.Year,... 'VariableNames',{'Growth','City','Year'});
Convert the data in a proper format to do repeated measures analysis.
t = unstack(t,'Growth','Year','NewDataVariableNames',... {'year1','year2','year3','year4','year5','year6'});
Add the mean employment level over the years as a predictor variable to the table t.
t(:,8) = table(grpstats(panelData.Employ,panelData.City));
t.Properties.VariableNames{'Var8'} = 'meanEmploy';Define the within-subjects variable.
Year = [1 2 3 4 5 6]';
Fit a repeated measures model, where the growth figures over the 6 years are the responses and the mean employment is the predictor variable.
rm = fitrm(t,'year1-year6 ~ meanEmploy','WithinDesign',Year);
Perform analysis of variance.
anovatbl = anova(rm,'WithinModel',Year)anovatbl=3×7 table
Within Between SumSq DF MeanSq F pValue
_________ __________ __________ __ __________ ________ _________
Contrast1 constant 588.17 1 588.17 0.038495 0.85093
Contrast1 meanEmploy 3.7064e+05 1 3.7064e+05 24.258 0.0026428
Contrast1 Error 91675 6 15279
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 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 do 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.
rm = fitrm(t,'t0-t8 ~ Gender','WithinDesign',Time);
Perform analysis of variance.
anovatbl = anova(rm)
anovatbl=3×7 table
Within Between SumSq DF MeanSq F pValue
________ ________ ______ __ ______ ______ __________
Constant constant 54702 1 54702 1079.2 1.1897e-14
Constant Gender 2251.7 1 2251.7 44.425 1.0693e-05
Constant Error 709.6 14 50.685
There are 2 genders and 16 observations, so the degrees of freedom for gender is (2 - 1) = 1 and for error it is (16 - 2)*(2 - 1) = 14. The small -value of 1.0693e-05 indicates that there is a significant effect of gender on blood pressure.
Repeat analysis of variance using orthogonal contrasts.
anovatbl = anova(rm,'WithinModel','orthogonalcontrasts')
anovatbl=15×7 table
Within Between SumSq DF MeanSq F pValue
________ ________ __________ __ __________ __________ __________
Constant constant 54702 1 54702 1079.2 1.1897e-14
Constant Gender 2251.7 1 2251.7 44.425 1.0693e-05
Constant Error 709.6 14 50.685
Time constant 310.83 1 310.83 31.023 6.9065e-05
Time Gender 13.341 1 13.341 1.3315 0.26785
Time Error 140.27 14 10.019
Time^2 constant 565.42 1 565.42 98.901 1.0003e-07
Time^2 Gender 1.4076 1 1.4076 0.24621 0.62746
Time^2 Error 80.039 14 5.7171
Time^3 constant 2.6127 1 2.6127 1.4318 0.25134
Time^3 Gender 7.8853e-06 1 7.8853e-06 4.3214e-06 0.99837
Time^3 Error 25.546 14 1.8247
Time^4 constant 2.8404 1 2.8404 0.47924 0.50009
Time^4 Gender 2.9016 1 2.9016 0.48956 0.49559
Time^4 Error 82.977 14 5.9269
Vandermonde matrix is the matrix where columns are the powers of the vector a, that is, V(i,j) = a(i)(n — j), where n is the length of a.
QR factorization of an m-by-n matrix A is the factorization that matrix into the product A = Q*R, where R is an m-by-n upper triangular matrix and Q is an m-by-m unitary matrix.
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