NG BB 34 Analysis of Variance (ANOVA)

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National Guard
Black Belt Training
Module 34

Analysis of Variance
(ANOVA)

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CPI Roadmap – Analyze
8-STEP PROCESS
6. See
1.Validate 2. Identify 3. Set 4. Determine 5. Develop 7. Confirm 8. Standardize
Counter-
the Performance Improvement Root Counter- Results Successful
Measures
Problem Gaps Targets Cause Measures & Process Processes
Through

Define Measure Analyze Improve Control

ACTIVITIES TOOLS
• Value Stream Analysis
• Identify Potential Root Causes • Process Constraint ID
• Reduce List of Potential Root • Takt Time Analysis
Causes • Cause and Effect Analysis
• Brainstorming
• Confirm Root Cause to Output
• 5 Whys
Relationship
• Affinity Diagram
• Estimate Impact of Root Causes • Pareto
on Key Outputs • Cause and Effect Matrix
• FMEA
• Prioritize Root Causes
• Hypothesis Tests
• Complete Analyze Tollgate • ANOVA
• Chi Square
• Simple and Multiple
Regression

Note: Activities and tools vary by project. Lists provided here are not necessarily all-inclusive. UNCLASSIFIED / FOUO

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Learning Objectives
 Gain a conceptual understanding of Analysis of Variance
(ANOVA) and the ANOVA table
 Be able to design and perform a one or two factor
experiment
 Recognize and interpret interactions
 Fully understand the ANOVA model assumptions and how
to validate them
 Understand and apply multiple pair-wise comparisons
 Establish a sound basis on which to learn more complex
experimental designs

Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 3

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Applications for ANOVA
 Administrative – A manager wants to understand how
different attendance policies may affect productivity.

 Transportation – An AAFES manager wants to know if
the average shipping costs are higher between three
distribution centers.


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When To Use ANOVA
Independent Variable (X)
Continuous Categorical
Categorical Continuous
Dependent Variable (Y)

Regression ANOVA

Logistic Chi-Square (2)
Regression Test

The tool depends on the data type. ANOVA is used with an
attribute (categorical) input and a continuous response.

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ANOVA Output
Boxplot of Processing Time by Facility

12

10

Processing Time
8

6

4

2

0
Facility A Facility B Facility C
Facility


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One-Way ANOVA vs. Two-Sample t-Test
Let’s compare sets of data taken on different methods
of processing invoices which vary a Factor A

A two-sample t-test: What if we compare several methods?
Old Method New Method Method 1 Method 2 Method 3 Method 4
13.6 15.3 16.3 17.2 19.4 20.5
14.9 17.6 15.2 17.3 17.9 18.8
15.2 15.6 14.9 16.0 18.1 21.3
13.2 16.2 19.2 20.5 22.8 25.0
19.5 21.7 20.1 22.6 24.7 26.4
13.2 15.1 13.2 14.3 17.3 18.5
15.8 17.2 15.8 17.6 19.7 23.2

Q: Is there a difference Q: Are there any statistically significant
in the average for differences in the averages for the
each method? methods?
Q: If so, which are different from which
others?

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Is There a Difference?

30

25
x
20 x
Response

x
15 x

10

5

Method 1 Method 2 Method 3 Method 4

Factor A
Plotting the averages for the different methods shows a
difference, but is it statistically significant?

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Is There a Difference Now?

30

25
x
20 x
Response

x
15 x

10

5


Factor A
Now that we have a bit more data, does
factor A make a difference? Why or why
not?

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What About Now?

30

25
x
20 x
Response

x
15 x

10

5


Factor A

Now what do you think? Does factor A
make a difference? Why or why not?

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One-Way ANOVA Fundamentals
 One-Way Analysis of Variance (ANOVA) is a statistical method
for comparing the means of more than two levels when a single
factor is varied
 The hypothesis tested is:
Ho: µ1 = µ2 = µ3 = µ4 =…= µk
Ha: At least one µ is different
 Simply speaking, an ANOVA tests whether any of the means are
different. ANOVA does not tell us which ones are different (we‟ll
supplement ANOVA with multiple comparison procedures for
that)


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Sources of Variability
 ANOVA looks at three sources of variability:
 Total – Total variability among all observations
 Between – Variation between subgroup means
(factor)
 Within – Random (chance) variation within each
subgroup (noise, or statistical error)

“Between “Within
Subgroup Subgroup
Variation” Variation”

Total = Between + Within

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Questions Asked by ANOVA

Ho : 1  2  3   4

Ha : At least one k is different

Are any of the 4 population means different?


