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Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 10
Analysis of Variance

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Chap 10-1
Chapter Goals
After completing this chapter, you should be able
to:


Recognize situations in which to use analysis of variance



Understand different analysis of variance designs



Perform a single-factor hypothesis test and interpret results



Conduct and interpret post-hoc multiple comparisons
procedures



Analyze two-factor analysis of variance tests

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-2
Chapter Overview
Analysis of Variance (ANOVA)
One-Way
ANOVA
F-test
TukeyKramer
test
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Two-Way
ANOVA
Interaction
Effects

Chap 10-3
General ANOVA Setting


Investigator controls one or more independent
variables





Observe effects on the dependent variable




Called factors (or treatment variables)
Each factor contains two or more levels (or groups or
categories/classifications)
Response to levels of independent variable

Experimental design: the plan used to collect
the data

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-4
Completely Randomized Design


Experimental units (subjects) are assigned
randomly to treatments




Only one factor or independent variable




Subjects are assumed homogeneous
With two or more treatment levels

Analyzed by one-factor analysis of variance
(one-way ANOVA)

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-5
One-Way Analysis of Variance


Evaluate the difference among the means of three
or more groups
Examples: Accident rates for 1st, 2nd, and 3rd shift
Expected mileage for five brands of tires



Assumptions
 Populations are normally distributed
 Populations have equal variances
 Samples are randomly and independently drawn

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-6
Hypotheses of One-Way ANOVA


H0 : μ1 = μ2 = μ3 =  = μc





All population means are equal
i.e., no treatment effect (no variation in means among
groups)

H1 : Not all of the population means are the same


At least one population mean is different



i.e., there is a treatment effect

Does not mean that all population means are different
(some pairs may be the same)
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Chap 10-7
Prentice-Hall, Inc.

One-Factor ANOVA
H0 : μ1 = μ2 = μ3 =  = μc
H1 : Not all μi are the same
All Means are the same:
The Null Hypothesis is True
(No Treatment Effect)

Statistics forμ = μ = μ
Managers Using
1
2
3
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-8
One-Factor ANOVA
(continued)

H0 : μ1 = μ2 = μ3 =  = μc
H1 : Not all μi are the same
At least one mean is different:
The Null Hypothesis is NOT true
(Treatment Effect is present)
or
Statistics for Managers Using
μ = μ2 4e 3
Microsoft1Excel, ≠ μ© 2004
Prentice-Hall, Inc.

μ1 ≠ μ2 ≠ μ3
Chap 10-9
Partitioning the Variation


Total variation can be split into two parts:

SST = SSA + SSW
SST = Total Sum of Squares
(Total variation)
SSA = Sum of Squares Among Groups
(Among-group variation)
SSW = Sum of Squares Within Groups
(Within-group variation)
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Chap 10-10
Prentice-Hall, Inc.
Partitioning the Variation
(continued)

SST = SSA + SSW
Total Variation = the aggregate dispersion of the individual
data values across the various factor levels (SST)
Among-Group Variation = dispersion between the factor
sample means (SSA)
Within-Group Variation = dispersion that exists among the
data values within a particular factor level (SSW)
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Chap 10-11
Prentice-Hall, Inc.
Partition of Total Variation
Total Variation (SST)

=





Variation Due to
Factor (SSA)

Commonly referred to as:
Sum of Squares Between
Sum of Squares Among
Sum of Squares Explained
Among Groups Variation

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Variation Due to Random
Sampling (SSW)

+





Commonly referred to as:
Sum of Squares Within
Sum of Squares Error
Sum of Squares Unexplained
Within Groups Variation

Chap 10-12
Total Sum of Squares
SST = SSA + SSW
c

nj

SST = ∑∑ ( Xij − X)
Where:

2

j=1 i =1

SST = Total sum of squares
c = number of groups (levels or treatments)
nj = number of observations in group j
Xij = ith observation from group j
Statistics for Managers Using
X = © 2004
Microsoft Excel, 4egrand mean (mean of all data values)
Chap 10-13
Prentice-Hall, Inc.
Total Variation
(continued)

SST = ( X11 − X)2 + ( X12 − X)2 + ... + ( Xcnc − X)2
Response, X

X
Statistics for Managers Using 2
Group 1
Group
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Group 3

