Two-way repeated-measures ANOVA (RBF-pq design)
TODO
- link to 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])Traditional univariate approach
Using aov() with data in long format
set.seed(123)
N <- 10
P <- 2
Q <- 3
muJK <- c(rep(c(1, -2), N), rep(c(2, 0), N), rep(c(3, 3), N))
dfRBFpqL <- data.frame(id =factor(rep(1:N, times=P*Q)),
IV1=factor(rep(rep(1:P, each=N), times=Q)),
IV2=factor(rep(rep(1:Q, each=N*P))),
DV =rnorm(N*P*Q, muJK, 2))
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 9 55.24 6.138
Error: id:IV1
Df Sum Sq Mean Sq F value Pr(>F)
IV1 1 8.229 8.229 2.569 0.143
Residuals 9 28.833 3.204
Error: id:IV2
Df Sum Sq Mean Sq F value Pr(>F)
IV2 2 122.4 61.22 7.944 0.00337 **
Residuals 18 138.7 7.71
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Error: id:IV1:IV2
Df Sum Sq Mean Sq F value Pr(>F)
IV1:IV2 2 4.34 2.170 0.895 0.426
Residuals 18 43.67 2.426 Using Anova() from package car with data in wide format
dfTemp <- reshape(dfRBFpqL, v.names="DV", timevar="IV1",
idvar=c("id", "IV2"), direction="wide")
dfRBFpqW <- reshape(dfTemp, v.names=c("DV.1", "DV.2"),
timevar="IV2", idvar="id", direction="wide")library(car)
fitRBFpq <- lm(cbind(DV.1.1, DV.2.1, DV.1.2, DV.2.2, DV.1.3, DV.2.3) ~ 1,
data=dfRBFpqW)
inRBFpq <- expand.grid(IV1=gl(P, 1), IV2=gl(Q, 1))
AnovaRBFpq <- Anova(fitRBFpq, idata=inRBFpq, idesign=~IV1*IV2)
summary(AnovaRBFpq, 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) 101.073 1 55.239 9 16.4677 0.002850 **
IV1 8.229 1 28.833 9 2.5685 0.143472
IV2 122.440 2 138.713 18 7.9442 0.003365 **
IV1:IV2 4.340 2 43.669 18 0.8945 0.426218
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Mauchly Tests for Sphericity
Test statistic p-value
IV2 0.70743 0.25045
IV1:IV2 0.99594 0.98385
Greenhouse-Geisser and Huynh-Feldt Corrections
for Departure from Sphericity
GG eps Pr(>F[GG])
IV2 0.77365 0.007503 **
IV1:IV2 0.99596 0.425903
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
HF eps Pr(>F[HF])
IV2 0.9039018 0.004723577
IV1:IV2 1.2784582 0.426217623Using anova.mlm() and mauchly.test() with data in wide format
Analysis of Variance Table
Contrasts orthogonal to
~1
Contrasts spanned by
~IV1
Greenhouse-Geisser epsilon: 1
Huynh-Feldt epsilon: 1
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 2.5685 1 9 0.14347 0.14347 0.14347
Residuals 9 Analysis of Variance Table
Contrasts orthogonal to
~IV1
Contrasts spanned by
~IV1 + IV2
Greenhouse-Geisser epsilon: 0.7737
Huynh-Feldt epsilon: 0.9039
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 7.9442 2 18 0.0033651 0.0075029 0.0047236
Residuals 9 Analysis of Variance Table
Contrasts orthogonal to
~IV1 + IV2
Contrasts spanned by
~IV1 + IV2 + IV1:IV2
Greenhouse-Geisser epsilon: 0.996
Huynh-Feldt epsilon: 1.278
Df F num Df den Df Pr(>F) G-G Pr H-F Pr
(Intercept) 1 0.8945 2 18 0.42622 0.4259 0.42622
Residuals 9 Mauchly-Test for IV1 is unnecessary here since P=2 -> sphericity holds automatically
Mauchly's test of sphericity
Contrasts orthogonal to
~1
Contrasts spanned by
~IV1
data: SSD matrix from lm(formula = cbind(DV.1.1, DV.2.1, DV.1.2, DV.2.2, DV.1.3, DV.2.3) ~ SSD matrix from 1, data = dfRBFpqW)
W = 1, p-value = 1
Mauchly's test of sphericity
Contrasts orthogonal to
~IV1
Contrasts spanned by
~IV1 + IV2
data: SSD matrix from lm(formula = cbind(DV.1.1, DV.2.1, DV.1.2, DV.2.2, DV.1.3, DV.2.3) ~ SSD matrix from 1, data = dfRBFpqW)
W = 0.70743, p-value = 0.2505
Mauchly's test of sphericity
Contrasts orthogonal to
~IV1 + IV2
Contrasts spanned by
~IV1 + IV2 + IV1:IV2
data: SSD matrix from lm(formula = cbind(DV.1.1, DV.2.1, DV.1.2, DV.2.2, DV.1.3, DV.2.3) ~ SSD matrix from 1, data = dfRBFpqW)