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Sums of Squares

yj = Mean of Group
70

y = Grand Mean of the
Response

experiment
65

60

yi,j = Individual measurement
55

1 2 3 4
i =represents a data point
Factor/Level within the jth group
j = represents the jth group


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Sums of Squares Formula

g nj g g nj

 (y ij  y )2   n j (y j  y )2   (y ij  y j )2
j 1 i 1 j 1 j 1 i 1

SS(Total)  SS(Factor)  SS(Error)

SS(Total) = Total Sum of Squares of the Experiment (individuals - Grand Mean)
SS(Factor) = Sum of Squares of the Factor (Group Mean - Grand Mean)
SS(Error) = Sum of Squares within the Group (individuals - Group Mean)

By comparing the Sums of Squares, we can tell if the observed
difference is due to a true difference or random chance


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ANOVA Sum of Squares
 We can separate the total sum of squares into two
components (“within” and “between”).
 If the factor we are interested in has little or no effect on
the average response, then these two estimates (within
and between) should be fairly equal and we will conclude
all subgroups could have come from one larger population.
 As these two estimates (within and between) become
significantly different, we will attribute this difference as
originating from a difference in subgroup means.
 Minitab will calculate this!


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Null and Alternate Hypothesis

Ho : 1   2  3   4
Ha : At least one  k is different

To determine whether we can reject the null hypothesis, or
not, we must calculate the Test Statistic (F-ratio) using the
Analysis of Variance table as described on the following slide


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Developing the ANOVA Table

SOURCE SS df MS (=SS/df) F {=MS(Factor)/MS(Error)}

BETWEEN SS(Factor) a-1 SS (factor)/df factor MS(Factor) / MS(Error)

 n  1
a

WITHIN SS(Error) j SS(Error) / df error
j 1

 a 
TOTAL SS(Total)   nj   1
 
 j 1 
i = represents a data point within
the jth group (factor level)
j = represents the jth group (factor
level)
a = total # of groups (factor levels)

Why is Source “Within” called the Error or Noise?
In practical terms, what is the F-ratio telling us?
What do you think large F-ratios mean?


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ANOVA Example: Invoice Processing CT
 A Six Sigma team wants to compare the invoice
processing times at three different facilities.
 If one facility is better than the others, they can look
for opportunities to implement the best practice
across the organization.
 Open the Minitab worksheet:
Invoice ANOVA.mtw.
 The data shows invoice processing cycle times at
Facility A, B, and C.


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Is One Facility Better Than The Others?

 How might we determine
which, if any, of the three
facilities has a shorter cycle
time?
 What other concerns might you
have about this experiment?
Minitab Tip: Minitab usually likes data in
columns (List the numerical response data
in one single column, and the factor you
want to investigate beside it). ANOVA is
one tool that breaks that rule – ANOVA
(unstacked) can analyze unstacked data.


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One-Way ANOVA in Minitab
Select
Stat>ANOVA>One-Way


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One-Way ANOVA in Minitab

Enter the Response and
the Factor

Select Graphs to go to
the Graphs dialog box


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ANOVA-Boxplots
Select >
Boxplots of data

Let‟s look at some
Boxplots
while we are here


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ANOVA – Multiple Comparisons
Select Comparisons>Tukey’s
We will get into more
detail on these later
in this session


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Boxplots – What Do You Think?
Boxplot of Processing Time

12

10
Processing Time

8

6

4

2

0
Facility A Facility B Facility C
Facility

What would you conclude? Which facility has the best cycle time?


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ANOVA Table – Session Window

What would we conclude from the ANOVA table?

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Pairwise Comparisons – Tukey
This is the output for the Tukey Test
Pairwise Comparisons
are simply confidence
intervals for the
difference between the
tabulated pairs,
with alpha being
determined by the
individual error rate

Tukey pairwise
comparisons
answer the question
“Which ones are
Statistically Significantly
Different?”

How do we interpret these paired tests?

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Tukey Pairwise Interpretation
Since we have 3 Facilities, there are 3 Two-Way Comparisons in this analysis

First: We subtract the mean for
cycle time for Facility A from the
means for Facilities B & C. Minitab
then calculates confidence
intervals around these differences.
If the interval contains zero,
then there is Not a
Statistically Significant
Difference between that pair.
Here the intervals for Facility B
and C do Not contain zero so
there is a Statistically Significant
Difference between Facility A and
the other two Facilities.

Facility A is Statistically Significantly Different from Facilities B & C

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Tukey Pairwise Interpretation (Cont.)

