STA 6208 – Spring 2012 – Exam 2
PRINT Name _______________________
Note: Conduct all tests at at  = 0.05 significance level
Q.1. For the 2-way ANOVA with one fixed factor, 1 random factor and a random interaction, considering the following model:
yijk     i  b j   ab ij  eijk
a

i 1
i

 0 b j ~ NID 0,  b2
i  1,..., a j  1,..., b k  1,..., r
  ab 
ij

2
~ NID 0,  ab


eijk ~ NID 0,  2
 b   ab   e 
j
ij
ijk
p.1.a. Derive E(Yij•), V(Yij•), Cov(Yij•, Yij’•)
p.1.b. Consider the following results:
 
V y i 

1
2
r b2  r ab
 2
br



COV y i , y i ' 
 
1 2
b
b
2
i  i ' E  MSAB   r ab
 2
Set-up a 95% Confidence Interval for (i - i’) based on these results (note that the variance components are unknown). Hint: Derive
the mean and variance of
yi  yi ' and make use of the estimated variance (standard error). Be specific (symbolically) and on
degrees of freedom being used.
Q.2. A 2-Factor (nested, fixed effects ANOVA) is fit, with factor A at 4 levels, and factor B at 3 levels (within each level of A). There
were 6 replicates for each combination of factor levels. Compute Bonferroni’s and Tukey’s minimum significant differences for
comparing all pairs of means (Factor A) with an experiment-wise error rate of 0.05. Note: This is the same as the margin of error for
the point estimates (half-width of simultaneous CI’s). Give your results as functions of only MSE. The model is:
yijk     i   j (i )  ek (ij )
4
i  0
i 1
3

j 1
j (i )
i  1, 2,3, 4
j  1, 2,3 k  1,..., 6

 0  i ek (ij ) ~ NID 0,  2

Bonferroni _______________________________________ Tukey _____________________________________________
Q.3. The following partial ANOVA table gives the results of a balanced 2-Way ANOVA with interaction. Fill in the following values
(for the mixed model, assume the unrestricted model, that is (ab)ij ~ NID(0,ab2)):
Source
A
B
AB
Error
Total
df
2
4
SS
600
820
256
75
2126
a  ___ b  ___ r  ___ FAB  ______
A Fixed, B Fixed: FA  _____
F (.05)  _____   _____
F (.05)  _____
A Random, B Random: FA  _____
A Fixed, B Random: FA  _____
^ 2
FB  _____
F (.05)  _____
F (.05)  _____
FB  _____
F (.05)  _____ FB  _____
F (.05)  _____
F (.05)  _____
Q.4. Consider the 1-Way Random Effects Model:
yij    ai  eij i  1,..., t j  1,..., r ai ~ NID  0,  a2  eij ~ NID  0,  e2 
a  e
p.4.a. Based on rules involving linear functions of RVs, derive the following values:
E  yij  , V  yij  , V  yi  , V  y  , COV  yij , yi 
p.4.b. In a population of individuals, 95% have mean values between 80 and 120. I, the Intraclass Correlation
is 0.80. Within individuals, 95% of individual observations lie within how many units from the individual
mean?
Q.5. For the 2-Way ANOVA with random effects and interactions,
yijk    ai  b j   ab ij  eijk
i  1,..., a; j  1,..., b; k  1,..., r
ai ~ NID  0,  a2  b j ~ NID  0,  b2 
 ab ij ~ NID  0,  ab2 
eijk ~ N  0,  2 
a  b  ab  e
Obtain the following quantities:
p.5.a. E  yijk  



p.5.b. COV  yijk , yi ' j ' k '   



 
p.5.c. V y ij  
i  i ', j  j ', k  k '
i  i ', j  j ', k  k '
i  i ', j  j ', k , k '
i  i ', j  j ', k , k '
i  i ', j  j ', k , k '
Q.6. In a large factory, with many Operators, Parts, and Devices, an experiment is conducted to measure the variation in measured
strengths of parts. Samples of 6 Operators, 8 Parts, and 4 Devices were obtained; with each combination of Operators, Parts, and
Instruments being replicated 3 times. The following model is fit (with all random effects independent).
yijkl    oi  p j  d k  opij  odik  pd jk  opdijk  eijkl
opij ~ N  0,  op2  odik ~ N  0,  od2 
oi ~ N  0,  o2 
p j ~ N  0,  p2  d k ~ N  0,  d2 
2
pd jk ~ N  0,  pd
 opdijk ~ N  0,  opd2  eijkl ~ N  0,  2 
p.6.a. Complete the following ANOVA Table:
Source
O
P
D
OP
OD
PD
OPD
Error
Total
df
SS
800
490
420
700
240
105
315
2
 opd
0
vs HA:
 op2  0
vs HA:
 o2  0
vs HA:
 op2
 op2  0
p.6.d.i Obtain an unbiased estimate of
p.6.d.ii. Test H0:
2
 opd
_________________________
2
 opd
0
p.6.c.i Obtain an unbiased estimate of
p.6.c.ii. Test H0:
E(MS)
4030
p.6.b.i Obtain an unbiased estimate of
p.6.b.ii. Test H0:
MS
 o2
 o2  0
Test Statistic: _____________ Rejection Region: ___________
_________________________
Test Statistic: _____________ Rejection Region: ___________
_________________________
Test Statistic: _____________ Rejection Region: ___________
2
 c

  gi MSi 
c
^

"Synthetic Mean Square" =  gi MSi   *   i 1
2
c
i 1
 gi MSi 

i 1
i
where  i  df  MSi 
Q.7. A study is conducted to compare 3 methods of teaching speed reading to adults. A sample of 9 adults (with
no prior exposure to speed-reading techniques) is taken, and randomly assigned so that 3 adults receive method
1, 3 receive method 2, and 3 receive method 3. Each adult is observed reading 4 texts of the same length, and y,
the amount of time needed to complete the text is measured. The model fit is:
yij     i  b j (i )  ek (ij )
a

i 1
i
i  1,..., a
j  1,..., b k  1,..., r
 0 b j (i ) ~ NID  0,  B2 ( A)  ek (ij ) ~ NID  0,  2 
b  e
The means (SDs) for each adult are given below (in minutes).
Method
Adult
Mean (SD)
1
1
120 (10)
1
2
125 (12)
1
3
115 (8)
2
4
105 (12)
2
5
110 (10)
2
6
115 (8)
3
7
135 (10)
3
8
130 (12)
3
9
125 (8)
Complete the following ANOVA table.
Source
Method
Adult(Method)
Error
Total
df
SS
MS
F
F(0.05)