Transcript Chapter 4: Random Variables and Probability Distributions

Chapter 10: Analysis of Variance:
Comparing More Than Two Means
Where We’ve Been


Presented methods for estimating and
testing hypotheses about a single
population mean
Presented methods for comparing two
population means
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
2
Where We’re Going



Discuss the critical elements in the design of
a sampling experiment
Investigate completely randomized,
randomized block, and factorial designs
Show how to analyze data using a
technique called analysis of variance
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
3
10.1: Elements of a Designed
Experiment
Elements of
Designed
Experiments
Response Variable
(or dependent variable)
Factors
(possible impacting the response
variable)
Quantitative
Factors
(Numerical)
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
Qualitative
Factors
(Non-numerical)
4
10.1: Elements of a Designed
Experiment
Elements of
Designed
Experiments
Response Variable
(or dependent variable)
Factor Levels are the
values of the factors
Factors
(possible impacting the response
variable)
Quantitative
Factors
(Numerical)
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
Qualitative
Factors
(Non-numerical)
5
10.1: Elements of a Designed
Experiment

Treatments are the factor-level
combinations

In the example above, a variety of
different GPA – Hours Studied
combinations could occur within each
subset (Yes or No) of the Study Group
factor
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
6
10.1: Elements of a Designed
Experiment

An experimental unit is the object on
which the response and factors are
observed or measured

In the example above, an individual
student would be the experimental unit
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
7
10.1: Elements of a Designed
Experiment
In a designed experiment
the analyst controls the
treatments and the selection
of experimental units to each
treatment
In an observational
experiment the analyst
observes the treatment and
response on a sample of
experimental units
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
8
10.1: Elements of a Designed
Experiment
In a designed experiment
the analyst controls the
treatments and the selection
of experimental
The
method by units to each
treatment
which the
experimental units
are selected
In an observational
determines the type
experiment the analyst
of experiment
observes the treatment
and
response on a sample of
experimental units
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
9
10.1: Elements of a Designed
Experiment
Population of Experimental Units
Sample of Experimental Units
Apply factor-level combinations
Treatment 1
Sample
Treatment 2
Sample
Treatment 3
Sample
…
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
Treatment k
Sample
10
10.2: The Completely
Randomized Design
The completely randomized design
is a design in which treatments are
randomly assigned to the experimental
units or in which independent random
samples of experimental units are
selected for each treatment.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
11
10.2: The Completely
Randomized Design
Randomly
assign
observations
to treatments
Completely
Randomized
Design
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
Randomly
assign
treatments to
observations
12
10.2: The Completely
Randomized Design

Very often the object is to determine
whether the varying treatments result
in different means:
H0: µ1 = µ2 = µ3 = µ4 = ··· = µk
Ha: At least two of the k treatment
means differ
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
13
10.2: The Completely
Randomized Design

Testing the equity of the means involves comparing the variability among
the different treatments as well as within the treatments, adjusted for
degrees of freedom.
Variation between the treatment means:
Sum of Squares for Treatments (SST)
k
SST   ni ( xi  x ) 2
i 1
Variation within the treatments:
Sum of Squares for Error (SSE)
n1
n2
SSE   ( x1 j  x1 )   ( x2 j  x2 ) 
j 1
2
j 1
 (n1  1) s12  (n2  1) s22 
2
nk
+ ( xkj  xk ) 2
j 1
 ( nk  1) sk2
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
14
10.2: The Completely
Randomized Design

Adjusting for degrees of freedom produces
comparable measures of variability
Mean Square for Treatments (MST)
SST
MST 
k 1
Mean Square for Error (MSE)
SSE
MSE 
nk
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
15
10.2: The Completely
Randomized Design
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
16
10.2: The Completely
Randomized Design

The ratio of the variability among the
treatment means to that within the
treatment means is an F -statistic:
Fk 1,n  k
MST

MSE
with k-1 numerator and n-k denominator
degrees of freedom.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
17
10.2: The Completely
Randomized Design
If F*  1, the difference
between the treatment
means may be attributable
to sampling error.
If F* > 1 (significantly),
there is support for the
alternative hypothesis
that the treatments
themselves produce
different results.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
18
10.2: The Completely
Randomized Design
ANOVA F -Test to Compare k Treatment Means:
Completely Randomized Design
H 0 : 1  2 
 k
H a : At least two treatment means differ.
MST
Test Statistic: F =
MSE
Rejection region: F *  F , with k  1 numerator
and n - k denominator degrees of freedom.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
19
10.2: The Completely
Randomized Design

Conditions required for a Valid ANOVA
F-Test: Completely Randomized Design
1.
2.
3.
The samples are randomly selected in an
independent manner from the k treatment
populations.
All k sampled populations have
distributions that are approximately normal.
The k population variances are equal.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
20
10.2: The Completely
Randomized Design

The USGA compares the driving distance of
four brands of golf balls.