Chap 10-14
Among-Group Variation
SST = SSA + SSW
c

SSA = ∑ n j ( X j − X)2
Where:

j =1

SSA = Sum of squares among groups
c = number of groups or populations
nj = sample size from group j
Xj = sample mean from group j
Statistics for Managers Using
X = © 2004
Microsoft Excel, 4egrand mean (mean of all data values)
Chap 10-15
Prentice-Hall, Inc.
Among-Group Variation
(continued)
c

SSA = ∑ n j ( X j − X)2
j =1

Variation Due to
Differences Among Groups

SSA
MSA =
k −1
Mean Square Among =
SSA/degrees of freedom

µi
µj
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-16
Among-Group Variation
(continued)

SSA = n1 ( x1 − x ) + n 2 ( x 2 − x ) + ... + nc ( x c − x )
2

2

Response, X

X3
X1
Statistics for Managers Using
Group 1
Group 2
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

X2

X

Group 3

Chap 10-17

2
Within-Group Variation
SST = SSA + SSW
c

SSW = ∑
j=1

nj

∑
i =1

( Xij − X j )

2

Where:

SSW = Sum of squares within groups
c = number of groups
nj = sample size from group j
Xj = sample mean from group j
Statistics for Managers Using
Microsoft Excel, 4e i© observation in group j
Xij = th 2004
Prentice-Hall, Inc.

Chap 10-18
Within-Group Variation
(continued)
c

SSW = ∑
j=1

nj

∑
i =1

( Xij − X j )2

Summing the variation
within each group and then
adding over all groups

SSW
MSW =
n−c
Mean Square Within =
SSW/degrees of freedom

Statistics for Managers Using
µi
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-19
Within-Group Variation
(continued)

SSW = ( x11 − X1 ) + ( X12 − X 2 ) + ... + ( Xcnc − Xc )
2

2

Response, X

X1
Statistics for Managers Using
Group
Microsoft Excel, 4e1 © 2004 Group 2
Prentice-Hall, Inc.

X2

X3

Group 3

Chap 10-20

2
Obtaining the Mean Squares
SSA
MSA =
c −1
SSW
MSW =
n−c
SST
MST =
n −1
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-21
One-Way ANOVA Table
Source of
Variation

SS

df

Among
Groups

SSA

c-1

Within
Groups

SSW

n-c

Total

SST =
SSA+SSW

MS
(Variance)

F ratio

SSA
MSA
MSA =
c - 1 F = MSW
SSW
MSW =
n-c

n-1

c = number of groups
for Managers Using of the sample sizes from all groups
n = sum
df
Excel, 4e © 2004 = degrees of freedom

Statistics
Microsoft
Prentice-Hall, Inc.

Chap 10-22
One-Factor ANOVA
F Test Statistic
H0: μ1= μ2 = … = μc
H1: At least two population means are different


Test statistic

MSA
F=
MSW

MSA is mean squares among variances
MSW is mean squares within variances


Degrees of freedom


df1 = c – 1

(c = number of groups)

Statistics for Managers Using

(n =
Microsoft df2 = n –4e © 2004sum of sample sizes from all populations)
Excel, c
Chap 10-23
Prentice-Hall, Inc.
Interpreting One-Factor ANOVA
F Statistic


The F statistic is the ratio of the among
estimate of variance and the within estimate
of variance




The ratio must always be positive
df1 = c -1 will typically be small
df2 = n - c will typically be large

Decision Rule:
 Reject H if F > F ,
0
U
otherwise do not
Statistics for Managers Using
reject H0
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

α = .05

0

Do not
reject H0

Reject H0

FU

Chap 10-24
One-Factor ANOVA
F Test Example
You want to see if three
different golf clubs yield
different distances. You
randomly select five
measurements from trials on
an automated driving
machine for each club. At the
.05 significance level, is there
a difference in mean
distance?
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Club 1
254
263
241
237
251

Club 2
234
218
235
227
216

Club 3
200
222
197
206
204

Chap 10-25
One-Factor ANOVA Example:
Scatter Diagram
Club 1
254
263
241
237
251

Club 2
234
218
235
227
216

Club 3
200
222
197
206
204

Distance
270
260
250
240
230

•
•
•
•
•

220

x1 = 249.2 x 2 = 226.0 x 3 = 205.8
x = 227.0

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

210

X1
•
•
•
•
•

X2

•
•
•
•
•

200

X
X3

190
1

2
3
Chap 10-26
Club
One-Factor ANOVA Example
Computations
Club 1
254
263
241
237
251

