⫺0.2 Jay L. Devore Probability and Statistics
Example 11.4
Example 11.3 continued
If H
0A
is true, MSA is an unbiased estimator of s
2
, so F is a ratio of two unbiased estimators of s
2
. When H
0A
is false, MSA tends to overestimate s
2
. Thus H
0A
should be rejected when the ratio F
A
is too large. Similar comments apply to MSB and H
0B
.
Multiple Comparisons
After rejecting either H
0A
or H
0B
, Tukey’s procedure can be used to identify signifi- cant differences between the levels of the factor under investigation.
1.
For comparing levels of factor A, obtain .
For comparing levels of factor B, obtain .
2.
Compute
Q
a ,I,I21J21
2MSEJ for factor A comparisons means being compared
w 5 Q estimated standard deviation of the sample
Q
a ,J,I21J21
Q
a ,I,I21J21
Identification of significant differences among the four washing treatments requires and
. The four factor B sample means column averages are now listed in increasing order, and any pair differing by less
than .340 is underscored by a line segment:
Washing treatment 1 appears to differ significantly from the other three treatments, but no other significant differences are identified. In particular, it is not apparent
which among treatments 2, 3, and 4 is best at removing marks. ■
Randomized Block Experiments
In using single-factor ANOVA to test for the presence of effects due to the I dif- ferent treatments under study, once the IJ subjects or experimental units have been
chosen, treatments should be allocated in a completely random fashion. That is, J
subjects should be chosen at random for the first treatment, then another sample of J chosen at random from the remaining
subjects for the second treat- ment, and so on.
It frequently happens, though, that subjects or experimental units exhibit het- erogeneity with respect to other characteristics that may affect the observed
responses. Then, the presence or absence of a significant F value may be due to this extraneous variation rather than to the presence or absence of factor effects. This is
why paired experiments were introduced in Chapter 9. The analogy to a paired exper- iment when
is called a randomized block experiment. An extraneous factor,
“blocks,” is constructed by dividing the IJ units into J groups with I units in each I .
2 IJ 2 J
x
4
x
2
.300 337 x
3
x
1
.423 .803 w 5
4.902.014473 5 .340 Q
.05,4,6
5 4.90
because, e.g., the standard deviation of is
.
3.
Arrange the sample means in increasing order, underscore those pairs differing by less than w, and identify pairs not underscored by the same line as correspon-
ding to significantly different levels of the given factor. s
1J X
i
Q
a ,J,I21J21
MSEI for factor B comparisons
e 5
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Example 11.5
group. This grouping or blocking should be done so that within each block, the I units are homogeneous with respect to other factors thought to affect the responses. Then
within each homogeneous block, the I treatments are randomly assigned to the I units or subjects.
A consumer product-testing organization wished to compare the annual power con- sumption for five different brands of dehumidifier. Because power consumption
depends on the prevailing humidity level, it was decided to monitor each brand at four different levels ranging from moderate to heavy humidity thus blocking on
humidity level. Within each level, brands were randomly assigned to the five selected locations. The resulting observations annual kWh appear in Table 11.2,
and the ANOVA calculations are summarized in Table 11.3.
Table 11.2 Power Consumption Data for Example 11.5
Treatments Blocks humidity level
brands 1
2 3
4 1
685 792
838 875
3190 797.50
2
722 806
893 953
3374 843.50
3
733 802
880 941
3356 839.00
4
811 888
952 1005
3656 914.00
5
828 920
978 1023
3749 937.25
3779 4208
4541 4797
17,325 755.80
841.60 908.20
959.40 866.25
x
j
x
j
x
i
x
i
Table 11.3 ANOVA Table for Example 11.5
Source of Variation df
Sum of Squares Mean Square
f
Treatments brands 4
53,231.00 13,307.75
Blocks 3
116,217.75 38,739.25
Error 12
1671.00 139.25
Total 19
171,119.75 f
B
5 278.20
f
A
5 95.57
Since and ,
H is rejected in favor of H
a
. Power consumption appears to depend on the brand of humidifier. To identify
significantly different brands, we use Tukey’s procedure. and
.
The underscoring indicates that the brands can be divided into three groups with respect to power consumption.
Because the block factor is of secondary interest, is not needed,
though the computed value of F
B
is clearly highly significant. Figure 11.4 shows SAS output for this data. At the top of the ANOVA table, the sums of squares SSs
for treatments brands and blocks humidity levels are combined into a single “model” SS.