H0: µ1 = µ2 = µ3 = µ4
Ha: The mean distances differ for at least
two of the brands
 = .10
Test Statistic: F = MST/MSE
Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
21
10.2: The Completely
Randomized Design
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
22
10.2: The Completely
Randomized Design

The USGA compares the driving distance of four brands of golf
balls.




H0: µ1 = µ2 = µ3 = µ4
Ha: The mean distances differ for at least two of the brands
 = .10Test Statistic: F = MST/MSE
Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36
Source
Degrees of
Freedom
Sum of
Squares
Mean
Square
F
p-value
Brands
3
2,794.39
931.46
43.99
.000
Error
36
762.30
21.18
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
23
10.2: The Completely
Randomized Design

The USGA compares the driving distance of four brands of golf
balls.




Source
Brands
Error
H0: µ1 = µ2 = µ3 = µ4
Ha: The mean distances differ for at least two of the brands
 = .10Test Statistic: F = MST/MSE
Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36
Degrees of
Sum of
Since
the
Freedom
Squares
calculated
3
F > 2.25,2,794.39
we
reject the
null
36
762.30
hypothesis.
Mean
Square
F
p-value
931.46
43.99
.000
21.18
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
24
10.2: The Completely
Randomized Design


If the conditions for ANOVA are not
met, a nonparametric procedure is
recommended (see Chapter 14).
If the null hypothesis is not rejected,
that is not conclusive proof that the
treatment means are all equal.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
25
10.3: Multiple Comparisons of
Means


Suppose the ANOVA F-test indicates
differences in the means. To determine the
differences, we would compare the
differences of the means.
With k treatment means, there are
c = k(k – 1)/2
pairs of means to be compared, and each
would have a significance level smaller than
the overall .
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
26
10.3: Multiple Comparisons of
Means

To retain the overall confidence level,
various techniques are available for
pair wise comparisons:



Tukey – treatment sample sizes are
equal
Bonferroni - treatment sample sizes may
be unequal
Scheffé – general procedure for all linear
combinations of treatment means
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
27
10.3: Multiple Comparisons of
Means

Let’s go back to the four brands of golf
balls in the previous example:


Rank the treatment means with an overall
95% level of confidence using Tukey’s
procedure.
Estimate the highest ranked golf ball's
mean driving distance.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
28
10.3: Multiple Comparisons of
Means
Pair wise Comparisons for Four Golf ball Brands
Based on a SAS ANOVA report (see pages 506-7)
Brand Comparison
95% Confidence Interval
µA - µB
-15.82 < µA - µB < - 4.74
µA - µC
-24.71 < µA - µC < -13.63
µA - µD
-4.08 < µA - µD < 7.00
µB - µC
-14.43 < µB - µC < -3.35
µB - µD
6.2 < µB - µD < 17.28
µC - µD
15.09 < µC - µD < 26.17
Brand C outperforms each of the other brands.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
29
10.3: Multiple Comparisons of
Means

To construct a confidence interval on Brand
C, we can use the descriptive statistics from
the ANOVA and a straightforward onesample t-based confidence interval (see
section 7.3):
95% sure   xC  t /2,df 36 s 1/ n ,
where s  MSE , n  10 and t /2
2,
  270  (2)(4.60) 1/10  270  2.9
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
30
10.4: The Randomized Block
Design

The randomized block design:

Blocks (matched sets of experimental
units) are formed.


