Three-factor factorial experiment
three_factor <- read_excel("three_factorial.xlsx")
three_factor <- three_factor %>%
mutate(Block = factor(Block),
A = factor(A, labels = c("A1", "A2")),
B = factor(B, labels = c("B1", "B2")),
C = factor(C, labels=c("C1", "C2", "C3")))
with(three_factor, fat3.rbd(factor1 = A,
factor2 = B,
factor3 = C,
block = Block,
resp = Y,
quali = c(TRUE, TRUE, TRUE),
mcomp = "tukey",
fac.names = c("A", "B", "C")))
## ------------------------------------------------------------------------
## Legend:
## FACTOR 1: A
## FACTOR 2: B
## FACTOR 3: C
## ------------------------------------------------------------------------
##
## ------------------------------------------------------------------------
## Analysis of Variance Table
## ------------------------------------------------------------------------
## DF SS MS Fc Pr>Fc
## Block 5 131.89347 26.37869 1.4099 0.2351
## A 1 313.35877 313.35877 16.749 1e-04
## B 1 1793.55576 1793.55576 95.8655 0
## C 2 283.16803 141.58402 7.5677 0.0012
## A*B 1 2.73058 2.73058 0.1459 0.7039
## A*C 2 129.18232 64.59116 3.4524 0.0387
## B*C 2 27.59502 13.79751 0.7375 0.483
## A*B*C 2 286.59562 143.29781 7.6593 0.0012
## Residuals 55 1028.99941 18.70908
## Total 66 3997.07898
## ------------------------------------------------------------------------
## CV = 5.57 %
##
## ------------------------------------------------------------------------
## Shapiro-Wilk normality test
## p-value: 0.1545518
## According to Shapiro-Wilk normality test at 5% of significance, residuals can be considered normal.
## ------------------------------------------------------------------------
##
##
##
## Significant A*B*C interaction: analyzing the interaction
## ------------------------------------------------------------------------
##
## Analyzing A inside of each level of B and C
## ------------------------------------------------------------------------
## ------------------------------------------------------------------------
## Analysis of Variance Table
## ------------------------------------------------------------------------
## DF SS MS Fc Pr>Fc
## A: B1 C1 1 10.779355 10.779355 0.576156 0.451064
## A: B1 C2 1 68.454804 68.454804 3.658908 0.060982
## A: B1 C3 1 65.603549 65.603549 3.506509 0.066448
## A: B2 C1 1 577.446016 577.446016 30.864479 1e-06
## A: B2 C2 1 4.080335 4.080335 0.218094 0.642341
## A: B2 C3 1 5.503246 5.503246 0.294148 0.589765
## Residuals 55 1028.999406 18.709080
## ------------------------------------------------------------------------
##
##
##
## A inside of the combination of the levels B1 of B and C1 of C
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 A1 76.05178
## 2 A2 74.15623
## ------------------------------------------------------------------------
##
##
## A inside of the combination of the levels B1 of B and C2 of C
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 A1 73.13877
## 2 A2 68.36193
## ------------------------------------------------------------------------
##
##
## A inside of the combination of the levels B1 of B and C3 of C
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 A1 74.45586
## 2 A2 69.77956
## ------------------------------------------------------------------------
##
##
## A inside of the combination of the levels B2 of B and C1 of C
## ------------------------------------------------------------------------
## Tukey's test
## ------------------------------------------------------------------------
## Groups Treatments Means
## a A1 92.73885
## b A2 78.86506
## ------------------------------------------------------------------------
##
##
## A inside of the combination of the levels B2 of B and C2 of C
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 A1 82.34154
## 2 A2 81.17530
## ------------------------------------------------------------------------
##
##
## A inside of the combination of the levels B2 of B and C3 of C
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 A1 79.68074