Club 2
234
218
235
227
216

Club 3
200
222
197
206
204

X1 = 249.2

n1 = 5

X2 = 226.0

n2 = 5

X3 = 205.8

n3 = 5

X = 227.0

n = 15

c=3
SSA = 5 (249.2 – 227)2 + 5 (226 – 227)2 + 5 (205.8 – 227)2 = 4716.4
SSW = (254 – 249.2)2 + (263 – 249.2)2 +…+ (204 – 205.8)2 = 1119.6

MSA = 4716.4 / (3-1) = 2358.2

Statistics 1119.6 / (15-3) =Using
MSW = for Managers 93.3
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

2358.2
F=
= 25.275
93.3
Chap 10-27
One-Factor ANOVA Example
Solution
Test Statistic:

H0: μ1 = μ2 = μ3
H1: μi not all equal

MSA 2358.2
F=
=
= 25.275
MSW
93.3

α = .05
df1= 2
df2 = 12
Critical
Value:

Decision:
Reject H0 at α = 0.05

FU = 3.89

Conclusion:
There is evidence that
0 Do not
at least one μi differs
Statistics for Managers Using
Reject H
reject H
F=
Microsoft Excel, 4e © 200425.275 from the rest
FU = 3.89
α = .05

0

0

Prentice-Hall, Inc.

Chap 10-28
ANOVA -- Single Factor:
Excel Output
EXCEL: tools | data analysis | ANOVA: single factor
SUMMARY
Groups

Count

Sum

Average

Variance

Club 1

5

1246

249.2

108.2

Club 2

5

1130

226

77.5

Club 3

5

1029

205.8

94.2

ANOVA
Source of
Variation

SS

df

MS

Between
Groups

4716.4

2

2358.2

Within
Groups

1119.6

12

F

P-value

F crit

93.3

Statistics for Managers Using
Total
5836.0
14
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

25.275

4.99E-05

3.89

Chap 10-29
The Tukey-Kramer Procedure


Tells which population means are significantly
different





e.g.: μ1 = μ2 ≠ μ3
Done after rejection of equal means in ANOVA

Allows pair-wise comparisons


Compare absolute mean differences with critical
range

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

μ1= μ2

μ3

x

Chap 10-30
Tukey-Kramer Critical Range

Critical Range = QU

MSW  1 1 
 + 
2  n j n j' 



where:
QU = Value from Studentized Range Distribution
with c and n - c degrees of freedom for
the desired level of α (see appendix E.9 table)
MSW = Mean Square Within
Statistics and Managers Using groups j and j’
ni for nj = Sample sizes from

Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-31
The Tukey-Kramer Procedure:
Example
Club 1
254
263
241
237
251

Club 2
234
218
235
227
216

Club 3
200
222
197
206
204

1. Compute absolute mean
differences:
x1 − x 2 = 249.2 − 226.0 = 23.2
x1 − x 3 = 249.2 − 205.8 = 43.4
x 2 − x 3 = 226.0 − 205.8 = 20.2

2. Find the QU value from the table in appendix E.9 with
c = 3 and (n – c) = (15 – 3) = 12 degrees of freedom
for the desired level of α (α = .05 used here):

Statistics for Managers Using U
Q
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

= 3.77
Chap 10-32
The Tukey-Kramer Procedure:
Example

(continued)

3. Compute Critical Range:
Critical Range = QU

MSW  1 1 
 +  = 3.77 93.3  1 + 1  = 16.285


2  n j n j' 
2 5 5



4. Compare:
5. All of the absolute mean differences
are greater than critical range.
Therefore there is a significant
difference between each pair of
means at 5% level of significance.
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

x1 − x 2 = 23.2
x1 − x 3 = 43.4
x 2 − x 3 = 20.2

Chap 10-33
Tukey-Kramer in PHStat

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-34
Two-Way ANOVA


Examines the effect of


Two factors of interest on the dependent
variable




e.g., Percent carbonation and line speed on soft drink
bottling process

Interaction between the different levels of these
two factors


e.g., Does the effect of one particular carbonation
level depend on which level the line speed is set?