Each of the b blocks has k experimental
units, one for each treatment.
One experimental unit from each block is
randomly assigned to each treatment, for
a total of n = bk responses.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
31
10.4: The Randomized Block
Design

To test the equity of the means, we use the
ratio MST/MSE ~ F
k
2
b
(
x

x
)
 Ti
SST i 1
MST 

k 1
k 1
SSE
SS (total )  SST  SSB
MSE 

n  b  k 1
n  b  k 1
n
k
b
2
(
x

x
)

b
(
x

x
)

k
(
x

x
)
 i
 Ti
 Bi
2

i 1
2
i 1
i 1
n  b  k 1
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
32
10.4: The Randomized Block
Design
ANOVA F -Test to Compare k Treatment Means:
Randomized Block Design
H 0 : 1   2 
 p
H a : At least two treatment means differ
MST
Test Statistic: F =
MSE
Rejection Region: F  F , with (k -1) numerator and
(n - b - k  1) [= (b -1)(k -1)] denominator degrees of freedom.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
33
10.4: The Randomized Block
Design
Conditions required for a valid ANOVA F – Test



The b blocks are randomly selected and all k
treatments are applied (in random order) to each
block.
The distribution of observations corresponding to
all bk block-treatment combinations are
approximately normal.
The bk block-treatment distributions have equal
variances.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
34
10.4: The Randomized Block
Design
Completely
Randomized Design
Randomized
Block Design
Sum of Squares for
Treatments
SST
df=k-1
Sum of Squares for
Treatments
SST
df=k-1
Total Sum of Squares
SS(Total)
df=n-1
Sum of Squares for
Error
SSE
df=n-k
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
Sum of Squares for
Blocks
SSB
df=b-1
Sum of Squares for
Error
SSE
df=n-b-k+1=(b-1)(k-1)
35
10.4: The Randomized Block
Design

Suppose the golf balls analyzed above are
analyzed again using ten real golfers
instead of a machine.



Each golfer is a block
Each brand is a treatment assigned in random
order to each golfer
The ten drives for each brand produce the
following means:
Brand A
Brand B
Brand C
Brand D
227 yards
233.2 yards
245.3 yards
220.7 yards
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
36
10.4: The Randomized Block
Design
ANOVA Table for the Golf Ball Tests
Source
df
SS
MS
F
p
Treatment (Brand)
3
2,298.7
1,099.6
54.31
.000
Block (Golfer)
9
12,073.9
1,341.5
Error
27
546.6
20.2
Total
39
15,919.2
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
37
10.4: The Randomized Block
Design
Equity of Means 95% Confidence Intervals for the Golf Balls’ Distance
µA - µB
µA - µC
µA - µD
(-11.9,--.4)
(-24.0, -2.6) (.6, 12.0)
µB - µC
µB - µD
(-17.9, -6.4) (6.7, 18.2)
µC - µD
(18.9, 30.3)
None of the confidence
intervals contain zero, so we
can be 95% certain all of the
brand means differ.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
38
10.4: The Randomized Block
Design
ANOVA Table for the Golf Ball Tests
Source
df
SS
MS
F
p
Treatment (Brand)
3
2,298.7
1,099.6
54.31
.000
Block (Golfer)
9
12,073.9
1,341.5
Error
27
546.6
20.2
Total
39
15,919.2
To test for block mean differences, use the ratio of MSB to MEE
F
MSB MS(Golfers) 1,341.5


 66.26
MSE
MS(Error)
20.2
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
39
10.5: Factorial Experiments

A complete factorial experiment is a
factorial experiment in which every
factor-level combination is utilized.
That is, the number of treatments in
the experiment equals the total
number of factor-level combinations.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
40
10.5: Factorial Experiments
Factor A
(only) matters
Factors A and B
both matter,
independently
Say two factors, A
and B, are involved.
Results could vary
among observations
because …
Factors A and B
both matter and
they interact
Factor B
(only) matters
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
41
10.5: Factorial Experiments
Main
Effect of
Factor B
Main
Effect of
Factor A
Interaction
between A
and B
Overall
Treatment
Variability
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
42
10.5: Factorial Experiments
Stage 1
Stage 2
Main effect sum of squares Factor A
SS(A)
df=a-1
Sum of Squares for
Treatments
SST
df=ab-1
Total Sum of
Squares
SS(Total)
df=n-1
Main effect sum of squares Factor B
SS(B)
df = b-1
Interaction sum of squares
Factors A and B
SS(AB)
df = (a-1)(b-1)
Sum of Squares for
Error
SSE
df=n-ab
Sum of Squares for Error
SSE
df= n - ab
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
43
10.5: Factorial Experiments

Tests Conducted in Analyses of Factorial
Experiments: Completely Randomized Design, r
Replicates per Treatment
Test for Treatment Means
H 0 : No difference among the ab treatment means
H a : At least two treatment means differ
MST
MSE
Rejection Region : F  F , with (ab - 1) numerator
Test Statistic: F =
and (n - ab) denominator degrees of freedom
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
44
10.5: Factorial Experiments

