Three-way split-plot-factorial ANOVA (SPF-pq.r and SPF-p.qr design)
TODO
- link to anovaSPFpq, anovaMixed, dfReshape
Traditional univariate analysis and multivariate approach.
Install required packages
wants <- c("car", "DescTools")
has <- wants %in% rownames(installed.packages())
if(any(!has)) install.packages(wants[!has])Three-way SPF-\(pq \cdot r\) ANOVA
Using aov() with data in long format
set.seed(123)
Njk <- 10
P <- 2
Q <- 2
R <- 3
DV_t1 <- rnorm(P*Q*Njk, -3, 2)
DV_t2 <- rnorm(P*Q*Njk, 1, 2)
DV_t3 <- rnorm(P*Q*Njk, 2, 2)
dfSPFpq.rL <- data.frame(id=factor(rep(1:(P*Q*Njk), times=R)),
IVbtw1=factor(rep(1:P, times=Q*R*Njk)),
IVbtw2=factor(rep(rep(1:Q, each=P*Njk), times=R)),
IVwth=factor(rep(1:R, each=P*Q*Njk)),
DV=c(DV_t1, DV_t2, DV_t3))aovSPFpq.r <- aov(DV ~ IVbtw1*IVbtw2*IVwth + Error(id/IVwth), data=dfSPFpq.rL)
summary(aovSPFpq.r)
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
IVbtw1 1 0.11 0.106 0.038 0.8467
IVbtw2 1 17.75 17.749 6.326 0.0165 *
IVbtw1:IVbtw2 1 0.05 0.048 0.017 0.8969
Residuals 36 101.00 2.806
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Error: id:IVwth
Df Sum Sq Mean Sq F value Pr(>F)
IVwth 2 540.0 269.99 80.859 <2e-16 ***
IVbtw1:IVwth 2 2.3 1.17 0.351 0.705
IVbtw2:IVwth 2 7.4 3.68 1.103 0.337
IVbtw1:IVbtw2:IVwth 2 11.5 5.77 1.728 0.185
Residuals 72 240.4 3.34
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Effect size estimates: generalized \(\hat{\eta}_{g}^{2}\)
library(DescTools)
EtaSq(aovSPFpq.r, type=1)Error in EtaSq.aovlist(aovSPFpq.r, type = 1): konnte Funktion "is" nicht findenUsing Anova() from package car with data in wide format
dfSPFpq.rW <- reshape(dfSPFpq.rL, v.names="DV", timevar="IVwth",
idvar=c("id", "IVbtw1", "IVbtw2"), direction="wide")library(car)
fitSPFpq.r <- lm(cbind(DV.1, DV.2, DV.3) ~ IVbtw1*IVbtw2, data=dfSPFpq.rW)
inSPFpq.r <- data.frame(IVwth=gl(R, 1))
AnovaSPFpq.r <- Anova(fitSPFpq.r, idata=inSPFpq.r, idesign=~IVwth)
summary(AnovaSPFpq.r, multivariate=FALSE, univariate=TRUE)
Univariate Type II Repeated-Measures ANOVA Assuming Sphericity
SS num Df Error SS den Df F Pr(>F)
(Intercept) 0.11 1 101.00 36 0.0408 0.8411
IVbtw1 0.11 1 101.00 36 0.0379 0.8467
IVbtw2 17.75 1 101.00 36 6.3261 0.0165 *
IVbtw1:IVbtw2 0.05 1 101.00 36 0.0170 0.8969
IVwth 539.98 2 240.41 72 80.8589 <2e-16 ***
IVbtw1:IVwth 2.34 2 240.41 72 0.3511 0.7051
IVbtw2:IVwth 7.37 2 240.41 72 1.1033 0.3373
IVbtw1:IVbtw2:IVwth 11.54 2 240.41 72 1.7283 0.1849
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Mauchly Tests for Sphericity
Test statistic p-value
IVwth 0.9932 0.88741
IVbtw1:IVwth 0.9932 0.88741
IVbtw2:IVwth 0.9932 0.88741
