Analysis of Variance

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-1

 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-analysis of variance pairwise
comparisons procedures

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-2

 General ANOVA Setting

Investigator controls one or more independent

variables

Called factors (or treatment variables)

Each factor contains two or more levels (or
categories/classifications)

Observe effects on dependent variable

Response to levels of independent variable

Experimental design: the plan used to test
hypothesis

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-3

 One-Way Analysis of Variance

Evaluate the difference among the means of

three or more populations
Examples: ● Accidentrates 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

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-4

 Completely Randomized Design

Experimental units (subjects) are assigned

randomly to treatments

Only one factor or independent variable

With two or more treatment levels

Analyzed by

One-factor analysis of variance (one-way ANOVA)

Called a Balanced Design if all factor levels
have equal sample size

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-5

 Hypotheses of One-Way
ANOVA

H0 : µ1 = µ2 = µ3 = L = µk

All population means are equal

i.e., no treatment effect (no variation in means among
groups)

HA : 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)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-6

 One-Factor ANOVA
H0 : µ1 = µ2 = µ3 = L = µk
HA : Not all µi are the same

All Means are the same:

The Null Hypothesis is True
(No Treatment Effect)

µ1 = µ2 = µ3
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-7
 One-Factor ANOVA
(continued)
H0 : µ1 = µ2 = µ3 = L = µk
HA : Not all µi are the same
At least one mean is different:
The Null Hypothesis is NOT true
(Treatment Effect is present)

or

µ1 = µ2 ≠ µ3 µ1 ≠ µ2 ≠ µ3
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-8
 Partitioning the Variation

Total variation can be split into two parts:

SST = SSB + SSW

SST = Total Sum of Squares

SSB = Sum of Squares Between
SSW = Sum of Squares Within

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-9

 Partitioning the Variation
(continued)

SST = SSB + SSW

Total Variation (SST) = the aggregate dispersion of the

individual data values across the various factor levels

Between-Sample Variation (SSB) = dispersion among the

factor sample means

Within-Sample Variation (SSW) = dispersion that exists

among the data values within a particular factor level

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-10

 Partition of Total Variation

Total Variation (SST)

Variation Due to Variation Due to Random

= Factor (SSB) + Sampling (SSW)

Commonly referred to as: Commonly referred to as:

 Sum of Squares Between  Sum of Squares Within
 Sum of Squares Among  Sum of Squares Error
 Sum of Squares Explained  Sum of Squares Unexplained
 Among Groups Variation  Within Groups Variation

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-11

 Total Sum of Squares

SST = SSB + SSW

k ni
SST = ∑∑ ( x ij − x )2
i=1 j =1
Where:
SST = Total sum of squares
k = number of populations (levels or treatments)
ni = sample size from population i
xij = jth measurement from population i
x = grand mean (mean of all data values)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-12
 Total Variation
(continued)

SST = ( x11 − x )2 + ( x12 − x )2 + ... + ( x knk − x )2

Response, X

Group 1 Group 2 Group 3

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-13
 Sum of Squares Between

SST = SSB + SSW

k
SSB = ∑ ni ( x i − x ) 2

i=1
i=
Where:
SSB = Sum of squares between
k = number of populations
ni = sample size from population i
xi = sample mean from population i
x = grand mean (mean of all data values)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-14
 Between-Group Variation

k
SSB = ∑ ni ( x i − x ) 2

i=1

Variation Due to SSB

Differences Among Groups
MSB =
k −1
Mean Square Between =
SSB/degrees of freedom

µi µj
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-15
 Between-Group Variation
(continued)

2 2 2
SSB = n1 ( x1 − x ) + n2 ( x 2 − x ) + ... + nk ( x k − x )

Response, X

x3
x2
x
x1

Group 1 Group 2 Group 3

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-16
 Sum of Squares Within

SST = SSB + SSW

k nj

SSW = ∑ ∑ ( x ij − x i ) 2

i =1 j=1
Where:
SSW = Sum of squares within
k = number of populations
ni = sample size from population i
xi = sample mean from population i
xij = jth measurement from population i
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-17
 Within-Group Variation

k nj

SSW = ∑ ∑ ( x ij − x i ) 2

i =1 j=1
SSW
Summing the variation
within each group and then
MSW =
adding over all groups
nT − k
Mean Square Within =
SSW/degrees of freedom

µi
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-18
 Within-Group Variation
(continued)