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-35
Two-Way ANOVA
(continued)


Assumptions


Populations are normally distributed



Populations have equal variances



Independent random samples are
drawn

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-36
Two-Way ANOVA
Sources of Variation
Two Factors of interest: A and B
r = number of levels of factor A
c = number of levels of factor B
n’ = number of replications for each cell
n = total number of observations in all cells
(n = rcn’)
Xijk = value of the kth observation of level i of
Statistics for Managers Using
factor A and
Microsoft Excel, 4e © 2004 level j of factor B
Prentice-Hall, Inc.

Chap 10-37
Two-Way ANOVA
Sources of Variation
SST = SSA + SSB + SSAB + SSE
SSA
Factor A Variation

SST
Total Variation

SSB
Factor B Variation

SSAB
n-1

Variation due to interaction
between A and B

(continued)
Degrees of
Freedom:
r–1

c–1

(r – 1)(c – 1)

SSE
rc(n’ – 1)
Statistics for Managers Using
Random variation (Error)
Microsoft Excel, 4e © 2004
Chap 10-38
Prentice-Hall, Inc.
Two Factor ANOVA Equations
Total Variation:

r

n′

c

SST = ∑∑∑ ( Xijk − X)

2

i =1 j=1 k =1

Factor A Variation:

r

′∑ ( Xi.. − X)2
SSA = cn
i=1

Factor B Variation:

c

SSB = rn′∑ ( X. j. − X)2

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

j=1

Chap 10-39
Two Factor ANOVA Equations
(continued)

Interaction Variation:
r

c

SSAB = n′∑∑ ( Xij. − Xi.. − X. j. + X)2
i =1 j =1

Sum of Squares Error:

r

c

n′

SSE = ∑∑∑ ( Xijk − Xij. )
i =1 j=1 k =1

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-40

2
Two Factor ANOVA Equations
r

where:

X=

Xi.. =

∑∑ X
j=1 k =1

i=1 j=1 k =1

ijk

rcn′

= Grand Mean

ijk

= Mean of ith level of factor A (i = 1, 2, ..., r)

cn′
r

X. j. =

(continued)

n′

∑∑∑ X

n′

c

c

n′

∑∑ X
i=1 k =1

rn′

ijk

= Mean of jth level of factor B (j = 1, 2, ..., c)

Xijk
Xij. = for
= Mean of cell
Statistics ∑ Managers Using ij
k =1 n′
n′

Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

r = number of levels of factor A
c = number of levels of factor B
n’ = number of replications in each cell

Chap 10-41
Mean Square Calculations
SSA
MSA = Mean square factor A =
r −1
SSB
MSB = Mean square factor B =
c −1

SSAB
MSAB = Mean square interaction =
(r − 1)(c − 1)
SSE
MSE = Mean square error =
Statistics for Managers Using
rc(n'−1)
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 10-42
Two-Way ANOVA:
The F Test Statistic
H0: μ1.. = μ2.. = μ3.. = • • •
H1: Not all μi.. are equal

H0: μ.1. = μ.2. = μ.3. = • • •
H1: Not all μ.j. are equal

H0: the interaction of A and B is
equal to zero

Statistics for Managers Using
H1: interaction of A and B is not
Microsoft Excel, 4e © 2004
zero
Prentice-Hall, Inc.

F Test for Factor A Effect

MSA
F=
MSE

Reject H0
if F > FU

F Test for Factor B Effect

MSB
F=
MSE

Reject H0
if F > FU

F Test for Interaction Effect

MSAB
F=
MSE

Reject H0
if F > FU

Chap 10-43
Two-Way ANOVA
Summary Table
Source of
Variation

Sum of
Squares

Degrees of
Freedom

Factor A

SSA

r–1

Factor B

SSB

c–1

AB
(Interaction)

SSAB

(r – 1)(c – 1)

Error

SSE

rc(n’ – 1)

Statistics for Managers Using
Total
SST
n–1
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Mean
Squares

F
Statistic

MSA

MSA
MSE

= SSA /(r – 1)

MSB
= SSB /(c – 1)

MSAB
= SSAB / (r – 1)(c – 1)

MSB
MSE
MSAB
MSE

MSE =
SSE/rc(n’ – 1)

Chap 10-44
Features of Two-Way ANOVA
F Test


Degrees of freedom always add up


n-1 = rc(n’-1) + (r-1) + (c-1) + (r-1)(c-1)



