STA 6207 – Exam 3 – Fall 2013
PRINT Name _____________________
Unless stated otherwise, all Questions are based on the following 2 regression models, where SIMPLE
REGRESSION refers to the case where p=1, and X is of full column rank (no linear dependencies among
predictor variables).
Model 1 : Yi   0  1 X i1     p X ip   i

i  1,..., n  i ~ NID 0,  2



Model 2 : Y  Xβ  ε X  n  p' β  p'1 ε ~ N 0,  2 I
_______________________________________________________________________________________
CONDUCT ALL TESTS AT  = 0.05 SIGNIFICANCE LEVEL.
Residuals
500
Plot 1 – Q.1.
400
300
200
100
0
-200 -100 0
Residuals
200
400
600
800
1000
-200
-300
Residuals
Plot 2 – Q.3.
4
3
2
1
Residuals
0
-1
-2
-3
0
5
10
15
Q.1. A regression model is fit, relating time to complete a task (Y, in minutes) to nationality of the team (X 1=1 if US, 0 if non-US)
and complexity (X2, on a TACOM scale) for nuclear power plant operators. The model fit is:
Yi  0  1 X i1  2 X i 2  3 X i1 X i 2   i
ANOVA
df
Regression
Residual
Total
3
66
69
SS
3234355
903715
4138069
MS
1078118
13693
F
Significance F
79
0.0000
INV(X'X)
0.7998
-0.7998
-0.1410
0.1410
-0.7998
1.5996
0.1410
-0.2820
-0.1410
0.1410
0.0258
-0.0258
0.1410
-0.2820
-0.0258
0.0515
Coefficients
Standard Error t Stat
P-value
Intercept
-406.1
104.7
-3.88
0.0002
Nation
-386.0
148.0
-2.61
0.0112
Complexity
117.2
18.8
6.24
0.0000
N*C
108.0
26.6
4.06
0.0001
p.1.a. Test whether the slopes (with respect to complexity scores) are equivalent for US and non-US power plants.
H0: ____________________ HA: __________________
Test Stat: _____________________ P-Value: _________________
p.1.b. Give the estimated mean time to complete a task with complexity of X 2 = 5 for US and non-US plants.
US: ______________________________________ non-US: ______________________________________
p.1.c. Compute a 95% Confidence Interval for the difference of the means estimated in p.1.c.
p.1.d. A plot of residuals versus fitted values (See Page 1) clearly displays non-constant error variance. We want to fit a power model,
relating the variance to the mean:
 i2  i
^
^ 
 ln  i2    *   ln  i  Regressing ln  ei2  on ln  Y i     0.75
 
Based on Bartlett’s method, suggest a power transformation to approximately stabilize the error variance.
 2  ( )

f ( )  
1
( )1 / 2
d
Q.2. A study was conducted to compare growth rates of Thai and Commercial Tilapia. The authors fit separate models for the two
strains, with samples of 20 fish of each type selected at 5 time points and weighed (then sacrificed ). Thus 100 fish of each strain
were measured. The model fit by unweighted nonlinear least squares was (where Y=Weight, and X=Age, in days):
Yi  i   i   0 e 1 X i   i
p.2.a. How would you interpret
p.2.b. Obtain
i
 0
and
 0 and e  ?
1
i
1
^
p.2.c. The estimates and standard errors for
^
 0 and  1
are given in the following table for each strain. Obtain 95% CI’s for all
model parameters.
Strain
Parameter
Beta0
Beta1
Thai
Estimate
30.5771
0.0132
Thai
StdError
3.2614
0.0008
Commercial
Estimate
21.6209
0.0168
Commercial
StdError
2.6555
0.0010
 0T : ________________ 1T : ________________  0C : ________________ 1C : ________________
p.2.d As a crude test of equality of Strains, based on the confidence intervals, do you reject the null hypothesis:
H 0 :  0T   0C , 1T  1C
Note that their estimates are independent as models were fit separately.
p.2.e. At what age (if any) do the two strains’ curves cross?
Yes / No
Q.3. A study related the spread in shotgun pellets (Y, sqrt(AREA)) to distance shot (X, in feet) for a particular shotgun and bullet
type (different from the in-class example).
p.3.a. Based on the residuals versus fitted plot (Plot 2 on page 1), which violation(s) if any is/are violated?
i) Linearity
ii) Equal Variances
iii) Independence
p.3.b. We wish to fit an Estimated Weighted Least Squares Regression of the following model:
Yi  0  1 xi  2 xi2   i
xi  X i  X
with weights wi 
1
where s j is SD of Y for distance group of shot i
s 2j
^
The data are given below, give the form of W used in estimating
X
10
20
30
40
50
x
-20
-10
0
10
20
x^2
400
100
0
100
400
 _____ I 6
 _____ I
6