Tests Conducted in Analyses of Factorial
Experiments: Completely Randomized Design, r
Replicates per Treatment
Test for Factor Interaction
H 0 : Factors A and B do not interact to affect the response mean
H a : Factors A and B do interact to affect the response mean
MS(AB)
Test Statistic: F =
MSE
Rejection Region : F  F , with (a - 1)(b - 1) numerator and
(n - ab) denominator degrees of freedom
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
45
10.5: Factorial Experiments

Tests Conducted in Analyses of Factorial
Experiments: Completely Randomized Design, r
Replicates per Treatment
Test for Main Effect of Factor A
H 0 : No difference among the a mean levels of factor A
H a : At least two factor A mean levels differ
MS(A)
Test Statistic: F =
MSE
Rejection Region : F  F , with (a - 1) numerator and
(n - ab) denominator degrees of freedom
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
46
10.5: Factorial Experiments

Tests Conducted in Analyses of Factorial
Experiments: Completely Randomized Design, r
Replicates per Treatment
Test for Main Effect of Factor B
H 0 : No difference among the b mean levels of factor B
H a : At least two factor B mean levels differ
MS(B)
Test Statistic: F =
MSE
Rejection Region : F  F , with (b - 1) numerator and
(n - ab) denominator degrees of freedom
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
47
10.5: Factorial Experiments

Tests Conducted in Analyses of Factorial
Experiments: Completely Randomized Design, r
Replicates per Treatment
 Conditions Required:



Response distribution for each factor-level
combination is normal.
Response variance is constant for all treatments.
Random and independent samples of
experimental units are associated with each
treatment.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
48
10.5: Factorial Experiments

The four brands of golf balls are tested
again, this time with a driver and a 5
iron. Each brand-club combination
(eight in all) is assigned randomly to
four experimental units in a sequence of
swings by Iron Byron.


Are the treatment means equal?
Do the factors “brand“ and “club” interact?
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
49
10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source
df
SS
MS
F
Model
7
33659.81
4808.54
140.35
Brand
1
32,092.11
32,093.11
936.75
Club
3
800.74
266.91
7.79
Interaction
3
765.96
255.32
7.45
Error
24
822.24
34.26
Total
31
34,482.05
H 0 : The treatment means are equal.
H a : At least two of the eight means differ.
Test Statistic: F 
MST 33, 659.81/ 7

 140.35; p  .0001
MSE
822.24 / 24
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
50
10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source
df
SS
MS
F
Model
7
33659.81
4808.54
140.35
Brand
1
32,092.11
32,093.11
936.75
Club
3
800.74
266.91
7.79
Interaction
3
765.96
255.32
7.45
Error
24
822.24
34.26
Total
31
34,482.05
H 0 : The treatment means are equal.
H a : At least two of the eight means differ.
Test Statistic: F 
Reject the null
hypothesis
MST 33, 659.81/ 7

 140.35; p  .0001
MSE
822.24 / 24
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
51
10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source
df
SS
MS
F
Model
7
33659.81
4808.54
140.35
Brand
1
32,092.11
32,093.11
936.75
Club
3
800.74
266.91
7.79
Interaction
3
765.96
255.32
7.45
Error
24
822.24
34.26
Total
31
34,482.05
H 0 : The factors Club and Brand do not interact to affect the mean response.
H a : The factors Club and Brand interact to affect the mean response
Test Statistic: F 
MS(AB) 255.32

 7.45; p  .0011
MSE
34.26
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
52
10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source
df
SS
MS
F
Model
7
33659.81
4808.54
140.35
Brand
1
32,092.11
32,093.11
936.75
Club
3
800.74
266.91
7.79
Interaction
3
765.96
255.32
Error
24
822.24
34.26
7.45
Reject the null
hypothesis
Total
31
34,482.05
H 0 : The factors Club and Brand do not interact to affect the mean response.
H a : The factors Club and Brand interact to affect the mean response
Test Statistic: F 
MS(AB) 255.32

 7.45; p  .0011
MSE
34.26
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
53
10.5: Factorial Experiments
Further analysis (see text) suggests that, although the factor
“Club” clearly has an impact on distance, the results for “Brand “
are more ambiguous: Brand B hit with a 5 iron outdistances the
others, but not when hit with the driver.
Statistics for Business and Economics, 11th ed.
Chapter 10: Analysis of Variance
54