Total = error + factor A + factor B + interaction



The denominator of the F Test is always the
same but the numerator is different



The sums of squares always add up


SST = SSE + SSA + SSB + SSAB

 Total = error
factor
Statistics for Managers+Using A + factor B + interaction
Microsoft Excel, 4e © 2004
Chap 10-45
Prentice-Hall, Inc.
Examples:
Interaction vs. No Interaction
No interaction:

Interaction is
present:

Factor B Level 1
Factor B Level 3
Factor B Level 2

Factor A Levels
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Mean Response

Mean Response





Factor B Level 1
Factor B Level 2
Factor B Level 3

Factor A Levels

Chap 10-46
Chapter Summary


Described one-way analysis of variance



ANOVA assumptions



F test for difference in c means





The logic of ANOVA

The Tukey-Kramer procedure for multiple comparisons

Described two-way analysis of variance


Examined effects of multiple factors

Examined interaction between factors
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.


Chap 10-47

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Chap10 anova

  • 1. Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 10 Analysis of Variance Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-1
  • 2. Chapter Goals After completing this chapter, you should be able to:  Recognize situations in which to use analysis of variance  Understand different analysis of variance designs  Perform a single-factor hypothesis test and interpret results  Conduct and interpret post-hoc multiple comparisons procedures  Analyze two-factor analysis of variance tests Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-2
  • 3. Chapter Overview Analysis of Variance (ANOVA) One-Way ANOVA F-test TukeyKramer test Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Two-Way ANOVA Interaction Effects Chap 10-3
  • 4. General ANOVA Setting  Investigator controls one or more independent variables    Observe effects on the dependent variable   Called factors (or treatment variables) Each factor contains two or more levels (or groups or categories/classifications) Response to levels of independent variable Experimental design: the plan used to collect the data Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-4
  • 5. Completely Randomized Design  Experimental units (subjects) are assigned randomly to treatments   Only one factor or independent variable   Subjects are assumed homogeneous With two or more treatment levels Analyzed by one-factor analysis of variance (one-way ANOVA) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-5
  • 6. One-Way Analysis of Variance  Evaluate the difference among the means of three or more groups Examples: Accident rates for 1st, 2nd, and 3rd shift Expected mileage for five brands of tires  Assumptions  Populations are normally distributed  Populations have equal variances  Samples are randomly and independently drawn Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-6
  • 7. Hypotheses of One-Way ANOVA  H0 : μ1 = μ2 = μ3 =  = μc    All population means are equal i.e., no treatment effect (no variation in means among groups) H1 : Not all of the population means are the same  At least one population mean is different  i.e., there is a treatment effect Does not mean that all population means are different (some pairs may be the same) Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 10-7 Prentice-Hall, Inc. 
  • 8. One-Factor ANOVA H0 : μ1 = μ2 = μ3 =  = μc H1 : Not all μi are the same All Means are the same: The Null Hypothesis is True (No Treatment Effect) Statistics forμ = μ = μ Managers Using 1 2 3 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-8
  • 9. One-Factor ANOVA (continued) H0 : μ1 = μ2 = μ3 =  = μc H1 : Not all μi are the same At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or Statistics for Managers Using μ = μ2 4e 3 Microsoft1Excel, ≠ μ© 2004 Prentice-Hall, Inc. μ1 ≠ μ2 ≠ μ3 Chap 10-9
  • 10. Partitioning the Variation  Total variation can be split into two parts: SST = SSA + SSW SST = Total Sum of Squares (Total variation) SSA = Sum of Squares Among Groups (Among-group variation) SSW = Sum of Squares Within Groups (Within-group variation) Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 10-10 Prentice-Hall, Inc.
