Download ST512 QUIZ 3 Exercises 1

ST512
QUIZ 3 Exercises
1
2.
An experimenter tries 6 different concentrations, X, of soap in a
dishwashing machine. The response is the amount of dirt, Y, left on
the dishes after washing. He has 9 observations so obviously some
concentrations were used more than once.
He fits a quadratic polynomial getting error sum of squares 1000 and
(corrected) regression sum of squares 500.
He also fits the highest possible degree polynomial getting error sum
of squares 600.
B. Give the numerator of the F that tests to see if either the linear or
quadratic (or both) terms can be omitted from the quadratic polynomial.
F = ______/MSE.
C. Give the calculated F that tests to see if a polynomial of degree
higher than 2 is needed. F = _______
D.
What will SAS report as the corrected regression sum of squares (SAS
calls it the Model SS) if we issue the statements
PROC GLM; CLASS X; MODEL Y = X;
E.
In the quadratic regression printout we know SAS will compute a
parameter estimate and t test for the linear term 1X . From the above
information, what is the maximum absolute value that t could have?
For linear term,
|t| must be no more than _________
3. Here are some data on the response Y to 3 drugs, A, B, and C and a
placebo, P each of which was used on 3 patients. This is a completely
randomized design with 12 patients. Your company produces drug A, while
B and C are produced by a competitor.
A. (12 points) Fill in the columns of the X matrix below in such a way
that the regression of Y on X will produce the ANOVA table for treatments.
Make it so that the X1 sum of squares tests placebo versus average of all
3 drugs, X2 tests your drug versus the competitor average, and the others
test as many more orthogonal contrasts as you need to get the right
treatment degrees of freedom.
treatment
drug
drug
drug
drug
drug
drug
drug
drug
drug
placebo
placebo
placebo
A*
A*
A*
B
B
B
C
C
C
P
P
P
Y
Y11
Y12
Y13
Y21
Y22
Y23
Y31
Y32
Y33
Y41
Y42
Y43
X0
1
1
1
1
1
1
1
1
1
1
1
1
X1
__
__
__
__
__
__
__
__
__
__
__
__
X2
__
__
__
__
__
__
__
__
__
__
__
__
* our company's drug
X3
X4
X5
X6
X7
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__
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ST512
QUIZ 3 Exercises
2
_
_
(8 points) Assuming treatment means Y1. = 50, Y2. = 80,
_
_
Y3. = 100, and Y4. = 200, compute if possible the Type I sum of squares
for X1 .
Solution
2.
Reduced: 2 500
6 1000
B.
C.
D.
E.
Full:
5
3
900
600
F = (500/2) / MSE (reduced model F test)
F = (1000-600)/3 divided by (600/3) so F=0.66667.
900
This one's a little harder: In the most extreme case, the entire
model SS (500) would be associated with X and that would give
F = 500/(1000/6) = 3 and so |t| < sqrt(3).
Details: Type II (and its F test) do not depend on order.
add to 500:
X*X
X
treatment
drug
drug
drug
drug
drug
drug
drug
drug
drug
placebo
placebo
placebo
A*
A*
A*
B
B
B
C
C
C
P
P
P
Type I
0
500
----------500
Type II
500
Y
Y11
Y12
Y13
Y21
Y22
Y23
Y31
Y32
Y33
Y41
Y42
Y43
Type I must
X0
1
1
1
1
1
1
1
1
1
1
1
1
--> F < 500/MSE --> |t| < sqrt(500/MSE)
X1
-1
-1
-1
-1
-1
-1
-1
-1
-1
3
3
3
X2
2
2
2
-1
-1
-1
-1
-1
-1
0
0
0
0
0
0
-1
-1
-1
1
1
1
0
0
0
X3
X4
X5
X6
X7
Just 3 df so 3 columns.
SS uses TOTALS, 150, 240, 300, 600
-1
-1
-1
3
--> Q = 1800 - 690 = 1110,
denom = (12)(3) = 36, SSq = 1110*1110/36 = 34225
ST512
QUIZ 3 Exercises
Quiz 2 St 512
3
Spring 2000
Dickey
I place several mesh containers, each with 100 insect eggs in different
locations of a large wooded region. Each container includes equipment
to measure LIGHT, MOIST (moisture), and TEMP (night temperature) at
that spot. The response, HATCH, is the number of hatched larvae.
I now fit this model:
HATCH = beta0 + beta1*MOIST + beta2*LIGHT +
beta3*TEMP + beta4*MT + e
where MT = MOIST*TEMP is an interaction term. Here is the
PROC REG output including Type I sums of squares.