## 2 A2 81.03514
## ------------------------------------------------------------------------
##
##
##
## Analyzing B inside of each level of A and C
## ------------------------------------------------------------------------
## ------------------------------------------------------------------------
## Analysis of Variance Table
## ------------------------------------------------------------------------
## DF SS MS Fc Pr>Fc
## B: A1 C1 1 835.37433 835.37433 44.650743 0
## B: A1 C2 1 254.07290 254.07290 13.580192 0.000524
## B: A1 C3 1 81.89790 81.89790 4.377441 0.041046
## B: A2 C1 1 66.51918 66.51918 3.555449 0.064636
## B: A2 C2 1 492.54799 492.54799 26.326681 4e-06
## B: A2 C3 1 380.06468 380.06468 20.314451 3.5e-05
## Residuals 55 1028.99941 18.70908
## ------------------------------------------------------------------------
##
##
##
## B inside of the combination of the levels A1 of A and C1 of C
## ------------------------------------------------------------------------
## Tukey's test
## ------------------------------------------------------------------------
## Groups Treatments Means
## a B2 92.73885
## b B1 76.05178
## ------------------------------------------------------------------------
##
##
## B inside of the combination of the levels A1 of A and C2 of C
## ------------------------------------------------------------------------
## Tukey's test
## ------------------------------------------------------------------------
## Groups Treatments Means
## a B2 82.34154
## b B1 73.13877
## ------------------------------------------------------------------------
##
##
## B inside of the combination of the levels A1 of A and C3 of C
## ------------------------------------------------------------------------
## Tukey's test
## ------------------------------------------------------------------------
## Groups Treatments Means
## a B2 79.68074
## b B1 74.45586
## ------------------------------------------------------------------------
##
##
## B inside of the combination of the levels A2 of A and C1 of C
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 B1 74.15623
## 2 B2 78.86506
## ------------------------------------------------------------------------
##
##
## B inside of the combination of the levels A2 of A and C2 of C
## ------------------------------------------------------------------------
## Tukey's test
## ------------------------------------------------------------------------
## Groups Treatments Means
## a B2 81.1753
## b B1 68.36193
## ------------------------------------------------------------------------
##
##
## B inside of the combination of the levels A2 of A and C3 of C
## ------------------------------------------------------------------------
## Tukey's test
## ------------------------------------------------------------------------
## Groups Treatments Means
## a B2 81.03514
## b B1 69.77956
## ------------------------------------------------------------------------
##
## Analyzing C inside of each level of A and B
## ------------------------------------------------------------------------
## ------------------------------------------------------------------------
## Analysis of Variance Table
## ------------------------------------------------------------------------
## DF SS MS Fc Pr>Fc
## C: A1 B1 2 25.53464 12.76732 0.682413 0.509623
## C: A1 B2 2 571.39621 285.69811 15.270559 5e-06
## C: A2 B1 2 109.47787 54.73893 2.925795 0.062015
## C: A2 B2 2 20.13228 10.06614 0.538035 0.586936
## Residuals 55 1028.99941 18.70908
## ------------------------------------------------------------------------
##
##
##
## C inside of the combination of the levels A1 of A and B1 of B
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 C1 76.05178
## 2 C2 73.13877
## 3 C3 74.45586
## ------------------------------------------------------------------------
##
##
## C inside of the combination of the levels A1 of A and B2 of B
## ------------------------------------------------------------------------
## Tukey's test
## ------------------------------------------------------------------------
## Groups Treatments Means
## a C1 92.73885
## b C2 82.34154
## b C3 79.68074
## ------------------------------------------------------------------------
##
##
## C inside of the combination of the levels A2 of A and B1 of B
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 C1 74.15623