  • 11. Partitioning the Variation (continued) SST = SSA + SSW Total Variation = the aggregate dispersion of the individual data values across the various factor levels (SST) Among-Group Variation = dispersion between the factor sample means (SSA) Within-Group Variation = dispersion that exists among the data values within a particular factor level (SSW) Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 10-11 Prentice-Hall, Inc.
  • 12. Partition of Total Variation Total Variation (SST) =     Variation Due to Factor (SSA) Commonly referred to as: Sum of Squares Between Sum of Squares Among Sum of Squares Explained Among Groups Variation Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Variation Due to Random Sampling (SSW) +     Commonly referred to as: Sum of Squares Within Sum of Squares Error Sum of Squares Unexplained Within Groups Variation Chap 10-12
  • 13. Total Sum of Squares SST = SSA + SSW c nj SST = ∑∑ ( Xij − X) Where: 2 j=1 i =1 SST = Total sum of squares c = number of groups (levels or treatments) nj = number of observations in group j Xij = ith observation from group j Statistics for Managers Using X = © 2004 Microsoft Excel, 4egrand mean (mean of all data values) Chap 10-13 Prentice-Hall, Inc.
  • 14. Total Variation (continued) SST = ( X11 − X)2 + ( X12 − X)2 + ... + ( Xcnc − X)2 Response, X X Statistics for Managers Using 2 Group 1 Group Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Group 3 Chap 10-14
  • 15. Among-Group Variation SST = SSA + SSW c SSA = ∑ n j ( X j − X)2 Where: j =1 SSA = Sum of squares among groups c = number of groups or populations nj = sample size from group j Xj = sample mean from group j Statistics for Managers Using X = © 2004 Microsoft Excel, 4egrand mean (mean of all data values) Chap 10-15 Prentice-Hall, Inc.
  • 16. Among-Group Variation (continued) c SSA = ∑ n j ( X j − X)2 j =1 Variation Due to Differences Among Groups SSA MSA = k −1 Mean Square Among = SSA/degrees of freedom µi µj Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-16
  • 17. Among-Group Variation (continued) SSA = n1 ( x1 − x ) + n 2 ( x 2 − x ) + ... + nc ( x c − x ) 2 2 Response, X X3 X1 Statistics for Managers Using Group 1 Group 2 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. X2 X Group 3 Chap 10-17 2
  • 18. Within-Group Variation SST = SSA + SSW c SSW = ∑ j=1 nj ∑ i =1 ( Xij − X j ) 2 Where: SSW = Sum of squares within groups c = number of groups nj = sample size from group j Xj = sample mean from group j Statistics for Managers Using Microsoft Excel, 4e i© observation in group j Xij = th 2004 Prentice-Hall, Inc. Chap 10-18
  • 19. Within-Group Variation (continued) c SSW = ∑ j=1 nj ∑ i =1 ( Xij − X j )2 Summing the variation within each group and then adding over all groups SSW MSW = n−c Mean Square Within = SSW/degrees of freedom Statistics for Managers Using µi Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-19
  • 20. Within-Group Variation (continued) SSW = ( x11 − X1 ) + ( X12 − X 2 ) + ... + ( Xcnc − Xc ) 2 2 Response, X X1 Statistics for Managers Using Group Microsoft Excel, 4e1 © 2004 Group 2 Prentice-Hall, Inc. X2 X3 Group 3 Chap 10-20 2
  • 21. Obtaining the Mean Squares SSA MSA = c −1 SSW MSW = n−c SST MST = n −1 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-21
  • 22. One-Way ANOVA Table Source of Variation SS df Among Groups SSA c-1 Within Groups SSW n-c Total SST = SSA+SSW MS (Variance) F ratio SSA MSA MSA = c - 1 F = MSW SSW MSW = n-c n-1 c = number of groups for Managers Using of the sample sizes from all groups n = sum df Excel, 4e © 2004 = degrees of freedom Statistics Microsoft Prentice-Hall, Inc. Chap 10-22
  • 23. One-Factor ANOVA F Test Statistic H0: μ1= μ2 = … = μc H1: At least two population means are different  Test statistic MSA F= MSW MSA is mean squares among variances MSW is mean squares within variances  Degrees of freedom  df1 = c – 1 (c = number of groups) Statistics for Managers Using  (n = Microsoft df2 = n –4e © 2004sum of sample sizes from all populations) Excel, c Chap 10-23 Prentice-Hall, Inc.