IVbtw1:IVbtw2:IVwth 0.9932 0.88741
Greenhouse-Geisser and Huynh-Feldt Corrections
for Departure from Sphericity
GG eps Pr(>F[GG])
IVwth 0.99324 <2e-16 ***
IVbtw1:IVwth 0.99324 0.7037
IVbtw2:IVwth 0.99324 0.3370
IVbtw1:IVbtw2:IVwth 0.99324 0.1851
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
HF eps Pr(>F[HF])
IVwth 1.051053 3.900060e-19
IVbtw1:IVwth 1.051053 7.051035e-01
IVbtw2:IVwth 1.051053 3.373222e-01
IVbtw1:IVbtw2:IVwth 1.051053 1.848719e-01Using anova.mlm() and mauchly.test() with data in wide format
anova(fitSPFpq.r, M=~1, X=~0, idata=inSPFpq.r, test="Spherical")Analysis of Variance Table
Contrasts orthogonal to
~0
Contrasts spanned by
~1
Greenhouse-Geisser epsilon: 1
Huynh-Feldt epsilon: 1
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 0.0408 1 36 0.84107 0.84107 0.84107
IVbtw1 1 0.0379 1 36 0.84668 0.84668 0.84668
IVbtw2 1 6.3261 1 36 0.01650 0.01650 0.01650
IVbtw1:IVbtw2 1 0.0170 1 36 0.89693 0.89693 0.89693
Residuals 36 anova(fitSPFpq.r, M=~IVwth, X=~1, idata=inSPFpq.r, test="Spherical")Analysis of Variance Table
Contrasts orthogonal to
~1
Contrasts spanned by
~IVwth
Greenhouse-Geisser epsilon: 0.9932
Huynh-Feldt epsilon: 1.0511
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 80.8589 2 72 0.00000 0.00000 0.00000
IVbtw1 1 0.3511 2 72 0.70510 0.70367 0.70510
IVbtw2 1 1.1033 2 72 0.33732 0.33704 0.33732
IVbtw1:IVbtw2 1 1.7283 2 72 0.18487 0.18512 0.18487
Residuals 36 mauchly.test(fitSPFpq.r, M=~IVwth, X=~1, idata=inSPFpq.r)
Mauchly's test of sphericity
Contrasts orthogonal to
~1
Contrasts spanned by
~IVwth
data: SSD matrix from lm(formula = cbind(DV.1, DV.2, DV.3) ~ IVbtw1 * IVbtw2, data = dfSPFpq.rW)
W = 0.9932, p-value = 0.8874Three-way SPF-\(p \cdot qr\) ANOVA
Using aov() with data in long format
Nj <- 10
P <- 2
Q <- 3
R <- 2
DV_t11 <- rnorm(P*Nj, 8, 2)
DV_t21 <- rnorm(P*Nj, 13, 2)
DV_t31 <- rnorm(P*Nj, 13, 2)
DV_t12 <- rnorm(P*Nj, 10, 2)
DV_t22 <- rnorm(P*Nj, 15, 2)
DV_t32 <- rnorm(P*Nj, 15, 2)
dfSPFp.qrL <- data.frame(id=factor(rep(1:(P*Nj), times=Q*R)),
IVbtw=factor(rep(LETTERS[1:P], times=Q*R*Nj)),
IVwth1=factor(rep(1:Q, each=P*R*Nj)),
IVwth2=factor(rep(rep(1:R, each=P*Nj), times=Q)),
DV=c(DV_t11, DV_t12, DV_t21, DV_t22, DV_t31, DV_t32))aovSPFp.qr <- aov(DV ~ IVbtw*IVwth1*IVwth2 + Error(id/(IVwth1*IVwth2)),
data=dfSPFp.qrL)
summary(aovSPFp.qr)
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
IVbtw 1 2.32 2.323 0.407 0.531
Residuals 18 102.64 5.702
Error: id:IVwth1
Df Sum Sq Mean Sq F value Pr(>F)
IVwth1 2 534.5 267.25 62.647 1.89e-12 ***
IVbtw:IVwth1 2 5.1 2.53 0.593 0.558
Residuals 36 153.6 4.27
