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ranova

Analysis of variance for within-subject effects in a repeated measures model

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Syntax

ranovatbl = ranova(rm)
ranovatbl = ranova(rm,'WithinModel',WM)
[ranovatbl,A,C,D] = ranova(___)

Description

ranovatbl = ranova(rm) returns the results of repeated measures analysis of variance for a repeated measures model rm in table ranovatbl.

example

ranovatbl = ranova(rm,'WithinModel',WM) returns the results of repeated measures analysis of variance using the responses specified by the within-subject model WM.

example

[ranovatbl,A,C,D] = ranova(___) also returns arrays A, C, and D for the hypotheses tests of the form A*B*C = D, where D is zero.

example

Examples

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Open Live Script

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);

Perform repeated measures analysis of variance. 

ranovatbl = ranova(rm)
ranovatbl=3×8 table
                                SumSq     DF      MeanSq       F         pValue        pValueGG       pValueHF       pValueLB  
                                ______    ___    ________    ______    ___________    ___________    ___________    ___________

    (Intercept):Measurements    1656.3      3      552.09    6873.3              0    9.4491e-279    2.9213e-283    2.5871e-125
    species:Measurements        282.47      6      47.078     586.1    1.4271e-206    4.9313e-156    1.5406e-158     9.0151e-71
    Error(Measurements)         35.423    441    0.080324

There are four measurements, three types of species, and 150 observations. So, degrees of freedom for measurements is (4–1) = 3, for species-measurements interaction it is (4–1)*(3–1) = 6, and for error it is (150–3)*(4–1) = 441. ranova computes the last three p-values using Greenhouse-Geisser, Huynh-Feldt, and Lower bound corrections, respectively. You can check the compound symmetry (sphericity) assumption using the mauchly method, and display the epsilon corrections using the epsilon method.

Open Live Script

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 the blood levels are the responses and gender is the predictor variable. 

rm = fitrm(t,'t0-t8 ~ Gender','WithinDesign',Time);

Perform repeated measures analysis of variance. 

ranovatbl = ranova(rm)
ranovatbl=3×8 table
                        SumSq     DF    MeanSq       F         pValue       pValueGG      pValueHF      pValueLB 
                        ______    __    ______    _______    __________    __________    __________    __________

    (Intercept):Time     881.7     4    220.43     37.539    3.0348e-15    4.7325e-09    2.4439e-10    2.6198e-05
    Gender:Time          17.65     4    4.4125    0.75146       0.56126        0.4877       0.50707       0.40063
    Error(Time)         328.83    56     5.872

There are 5 time points, 2 genders, and 16 observations. So, the degrees of freedom for time is (5–1) = 4, for gender-time interaction it is (5–1)*(2–1) = 4, and for error it is (16–2)*(5–1) = 56. The small p-value of 2.6198e–05 indicates that there is a significant effect of time on blood pressure. The p -value of 0.40063 indicates that there is no significant gender-time interaction.

Open Live Script

Load the sample data. 

load repeatedmeas

The table between includes the between-subject variables age, IQ, group, gender, and eight repeated measures y1 through y8 as responses. The table within includes the within-subject variables w1 and w2. This is simulated data. Hypothetically, the response can be results of a memory test. The within-subject variable w1 can be the type of exercise the subject does before the test and w2 can be the different points in the day the subject takes the memory test. So, one subject does two different type of exercises A and B before taking the test and takes the test at four different times on different days. For each subject, the measurements are taken under these conditions:

Exercise to perform before the test: A B A B A B A B

Test time: 1 1 2 2 3 3 4 4

Fit a repeated measures model, where the repeated measures y1 through y8 are the responses, and age, IQ, group, gender, and the group-gender interaction are the predictor variables. Also specify the within-subject design matrix.

rm = fitrm(between,'y1-y8 ~ Group*Gender + Age + IQ','WithinDesign',within);

Perform repeated measures analysis of variance. 

ranovatbl = ranova(rm)
ranovatbl=7×8 table
                         SumSq     DF     MeanSq       F        pValue      pValueGG    pValueHF     pValueLB
                         ______    ___    ______    _______    _________    ________    _________    ________

    (Intercept):Time     6645.2      7    949.31     2.2689     0.031674    0.071235     0.056257     0.14621
    Age:Time             5824.3      7    832.05     1.9887     0.059978     0.10651     0.090128     0.17246
    IQ:Time              5188.3      7    741.18     1.7715     0.096749     0.14492      0.12892     0.19683
    Group:Time            15800     14    1128.6     2.6975    0.0014425    0.011884    0.0064346    0.089594
    Gender:Time          4455.8      7    636.55     1.5214      0.16381     0.20533      0.19258     0.23042
    Group:Gender:Time    4247.3     14    303.38    0.72511      0.74677       0.663      0.69184     0.49549
    Error(Time)           64433    154    418.39

Specify the model for the within-subject factors. Also display the matrices used in the hypothesis test. 