W   _____ I 6

 _____ I 6
 _____ I 6
y1
3.01
5.57
8.09
10.81
16.07
_____ I 6
_____ I 6
_____ I 6
_____ I 6
_____ I 6
y2
3.02
5.00
6.80
10.19
14.90
βW =  X'WX  X'WY
y3
3.29
5.42
7.95
13.01
17.47
_____ I 6
_____ I 6
_____ I 6
_____ I 6
_____ I 6
-1
y4
3.00
5.73
8.62
11.17
14.21
_____ I 6
_____ I 6
_____ I 6
_____ I 6
_____ I 6
y5
3.20
5.29
8.41
11.33
13.13
y6
3.11
5.10
8.62
9.35
11.93
SD(grp)
0.119
0.278
0.685
1.232
1.996
_____ I 6 
_____ I 6 
_____ I 6 

_____ I 6 
_____ I 6 
p.3.c. From the results below, complete the following table. (SSE* based on transformed residuals)
X'WX
519.5807125 -9180.684522 178240.8398
-9180.684522 178240.8398 -3451225.692
178240.8398 -3451225.692 68848673.11
INV(X'WX)
0.02146079
0.00100725
-0.00000507
SSE*
0.00100725
0.00023816
0.00000933
s^2
25.09147082
-0.00000507
0.00000933
0.00000050
X'WY
1899.813
-29592.3
580927.4
Beta_W
SE(B_W)
8.020371
0.286385
0.00203
Q.4. For the simple regression model (scalar form):
X X
1   i
S XX
i 1 
^
we get:
p.4.a. Derive
p.4.b. Derive
n
Yi   0  1 X i   i
i  1,..., n  i ~ NID  0,  2 
n
^
^

1
Yi , Y    Yi ,  0  Y   1 X
i 1  n 

 
^ 
^ 
E  1 , E Y , E   0 
 
 
 
^ 
^

^ 
^ ^ 
V   1  , V Y , Cov   1 , Y  , V   0  , Cov   1 ,  0 
 


 


Q.5. A simple linear regression model is fit, based on n=5 individuals. The data and the projection matrix are given below:
X
1
1
1
1
1
2
4
25
6
8
Y
8
12
24
14
18
P
0.344
0.303
-0.129
0.262
0.221
0.303
0.274
-0.035
0.244
0.215
-0.129
-0.035
0.953
0.059
0.153
0.262
0.244
0.059
0.226
0.209
0.221
0.215
0.153
0.209
0.203
p.5.a. Give the leverage values for each observation. Do any exceed twice the average of the leverage values?
Observation 1 ________
p.5.b. Give
X'X
5
45
Obs 2 ______________ Obs 3 ___________ Obs 4 ____________ Obs 5 ______________
β , based on the following results
45
745
X'Y
76
892
INV(X'X)
0.438
-0.026
-0.026
0.003
p.5.c. Compute SSE and Se (Note: Y'Y  1304
Y'PY  1282.45 )
p.5.d. The following table contains the fitted values with and without each observation, residual standard deviation when that
observation was not included in the regression, and the regression coefficients when that observation was not included in the
regression.
Y-hat
10.92
12.14
24.99
13.36
14.59
Y-hat(-i)
12.45
12.19
45.00
9.46
13.72
S_i
2.07
3.28
0.63
3.24
1.86
beta0_i
11.41
9.76
5.00
9.46
8.72
beta1_i
0.52
0.61
1.60
0.62
0.62
p.5.d.i. Compute DFFITS for the fifth observation
p.5.d.ii. Compute DFBETAS0 for the first observation
p.5.d.iii. Compute DFBETAS1 for the third observation