Chapters 8, 9:
Miscellaneous topics
Chapter 14:
Experimental design concepts
Chapter 15:
Randomized Complete Block Design
(15.3)
1

1-Factor ANOVA Model y ij
= m i
+ e ij or y ij
= m + a i
+ e ij observed data mean for i th treatment unexplained part
2

The hypotheses:
H
0
: m
1
 m
2
  m t
H a
: at least 2 means a unequal were rewritten as:
H
0
: a
1
 a
2
  a t

H a
: at least one a i

0
0
3

t n
 i
 
1
( y ij
 y
..
)
2  n ( y i .
 y
..
i t


1
)
2 + t n
 i
 
1
( y ij
 y i .
)
2
Notation: TSS

SSB
+
SSW
In words:
TSS (total SS) = total sample variability among y ij values
SSB (SS “between”) = variability explained by differences in group means
SSW (SS “within”) = unexplained variability
(within groups)
4

We reject H
0 a
F
 s
2
B s
2
W
 a (

1, n
T
 t )
Note: unequal sample sizes allowed
5

Extracted from From Ex. 8.2, page 390-391
3 Methods for Reducing Hostility
12 students displaying similar hostility were randomly assigned to 3 treatment methods.
Scores (HLT) at end of study recorded.
Method 1 96 79 91 85
Method 2 77 76 74 73
Method 3 66 73 69 66
Test: H
0
: m
1
 m
2
 m
3
6

ANOVA Table Output – extracted hostility data
- calculations done in class
Source SS df MS F
Between samples p -value
767.17 2 383.58 16.7 <.001
Within samples
205.74 9 22.86
Totals 972.91
7

Fisher’s Least Significant
Difference (LSD) y and y are significantly different if
|

2
|

LSD
where
LSD = t
α/ 2 s
2
W



1 1
+ n
1 n
2



and
 
within (error) df
Protected LSD: Preceded by an F-test for overall significance.
Only use the LSD if F is significant.
X
Not preceded by an F-test (like individual t-tests) .

Hostility Data - Completely Randomized Design
The GLM Procedure t Tests (LSD) for score
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 9
Error Mean Square 22.86111
Critical Value of t 2.26216
Least Significant Difference 7.6482
Means with the same letter are not significantly different.
t Grouping Mean N method
A 87.750 4 1
B 75.000 4 2
B
B 68.500 4 3
9

Ex. 8.2, page 390-391
3 Methods for Reducing Hostility
24 students displaying similar hostility were randomly assigned to 3 treatment methods.
Scores (HLT) at end of study recorded.
Method 1 96 79 91 85 83 91 82 87
Method 2 77 76 74 73 78 71 80
Method 3 66 73 69 66 77 73 71 70 74
Notice unequal sample sizes
Test: H
0
: m
1
 m
2
 m
3
10

ANOVA Table Output – full hostility data
Source SS df MS F p -value
Between samples
1090.6 2 545.3 29.57 <.0001
Within samples
387.2 21 18.4
Totals 1477.8 23
11

The GLM Procedure t Tests (LSD) for score
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 21
Error Mean Square 18.43878
Critical Value of t 2.07961
Comparisons significant at the 0.05 level are indicated by ***.
Difference method Between 95% Confidence
Comparison Means Limits
1 - 2 11.179 6.557 15.800 ***
1 - 3 15.750 11.411 20.089 ***
2 - 1 -11.179 -15.800 -6.557 ***
2 - 3 4.571 0.071 9.072 ***
3 - 1 -15.750 -20.089 -11.411 ***
3 - 2 -4.571 -9.072 -0.071 ***
Notice the different format since there is not one LSD value with which to make all pairwise comparisons.
12

Duncan's Multiple Range Test for score
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 21
Error Mean Square 18.43878
Harmonic Mean of Cell Sizes 7.91623
NOTE: Cell sizes are not equal.
Number of Means 2 3
Critical Range 4.489 4.712
Means with the same letter are not significantly different.
Duncan Grouping Mean N method
A 86.750 8 1
B 75.571 7 2
C 71.000 9 3
Note: Duncan’s test (another multiple comparison test) avoids the issue of different sample sizes by using the harmonic mean of the n i
’ s .
13

