Mark Breunig
NRES 797
Project 3
Problem 1
(a) The Generalized Randomized Block Design Model:
yijv =  + i+ j+ ()ij + eijv
yijv = the response of the vth replicate of the jth pesticide treatment in area i (v=1,2,..5 j=1,2,3 i=1,2)
 = overall mean soil pH
th
i= i area effect
 j = jth pesticide treatment effect (averaged over areas)
()ij = area-pesticide treatment interaction of the ith area and the jth pesticide treatment
eijv = the error associated with the vth replicate of the jth pesticide treatment in the ith area
(b) First, the initial assumptions of normality and constant variance must be satisfied.
Test for Normality of pH:
Ho: errors from the ANOVA are normal
H1: errors from the ANOVA are non-normal
α =0.05
Shapiro-Wilk test
The p-value for this test is 0.3237; therefore, I fail to reject the null hypothesis and conclude the pH
values are normally distributed.
Test for Constant Variance:
H0: σ²1 = σ²2 = σ²3
H1: at least one not equal
α=0.05
Levene’s Test
The p-value for this test is 0.4892; therefore, I fail to reject the null hypothesis and conclude
homogeneity of variance.
Since the initial assumptions of normality by treatment and constant variance are satisfied, the following
test for Block Treatment Interaction is valid:
H0: ()ij=0  i,j
H1: at least one not equal
F test: MS(BT)/MS(E)
GLM
The p-value for this test is 0.2705; therefore, I fail to reject the null hypothesis and conclude there is no
significant interaction between area and pesticide treatment.
Test for Difference in Treatment:
H0: 1=2=...=3
H1: at least one not equal
F test: MS(T)/MS(E)
GLM
The p-value for this test is 0.003; therefore, I fail reject the null hypothesis and conclude there is a
significant difference pesticide treatment effects on soil pH values.
(c) Multiple Range Test for Differences in pesticide treatments:
µ = mean soil pH
H0: µpesticide treatment 1 = µpesticide treatment 2= µpesticide treatment 3
H1: at least one not equal
α =0.05
Tukey’s test
Tukey Grouping
Mean
N
treat
A
A
A
6.4700
10
1
6.2100
10
2
B
5.6200
10
3
Conclusion: The mean soil pH, while no different between pesticide treatments 1 and 2, is significantly
lower for pesticide treatment 3.
(d) Pesticide treatment 3 had the most impact on soil pH.
Problem 2
A
B C
D
AB
AC
AD
BC
BD CD
ABC
ABD
ACD
BCD
yijklm = μ + αi + βj +γk + δl + αβij + αγik + αδil + βγjk + βδjl + γδkl + αβγijk + αβδijl + αγδikl + βγδjkl +
ABCD E
αβγδijkl + εijklm
(b)
(a)
A: A + AB + AD + ABD + E
BC: BC + BCD + E
ACD: ACD + ABCD + E
BCD: BCD + E
BC: Βγjk + βγδjkl + εijklm
ACD: αγδikl + αβγδijkl + εijklm
C: γk + βγjk + γδkl + βγδjkl + εijklm
BCD: βγδjkl + εijklm