Split-plot-factorial ANOVA (SPF-p.q design)
TODO
- link to anovaSPFpqr, anovaMixed, dfReshape
Traditional univariate analysis and multivariate approach.
Install required packages
wants <- c("car", "DescTools", "multcomp")
has <- wants %in% rownames(installed.packages())
if(any(!has)) install.packages(wants[!has])Two-way SPF-\(p \cdot q\) ANOVA
Using aov() with data in long format
set.seed(123)
Nj <- 10
P <- 3
Q <- 3
muJK <- c(rep(c(1,-1,-2), Nj), rep(c(2,1,-1), Nj), rep(c(3,3,0), Nj))
dfSPFpqL <- data.frame(id=factor(rep(1:(P*Nj), times=Q)),
IVbtw=factor(rep(LETTERS[1:P], times=Q*Nj)),
IVwth=factor(rep(1:Q, each=P*Nj)),
DV=rnorm(Nj*P*Q, muJK, 3))aovSPFpq <- aov(DV ~ IVbtw*IVwth + Error(id/IVwth), data=dfSPFpqL)
summary(aovSPFpq)
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
IVbtw 2 178.4 89.21 14.92 4.33e-05 ***
Residuals 27 161.5 5.98
---
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 2 131.0 65.50 8.446 0.000643 ***
IVbtw:IVwth 4 43.4 10.85 1.399 0.246589
Residuals 54 418.8 7.75
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Using Anova() from package car with data in wide format
dfSPFpqW <- reshape(dfSPFpqL, v.names="DV", timevar="IVwth",
idvar=c("id", "IVbtw"), direction="wide")library(car)
fitSPFpq <- lm(cbind(DV.1, DV.2, DV.3) ~ IVbtw, data=dfSPFpqW)
inSPFpq <- data.frame(IVwth=gl(Q, 1))
AnovaSPFpq <- Anova(fitSPFpq, idata=inSPFpq, idesign=~IVwth)
summary(AnovaSPFpq, multivariate=FALSE, univariate=TRUE)
Univariate Type II Repeated-Measures ANOVA Assuming Sphericity
SS num Df Error SS den Df F Pr(>F)
(Intercept) 60.859 1 161.46 27 10.1769 0.0035871 **
IVbtw 178.416 2 161.46 27 14.9174 4.326e-05 ***
IVwth 130.999 2 418.76 54 8.4464 0.0006433 ***
IVbtw:IVwth 43.407 4 418.76 54 1.3994 0.2465888
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Mauchly Tests for Sphericity
Test statistic p-value
IVwth 0.98096 0.77892
IVbtw:IVwth 0.98096 0.77892
Greenhouse-Geisser and Huynh-Feldt Corrections
for Departure from Sphericity
GG eps Pr(>F[GG])
IVwth 0.98132 0.0007035 ***
IVbtw:IVwth 0.98132 0.2473928
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
HF eps Pr(>F[HF])
IVwth 1.057498 0.0006432656
IVbtw:IVwth 1.057498 0.2465887981Using anova.mlm() and mauchly.test() with data in wide format
anova(fitSPFpq, M=~1, X=~0, idata=inSPFpq, 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 10.177 1 27 0.0035871 0.0035871 0.0035871
IVbtw 2 14.917 2 27 0.0000433 0.0000433 0.0000433
Residuals 27 anova(fitSPFpq, M=~IVwth, X=~1, idata=inSPFpq, test="Spherical")Analysis of Variance Table
Contrasts orthogonal to
~1
Contrasts spanned by
~IVwth
Greenhouse-Geisser epsilon: 0.9813
Huynh-Feldt epsilon: 1.0575
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 8.4464 2 54 0.000643 0.000703 0.000643
IVbtw 2 1.3994 4 54 0.246589 0.247393 0.246589
Residuals 27 mauchly.test(fitSPFpq, M=~IVwth, X=~1, idata=inSPFpq)
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) ~ IVbtw, data = dfSPFpqW)
W = 0.981, p-value = 0.7789Effect size estimates: generalized \(\hat{\eta}_{g}^{2}\)
library(DescTools)
EtaSq(aovSPFpq, type=1)Error in EtaSq.aovlist(aovSPFpq, type = 1): konnte Funktion "is" nicht findenSimple effects
Separate error terms
Between-subjects effect at a fixed level of the within-subjects factor
summary(aov(DV ~ IVbtw, data=dfSPFpqL, subset=(IVwth==1))) Df Sum Sq Mean Sq F value Pr(>F)
IVbtw 2 81.81 40.90 4.694 0.0178 *
Residuals 27 235.26 8.71
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(aov(DV ~ IVbtw, data=dfSPFpqL, subset=(IVwth==2))) Df Sum Sq Mean Sq F value Pr(>F)
IVbtw 2 37.15 18.577 2.814 0.0776 .
Residuals 27 178.24 6.601
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(aov(DV ~ IVbtw, data=dfSPFpqL, subset=(IVwth==3))) Df Sum Sq Mean Sq F value Pr(>F)
IVbtw 2 102.9 51.43 8.329 0.00152 **
Residuals 27 166.7 6.17
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Within-subjects effect at a fixed level of the between-subjects factor
summary(aov(DV ~ IVwth + Error(id/IVwth), data=dfSPFpqL,
subset=(IVbtw=="A")))
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 9 22.6 2.511
Error: id:IVwth
Df Sum Sq Mean Sq F value Pr(>F)
IVwth 2 46.99 23.49 3.512 0.0516 .
Residuals 18 120.41 6.69
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(aov(DV ~ IVwth + Error(id/IVwth), data=dfSPFpqL,
subset=(IVbtw=="B")))
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 9 23.14 2.571
Error: id:IVwth
Df Sum Sq Mean Sq F value Pr(>F)
IVwth 2 111.4 55.71 8.153 0.00301 **
Residuals 18 123.0 6.83
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(aov(DV ~ IVwth + Error(id/IVwth), data=dfSPFpqL,
subset=(IVbtw=="C")))
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 9 115.7 12.86
Error: id:IVwth
Df Sum Sq Mean Sq F value Pr(>F)
IVwth 2 16.0 7.998 0.821 0.456
Residuals 18 175.3 9.742 Planned comparisons for the between-subjects factor
mDf <- aggregate(DV ~ id + IVbtw, data=dfSPFpqL, FUN=mean)
aovRes <- aov(DV ~ IVbtw, data=mDf)cMat <- rbind("-0.5*(A+B)+C"=c(-1/2, -1/2, 1),
"A-C"=c(-1, 0, 1))library(multcomp)
summary(glht(aovRes, linfct=mcp(IVbtw=cMat), alternative="greater"),
test=adjusted("none"))
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: User-defined Contrasts
Fit: aov(formula = DV ~ IVbtw, data = mDf)
Linear Hypotheses:
Estimate Std. Error t value Pr(>t)
-0.5*(A+B)+C <= 0 -2.3750 0.5468 -4.343 1
A-C <= 0 -3.4206 0.6314 -5.418 1
(Adjusted p values reported -- none method)Detach (automatically) loaded packages (if possible)
try(detach(package:DescTools))
try(detach(package:multcomp))
try(detach(package:mvtnorm))
try(detach(package:car))
try(detach(package:survival))
try(detach(package:splines))
try(detach(package:TH.data))Get the article source from GitHub
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