Some Multiple Comparison Techniques in SAS
FISHER’S LSD (LSD)
BONFERONNI (BON)
DUNCAN
STUDENT-NEWMAN-KEULS (SNK)
DUNNETT
RYAN-EINOT-GABRIEL-WELCH (REGWQ)
SCHEFFE
TUKEY
14

17218.3
18117.5
19418.7
20322.9
21116.3
22414.0
23416.6
24218.1
25218.9
26416.0
27220.1
28322.5
29316.0
30119.3
31115.9
1122.4
2324.6
3120.3
4419.8
5324.3
6222.2
7228.5
8225.7
9320.2
10119.6
11228.8
12424.0
13417.1
14419.3
15324.2
16115 .8
32320.3
Balloon Data
Col. 1-2 - observation number
Col. 3 - color (1=pink, 2=yellow, 3=orange, 4=blue)
Col. 4-7 - inflation time in seconds
15

17218.3
18117.5
19418.7
20322.9
21116.3
22414.0
23416.6
24218.1
25218.9
26416.0
27220.1
28322.5
29316.0
30119.3
31115.9
1122.4
2324.6
3120.3
4419.8
5324.3
6222.2
7228.5
8225.7
9320.2
10119.6
11228.8
12424.0
13417.1
14419.3
15324.2
16115 .8
32320.3
Balloon Data
Col. 1-2 - observation number
Col. 3 - color (1=pink, 2=yellow, 3=orange, 4=blue)
Col. 4-7 - inflation time in seconds
16

ANOVA --- Balloon Data
General Linear Models Procedure
Dependent Variable: TIME
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 126.15125000 42.05041667 3.85 0.0200
Error 28 305.64750000 10.91598214
Corrected Total 31 431.79875000
R-Square C.V. Root MSE TIME Mean
0.292153 16.31069 3.3039343 20.256250
Mean
Source DF Type I SS Square F Value Pr > F
Color 3 126.15125000 42.05041667 3.85 0.0200
17

ANOVA --- Balloon Data
The GLM Procedure t Tests (LSD) for time
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha 0.05
Error Degrees of Freedom 28
Error Mean Square 10.91598
Critical Value of t 2.04841
Least Significant Difference 3.3839
Means with the same letter are not significantly different.
t Grouping Mean N color
A 22.575 8 2
A
A 21.875 8 3
B 18.388 8 1
B
B 18.188 8 4
18

Experimental Design:
Concepts and Terminology
Designed Experiment
- an investigation in which a specified framework is used to compare groups or treatments
Factors
- any feature of the experiment that can be varied from trial to trial
up to this point we’ve only looked at experiments with a single factor
19

Treatments
- conditions constructed from the factors
(levels of the factor considered, etc.)
Experimental Units
- subjects, material, etc. to which treatment factors are randomly assigned
- there is inherent variability among these units irrespective of the treatment imposed
Replication
- we usually assign each treatment to several experimental units
- these are called replicates
20

Car Data
1. factor
2. treatments
3. experimental units
4. replicates
Hostility Data
Balloon Data
21

17218.3
18117.5
19418.7
20322.9
21116.3
22414.0
23416.6
24218.1
25218.9
26416.0
27220.1
28322.5
29316.0
30119.3
31115.9
1122.4
2324.6
3120.3
4419.8
5324.3
6222.2
7228.5
8225.7
9320.2
10119.6
11228.8
12424.0
13417.1
14419.3
15324.2
16115 .8
32320.3
Balloon Data
Col. 1-2 - observation number (run order)
Col. 3 - color (1=pink, 2=yellow, 3=orange, 4=blue)
Col. 4-7 - inflation time in seconds
Question:
Why randomize run order? i.e. why not blow up all the pink balloons first, blue balloons next, etc?
22