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Error: id:IVwth2
Df Sum Sq Mean Sq F value Pr(>F)
IVwth2 1 109.87 109.87 17.043 0.000631 ***
IVbtw:IVwth2 1 10.23 10.23 1.587 0.223859
Residuals 18 116.03 6.45
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Error: id:IVwth1:IVwth2
Df Sum Sq Mean Sq F value Pr(>F)
IVwth1:IVwth2 2 6.93 3.466 0.846 0.437
IVbtw:IVwth1:IVwth2 2 7.80 3.899 0.952 0.396
Residuals 36 147.50 4.097 Effect size estimates: generalized \(\hat{\eta}_{g}^{2}\)
library(DescTools)
EtaSq(aovSPFp.qr, type=1)Error in EtaSq.aovlist(aovSPFp.qr, type = 1): konnte Funktion "is" nicht findenUsing Anova() from package car with data in wide format
dfW1 <- reshape(dfSPFp.qrL, v.names="DV", timevar="IVwth1",
idvar=c("id", "IVbtw", "IVwth2"), direction="wide")
dfSPFp.qrW <- reshape(dfW1, v.names=c("DV.1", "DV.2", "DV.3"),
timevar="IVwth2", idvar=c("id", "IVbtw"), direction="wide")library(car)
fitSPFp.qr <- lm(cbind(DV.1.1, DV.2.1, DV.3.1, DV.1.2, DV.2.2, DV.3.2) ~ IVbtw,
data=dfSPFp.qrW)
inSPFp.qr <- expand.grid(IVwth1=gl(Q, 1), IVwth2=gl(R, 1))
AnovaSPFp.qr <- Anova(fitSPFp.qr, idata=inSPFp.qr, idesign=~IVwth1*IVwth2)
summary(AnovaSPFp.qr, multivariate=FALSE, univariate=TRUE)
Univariate Type II Repeated-Measures ANOVA Assuming Sphericity
SS num Df Error SS den Df F Pr(>F)
(Intercept) 18082.5 1 102.64 18 3170.9826 < 2.2e-16 ***
IVbtw 2.3 1 102.64 18 0.4075 0.5313035
IVwth1 534.5 2 153.58 36 62.6468 1.889e-12 ***
IVbtw:IVwth1 5.1 2 153.58 36 0.5930 0.5579974
IVwth2 109.9 1 116.03 18 17.0433 0.0006311 ***
IVbtw:IVwth2 10.2 1 116.03 18 1.5869 0.2238586
IVwth1:IVwth2 6.9 2 147.50 36 0.8460 0.4374751
IVbtw:IVwth1:IVwth2 7.8 2 147.50 36 0.9516 0.3956300
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Mauchly Tests for Sphericity
Test statistic p-value
IVwth1 0.80930 0.16556
IVbtw:IVwth1 0.80930 0.16556
IVwth1:IVwth2 0.94011 0.59160
IVbtw:IVwth1:IVwth2 0.94011 0.59160
Greenhouse-Geisser and Huynh-Feldt Corrections
for Departure from Sphericity
GG eps Pr(>F[GG])
IVwth1 0.83985 8.499e-11 ***
IVbtw:IVwth1 0.83985 0.5308
IVwth1:IVwth2 0.94350 0.4320
IVbtw:IVwth1:IVwth2 0.94350 0.3915
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
HF eps Pr(>F[HF])
IVwth1 0.9164691 1.373942e-11
IVbtw:IVwth1 0.9164691 5.443499e-01
IVwth1:IVwth2 1.0504830 4.374751e-01
IVbtw:IVwth1:IVwth2 1.0504830 3.956300e-01Using anova.mlm() and mauchly.test() with data in wide format
anova(fitSPFp.qr, M=~1, X=~0,
idata=inSPFp.qr, test="Spherical")Analysis of Variance Table
Contrasts orthogonal to
~0
Contrasts spanned by
~1
Greenhouse-Geisser epsilon: 1
Huynh-Feldt epsilon: 1