[ranovatbl,A,C,D] = ranova(rm,'WithinModel','w1+w2')
ranovatbl=21×8 table
                       SumSq     DF    MeanSq       F         pValue      pValueGG     pValueHF     pValueLB 
                       ______    __    ______    ________    _________    _________    _________    _________

    (Intercept)        3141.7     1    3141.7      2.5034      0.12787      0.12787      0.12787      0.12787
    Age                537.48     1    537.48     0.42828      0.51962      0.51962      0.51962      0.51962
    IQ                 2975.9     1    2975.9      2.3712      0.13785      0.13785      0.13785      0.13785
    Group               20836     2     10418      8.3012    0.0020601    0.0020601    0.0020601    0.0020601
    Gender             3036.3     1    3036.3      2.4194      0.13411      0.13411      0.13411      0.13411
    Group:Gender        211.8     2     105.9    0.084385      0.91937      0.91937      0.91937      0.91937
    Error               27609    22      1255           1          0.5          0.5          0.5          0.5
    (Intercept):w1     146.75     1    146.75     0.23326      0.63389      0.63389      0.63389      0.63389
    Age:w1             942.02     1    942.02      1.4974      0.23402      0.23402      0.23402      0.23402
    IQ:w1              11.563     1    11.563     0.01838      0.89339      0.89339      0.89339      0.89339
    Group:w1           4481.9     2    2240.9       3.562     0.045697     0.045697     0.045697     0.045697
    Gender:w1          270.65     1    270.65      0.4302      0.51869      0.51869      0.51869      0.51869
    Group:Gender:w1    240.37     2    120.19     0.19104      0.82746      0.82746      0.82746      0.82746
    Error(w1)           13841    22    629.12           1          0.5          0.5          0.5          0.5
    (Intercept):w2     3663.8     3    1221.3      3.8381     0.013513     0.020339      0.01575     0.062894
    Age:w2             1199.9     3    399.95      1.2569       0.2964      0.29645      0.29662      0.27432
      ⋮


A=6×1 cell array
    {[1 0 0 0 0 0 0 0]}
    {[0 1 0 0 0 0 0 0]}
    {[0 0 1 0 0 0 0 0]}
    {2x8 double       }
    {[0 0 0 0 0 1 0 0]}
    {2x8 double       }


C=1×3 cell array
    {8x1 double}    {8x1 double}    {8x3 double}


D = 
0

Display the contents of A. 

[A{1};A{2};A{3};A{4};A{5};A{6}]
ans = 8×8

     1     0     0     0     0     0     0     0
     0     1     0     0     0     0     0     0
     0     0     1     0     0     0     0     0
     0     0     0     1     0     0     0     0
     0     0     0     0     1     0     0     0
     0     0     0     0     0     1     0     0
     0     0     0     0     0     0     1     0
     0     0     0     0     0     0     0     1

Display the contents of C. 

[C{1} C{2} C{3}]
ans = 8×5

     1     1     1     0     0
     1     1     0     1     0
     1     1     0     0     1
     1     1    -1    -1    -1
     1    -1     1     0     0
     1    -1     0     1     0
     1    -1     0     0     1
     1    -1    -1    -1    -1
Open Live Script

Load the fisheriris sample data set.

load fisheriris

The column vector species contains three iris flower species: setosa, versicolor, and virginica. The matrix meas contains four types of measurements for the flower: the length and width of sepals and petals in centimeters.

Convert the data in species to categorical string vectors by using the string, " " and categorical functions. Create a matrix corresponding to the measurements for setosa flowers and use the array2table function to convert the matrix to a table.

species = categorical(string(species));
m = meas(species=="setosa",:);
tbl = array2table(m,VariableNames=["y1","y2","y3","y4"])
tbl=50×4 table
    y1     y2     y3     y4 
    ___    ___    ___    ___

    5.1    3.5    1.4    0.2
    4.9      3    1.4    0.2
    4.7    3.2    1.3    0.2
    4.6    3.1    1.5    0.2
      5    3.6    1.4    0.2
    5.4    3.9    1.7    0.4
    4.6    3.4    1.4    0.3
      5    3.4    1.5    0.2
    4.4    2.9    1.4    0.2
    4.9    3.1    1.5    0.1
    5.4    3.7    1.5    0.2
    4.8    3.4    1.6    0.2
    4.8      3    1.4    0.1
    4.3      3    1.1    0.1
    5.8      4    1.2    0.2
    5.7    4.4    1.5    0.4
      ⋮

Each of the four response variables in tbl corresponds to the length or width of a sepal or petal.

Create a design for the within-subject factors by using the table function. Specify the first within-subject factor as the part of the flower being measured and the second as the direction in which the measurement was taken.

part = ["sepal";"sepal";"petal";"petal"];
direction = ["length";"width";"length";"width"];
w2design = table(part,direction,VariableNames=["part","direction"])
w2design=4×2 table
     part      direction
    _______    _________

    "sepal"    "length" 
    "sepal"    "width"  
    "petal"    "length" 
    "petal"    "width"

Each of the rows in w2design corresponds to a response variable in tbl.

