1 Problem 1

  1. Model equation:

\[ y_{ijkl} = \mu + \alpha_i + \beta_j + \gamma_k + \alpha\beta_{ij} + \alpha\gamma_{ik} + \beta\gamma_{jk} + \alpha\beta\gamma_{ijk} + e_{ijkl} \]

  1. Significant Factors:

Step 1: Full factorial

dat <- data.frame(
  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))

library(GAD)

dat$Ammonium <- as.fixed(factor(dat$Ammonium))
dat$StirRate <- as.fixed(factor(dat$StirRate))
dat$Temperature <- as.fixed(factor(dat$Temperature))

model <- aov(
  Density ~ Ammonium * StirRate * Temperature,
  data = dat
)

summary(model)
##                               Df Sum Sq Mean Sq F value  Pr(>F)   
## Ammonium                       1  44.39   44.39  11.180 0.01018 * 
## StirRate                       1  70.69   70.69  17.804 0.00292 **
## Temperature                    1   0.33    0.33   0.083 0.78117   
## Ammonium:StirRate              1  28.12   28.12   7.082 0.02875 * 
## Ammonium:Temperature           1   0.02    0.02   0.005 0.94281   
## StirRate:Temperature           1  10.13   10.13   2.551 0.14889   
## Ammonium:StirRate:Temperature  1   1.52    1.52   0.383 0.55341   
## Residuals                      8  31.76    3.97                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Step 2: Remove non-significant terms

Removed terms:

  • Temperature

  • Ammonium:Temperature

  • StirRate:Temperature

  • Ammonium:StirRate:Temperature

Step 3: Final ANOVA and Interaction Plot

final_model <- aov(
  Density ~ Ammonium * StirRate,
  data = dat
)

summary(final_model)
##                   Df Sum Sq Mean Sq F value   Pr(>F)    
## Ammonium           1  44.39   44.39   12.17 0.004472 ** 
## StirRate           1  70.69   70.69   19.38 0.000861 ***
## Ammonium:StirRate  1  28.12   28.12    7.71 0.016751 *  
## Residuals         12  43.76    3.65                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
interaction.plot(
  x.factor = dat$Ammonium,
  trace.factor = dat$StirRate,
  response = dat$Density,
  xlab = "Ammonium",
  ylab = "Mean Density",
  trace.label = "Stir Rate"
)

The full factorial ANOVA shows that Ammonium, Stir Rate, and their interaction (AB) are significant at the 0.05 level of significance. The ABC, AC, and BC interactions are not significant and are removed from the model. Temperature is also removed because it is not significant.

The final model includes Ammonium (p = 0.0045), Stir Rate (p = 0.0009), and the AB interaction (p = 0.0168). The interaction plot shows that the effect of Ammonium on Density depends on the Stir Rate. Therefore, both factors and their interaction should be included in the final model.

2 Problem 2

library(agricolae)

design <- design.ab(
  trt = c(2, 2, 2),
  r = 3,
  design = "rcbd",
  seed = 711
)

design$book
##    plots block A B C
## 1    101     1 2 1 2
## 2    102     1 2 2 2
## 3    103     1 1 2 2
## 4    104     1 1 1 2
## 5    105     1 1 1 1
## 6    106     1 1 2 1
## 7    107     1 2 1 1
## 8    108     1 2 2 1
## 9    109     2 2 2 1
## 10   110     2 1 2 1
## 11   111     2 2 2 2
## 12   112     2 1 2 2
## 13   113     2 2 1 2
## 14   114     2 1 1 2
## 15   115     2 2 1 1
## 16   116     2 1 1 1
## 17   117     3 1 2 1
## 18   118     3 1 2 2
## 19   119     3 1 1 1
## 20   120     3 2 1 2
## 21   121     3 2 1 1
## 22   122     3 2 2 1
## 23   123     3 1 1 2
## 24   124     3 2 2 2

A full (2^3) factorial experiment was designed using a randomized complete block design (RCBD) with three replications. The experiment consists of eight treatment combinations and 24 experimental units. The design.ab() function from the agricolae package was used to randomly assign the treatment combinations within each block. Each block contains all eight treatment combinations.