F
.05,3,12
x
1
x
3
x
2
x
4
x
5
797.50 839.00 843.50 914.00 937.25 w 5
4.512139.254 5 26.6 Q
.05,5,12
5 4.51
f
A
5 95.57 3.26
F
.05,4,12
5 3.26
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Example 11.6
Analysis of Variance Procedure Dependent Variable: POWERUSE
Sum of Mean
Source DF
Squares Square
F Value Pr . F
Model 7
169448.750 24206.964
173.84 0.0001
Error 12
1671.000 139.250
Corrected Total 19
171119.750 R-Square
C.V. Root MSE
POWERUSE Mean 0.990235
1.362242 11.8004
866.25000 Source
DF Anova SS
Mean Square F Value
PR . F BRAND
4 53231.000
13307.750 95.57
0.0001 HUMIDITY
3 116217.750
38739.250 278.20
0.0001 Alpha
⫽ 0.05 df ⫽ 12 MSE ⫽ 139.25 Critical Value of Studentized Range
⫽ 4.508 Minimum Significant Difference
⫽ 26.597 Means with the same letter are not significantly different.
Tukey Grouping Mean
N BRAND
A 937.250
4 5
A A
914.000 4
4 B
843.500 4
2 B
B 839.000
4 3
C 797.500
4 1
How does string tension in tennis rackets affect the speed of the ball coming off the racket? The article “Elite Tennis Player Sensitivity to Changes in String
Tension and the Effect on Resulting Ball Dynamics” Sports Engr., 2008: 31–36 described an experiment in which four different string tensions N were used,
and balls projected from a machine were hit by 18 different players. The rebound speed kmh was then determined for each tension-player combination. Consider
the following data in Table 11.4 from a similar experiment involving just six play- ers the resulting ANOVA is in good agreement with what was reported in the
article.
The ANOVA calculations are summarized in Table 11.5. The P-value for testing to see whether true average rebound speed depends on string tension is .049. Thus
is barely rejected at significance level .05 in favor of the conclusion that true average speed does vary with tension
. Application of Tukey’s procedure to identify significant differences among tensions
requires . Then
. The difference between the largest and smallest sample mean tensions is 6.87. So although the F test is significant, Tukey’s
w 5 7.464
Q
.05,4,15
5 4.08
F
.05,3,15
5 3.29
H : a
1
5 a
2
5 a
3
5 a
4
5
Figure 11.4 SAS output for power consumption data
■ In many experimental situations in which treatments are to be applied to sub-
jects, a single subject can receive all I of the treatments. Blocking is then often done on the subjects themselves to control for variability between subjects; each subject
is then said to act as its own control. Social scientists sometimes refer to such exper- iments as repeated-measures designs. The “units” within a block are then the differ-
ent “instances” of treatment application. Similarly, blocks are often taken as different time periods, locations, or observers.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Table 11.4 Rebound Speed Data for Example 11.6
Player Tension
1 2
3 4
5 6
210 105.7
116.6 106.6
113.9 119.4
123.5 114.28
235 113.3
119.9 120.5
119.3 122.5
124.0 119.92
260 117.2
124.4 122.3
120.0 115.1
127.9 121.15
285 110.0
106.8 110.0
115.3 122.6
128.3 115.50
111.55 116.93
114.85 117.13
119.90 125.93
x
.j
x
i
.
Table 11.5 ANOVA Table for Example 11.6
Source df
SS MS
f P
Tension 3
199.975 66.6582
3.32 0.049
Player 5
477.464 95.4928
4.76 0.008
Error 15
301.188 20.0792
Total 23
978.626
method does not identify any significant differences. This occasionally happens when the null hypothesis is just barely rejected. The configuration of sample means in the
cited article is similar to ours. The authors commented that the results were contrary to previous laboratory-based tests, where higher rebound speeds are typically
associated with low string tension.
■ In most randomized block experiments in which subjects serve as blocks, the
subjects actually participating in the experiment are selected from a large population. The subjects then contribute random rather than fixed effects. This does not affect
the procedure for comparing treatments when one observation per “cell,” as
in this section, but the procedure is altered if . We will shortly consider
two-factor models in which effects are random.
More on Blocking
When , either the F test or the paired differences t test can
be used to analyze the data. The resulting conclusion will not depend on which procedure is used, since
and .