W = 0.99594, p-value = 0.9839Effect size estimates: generalized \(\hat{\eta}_{g}^{2}\)
Group | Parameter | Eta2 | 90% CI
--------------------------------------------
id:IV1 | IV1 | 0.02 | [0.00, 0.31]
id:IV2 | IV2 | 0.30 | [0.01, 0.52]
id:IV1:IV2 | IV1:IV2 | 0.01 | [0.00, 0.07]Group | Parameter | Eta2 (partial) | 90% CI
------------------------------------------------------
id:IV1 | IV1 | 0.22 | [0.00, 0.55]
id:IV2 | IV2 | 0.47 | [0.15, 0.65]
id:IV1:IV2 | IV1:IV2 | 0.09 | [0.00, 0.29]Group | Parameter | Eta2 (generalized) | 90% CI
----------------------------------------------------------
id:IV1 | IV1 | 0.03 | [0.00, 0.33]
id:IV2 | IV2 | 0.31 | [0.02, 0.53]
id:IV1:IV2 | IV1:IV2 | 0.02 | [0.00, 0.11]Simple effects
Separate error terms
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 9 122.4 13.6
Error: id:IV1
Df Sum Sq Mean Sq F value Pr(>F)
IV1 1 0.359 0.3591 0.213 0.655
Residuals 9 15.183 1.6870
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 9 31.95 3.55
Error: id:IV1
Df Sum Sq Mean Sq F value Pr(>F)
IV1 1 11.15 11.148 3.542 0.0925 .
Residuals 9 28.33 3.147
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Error: id
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 9 39.6 4.4
Error: id:IV1
Df Sum Sq Mean Sq F value Pr(>F)
IV1 1 1.062 1.062 0.33 0.58
Residuals 9 28.993 3.221 Multivariate approach
Type III Repeated Measures MANOVA Tests:
------------------------------------------
Term: (Intercept)
Response transformation matrix:
(Intercept)
DV.1.1 1
DV.2.1 1
DV.1.2 1
DV.2.2 1
DV.1.3 1
DV.2.3 1
Sum of squares and products for the hypothesis:
(Intercept)
(Intercept) 606.4372
Multivariate Tests: (Intercept)
Df test stat approx F num Df den Df Pr(>F)
Pillai 1 0.6466107 16.46767 1 9 0.0028503 **
Wilks 1 0.3533893 16.46767 1 9 0.0028503 **
Hotelling-Lawley 1 1.8297405 16.46767 1 9 0.0028503 **
Roy 1 1.8297405 16.46767 1 9 0.0028503 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------
Term: IV1
Response transformation matrix:
IV11
DV.1.1 1
DV.2.1 -1
DV.1.2 1
DV.2.2 -1
DV.1.3 1
DV.2.3 -1
Sum of squares and products for the hypothesis:
IV11
IV11 49.37295
Multivariate Tests: IV1
Df test stat approx F num Df den Df Pr(>F)
Pillai 1 0.2220279 2.568538 1 9 0.14347
Wilks 1 0.7779721 2.568538 1 9 0.14347
Hotelling-Lawley 1 0.2853931 2.568538 1 9 0.14347
Roy 1 0.2853931 2.568538 1 9 0.14347
------------------------------------------
Term: IV2
Response transformation matrix:
IV21 IV22
DV.1.1 1 0
DV.2.1 1 0
DV.1.2 0 1
DV.2.2 0 1
DV.1.3 -1 -1
DV.2.3 -1 -1
Sum of squares and products for the hypothesis:
IV21 IV22
IV21 470.5200 317.6588
IV22 317.6588 214.4588
Multivariate Tests: IV2
Df test stat approx F num Df den Df Pr(>F)
Pillai 1 0.5493664 4.87639 2 8 0.041238 *
Wilks 1 0.4506336 4.87639 2 8 0.041238 *
Hotelling-Lawley 1 1.2190976 4.87639 2 8 0.041238 *
Roy 1 1.2190976 4.87639 2 8 0.041238 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------
Term: IV1:IV2
Response transformation matrix:
IV11:IV21 IV11:IV22
DV.1.1 1 0
DV.2.1 -1 0
DV.1.2 0 1
DV.2.2 0 -1
DV.1.3 -1 -1
DV.2.3 1 1
Sum of squares and products for the hypothesis:
IV11:IV21 IV11:IV22
IV11:IV21 0.3717586 -1.990582
IV11:IV22 -1.9905822 10.658576
Multivariate Tests: IV1:IV2
Df test stat approx F num Df den Df Pr(>F)
Pillai 1 0.1725701 0.8342466 2 8 0.46873
Wilks 1 0.8274299 0.8342466 2 8 0.46873
Hotelling-Lawley 1 0.2085617 0.8342466 2 8 0.46873
Roy 1 0.2085617 0.8342466 2 8 0.46873Detach (automatically) loaded packages (if possible)
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