Exam 1 Review
Chapter 1
Probability: a measure of how many likely an event is to occur
o Prob = 1, event will happen
o Prob =0, will not happen
Population: all individuals who are of interest to a researcher
Sample: the individuals who were actually co
5.5
109
Checking Independence of the Error Terms
zit
T
3
2
1
0
1
2
Figure 5.3
Residual plot after
data correction for the
battery experiment
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
3
E
1
2
3
4
Battery type
value has been copied incorrectly at some stage. If the
5.3
Checking the Fit of the Model
107
residuals for treatment 2 seems a little larger than the spread for the other three treatments.
This could be interpreted as a sign of unequal variances of the error variables or that the data
values having standardiz
108
Chapter 5
Checking Model Assumptions
zit
T
3
2
1
0
1
b
b
b
2
Figure 5.2
Original residual plot
for the battery
experiment
b
b
b
b
b
b
b
b
bb
b
b
3
1
2
E
3
4
Battery type
and +2, and approximately 99.7% between 3 and +3. If there are more outliers t
104
Chapter 5
Checking Model Assumptions
the adequacy of the model can be checked. Even if a pilot experiment has been used to
help select the model, it is still important to check that the chosen model is a reasonable
description of the data arising from
5
Checking Model Assumptions
5.1 Introduction
5.2 Strategy for Checking Model Assumptions
5.3 Checking the Fit of the Model
5.4 Checking for Outliers
5.5 Checking Independence of the Error Terms
5.6 Checking the Equal Variance Assumption
5.7 Checking the
643
Exercises
7. Consider the following mixed model:
+ i + Bj + Ck + m + (B)ij + ()im
Yij kmt
i
+ (B)j m + (C)km + (B)ij m + ij kmt ,
1, . . . , a, j 1, . . . , b, k 1, . . . , c,
m
1, . . . , d, t
1, . . . , r,
2
2
2
Bj N (0, B ), Ck N (0, C ), (B)ij N
639
Exercises
Table 17.15
SAS analysis of variance for the ice cream experiment
The SAS System
General Linear Models Procedure
Dependent Variable: MELTTIME
Sum of
Mean
Source
DF
Squares
Square F Value
Model
4
250538.12
62634.53
13.93
Error
28
125927.94
44
5.6
Checking the Equal Variance Assumption
111
2
If an analysis is conducted under the assumptions of model (3.3.1) when, in fact, the
error variables are dependent, the true signicance levels of hypothesis tests can be much
higher than stated and the tru
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5.6
Checking the Equal Variance Assumption
113
2
2
ratio of the largest of the v treatment variance estimates to the smallest, smax /smin , does
not exceed three. The rule of thumb is suggested by simulation studies in which the true
variances i2 are spec
112
Chapter 5
Checking Model Assumptions
zit
T
3
2
b
b
b
b
1
0
1
2
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
bb
bb
b
b
b
b
b
b
3
b
b
b
bb
b
b
b
b
bb
b
b
b
b
b
b
b
b
Figure 5.6
E
Megaphoneshaped
residual plot
30
60
90
120
yit
are higher). Thus, unle
110
Chapter 5
Checking Model Assumptions
zit
T
3
2
b
1
0
b
b b
b
1
2
b
b
b
b
b
b
b
b
b
b
b
3
Figure 5.4
E
Residual plot for the
battery experiment
5
10
15
Run Order
the plot were to exhibit a strong pattern, then this would indicate a serious violation
638
Chapter 17
Random Effects and Variance Components
Plot of Z*ORDER.
Figure 17.6
Plot of the
standardized residuals
against order of
observation for the ice
cream experiment
The SAS System
Legend: A = 1 obs, B = 2 obs, etc.
2 +
A

A

A
A A
Z 

A
A

626
Chapter 17
Random Effects and Variance Components
been discussed throughout this subsection):
Yij kt
+ i + j + Dk + ()ij + (D)j k +
ij kt .
The xed part of the model is
+ i + j + ()ij ,
which looks exactly like one of the twoway analysis of varianc
17.7
625
Mixed Models
From Table 17.8 we can construct tests for the other relevant hypotheses in a similar manner.
For example, to test the hypothesis
AB
H0 : cfw_()ij ()i. ().j + ().
0 , for all i, j
against the alternative hypothesis that the interact
17.7
623
Mixed Models
though all the factors were random. We then collect all of the xed effects and list them
together as one quadratic form. The quadratic form is a function of xedeffect parameters
such as i i + ()i. (see Example 17.7.1) that typically
622
Chapter 17
Example 17.6.6
Random Effects and Variance Components
Ammunition experiment, continued
An unbiased estimate for the variance of the muzzle velocities due to the population of charge
lots (factor A) was calculated to be u 24.3 (feet per seco
17.6
621
Two or More Random Effects
A
2
A
2
Testing H0 : A 0 against HA : A > 0 is more complicated. Until now, we have used
the same test statistics as we used in the xedeffects case. But if we try to use msA/msE
A
A
2
0, the expected value of the
to te