  • 24. Interpreting One-Factor ANOVA F Statistic  The F statistic is the ratio of the among estimate of variance and the within estimate of variance    The ratio must always be positive df1 = c -1 will typically be small df2 = n - c will typically be large Decision Rule:  Reject H if F > F , 0 U otherwise do not Statistics for Managers Using reject H0 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. α = .05 0 Do not reject H0 Reject H0 FU Chap 10-24
  • 25. One-Factor ANOVA F Test Example You want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the .05 significance level, is there a difference in mean distance? Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204 Chap 10-25
  • 26. One-Factor ANOVA Example: Scatter Diagram Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204 Distance 270 260 250 240 230 • • • • • 220 x1 = 249.2 x 2 = 226.0 x 3 = 205.8 x = 227.0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 210 X1 • • • • • X2 • • • • • 200 X X3 190 1 2 3 Chap 10-26 Club
  • 27. One-Factor ANOVA Example Computations Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204 X1 = 249.2 n1 = 5 X2 = 226.0 n2 = 5 X3 = 205.8 n3 = 5 X = 227.0 n = 15 c=3 SSA = 5 (249.2 – 227)2 + 5 (226 – 227)2 + 5 (205.8 – 227)2 = 4716.4 SSW = (254 – 249.2)2 + (263 – 249.2)2 +…+ (204 – 205.8)2 = 1119.6 MSA = 4716.4 / (3-1) = 2358.2 Statistics 1119.6 / (15-3) =Using MSW = for Managers 93.3 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 2358.2 F= = 25.275 93.3 Chap 10-27
  • 28. One-Factor ANOVA Example Solution Test Statistic: H0: μ1 = μ2 = μ3 H1: μi not all equal MSA 2358.2 F= = = 25.275 MSW 93.3 α = .05 df1= 2 df2 = 12 Critical Value: Decision: Reject H0 at α = 0.05 FU = 3.89 Conclusion: There is evidence that 0 Do not at least one μi differs Statistics for Managers Using Reject H reject H F= Microsoft Excel, 4e © 200425.275 from the rest FU = 3.89 α = .05 0 0 Prentice-Hall, Inc. Chap 10-28
  • 29. ANOVA -- Single Factor: Excel Output EXCEL: tools | data analysis | ANOVA: single factor SUMMARY Groups Count Sum Average Variance Club 1 5 1246 249.2 108.2 Club 2 5 1130 226 77.5 Club 3 5 1029 205.8 94.2 ANOVA Source of Variation SS df MS Between Groups 4716.4 2 2358.2 Within Groups 1119.6 12 F P-value F crit 93.3 Statistics for Managers Using Total 5836.0 14 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 25.275 4.99E-05 3.89 Chap 10-29
  • 30. The Tukey-Kramer Procedure  Tells which population means are significantly different    e.g.: μ1 = μ2 ≠ μ3 Done after rejection of equal means in ANOVA Allows pair-wise comparisons  Compare absolute mean differences with critical range Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. μ1= μ2 μ3 x Chap 10-30
  • 31. Tukey-Kramer Critical Range Critical Range = QU MSW  1 1   +  2  n j n j'    where: QU = Value from Studentized Range Distribution with c and n - c degrees of freedom for the desired level of α (see appendix E.9 table) MSW = Mean Square Within Statistics and Managers Using groups j and j’ ni for nj = Sample sizes from Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-31
  • 32. The Tukey-Kramer Procedure: Example Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204 1. Compute absolute mean differences: x1 − x 2 = 249.2 − 226.0 = 23.2 x1 − x 3 = 249.2 − 205.8 = 43.4 x 2 − x 3 = 226.0 − 205.8 = 20.2 2. Find the QU value from the table in appendix E.9 with c = 3 and (n – c) = (15 – 3) = 12 degrees of freedom for the desired level of α (α = .05 used here): Statistics for Managers Using U Q Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. = 3.77 Chap 10-32
  • 33. The Tukey-Kramer Procedure: Example (continued) 3. Compute Critical Range: Critical Range = QU MSW  1 1   +  = 3.77 93.3  1 + 1  = 16.285   2  n j n j'  2 5 5   4. Compare: 5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. x1 − x 2 = 23.2 x1 − x 3 = 43.4 x 2 − x 3 = 20.2 Chap 10-33
  • 34. Tukey-Kramer in PHStat Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-34
  • 35. Two-Way ANOVA  Examines the effect of  Two factors of interest on the dependent variable   e.g., Percent carbonation and line speed on soft drink bottling process Interaction between the different levels of these two factors  e.g., Does the effect of one particular carbonation level depend on which level the line speed is set? Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-35
