assignment16
Wilbert Nicolas
Last compiled on November 09, 2025 at 4:43 PM - CST
#16 part 1
# Enter data
Ammonium <- c(2,2,30,30, 2,2,30,30, 2,2,30,30, 2,2,30,30)
StirRate <- c(100,100,100,100, 150,150,150,150, 100,100,100,100, 150,150,150,150)
Temperature <- c(8,8,8,8, 8,8,8,8, 40,40,40,40, 40,40,40,40)
Density <- c(14.68,15.18,15.12,17.48, 7.54,6.66,12.46,12.62,
10.95,17.68,12.65,15.96, 8.03,8.84,14.96,14.96)#16a
Model: Y = μ + A + B + C + AB + AC + BC + ABC + ε
Where: - Y = density - μ = mean - A = Ammonium - B = Stir Rate - C = Temperature - AB, AC, BC = two-way interactions - ABC = three-way interaction - ε = error
#16b
#model
model <- aov(Density ~ factor(Ammonium) * factor(StirRate) * factor(Temperature))
summary(model)## Df Sum Sq Mean Sq F value
## factor(Ammonium) 1 44.39 44.39 11.180
## factor(StirRate) 1 70.69 70.69 17.804
## factor(Temperature) 1 0.33 0.33 0.083
## factor(Ammonium):factor(StirRate) 1 28.12 28.12 7.082
## factor(Ammonium):factor(Temperature) 1 0.02 0.02 0.005
## factor(StirRate):factor(Temperature) 1 10.13 10.13 2.551
## factor(Ammonium):factor(StirRate):factor(Temperature) 1 1.52 1.52 0.383
## Residuals 8 31.76 3.97
## Pr(>F)
## factor(Ammonium) 0.01018 *
## factor(StirRate) 0.00292 **
## factor(Temperature) 0.78117
## factor(Ammonium):factor(StirRate) 0.02875 *
## factor(Ammonium):factor(Temperature) 0.94281
## factor(StirRate):factor(Temperature) 0.14889
## factor(Ammonium):factor(StirRate):factor(Temperature) 0.55341
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#interaction plots
interaction.plot(Ammonium, StirRate, Density)
interaction.plot(Ammonium, Temperature, Density)
interaction.plot(StirRate, Temperature, Density)
Answer:
At α = 0.05, the significant factors are Ammonium (p = 0.010), Stir Rate (p = 0.003), and the Ammonium×Stir Rate interaction (p = 0.029). Temperature is not significant (p = 0.781) and can be dropped. The Ammonium×Temperature interaction (p = 0.943), Stir Rate×Temperature interaction (p = 0.149), and the three-way interaction (p = 0.553) can all be dropped because their p-values are greater than 0.05. The Ammonium×Stir Rate interaction plot shows non-parallel lines, confirming this interaction is real.
#16 Part 2
library(agricolae)
#Factors
A <- c("Organic", "Chemical")
B <- c("Once", "Twice")
C <- c("22C", "28C")
#treatment combinations
treatments <- expand.grid(A = A, B = B, C = C)
treatments$treatment <- paste(treatments$A, treatments$B, treatments$C, sep = "-")
#treatment combinations
print(treatments)## A B C treatment
## 1 Organic Once 22C Organic-Once-22C
## 2 Chemical Once 22C Chemical-Once-22C
## 3 Organic Twice 22C Organic-Twice-22C
## 4 Chemical Twice 22C Chemical-Twice-22C
## 5 Organic Once 28C Organic-Once-28C
## 6 Chemical Once 28C Chemical-Once-28C
## 7 Organic Twice 28C Organic-Twice-28C
## 8 Chemical Twice 28C Chemical-Twice-28C#RCBD Design
design <- design.ab(
trt = c(2, 2, 2), # 2 levels for each of 3 factors
r = 3, # 3 replications
serie = 2,
design = "rcbd"
)
#display
print(design$book)## plots block A B C
## 1 101 1 1 2 1
## 2 102 1 1 1 1
## 3 103 1 2 2 1
## 4 104 1 1 1 2
## 5 105 1 2 1 1
## 6 106 1 1 2 2
## 7 107 1 2 1 2
## 8 108 1 2 2 2
## 9 109 2 2 2 2
## 10 110 2 1 2 1
## 11 111 2 2 1 2
## 12 112 2 1 1 2
## 13 113 2 2 2 1
## 14 114 2 1 2 2
## 15 115 2 1 1 1
## 16 116 2 2 1 1
## 17 117 3 2 1 2
## 18 118 3 2 1 1
## 19 119 3 1 2 1
## 20 120 3 1 1 2
## 21 121 3 2 2 2
## 22 122 3 1 2 2
## 23 123 3 2 2 1
## 24 124 3 1 1 1Experimental Layout:
The design creates 8 treatment combinations (2³ = 8) arranged in 3 blocks (replications). Each block contains all 8 treatments in random order to control for variation between greenhouse benches.