One-way repeated measures ANOVA (RB-p design)

TODO

• link to anovaMixed, dfReshape

Traditional univariate analysis and multivariate approach.

Install required packages

car, effectsize, DescTools

wants <- c("car", "effectsize", "DescTools")
has   <- wants %in% rownames(installed.packages())
if(any(!has)) install.packages(wants[!has])

Traditional univariate approach

Using aov() with data in long format

set.seed(123)
N      <- 10
P      <- 4
muJ    <- rep(c(-1, 0, 1, 2), each=N)
dfRBpL <- data.frame(id=factor(rep(1:N, times=P)),
                     IV=factor(rep(1:P,  each=N)),
                     DV=rnorm(N*P, muJ, 3))

aovRBp <- aov(DV ~ IV + Error(id/IV), data=dfRBpL)
summary(aovRBp)

Error: id
          Df Sum Sq Mean Sq F value Pr(>F)
Residuals  9  60.81   6.757               

Error: id:IV
          Df Sum Sq Mean Sq F value Pr(>F)  
IV         3  82.51  27.504   3.851 0.0205 *
Residuals 27 192.86   7.143                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Effect size estimate: generalized \(\hat{\eta}_{g}^{2}\)

library(effectsize)
eta_squared(aovRBp, partial=FALSE)

Group | Parameter | Eta2 |       90% CI
---------------------------------------
id:IV |        IV | 0.25 | [0.00, 0.42]

eta_squared(aovRBp, partial=TRUE)

Group | Parameter | Eta2 (partial) |       90% CI
-------------------------------------------------
id:IV |        IV |           0.30 | [0.03, 0.48]

eta_squared(aovRBp, generalized=TRUE)

Group | Parameter | Eta2 (generalized) |       90% CI
-----------------------------------------------------
id:IV |        IV |               0.25 | [0.00, 0.42]

Using Anova() from package car with data in wide format

dfRBpW <- reshape(dfRBpL, v.names="DV", timevar="IV", idvar="id",
                  direction="wide")

library(car)
fitRBp   <- lm(cbind(DV.1, DV.2, DV.3, DV.4) ~ 1, data=dfRBpW)
inRBp    <- data.frame(IV=gl(P, 1))
AnovaRBp <- Anova(fitRBp, idata=inRBp, idesign=~IV)
summary(AnovaRBp, multivariate=FALSE, univariate=TRUE)

Univariate Type III Repeated-Measures ANOVA Assuming Sphericity

            Sum Sq num Df Error SS den Df F value  Pr(>F)  
(Intercept) 16.157      1   60.813      9  2.3912 0.15643  
IV          82.512      3  192.859     27  3.8505 0.02047 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Mauchly Tests for Sphericity

   Test statistic p-value
IV        0.30399 0.10403


Greenhouse-Geisser and Huynh-Feldt Corrections
 for Departure from Sphericity

    GG eps Pr(>F[GG])  
IV 0.58505    0.04805 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

      HF eps Pr(>F[HF])
IV 0.7155256 0.03663577

Using anova.mlm() and mauchly.test() with data in wide format

anova(fitRBp, M=~IV, X=~1, idata=inRBp, test="Spherical")

Analysis of Variance Table


Contrasts orthogonal to
~1


Contrasts spanned by
~IV

Greenhouse-Geisser epsilon: 0.5851
Huynh-Feldt epsilon:        0.7155

            Df      F num Df den Df   Pr(>F)   G-G Pr   H-F Pr
(Intercept)  1 3.8505      3     27 0.020472 0.048054 0.036636
Residuals    9                                                

mauchly.test(fitRBp, M=~IV, X=~1, idata=inRBp)

    Mauchly's test of sphericity
    Contrasts orthogonal to
    ~1

    Contrasts spanned by
    ~IV


data:  SSD matrix from lm(formula = cbind(DV.1, DV.2, DV.3, DV.4) ~ 1, data = dfRBpW)
W = 0.30399, p-value = 0.104

Multivariate approach

Hotelling’s \(T^{2}\)-test using HotellingsT2Test() from package DescTools

DVw     <- data.matrix(subset(dfRBpW,
                       select=c("DV.1", "DV.2", "DV.3", "DV.4")))
diffMat <- combn(1:P, 2, function(x) { DVw[ , x[1]] - DVw[ , x[2]] } )
DVdiff  <- diffMat[ , 1:(P-1), drop=FALSE]
muH0    <- rep(0, ncol(DVdiff))

library(DescTools)
HotellingsT2Test(DVdiff, mu=muH0)

    Hotelling's one sample T2-test

data:  DVdiff
T.2 = 12.267, df1 = 3, df2 = 7, p-value = 0.003555
alternative hypothesis: true location is not equal to c(0,0,0)

Using Anova() from package car

library(car)
summary(AnovaRBp, multivariate=TRUE, univariate=FALSE)

Type III Repeated Measures MANOVA Tests:

------------------------------------------
 
Term: (Intercept) 

 Response transformation matrix:
     (Intercept)
DV.1           1
DV.2           1
DV.3           1
DV.4           1

Sum of squares and products for the hypothesis:
            (Intercept)
(Intercept)     64.6278

Multivariate Tests: (Intercept)
                 Df test stat approx F num Df den Df  Pr(>F)
Pillai            1 0.2099135 2.391158      1      9 0.15643
Wilks             1 0.7900865 2.391158      1      9 0.15643
Hotelling-Lawley  1 0.2656842 2.391158      1      9 0.15643
Roy               1 0.2656842 2.391158      1      9 0.15643

------------------------------------------
 
Term: IV 

 Response transformation matrix:
     IV1 IV2 IV3
DV.1   1   0   0
DV.2   0   1   0
DV.3   0   0   1
DV.4  -1  -1  -1

Sum of squares and products for the hypothesis:
          IV1      IV2       IV3
IV1 140.04485 87.57883 121.24202
IV2  87.57883 54.76853  75.82024
IV3 121.24202 75.82024 104.96371

Multivariate Tests: IV
                 Df test stat approx F num Df den Df    Pr(>F)   
Pillai            1  0.840184 12.26683      3      7 0.0035548 **
Wilks             1  0.159816 12.26683      3      7 0.0035548 **
Hotelling-Lawley  1  5.257214 12.26683      3      7 0.0035548 **
Roy               1  5.257214 12.26683      3      7 0.0035548 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Detach (automatically) loaded packages (if possible)

try(detach(package:car))
try(detach(package:carData))
try(detach(package:DescTools))
try(detach(package:effectsize))

Get the article source from GitHub

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