SSW = ( x11 − x1 )2 + ( x12 − x 2 )2 + ... + ( x knk − x k )2

Response, X

x3
x2
x1

Group 1 Group 2 Group 3

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-19
 One-Way ANOVA Table

Source of SS df MS F ratio
Variation
Between SSB MSB
SSB k-1 MSB =
Samples k - 1 F = MSW
Within SSW
SSW nT - k MSW =
Samples nT - k
SST =
Total nT - 1
SSB+SSW
k = number of populations
nT = sum of the sample sizes from all populations
df = degrees of freedom
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-20
 One-Factor ANOVA
F Test Statistic
H0: µ1= µ2 = … = µ k
HA: At least two population means are different

Test statistic MSB

F=
MSW
MSB is mean squares between variances
MSW is mean squares within variances

Degrees of freedom

df1 = k – 1 (k = number of populations)

df2 = nT – k (nT = sum of sample sizes from all populations)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-21
 Interpreting One-Factor ANOVA
F Statistic

The F statistic is the ratio of the between
estimate of variance and the within estimate
of variance

The ratio must always be positive

df1 = k -1 will typically be small

df2 = nT - k will typically be large

The ratio should be close to 1 if

H0: µ1= µ2 = … = µk is true

The ratio will be larger than 1 if

H0: µ1= µ2 = … = µk is false
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-22
 One-Factor ANOVA
F Test Example

You want to see if three Club 1 Club 2 Club 3

different golf clubs yield 254 234 200
different distances. You 263 218 222
randomly select five 241 235 197
measurements from trials on 237 227 206
an automated driving 251 216 204
machine for each club. At the
.05 significance level, is there
a difference in mean
distance?

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-23

 One-Factor ANOVA Example:
Scatter Diagram
Distance
Club 1 Club 2 Club 3 270
254 234 200 260 •
263 218 222 ••
250 x1
241 235 197 240 •
237 227 206 • ••
230
251 216 204
220
• x2 • x
••
210
x1 = 249.2 x 2 = 226.0 x 3 = 205.8
•• x3
200 ••
x = 227.0 190

1 2 3
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-24
Club
 One-Factor ANOVA Example
Computations
Club 1 Club 2 Club 3 x1 = 249.2 n1 = 5
254 234 200 x2 = 226.0 n2 = 5
263 218 222 x3 = 205.8 n3 = 5
241 235 197
nT = 15
237 227 206 x = 227.0
251 216 204 k=3

SSB = 5 [ (249.2 – 227)2 + (226 – 227)2 + (205.8 – 227)2 ] = 4716.4

SSW = (254 – 249.2)2 + (263 – 249.2)2 +…+ (204 – 205.8)2 = 1119.6

MSB = 4716.4 / (3-1) = 2358.2 2358.2

F= = 25.275
MSW = 1119.6 / (15-3) = 93.3 93.3
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-25
 One-Factor ANOVA Example
Solution
H0: µ1 = µ2 = µ3 Test Statistic:
HA: µi not all equal
MSB 2358.2
α = .05 F= = = 25.275
df1= 2 df2 = 12 MSW 93.3

Critical Decision:
Value:
Reject H0 at α = 0.05
Fα = 3.885
α = .05 Conclusion:
There is evidence that
0 Do not Reject H0 at least one µi differs
reject H0 F = 25.275
F.05 = 3.885 from the rest
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-26
 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

µ1= µ2 µ3 x
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-27
 Tukey-Kramer Critical Range

MSW  1 1 
Critical Range = qα +
2  ni n j 

where:
qα = Value from standardized range table
with k and nT - k degrees of freedom for
the desired level of α
MSW = Mean Square Within
ni and nj = Sample sizes from populations (levels) i and j

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-28

 The Tukey-Kramer Procedure:
Example
1. Compute absolute mean
Club 1 Club 2 Club 3 differences:
254 234 200
263 218 222 x1 − x 2 = 249.2 − 226.0 = 23.2
241 235 197 x1 − x 3 = 249.2 − 205.8 = 43.4
237 227 206
251 216 204 x 2 − x 3 = 226.0 − 205.8 = 20.2

2. Find the q value from the table in appendix J with

k and nT - k degrees of freedom for
the desired level of α

qα = 3.77
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-29
 The Tukey-Kramer Procedure:
Example
3. Compute Critical Range:
MSW  1 1  93.3  1 1 
Critical Range = qα

+

= 3.77  +  = 16.285
2  ni n j  2 5 5

4. Compare:
5. All of the absolute mean differences
x1 − x 2 = 23.2
are greater than critical range.
Therefore there is a significant x1 − x 3 = 43.4
difference between each pair of
means at 5% level of significance. x 2 − x 3 = 20.2

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 12-30