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 3170.9826 1 18 0.0000 0.0000 0.0000
IVbtw 1 0.4075 1 18 0.5313 0.5313 0.5313
Residuals 18 anova(fitSPFp.qr, M=~IVwth1, X=~1,
idata=inSPFp.qr, test="Spherical")Analysis of Variance Table
Contrasts orthogonal to
~1
Contrasts spanned by
~IVwth1
Greenhouse-Geisser epsilon: 0.8398
Huynh-Feldt epsilon: 0.9165
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 62.647 2 36 0.000 0.00000 0.00000
IVbtw 1 0.593 2 36 0.558 0.53075 0.54435
Residuals 18 anova(fitSPFp.qr, M=~IVwth1 + IVwth2, X=~IVwth1,
idata=inSPFp.qr, test="Spherical")Analysis of Variance Table
Contrasts orthogonal to
~IVwth1
Contrasts spanned by
~IVwth1 + IVwth2
Greenhouse-Geisser epsilon: 1
Huynh-Feldt epsilon: 1
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 17.0433 1 18 0.000631 0.000631 0.000631
IVbtw 1 1.5869 1 18 0.223859 0.223859 0.223859
Residuals 18 anova(fitSPFp.qr, M=~IVwth1 + IVwth2 + IVwth1:IVwth2, X=~IVwth1 + IVwth2,
idata=inSPFp.qr, test="Spherical")Analysis of Variance Table
Contrasts orthogonal to
~IVwth1 + IVwth2
Contrasts spanned by
~IVwth1 + IVwth2 + IVwth1:IVwth2
Greenhouse-Geisser epsilon: 0.9435
Huynh-Feldt epsilon: 1.0505
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 0.8460 2 36 0.43748 0.43201 0.43748
IVbtw 1 0.9516 2 36 0.39563 0.39154 0.39563
Residuals 18 mauchly.test(fitSPFp.qr, M=~IVwth1, X=~1,
idata=inSPFp.qr)
Mauchly's test of sphericity
Contrasts orthogonal to
~1
Contrasts spanned by
~IVwth1
data: SSD matrix from lm(formula = cbind(DV.1.1, DV.2.1, DV.3.1, DV.1.2, DV.2.2, DV.3.2) ~ SSD matrix from IVbtw, data = dfSPFp.qrW)
W = 0.8093, p-value = 0.1656Mauchly-Test for IVwth2 is unnecessary here since R=2 -> sphericity holds automatically
mauchly.test(fitSPFp.qr, M=~IVwth1 + IVwth2, X=~IVwth1,
idata=inSPFp.qr)
Mauchly's test of sphericity
Contrasts orthogonal to
~IVwth1
Contrasts spanned by
~IVwth1 + IVwth2
data: SSD matrix from lm(formula = cbind(DV.1.1, DV.2.1, DV.3.1, DV.1.2, DV.2.2, DV.3.2) ~ SSD matrix from IVbtw, data = dfSPFp.qrW)
W = 1, p-value = 1mauchly.test(fitSPFp.qr, M=~IVwth1 + IVwth2 + IVwth1:IVwth2, X=~IVwth1 + IVwth2,
idata=inSPFp.qr)
Mauchly's test of sphericity
Contrasts orthogonal to
~IVwth1 + IVwth2
Contrasts spanned by
~IVwth1 + IVwth2 + IVwth1:IVwth2
data: SSD matrix from lm(formula = cbind(DV.1.1, DV.2.1, DV.3.1, DV.1.2, DV.2.2, DV.3.2) ~ SSD matrix from IVbtw, data = dfSPFp.qrW)
W = 0.9401, p-value = 0.5916Detach (automatically) loaded packages (if possible)
try(detach(package:car))
try(detach(package:DescTools))Get the article source from GitHub
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