Fit a repeated measures model to the measurements in tbl, using w2design as the design for the within-subject factors.

rm = fitrm(tbl,"y1-y4~1",WithinDesign=w2design)
rm = 
  RepeatedMeasuresModel with properties:

   Between Subjects:
         BetweenDesign: [50x4 table]
         ResponseNames: {'y1'  'y2'  'y3'  'y4'}
    BetweenFactorNames: {1x0 cell}
          BetweenModel: '1'

   Within Subjects:
          WithinDesign: [4x2 table]
     WithinFactorNames: {'part'  'direction'}
           WithinModel: 'separatemeans'

   Estimates:
          Coefficients: [1x4 table]
            Covariance: [4x4 table]

rm is a RepeatedMeasuresModel object that contains the results of fitting the repeated measures model to the data.

Perform an RANOVA to determine if the part of the flower and direction of measurement have a statistically significant effect on the measurement value at the 95% confidence level.

ranovatbl = ranova(rm,WithinModel="part+direction");
disp(ranovatbl)
                             SumSq     DF    MeanSq       F         pValue       pValueGG      pValueHF      pValueLB 
                             ______    __    _______    ______    __________    __________    __________    __________

    (Intercept)              1285.8     1     1285.8    8360.7    2.0347e-56    2.0347e-56    2.0347e-56    2.0347e-56
    Error                    7.5354    49    0.15378                                                                  
    (Intercept):part         565.49     1     565.49    5329.6    1.1612e-51    1.1612e-51    1.1612e-51    1.1612e-51
    Error(part)               5.199    49     0.1061                                                                  
    (Intercept):direction     97.58     1      97.58    3686.4    8.8091e-48    8.8091e-48    8.8091e-48    8.8091e-48
    Error(direction)         1.2971    49    0.02647

ranovatbl is an RANOVA table that includes the p-values for each term in the within-subjects model. The small p-values for the (Intercept):part and (Intercept):direction terms indicate that both the part of the flower and direction of measurement have a statistically significant effect on the measurement values.

Input Arguments

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Repeated measures model, returned as a RepeatedMeasuresModel object.

For properties and methods of this object, see RepeatedMeasuresModel.

Model specifying the responses, specified as one of the following:

  • 'separatemeans' — Compute a separate mean for each group.

  • C — r-by-nc contrast matrix specifying the nc contrasts among the r repeated measures. If Y represents a matrix of repeated measures, ranova tests the hypothesis that the means of Y*C are zero.

  • 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 the modelspec argument of fitrm. Also see Model Specification for Repeated Measures Models.

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: single | double | char | string

Output Arguments

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Results of repeated measures anova, returned as a table.

ranovatbl includes a term representing all differences across the within-subjects factors. This term has either the name of the within-subjects factor if specified while fitting the model, or the name Time if the name of the within-subjects factor is not specified while fitting the model or there are more than one within-subjects factors. ranovatbl also includes all interactions between the terms in the within-subject model and all between-subject model terms. It contains the following columns.

Column NameDefinition
SumSqSum of squares.
DFDegrees of freedom.
MeanSqMean squared error.
FF-statistic.
pValuep-value for the corresponding F-statistic. A small p-value indicates significant term effect.
pValueGGp-value with Greenhouse-Geisser adjustment.
pValueHFp-value with Huynh-Feldt adjustment.
pValueLBp-value with Lower bound adjustment.

The last three p-values are the adjusted p-values for use when the compound symmetry assumption is not satisfied. For details, see Compound Symmetry Assumption and Epsilon Corrections. The mauchy method tests for sphericity (hence, compound symmetry) and epsilon method returns the epsilon adjustment values.

Specification based on the between-subjects model, returned as a matrix or a cell array. It permits the hypothesis on the elements within given columns of B (within time hypothesis). If ranovatbl contains multiple hypothesis tests, A might be a cell array.

Data Types: single | double | cell

Specification based on the within-subjects model, returned as a matrix or a cell array. It permits the hypotheses on the elements within given rows of B (between time hypotheses). If ranovatbl contains multiple hypothesis tests, C might be a cell array.

Data Types: single | double | cell

Hypothesis value, returned as 0.

Algorithms

ranova computes the regular p-value (in the pValue column of the rmanova table) using the F-statistic cumulative distribution function:

p-value = 1 – fcdf(F,v1,v2).

When the compound symmetry assumption is not satisfied, ranova uses a correction factor epsilon, ε, to compute the corrected p-values as follows:

p-value_corrected = 1 – fcdf(F,ε*v1,ε*v2).

The mauchly method tests for sphericity (hence, compound symmetry) and epsilon method returns the epsilon adjustment values.

Version History

Introduced in R2014a

See Also

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