Model: MODEL1
Dependent Variable: HATCH
Analysis of Variance
Source
Sum of
Squares
DF
Model
Error
C Total
{___}
{___}
14
Root MSE
Mean
Square
F Value
{_______}
144.76551
175.87128
{_______}
754.93333
4.19370
{______}
R-square
Prob>F
0.0033
{_______}
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
INTERCEP
MOIST
LIGHT
TEMP
MT
1
1
1
1
1
184.918359
-2.298055
0.580560
-6.098898
0.102716
204.62581039
3.25467600
0.29250931
6.45088664
0.10211402
Variable
INTERCEP
MOIST
LIGHT
TEMP
MT
DF
1
1
1
1
1
T for H0:
Parameter=0
0.904
{_______}
1.985
-0.945
1.006
Prob > |T|
0.3874
0.4963
0.0753
0.3667
0.3382
Type I SS
67469
502.658879
45.969567
12.638585
17.795023
(A)
(42 pts.) Fill in the 7 missing entries in the output.
(B)
(5 pts.) How many containers did I use in my experiment?
______
(C)
(12 pts.) Compute the increase in predicted HATCH _______
associated with a 1 unit increase in moisture when LIGHT=10
and TEMP=30. How does this answer change if we replace
LIGHT=10 with LIGHT=15?_____
(D) (18 pts.)
I see a t-test,
t=1.985, associated with LIGHT
ST512
QUIZ 3 Exercises
4
in the middle portion of my output.
(i) If I plot t=1.985 on the horizontal axis of
the corresponding t distribution,
what is the area____ to its left?
(ii) Which of these null hypotheses is being tested
by t? Select all that are right.
H0:
H0:
H0:
H0:
H0:
(iii)
(E)
X2=0
LIGHT=0
Beta2=0
Beta2=0.580560
LIGHT=0.580560
Do I reject, or fail to reject this null hypothesis
with a 5% significance level test?
(12 pts.) Maybe temperature does not affect HATCH in any way.
(i) Give a calculated F test statistic to test the null
hypothesis that both terms involving temperature (TEMP
and MT) can be simultaneously dropped from the model
F = ______
(ii) I notice that all my t statistics are insignificant.
Maybe I should say that none of the things I used to
predict HATCH matter. What do you think of that idea?
(support your answer with numbers, of course!)
(iii) If I drop both TEMP and MT from my model, my model
sum of squares will (increase, decrease) by ________
(F) (5 pts.) What would be my R-square if I did a simple linear
regression of HATCH on just an intercept and MOIST?
(G)
(6 pts.) Compute, if possible, the square root of c(MSE)= ______
where c is the third element (the one associated with LIGHT) on the
diagonal of the inverse of my (X'X) matrix.
***********************************ANSWERS**************************
Source
DF
Model
Error
C Total
10
14
4
Sum of
Squares
755-176=
579
175.87128
754.93333
Mean
Square
144.76551
17.5871
F Value
144.7/17.58
Prob>F
0.0033
ST512
QUIZ 3 Exercises
Root MSE
4.19370
R-square
5
579/755
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
INTERCEP
MOIST
LIGHT
TEMP
MT
1
1
1
1
1
184.918359
-2.298055
0.580560
-6.098898
0.102716
204.62581039
3.25467600
0.29250931
6.45088664
0.10211402
T for H0:
Parameter=0
Prob > |T|
0.904
-2.298/3.25
1.985
-0.945
1.006
0.3874
0.4963
0.0753
0.3667
0.3382
15 containers
(C) (-2.298 + 0.10276(30))MOIST = 0.78 MOIST so 0.78 per unit
increase (7.8 more hatch with each increase of 10 etc.) There is no
interaction involving LIGHT. This increase is the same for all
values of light. As in the class notes, of course, this is a
property of the MODEL.
(D) Area 0.0753 split in two tails.
0.96235 to left.
H0: Beta2=0
(E)
Variable
INTERCEP
MOIST
LIGHT
TEMP
MT
DF
1
1
1
1
1
Thus 0.03765 to right,
Do not reject (P>0.05)
Type I SS
67469
502.658879
45.969567
12.638585
17.795023
<--- for R-square question
F=(12.63+17.80)/2 divided by MSE
F=15.22/17.59 = 0.865
Clearly this F=0.865 is insignificant. We see that in fact
everything but MOIST can be omitted.
Having done that, MOIST will become highly significant.
The number 12.63+17.8 is the DECREASE in model sum of squares
that would result from omitting TEMP and MT.
Of course we cannot leave out everything! That is exactly
what the model F test is testing and it is highly significant.
(E)
502.66/755 would be simple linear regression R-square.
(F)
This is just the standard error formula for that beta.
0.29250931