  • 36. Two-Way ANOVA (continued)  Assumptions  Populations are normally distributed  Populations have equal variances  Independent random samples are drawn Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-36
  • 37. Two-Way ANOVA Sources of Variation Two Factors of interest: A and B r = number of levels of factor A c = number of levels of factor B n’ = number of replications for each cell n = total number of observations in all cells (n = rcn’) Xijk = value of the kth observation of level i of Statistics for Managers Using factor A and Microsoft Excel, 4e © 2004 level j of factor B Prentice-Hall, Inc. Chap 10-37
  • 38. Two-Way ANOVA Sources of Variation SST = SSA + SSB + SSAB + SSE SSA Factor A Variation SST Total Variation SSB Factor B Variation SSAB n-1 Variation due to interaction between A and B (continued) Degrees of Freedom: r–1 c–1 (r – 1)(c – 1) SSE rc(n’ – 1) Statistics for Managers Using Random variation (Error) Microsoft Excel, 4e © 2004 Chap 10-38 Prentice-Hall, Inc.
  • 39. Two Factor ANOVA Equations Total Variation: r n′ c SST = ∑∑∑ ( Xijk − X) 2 i =1 j=1 k =1 Factor A Variation: r ′∑ ( Xi.. − X)2 SSA = cn i=1 Factor B Variation: c SSB = rn′∑ ( X. j. − X)2 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. j=1 Chap 10-39
  • 40. Two Factor ANOVA Equations (continued) Interaction Variation: r c SSAB = n′∑∑ ( Xij. − Xi.. − X. j. + X)2 i =1 j =1 Sum of Squares Error: r c n′ SSE = ∑∑∑ ( Xijk − Xij. ) i =1 j=1 k =1 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-40 2
  • 41. Two Factor ANOVA Equations r where: X= Xi.. = ∑∑ X j=1 k =1 i=1 j=1 k =1 ijk rcn′ = Grand Mean ijk = Mean of ith level of factor A (i = 1, 2, ..., r) cn′ r X. j. = (continued) n′ ∑∑∑ X n′ c c n′ ∑∑ X i=1 k =1 rn′ ijk = Mean of jth level of factor B (j = 1, 2, ..., c) Xijk Xij. = for = Mean of cell Statistics ∑ Managers Using ij k =1 n′ n′ Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. r = number of levels of factor A c = number of levels of factor B n’ = number of replications in each cell Chap 10-41
  • 42. Mean Square Calculations SSA MSA = Mean square factor A = r −1 SSB MSB = Mean square factor B = c −1 SSAB MSAB = Mean square interaction = (r − 1)(c − 1) SSE MSE = Mean square error = Statistics for Managers Using rc(n'−1) Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-42
  • 43. Two-Way ANOVA: The F Test Statistic H0: μ1.. = μ2.. = μ3.. = • • • H1: Not all μi.. are equal H0: μ.1. = μ.2. = μ.3. = • • • H1: Not all μ.j. are equal H0: the interaction of A and B is equal to zero Statistics for Managers Using H1: interaction of A and B is not Microsoft Excel, 4e © 2004 zero Prentice-Hall, Inc. F Test for Factor A Effect MSA F= MSE Reject H0 if F > FU F Test for Factor B Effect MSB F= MSE Reject H0 if F > FU F Test for Interaction Effect MSAB F= MSE Reject H0 if F > FU Chap 10-43
  • 44. Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Factor A SSA r–1 Factor B SSB c–1 AB (Interaction) SSAB (r – 1)(c – 1) Error SSE rc(n’ – 1) Statistics for Managers Using Total SST n–1 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Mean Squares F Statistic MSA MSA MSE = SSA /(r – 1) MSB = SSB /(c – 1) MSAB = SSAB / (r – 1)(c – 1) MSB MSE MSAB MSE MSE = SSE/rc(n’ – 1) Chap 10-44
  • 45. Features of Two-Way ANOVA F Test  Degrees of freedom always add up  n-1 = rc(n’-1) + (r-1) + (c-1) + (r-1)(c-1)  Total = error + factor A + factor B + interaction  The denominator of the F Test is always the same but the numerator is different  The sums of squares always add up  SST = SSE + SSA + SSB + SSAB  Total = error factor Statistics for Managers+Using A + factor B + interaction Microsoft Excel, 4e © 2004 Chap 10-45 Prentice-Hall, Inc.
  • 46. Examples: Interaction vs. No Interaction No interaction: Interaction is present: Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Mean Response Mean Response   Factor B Level 1 Factor B Level 2 Factor B Level 3 Factor A Levels Chap 10-46
  • 47. Chapter Summary  Described one-way analysis of variance   ANOVA assumptions  F test for difference in c means   The logic of ANOVA The Tukey-Kramer procedure for multiple comparisons Described two-way analysis of variance  Examined effects of multiple factors Examined interaction between factors Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  Chap 10-47