Transcript Ch 13B More Analysis of Variance

Chapter 13B
Chapter 13 ANOVA
The Design and Analysis of Single
Factor Experiments - Part II
Class will begin in a few
minutes.
Reaching out to the internet
Today’s Special
My gosh, this
does look
promising.
Fixed versus Random Factors


Fixed Effects Model – conclusions only apply to factor levels
included in the study
Random-Effects Model – conclusions extend to the
population of factor levels where the levels in the study are a
sample
 assume population of factor levels is either infinite or large
enough to be treated as infinite
almost the
same model as before
NID  0,  2  NID  0,  2 
Var Yij    2   2
but this is different
The Model
For a random-effects model, the appropriate
hypotheses to test are:
The ANOVA decomposition of total variability is
still valid:
Expected Values of the
Mean Squares
under H0:
MSTreatments
F0 
MS E
The Estimators of the
Variance Components
The Problem (Example 13-4)
DOE in the Textile Industry
The Data and ANOVA
The Variances
Figure 13-8
LSL – Lower Specification Limit
The distribution of fabric strength. (a) Current process, (b)
improved process.
Randomized Block Designs
(RBD)



Variability among subjects may mask or
obscure treatment effect of interest
Nuisance variable of individualized
differences can be minimized using a RBD
Appropriate when



one treatment with r > 2 levels
assignment of subjects to blocks so that
variability among subjects within any block is
less than the variability among the blocks
random assignment of treatments to subjects
within blocks
Our Very First Blocking
Example




The factor levels: 4 different cockpit designs
The experimental units: 12 pilots with varying years
of experiences
The response variable: Number of pilot errors in
flying a simulated program
Block on years experience




1-2
3-4
5 or more
Randomly assign treatment levels (i.e. cockpit
design) to pilots in each block.
CRD versus RBD
CRD Design
subjects
A
Joe
Ted
B
Teresa
Bill
C
Jim
Barbara
D
John
Tom
RBD
Design
A
B
C
D
Terrie
Bob
Jill
Betty
yrs 1-2
yrs 3-4
yrs 5+
Joe
Jill
John
Jim
Tom
Ted
Terrie
Teresa
Bob
Barbara
Bill
Betty
Complete Confounding
CRD
Design
yrs 1-2
A
Joe
Jill
John
yrs 3-4
B
Tom
Ted
Terrie
yrs 5+
C
Bob
Barbara
Bill
The devil made me do it
this way.
13-4 Randomized Complete
Block Designs
The randomized block design is an extension of the
paired t-test to situations where the factor of interest has
more than two levels.
Figure 13-9 A randomized complete block design.
13-4.1 Design and Statistical
Analyses
For example, consider the situation of Example 10-9,
where two different methods were used to predict the
shear strength of steel plate girders. Say we use four
girders as the experimental units.
13-4.1 Design and Statistical
Analyses
General procedure for a randomized complete block
design:
The Linear Statistical Model
We assume
• treatments and blocks are initially fixed effects
• blocks do not interact
•
The Hypothesis
The partitioning of the sums of squares
SST
d.f.’s
= SSTreatements
ab – 1 =
(a – 1)
+ SSBlocks
+ SSE
+ (b – 1) + (a – 1)(b – 1)
Mean Squares
Notice that the effect of blocks is to reduce the sums of
squares left as error. This means that the calculation of F0
will always produce a larger quantity than otherwise. The
test on the i becomes more sensitive.
SSE = SSTotals - SSTreatments - SSBlocks
The effect of
the blocks may
or may not be
interesting in
and of itself.
Expected Values Of the Mean
Squares
If the i are
zero, this is an
unbiased
estimate of 2.
A Mean
Square
Computational Formulae
ANOVA
Reject H0 if
F0  f ,a 1,( a 1)(b 1)
Example 13-5
SS for Example 13-5
More SS for Example 13-5
ANOVA for Example 13-5
Minitab Output for Example 13-5
Fisher’s Least Significant
Difference for Example 13-5
Figure 13-10 Results of Fisher’s LSD method.
LSD  t.05/2,12
2MSE
2(.0792)
  2.1788
 .3878
n
5
CI on chemical
Individual 95% CI
Chemical
1
2
3
4
Mean
1.14
1.76
1.38
3.56
LSD  t.05/2,12
----------+---------+---------+---------+(--*---)
(--*--)
(--*---)
(---*--)
----------+---------+---------+---------+-
2MSE
2(.0792)
  2.1788
 .3878
n
5
CI on samples
Individual 95% CI
sample
Mean
1
2.30
2
2.53
3
0.87
4
2.20
5
1.90
-+---------+---------+---------+---------+
(----*----)
(----*----)
(-----*----)
(----*----)
(----*----)
-+---------+---------+---------+---------+
0.60
LSD  t.05/2,12
1.20
1.80
2.40
2MSE
2(.0792)
  2.1788
 .4336
n
4
3.00
An Excel Comparison


A computer model simulates depot overhaul of jet
engines in order to estimate repair cycle times in days
Three probability distributions are used to model task
times





gamma
lognormal
Weibull
Blocking occurs by using the same random number
stream with each treatment (distribution)
Go to the next slide to observe the highly interesting
results
Highly Interesting Results
treatment
gamma
lognormal
Weibull
1
13
16
5
random number seed
2
3
22
18
24
17
4
1
4
39
44
22
engine overhaul days
ANOVA
Source of Variation
Rows
Columns
Error
Total
SS
703.5
1106.917
51.83333
1862.25
df
MS
F
P-value
F crit
2
351.75 40.71704 0.000323 5.143253
3 368.9722 42.71061 0.000192 4.757063
6 8.638889
11
CI on distributions
Individual 95% CI
distribution
Mean
gamma
23.0
lognorma
25.2
Weibull
8.0
---+---------+---------+---------+-------(-----*-----)
(-----*-----)
(-----*-----)
---+---------+---------+---------+-------6.0
LSD  t.05/2,12
12.0
18.0
24.0
2MSE
2(8.639)
  2.4469 
 5.0855
n
4
CI on Random Number Seeds
Individual 95% CI
seed
Mean
1
11.3
2
16.7
3
12.0
4
35.0
--+---------+---------+---------+--------(----*----)
(----*----)
(----*----)
(----*----)
--+---------+---------+---------+--------8.0
LSD  t.05/2,12
16.0
24.0
32.0
2MSE
2(8.639)
  2.4469 
 5.8722
n
3
Even Higher Interesting Results
One-Way ANOVA without blocking
ANOVA
Source of Variation
Between Groups
Within Groups
SS
703.5
1158.75
Total
1862.25
Why it appears that the
variability introduced by the
random number stream is
masking the effect of the
task time distributions.
This is nothing but garbage.
df
2
9
11
F.01,2,9 = 8.02
F.05,2,9 = 4.26
MS
F
P-value
F crit
351.75 2.732039 0.118245 4.256495
128.75
Randomized Complete
Block Design is Now
Over
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