Midterm Stat 505 Fall 2013
October 19, 2013
Name:
100 pts total
We’ll use the linear model: y = Xβ + with ∼ (0, σ 2 V ) unless otherwise noted below.
80
weight
A study was done to determine the effect of
two different diets on weight gain of pigs. A
group of 48 freshly weaned pigs was selected
at random from young pigs available at a large
hog farm. They were randomly assigned to one
of two rations which they are fed throughout
the study. Each pig was weighed at 1, 2, . . . , 9
weeks and results are shown in Kg. Researchers
want to know if one ration increases gain over
the other.
diet
60
Ration1
Ration2
40
20
2.5
5.0
7.5
week
1. Use this model for the kth row:
yk = µ + τk[j] + βk[j] xk + k
where k[j] = 1 if the pig in row k is fed ration 1, and is 2 if this pig is eating ration 2.
To see if growth rates differ, researchers are interested in β1 − β2 .
See the Appendix for part of the X matrix with row numbers and columns labeled with
the associated components of β.
(a) Find a linear combination of rows of y which has expectation β1 . I want to see
E(aym + byn ) = β1 , you only have to pick a, b, m, n.
(6 pts)
(b) Find a linear combination of rows of y which has expectation β2 .
(6 pts)
(c) Find a linear combination of rows of y which has expectation β1 − β2 .
(6 pts)
(d) Is β1 − β2 estimable? Explain.
(8 pts)
Stat 505 Midterm F12 Page 2
2. Now add a random adjustment for each pig:
y i = X i β + bi 1 + i , i = 1, . . . , 48 and assume i ∼ N (0, σ 2 I).
Write out a normal distribution for the bi ’s.
(6 pts)
Does adding the random adjustment change the expectations used just above? Explain.
(6 pts)
3. What is now the distribution of y i ? (In general, do not assume bi is known.)
(6 pts)
4. When we stack up the response vectors for all 48 pigs into one big vector y of size 48 × 9,
what is the variance-covariance of y?
(6 pts)
Stat 505 Midterm F12 Page 3
5. Label the variance covariance from part 4 as V and pretend that we know V up to a
constant. Explain how to obtain the BLUE of β1 −β2 . Include a one sentence justification
to explain why it is optimal.
(12 pts)
6. Of course that wasn’t the whole story. What conclusions do you draw from the three
diagnostic plots and the Box-Cox plot below?
(12 pts)
Box−Cox Plot
Scale−Location
30
40
50
60
Fitted values
70
−2
0.0
−1
0
1
Theoretical Quantiles
2
3
30
40
50
60
Fitted values
−220
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70
−225
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95%
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Standardized residuals
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Residuals
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Normal Q−Q
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Residuals vs Fitted
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λ
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0.4
Stat 505 Midterm F12 Page 4
7. The estimated BLUE of β2 − β1 is shown in the appendix. Write out your conclusion
about the ration effects.
(12 pts)
8. What is the scope of inference?
(12 pts)
Appendix
row
1
2
3
4
5
6
7
8
9
217
218
219
220
221
222
223
224
225
µ
1
1
1
1
1
1
1
1
1
..
.
1
1
1
1
1
1
1
1
1
..
.
τ1
1
1
1
1
1
1
1
1
1
τ2
0
0
0
0
0
0
0
0
0
β1
1
2
3
4
5
6
7
8
9
β2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
2
3
4
5
6
7
8
9
Linear mixed-effects model fit by REML
Data: myPigs
AIC BIC logLik
1874 1902
-930
Random effects:
Formula: ~1 | pigid
(Intercept) Residual
StdDev:
3.23
0.676
Variance function:
Structure: Power of variance covariate
Formula: ~fitted(.)
Parameter estimates:
power
0.24
Fixed effects: weight ~ diet * week
Value Std.Error DF t-value p-value
(Intercept)
19.47
0.698 382
27.9
0.000
dietRation2
-0.22
0.988 46
-0.2
0.824
week
5.77
0.044 382
130.7
0.000
dietRation2:week 0.87
0.063 382
13.9
0.000
Correlation:
(Intr) dtRtn2 week
dietRation2
-0.707
week
-0.290 0.205
dietRation2:week 0.203 -0.291 -0.701
Standardized Within-Group Residuals:
Min
Q1
Med
Q3
Max
-3.4321 -0.5483 0.0379 0.6426 3.4741
Number of Observations: 432
Number of Groups: 48