## 2 C2 68.36193
## 3 C3 69.77956
## ------------------------------------------------------------------------
##
##
## C inside of the combination of the levels A2 of A and B2 of B
##
## According to the F test, the means of this factor are statistical equal.
## ------------------------------------------------------------------------
## Levels Means
## 1 C1 78.86506
## 2 C2 81.17530
## 3 C3 81.03514
## ------------------------------------------------------------------------
ANOVA Table using aov()
threefactor.model <- aov(Y ~ Block + A*B*C,
data = three_factor)
anova(threefactor.model)
## Analysis of Variance Table
##
## Response: Y
## Df Sum Sq Mean Sq F value Pr(>F)
## Block 5 131.89 26.38 1.4099 0.2351306
## A 1 313.36 313.36 16.7490 0.0001408 ***
## B 1 1793.56 1793.56 95.8655 1.182e-13 ***
## C 2 283.17 141.58 7.5677 0.0012494 **
## A:B 1 2.73 2.73 0.1459 0.7039086
## A:C 2 129.18 64.59 3.4524 0.0386862 *
## B:C 2 27.60 13.80 0.7375 0.4829890
## A:B:C 2 286.60 143.30 7.6593 0.0011629 **
## Residuals 55 1029.00 18.71
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Compute 3-way treatment combination means and assign letters
abc.means <- three_factor %>%
group_by(A,B,C) %>%
dplyr::summarise(Mean = mean(Y),
SD = sd(Y),
n = length(Y)) %>%
dplyr::mutate(SE = SD/sqrt(n))
abc.means
## # A tibble: 12 × 7
## # Groups: A, B [4]
## A B C Mean SD n SE
## <fct> <fct> <fct> <dbl> <dbl> <int> <dbl>
## 1 A1 B1 C1 76.1 3.85 6 1.57
## 2 A1 B1 C2 73.1 4.77 6 1.95
## 3 A1 B1 C3 74.5 4.89 6 2.00
## 4 A1 B2 C1 92.7 5.12 6 2.09
## 5 A1 B2 C2 82.3 5.00 6 2.04
## 6 A1 B2 C3 79.7 4.05 6 1.65
## 7 A2 B1 C1 74.2 3.69 6 1.51
## 8 A2 B1 C2 68.4 4.08 6 1.67
## 9 A2 B1 C3 69.8 2.72 6 1.11
## 10 A2 B2 C1 78.9 5.32 6 2.17
## 11 A2 B2 C2 81.2 4.62 6 1.89
## 12 A2 B2 C3 81.0 3.98 6 1.63
ABC.means <- emmeans(threefactor.model, specs=c("A", "B", "C"))
ABC.means
## A B C emmean SE df lower.CL upper.CL
## A1 B1 C1 76.1 1.77 55 72.5 79.6
## A2 B1 C1 74.2 1.77 55 70.6 77.7
## A1 B2 C1 92.7 1.77 55 89.2 96.3
## A2 B2 C1 78.9 1.77 55 75.3 82.4
## A1 B1 C2 73.1 1.77 55 69.6 76.7
## A2 B1 C2 68.4 1.77 55 64.8 71.9
## A1 B2 C2 82.3 1.77 55 78.8 85.9
## A2 B2 C2 81.2 1.77 55 77.6 84.7
## A1 B1 C3 74.5 1.77 55 70.9 78.0
## A2 B1 C3 69.8 1.77 55 66.2 73.3
## A1 B2 C3 79.7 1.77 55 76.1 83.2
## A2 B2 C3 81.0 1.77 55 77.5 84.6
##
## Results are averaged over the levels of: Block
## Confidence level used: 0.95
#AGenerate the letters
ABC.means.cld <- cld(ABC.means,
Letters=letters,
decreasing = TRUE)
ABC.means.cld <- ABC.means.cld %>%
arrange(A, B, C) %>%
mutate(SE1 = abc.means$SE)
ABC.means.cld
## A B C emmean SE df lower.CL upper.CL .group SE1
## 1 A1 B1 C1 76.05178 1.765837 55 72.51297 79.59060 bcd 1.573747
## 2 A1 B1 C2 73.13877 1.765837 55 69.59996 76.67759 cd 1.945328
## 3 A1 B1 C3 74.45586 1.765837 55 70.91705 77.99468 bcd 1.995871
## 4 A1 B2 C1 92.73885 1.765837 55 89.20003 96.27766 a 2.088637
## 5 A1 B2 C2 82.34154 1.765837 55 78.80273 85.88036 b 2.041608
## 6 A1 B2 C3 79.68074 1.765837 55 76.14192 83.21955 bc 1.651725
## 7 A2 B1 C1 74.15623 1.765837 55 70.61741 77.69505 bcd 1.506547
## 8 A2 B1 C2 68.36193 1.765837 55 64.82311 71.90074 d 1.666405
## 9 A2 B1 C3 69.77956 1.765837 55 66.24074 73.31837 d 1.110290
## 10 A2 B2 C1 78.86506 1.765837 55 75.32624 82.40387 bc 2.170447
## 11 A2 B2 C2 81.17530 1.765837 55 77.63649 84.71412 bc 1.885857
## 12 A2 B2 C3 81.03514 1.765837 55 77.49633 84.57396 bc 1.626823
#Comparing the levels of A for each level of B
ggplot(ABC.means.cld, aes(x = A,
y = emmean,
fill = B,
label = .group)) +
geom_col(position = "dodge") +
geom_errorbar(aes(ymin = emmean - SE1,
ymax = emmean + SE1),
width = 0.2,
size = 0.7,
position = position_dodge(0.8)) +
geom_text(vjust=-2.5,
position = position_dodge(0.9),
size = 2.5) +
labs(x= "A", y= "Mean Yield") +
lims(y=c(0,100))+
facet_wrap(. ~ C) +
theme_classic() +
theme(legend.position = "bottom") +
theme(legend.title=element_blank())