Just as with pairing, blocking entails both a potential gain and a potential loss in precision. If there is a great deal of heterogeneity in experimental units, the value
of the variance parameter s
2
in the one-way model will be large. The effect of block- ing is to filter out the variation represented by s
2
in the two-way model appropriate for a randomized block experiment. Other things being equal, a smaller value of s
2
results in a test that is more likely to detect departures from H i.e., a test with
greater power. However, other things are not equal here, since the single-factor F test is based on
degrees of freedom df for error, whereas the two-factor F test is based on df for error. Fewer error df results in a decrease in power, essentially
because the denominator estimator of s
2
is not as precise. This loss in df can be especially serious if the experimenter can afford only a small number of observations.
Nevertheless, if it appears that blocking will significantly reduce variability, the sacrifice of error df is sensible.
I 2 1J 2 1 I
J 2 1 t
a 2,n
2
5 F
a ,1,n
T
2
5 F I 5
2 K
ij
5 K . 1
K
ij
5 1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Models with Random and Mixed Effects
In many experiments, the actual levels of a factor used in the experiment, rather than being the only ones of interest to the experimenter, have been selected from a much
larger population of possible levels of the factor. If this is true for both factors in a two-factor experiment, a random effects model is appropriate. The case in which
the levels of one factor are the only ones of interest and the levels of the other fac- tor are selected from a population of levels leads to a mixed effects model. The two-
factor random effects model when
is The
, and ’s are all independent, normally distributed rv’s with mean 0 and variances ,
and s
2
, respectively. The hypotheses of interest are then level of factor A does not contribute to variation in the response
versus and versus . Whereas
as before, the expected mean squares for factors A and B are now Thus when
is true, is still a ratio of two unbiased estimators of s
2
. It can be shown that a level a test for H
0A
versus H
aA
still rejects if
, and, similarly, the same procedure as before is used to decide between H
0B
and H
aB
. If factor A is fixed and factor B is random, the mixed model is
where and the
B
j
’s and are normally distributed with mean 0 and vari-
ances and s
2
, respectively. Now the two null hypotheses are with expected mean squares
The test procedures for H
0A
versus H
aA
and H
0B
versus H
aB
are exactly as before. For example, in the analysis of the color-change data in Example 11.1, if the
four wash treatments were randomly selected, then because and
is rejected in favor of . An estimate of the
“variance component” is then given by
. Summarizing, when
, although the hypotheses and expected mean squares differ from the case of both effects fixed, the test procedures are identical.
K
ij
5 1
MSB 2 MSEI 5 .0485 s
B 2
H
aB
: s
B 2
. F
.05,3,6
5 4.76, H
0B
: s
B 2
5 f
B
5 11.05
E MSE 5 s
2
EMSA 5 s
2
1 J
I 2 1
g
a
i 2
EMSB 5 s
2
1 Is
B 2
H
0A
: a
1
5 c5 a
I
5 0 and H
0B
: s
B 2
5 s
B 2
P
ij
’s ga
i
5 X
ij
5 m 1 a
i
1 B
j
1 P
ij
i 5 1, c
, I, j 5 1, c
, J f
A
F
a ,I21,I21J21
H
0A
F
A
F
B
H
0A
H
0B
E MSA 5 s
2
1 Js
A 2
E MSB 5 s
2
1 Is
B 2
E MSE 5 s
2
H
aB
: s
B 2
. H
0B
: s
B 2
5 H
aA
: s
A 2
. H
0A
: s
A 2
5 s
A 2
, s
B 2
P
ij
A
i
’s, B
j
’s X
ij
5 m 1 A
i
1 B
j
1 P
ij
i 5 1, c
, I, j 5 1, c
, J K
ij
5 1
EXERCISES
Section 11.1 1–15
1.
The number of miles of useful tread wear in 1000s was determined for tires of each of five different makes of sub-
compact car factor A, with in combination with each
of four different brands of radial tires factor B, with , resulting in
observations. The values , and
were then computed. Assume that an additive model is appropriate.
SSE 5 59.2 SSB 5 44.1
SSA 5 30.6, IJ 5
20 J 5
4 I 5
5
a.
Test no differences
in true average tire lifetime due to makes of cars versus H
a
: at least one using a level .05 test.
b.
no differences in true aver- age tire lifetime due to brands of tires versus H
a
: at least one
using a level .05 test. b
j