Scatterplot Using GPLOT
Time
1 9
1 8
1 7
1 6
1 5
1 4
2 4
2 3
2 2
2 1
2 0 t i me
2 9
2 8
2 7
2 6
2 5
0 1 0 2 0 3 0
What do we learn from this plot?
4 0
23

RECALL: 1-Factor ANOVA Model y ij m a e ij
( m m a i
) e ij s NID
 2
' are (0, )
- random errors follow a Normal ( N ) distribution, are independently distributed ( ID ), and have zero mean and constant variance
-- i.e. variability does not change from group to group
24

Model Assumptions:
- equal variances
- normality
Checking Validity of Assumptions
Equal Variances
1. F-test similar to 2-sample case
Hartley’s test (p.366 text)
- not recommended
2. Graphical
- side-by-side box plots
25

Graphical Assessment of Equal Variance Assumption
26

Optional approaches if equal variance assumption is violated:
1. Use Kruskal Wallis nonparametric procedure – Section 8.6
2. Transform the data to induce more nearly equal variances – Section 8.5
-- log
-- square root
Note: These transformations may also help induce normality
27

Assessing Normality of Errors y ij
= m + a i
+ e ij so e ij
= y ij

( m + a i
)
= y ij
 m i e ij is estimated by e ij
 y ij
 y i .
The e ij
’ s are called residuals .
28

SAS Code for Balloon Data proc glm; class color; model time=color; title 'ANOVA --- Balloon Data'; output out=new r=resball ; means color/lsd; run; proc sort; run; by color; proc boxplot; plot time*color; title 'Side-by-Side Box Plots for Balloon Data'; run; proc univariate; var resball; histogram resball/normal; title 'Histogram of Residuals -- Balloon Data'; run;
proc univariate normal plot; var resball; title 'Normal Probability Plot for Residuals -
Balloon Data'; run; proc gplot; plot time*id; title 'Scatterplot of Time vs ID (Run Order)'; run;
29

Graphical Assessment of Normality
of Residuals e ij
 y ij
 y i .
3 0 c e n
P e r t
2 5
2 0
1 5
1 0
5
4 0
3 5
0
- 6 - 3 0 r e s b a l l
Normal Probability Plot
6.5+ +*+
| * *+++
| *+++
| +*+
| ***
| ****
0.5+ ***+
| ++**
| ++***
| *****
| +*+
| *+*+*
-5.5+ * ++++
+----+----+----+----+----+----+----+----+----+---
-2 -1 0 +1 +2
3 6
30

Chapter 15 introduces some new notation
- i.e. changes notation already defined
31

Recall: Sum-of-Squares Identity
1-Factor ANOVA t n
 i
 
1
( y ij
 y
..
)
2  n ( y i .
 y
..
i t


1
)
2 + t n
 i
 
1
( y ij
 y i .
)
2
Notation: TSS

SSB
+
SSW
In words:
T otal SS = SS between samples + within sample SS
32

Recall: Sum-of-Squares Identity
1-Factor ANOVA
- new notation for Chapter 15 t n
 i
 
1
( y ij
 y
..
)
2  n ( y i .
 y
..
i t


1
)
2 + t n
 i
 
1
( y ij
 y i .
)
2
Notation: TSS

SSB
+
SSW
33

Recall: Sum-of-Squares Identity
1-Factor ANOVA
- new notation for Chapter 15 t n
 i
 
1
( y ij
 y
..
)
2  n ( y i .
 y
..
i t


1
)
2 + t n
 i
 
1
( y ij
 y i .
)
2
Notation: TSS

S ST
+
SSW
34

Recall: Sum-of-Squares Identity
1-Factor ANOVA
- new notation for Chapter 15 t n
 i
 
