STA 6208 – Spring 2014 – 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. For the 2-Way Random Effects model, with a = 2, b = 4, and r = 3; DERIVE the following Variances:
yijk    ai  b j   ab ij  eijk
i  1, 2; j  1, 2,3, 4; k  1, 2,3
ai ~ NID  0,  a2  b j ~ NID  0,  b2 
 
 
 ab ij ~ NID  0,  ab2 
 
 
V  yijk  , V y ij  , V y i , V y  j  , V y 
eijk ~ N  0,  2 
a  b  ab  e
Q.3 Researchers conducted an experiment measuring acoustic metric values in a = 3 habitats (1=Cliff, 2=Mud,
3=Gravel). Nested within each habitat, there were b = 3 random patches. Replicates representing r = 5 sites within each
patch were obtained. The habitats are considered to be fixed levels, while patches within habitats are considered to be
random. The response measured was snap amplitude. The model is given below, numbers in the ANOVA are rounded.
yijk     i  b j (i )  ek (ij )
a

i 1
i
i  1,..., a

 0 b j (i ) ~ NID 0,  P2

j  1,..., b k  1,..., r

ek (ij ) ~ NID 0,  2

b  e
p.3.a. Complete the following ANOVA Table and give the expected mean square for each row. Note that for the F-Stats, you are
testing: H 0 : 1  ...   a  0
H
Source
Habitat
Patch(Hab)
Error
Total
df
H AH : Not all  i  0
SS
MS
H 0P :  P2  0 H AJ :  P2  0
F-Stat
F(.05)
E(MS)
400.0
300.0
1100.0
p.3.b Compute Tukey’s and Bonferroni’s minimum significant difference for comparing pairs of habitat means.
Tukey’s HSD: __________________________________ Bonferroni’s MSD: ___________________________
p.3.c Obtain point estimate for P2 and 2
Q.4. A study is conducted to measure intra- and inter-observer reliability in tasting experiments. A random sample of a = 10 Varieties
of wine were selected at a wine store. Further, a random sample of b = 5 trained Judges were obtained from a society of wine
aficionados. Each judge rated each wine r = 3 times. The judge gave the wine an evaluation for a particular attribute on a visual
analogue scale from 0-100. (Note: The order of the 30 tastes for each judge were randomized and blinded, and the judges were given
cab rides home). The model fit is:
yijk    ai  b j   ab ij  eijk
i  1,..., a; j  1,..., b; k  1,..., r
ai ~ NID  0,  V2  b j ~ NID  0,  J2 
 ab ij ~ NID  0, VJ2 
eijk ~ N  0,  2 
a  b  ab  e
p.4.a. Complete the following ANOVA Table and give the expected mean square for each row. Note that for the F-Stats, you are
testing: H 0 :  V  0
V
2
H VA :  V2  0
Source
Wine Variety (V)
Judge (J)
Variety x Judge (VJ)
Error
Total
df
H 0J :  J2  0 H AJ :  J2  0
SS
MS
2
H 0VJ :  VJ2  0 H VJ
A :  VJ  0
F-Stat
F(.05)
E(MS)
8100
2000
1440
12740
p.4.b. Give unbiased (ANOVA) estimates of each of the four variance components:
^2
^2
^2
^2
 V  _____________  J  _____________  VJ  _____________   _____________
p.4.c. For this type of reliability study, there are two types of correlations: Inter-Judge (Among), and Intra-Judge (Within). The InterJudge and Intra-Judge Covariances, as well as Observation measurement Variance are used to obtain the two correlations. Give these
terms (based on actual variance components) and their point estimates:
Measurement: V Yijk   ________________________ V Yijk   ________________________
^
Inter-Judge: COV Yijk , Yij ' k '   ________________
j  j '  k , k ' COV Yijk , Yij ' k '   ________________
Intra-Judge: COV Yijk , Yijk '   ________________________
^
k  k ' COV Yijk , Yijk '   ___________________
^
p.4.d. The Inter-Judge and Intra-Judge Correlations are defined below, give “simple” point estimates of each:
ICCInter 
ICCIntra 
COV Yijk , Yij ' k ' 
V Yijk 
COV Yijk , Yijk ' 
V Yijk 
^
ICC Inter  __________________________________________
^
ICC Intra  ______________________________________________
Q.5. 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 4 Operators, 10 Parts, and 3 Devices were obtained; with each combination of Operators, Parts, and
Instruments being replicated 5 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
1200
450
160
540
60
144
270
2
 opd
0
vs HA:
 op2  0
vs HA:
 o2  0
vs HA:
 op2
 op2  0
p.5.d.i Obtain an unbiased estimate of
p.5.d.ii. Test H0:
2
 opd
_________________________
2
 opd
0
p.5.c.i Obtain an unbiased estimate of
p.5.c.ii. Test H0:
E(MS)
5224
p.5.b.i Obtain an unbiased estimate of
p.5.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.6. An experiment was conducted to determine the effects of food appearance on the plate (balanced/unbalanced and
color/monochrome) on consumers hedonic liking of food. A sample of 68 restaurant customers was obtained, and
assigned such that 17 received each combination of balance and color. The following table gives the means (SDs) for each
treatment. The response (Y) was hedonic liking of the food dish (on a scale of -100 to +100).
Model: yijk    i   j     eijk
ij
i  1, 2 j  1, 2 k  1,...,17
2
2
2
2
           
i
i 1
j 1
j
ij
i 1
j 1
ij
 0  ijk ~ N  0,  2 
p.6.a. The following table gives the means (SDs) for each treatment:
Balance\Color
1=Balanced
2=Unbalanced
Col Mean
1=Mono
13.3 (53)
13.6 (41)
2=Color
-1.7 (69)
2.2 (58)
Row Mean
Complete the following ANOVA table:
ANOVA
Source
Balance
Color
Bal*Col
Error
Total
df
SS
MS
F*
F(0.95)
#N/A
#N/A
#N/A
#N/A
#N/A
p.6.b. Test H0: No Interaction between Balance and Color
p.6.b.i. Test Stat: _______
p.6.b.ii. Reject H0 if Test Stat is in the range _________
p.6.b.iii. P-value > or < .05?
p.6.c. Test H0: No Balance effect
p.6.c.i. Test Stat: _______
p.6.c.ii. Reject H0 if Test Stat is in the range _________
p.6.c.iii. P-value > or < .05?
p.6.d. Test H0: No Color effect
p.6.d.i. Test Stat: _________
p.6.d.ii. Reject H0 if Test Stat is in the range ________
p.6.d.iii. P-value > or < .05?