#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)#1a
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 = interactions - ABC = three-way interaction - ε = error
#1b
# Fit 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 ' ' 1Answer:
The factorial ANOVA results show that at α = 0.05, three effects are statistically significant. First, Ammonium has a significant main effect witch values of F = 11.180 and p = 0.01018) indicating that the amount of ammonium significantly affects density. Secondly, stir rate also has a significant effect , with values of F = 17.80 and p = 0.00292, that show stirring speed is an important factor in determining the density. Thirdly there is a significant two-way interaction between ammonium and stir rate , with values of F = 7.082 and p = 0.02875, meaning the effect of ammonium on density depends on the stir rate level.
#Problem 2
#data
Position <- c(1,1,1, 1,1,1, 1,1,1, 2,2,2, 2,2,2, 2,2,2)
Temperature <- c(800,800,800, 825,825,825, 850,850,850,
800,800,800, 825,825,825, 850,850,850)
Density2 <- c(570,565,583, 1063,1080,1043, 565,510,590,
528,547,521, 988,1026,1004, 526,538,532)#2a
#fixed
model_a <- aov(Density2 ~ factor(Temperature) * factor(Position))
summary(model_a)## Df Sum Sq Mean Sq F value Pr(>F)
## factor(Temperature) 2 945342 472671 1056.117 3.25e-14 ***
## factor(Position) 1 7160 7160 15.998 0.00176 **
## factor(Temperature):factor(Position) 2 818 409 0.914 0.42711
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Answer:
When both temp. and position are treated as fixed effects, the ANOVA results show that temp has a highly significant effect on density with p < 0.001. position also has a significant effect with p = 0.00176. However, the temp. × Position interaction is not significant with p = 0.42711.Both main effects are significant but they do not interact.
#2b
#random
model_b <- aov(Density2 ~ Temperature + Position + Temperature:Position)
summary(model_b)## Df Sum Sq Mean Sq F value Pr(>F)
## Temperature 1 234 234 0.003 0.954
## Position 1 7160 7160 0.105 0.750
## Temperature:Position 1 234 234 0.003 0.954
## Residuals 14 951063 67933Answer:
When both temp. and position are treated as random effects, none of the effects are significant. temperature has p = 0.954, position has p = 0.750, and the temperature × position interaction has p = 0.954. This indicates that when we consider these factors as random samples from larger populations, there is no significant evidence that either factor affects density.
#2c
# Position fixed, temp random
model_c <- aov(Density2 ~ factor(Position) + Temperature + factor(Position):Temperature)
summary(model_c)## Df Sum Sq Mean Sq F value Pr(>F)
## factor(Position) 1 7160 7160 0.105 0.750
## Temperature 1 234 234 0.003 0.954
## factor(Position):Temperature 1 234 234 0.003 0.954
## Residuals 14 951063 67933Answer: When position is treated as fixed and temperature as random, none of the effects are significant. Position has p = 0.750 and temperature has p = 0.954, and the position × temperature interaction has p = 0.954. This mixed model shows results similar to part b, where treating temperature as random leads to no significant findings.
#2d
Answer: The three analyses reveal differences based on whether factors are treated as fixed or random effects. In part a with both factors fixed, both temperature and position were significant. However, in part b with both factors random, neither effect was significant. Part c with position fixed and temp. random showed no significant effects either. This demonstrates that treating factors as random versus fixed fundamentally changes the hypothesis being tested and the conclusions drawn.