1
( y ij
 y
..
)
2  n ( y i .
 y
..
i t


1
)
2 + t n
 i
 
1
( y ij
 y i .
)
2
Notation: TSS

SST
+
SSE
In words:
T otal SS = SS for “ treatments” + SS for “ error”
35

Revised ANOVA Table for 1-Factor ANOVA
(Ch. 15 terminology - p.857)
Source SS df MS F
Treatments SST t

1 
/(

1)
Error SSE N
 t MSE

/(
 t )
Total TSS N
 1
N

total # of observations
 nt (if equal # obs.)
 + n
2
+ + n t
36

Recall 1-factor ANOVA (CRD) Model for Gasoline Octane Data y ij
= m i
+ e ij or y ij
= m + a i
+ e ij observed octane mean for i th gasoline unexplained part
-- car-to-car differences
-- temperature
-- etc.
37

Gasoline Octane Data
Question:
What if car differences are obscuring gasoline differences?
Similar to diet t-test example:
Recall: person-to-person differences obscured effect of diet
38

Test all 5 gasolines on the same car
- in essence we test the gasoline effect directly and remove effect of car-to-car variation
Question:
How would you randomize an experiment with 4 cars?
39

Blocking an Experiment
- dividing the observations into groups (called blocks) where the observations in each block are collected under relatively similar conditions
- comparisons can many times be made more precisely this way
40

Terminology is based on
Agricultural Experiments
Consider the problem of testing fertilizers on a crop
- t fertilizers
- n observations on each
41

B
A
C
A
B
C
A
B
A
C
C
B
C t = 3 fertilizers n = 5 replications
- randomly select 15 plots
- randomly assign fertilizers to the 15 plots
B
A
42

A | C | B
B | A | C
A | B | C t = 3 fertilizers
select 5 “blocks”
- randomly assign the 3 treatments to each block
C | A | B
Note: The 3 “plots” within each block are similar
- similar soil type, sun, water, etc
C | B | A
43

Randomized Complete Block Design
Randomly assign each treatment once to every block
Car Example
Car 1: randomly assign each gas to this car
Car 2: ....
etc.
Agricultural Example
Randomly assign each fertilizer to one of the 3 plots within each block
44

Model For Randomized
Complete Block (RCB) Design y ij
= m + a i
+ b j
+ e ij effect of i th treatment
(gasoline) effect of j th block
(car)
As before: i t a i
 b
 

1 j

1 b j

0 unexplained error
-- temperature
-- etc.
45

Previous Data Table from Chapter 8 for 1-factor ANOVA column averages don’t make any sense
46

Gas
Back to Octane data:
Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car.
Old Data Format
A 91.7 91.2 90.9 90.6
B 91.7 91.9 90.9 90.9
C 92.4 91.2 91.6 91.0
D 91.8 92.2 92.0 91.4
E 93.1 92.9 92.4 92.4
“Restructured” Data
Gas
Car
1 2 3 4
A 91.7 91.2 90.9 90.6
B 91.7 91.9 90.9 90.9
C 92.4 91.2 91.6 91.0
D 91.8 92.2 92.0 91.4
E 93.1 92.9 92.4 92.4
47

Recall: Sum-of-Squares Identity
1-Factor ANOVA
- using new notation for Chapter 15 t n
 i
 
1
( y ij
 y
..
)
2  n ( y i .
 y
..
i t


1
)
2 + t n
 i
 
1
( y ij
 y i .
)
2
Notation: TSS

SST
+
SSE
In words:
T otal SS = SS for “ treatments” + SS for “ error”
48

A New Sum-of-Squares Identity t b
 i
 
1
( y ij
 y
..
)
2  b ( y i .
 y
..
i t


1
)
2 + t ( y
.
j
 y
..
j b


1
)
2 + t b
 i
 
1
( y ij
 y i .
 y
.
j
+ y
..
)
2
Not atio n: TSS

SST
+
SSB
+
SSE
In words:
T otal SS = SS for treatments + SS for blocks + SS for error
49

To test for treatment effects
- i.e. gas differences we test
H
0
: a
1
 a
2
  a t
To test for block effects
- i.e. car differences
(not usually the research hypothesis) we test
H
0
: b
1
 b
2
  b b
50

Randomized Complete Block Design
ANOVA Table
Source SS df MS F
Treatments SST t

1
Blocks SSB b

1
MST

/(

1)
MSB

/(

1)
Error SSE ( b

1)( t

1) MSE

/(

1)( t

1)
Total TSS bt
 1
See page 866
51

H
H a
0
:
: a
1
 a
2
  a t
at least one a i

0
We reject H
0 a
F

MST
MSE
 a (

1,( b

1)( t

1))
Note:
MSE
MST
estimates
estimates

 e
2 e
2 + t
1

1 t
 a i
2 i

1
- if no treatment effects, we expect F

1 ;
- if treatment effects, we expect F

1
52

H
H a
0
: b
1
 b
2
 
: b
at least one b b j

0
We reject H
0 a
F

MSB
MSE
 a (

1,( b

1)( t

1))
53

“Restructured” CAR Data
- SAS Format
A B1 91.7
A B2 91.2
A B3 90.9
A B4 90.6
B B1 91.7
B B2 91.9
B B3 90.9
B B4 90.9
C B1 92.4
C B2 91.2
C B3 91.6
C B4 91.0
D B1 91.8
D B2 92.2
D B3 92.0
D B4 91.4
E B1 93.1
E B2 92.9
E B3 92.4
E B4 92.4
The first variable (A - E) indicates gas as it did with the Completely
Randomized Design. The second variable (B1 - B4) indicates car.
54

SAS file - Randomized Complete Block Design for CAR Data
INPUT gas$ block$ octane;
PROC GLM;
CLASS gas block;
MODEL octane=gas block;
TITLE 'Gasoline Example -Randomized Complete Block Design';
MEANS gas/LSD;
RUN;
55

1-Factor ANOVA Table Output - octane data
Source SS df MS F p -value
Gas 6.108 4 1.527 6.80 0.0025
(treatments)
Error 3.370 15 0.225
Totals 9.478 19
56

1-Factor ANOVA Table Output - car data
Source SS df MS F p -value
Gas 6.108 4 1.527 15.58 0.0001
(treatments)
Cars 2.194 3 0.731 7.46 0.0044
(blocks)
Error 1.176 12 0.098
Totals 9.478 19
57

SAS Output -- RCB CAR Data
Dependent Variable: OCTANE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 7 8.30200000 1.18600000 12.10 0.0001
Error 12 1.17600000 0.09800000
Corrected Total 19 9.47800000
R-Square C.V. Root MSE OCTANE Mean
0.875923 0.341347 0.3130495 91.710000
Source DF Anova SS Mean Square F Value Pr > F
GAS 4 6.10800000 1.52700000 15.58 0.0001
BLOCK 3 2.19400000 0.73133333 7.46 0.0044
58

Multiple Comparisons in RCB Analysis y and y are significantly different if
|
2
|
 t
α/ 2
2 MSE
(LSD) b t
α/ (2 )
2 MSE
(Bonferroni) b

CAR Data -- LSD Results
CRD Analysis t Grouping Mean N gas
A 92.7000 4 E
B 91.8500 4 D
B
C B 91.5500 4 C
C B
C B 91.3500 4 B
C
C 91.1000 4 A
RCB Analysis t Grouping Mean N gas
A 92.7000 4 E
B 91.8500 4 D
B
C B 91.5500 4 C
C
C 91.3500 4 B
C
C 91.1000 4 A
60

CAR Data -- Bonferroni Results
CRD Analysis
Bon Grouping Mean N gas
A 92.7000 4 E
A
B A 91.8500 4 D
B
B 91.5500 4 C
B
B 91.3500 4 B
B
B 91.1000 4 A
RCB Analysis
Bon Grouping Mean N gas
A 92.7000 4 E
B 91.8500 4 D
B
B 91.5500 4 C
B
B 91.3500 4 B
B
B 91.1000 4 A
61