Second: We subtract the mean
for cycle time for Facility B from
the mean for Facility C. Minitab
then calculates the confidence
interval around that difference.
If the interval contains zero,
then there is Not a
Difference between the pair.
Here the interval Does Not
contain zero so there is a
Difference between B and C.

Facility B is Statistically Significantly Different from Facility C


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Example: Pay for Performance
 In this study, the number of 411 calls processed in a
given day was measured under one of five different
pay-for-performance incentive plans
 The null hypothesis would be that the different pay
plans would have no significant effect on productivity
levels.
 Open the data set:
One Way ANOVA Example.mtw.


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Example Data – Pay for Performance

 We want to determine if
there is a significant
difference in the level of
production between the
different plans.
 What concerns might you
have about this experiment?


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One Way ANOVA in Minitab
Select Stat>ANOVA>One-way


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Boxplots in Minitab
Let‟s start with Graphs > Boxplots


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Production by Plan Boxplots
Boxplot of production
1250

1200

1150
If you
production

were the
1100
manager,
what would
1050
you do?
1000
A B C D E
plan

Does the incentive plan seem to matter?

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ANOVA Table – Pay for Performance

Do we have any evidence that the incentive plan matters?
Who can explain the ANOVA table to the class?

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Tukey Pairwise Comparisons – Pay for Perf

Which plans are different?

The ANOVA Table answers
the question “Are all the
subgroup averages the same?”
Tukey Pairwise Comparisons
answer the question “Which
ones are different?”


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Tukey Pairwise Comparisons – Pay for Perf

Which pairs are
different?

Which intervals
do not contain
zero?

Is it possible for
the ANOVA Table and
the Tukey
pairs to conflict?


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ANOVA – Main Effects Plot
Select Stat>ANOVA>Main Effects Plot


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ANOVA – Main Effects Plot (cont.)
Enter the Responses and Factors, then click on OK
to go to Main Effects Plot


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Graphical Analysis – Main Effects Plots
Main Effects Plot for production
Data Means
1200

1175

1150
Mean

1125

1100

1075

1050
A B C D E
plan

What does the plot tell us?

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Graphical Analysis – Interval Plots
Select Stat>ANOVA>Interval Plot

What do the Interval plots tell us about our experiment?

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ANOVA – Interval Plots
First select With Groups since we
have five groups, and then click on
OK to go to the next dialog box

Then enter Graph variable
And Categorical variable and
click on OK to go to the
Interval Plot

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ANOVA – Interval Plots
Another graphical way to present your findings !

Interval Plot of production
95% CI for the Mean

1200

1150
production

1100

1050

A B C D E
plan

How might the interval plot have looked differently if the
confidence interval level (percent) were changed?

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ANOVA Table – A Quick Quiz

Source DF SS MS F p

Factor 3 ? 1542.0 ? 0.000

Error ? 2,242 ?
Total 23 6,868

Could you complete the above ANOVA table?


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Exercise: Degrees of Freedom
Step One:
 Let‟s go around the room and have everyone give a number
which we will flipchart
 The numbers need to add up to 100
Step Two:
 How many degrees of freedom did I have?
 How many would I have if, in addition to adding to 100, I
added one more mathematical requirement?


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What Are “Degrees of Freedom?”

degrees of freedom  currency in statistics

We earn a degree of freedom for
every data point we collect
We spend a degree of freedom for
each parameter we estimate


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What Are “Degrees of Freedom”?
 In ANOVA, the degrees of freedom are as follows:
 dftotal = N-1 = # of observations - 1
 dffactor = L-1 = # of levels - 1
 dfinteraction = dffactorA X dffactorB
 dferror = dftotal - dfeverything else

Let‟s say we are testing a factor that has five levels and we collect seven data points
at each factor level…
How many observations would we have? 5 levels x 7 observations per level =35 total
observations
How many total degrees of freedom would we have? 35 - 1 = 34
How many degrees of freedom to estimate the factor effect? 5 levels - 1 = 4
How many degrees of freedom do we have to estimate error? 34 total - 4 factor =
30 degrees of freedom


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Key ANOVA Assumptions
 Model errors are assumed to be normally distributed
with a mean of zero, and are to be randomly
distributed (no patterns).
 The samples are assumed to come from normally
distributed populations.
We can investigate these assumptions with residual plots.

 The variance is assumed constant for all factor levels.
We can investigate this assumption with a statistical test
for equal variances.


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ANOVA – Residual Analysis
 Residual plots should show no pattern relative to any
factor, including the fitted response.
 Residuals vs. the fitted response should have an
average of about zero.
 Residuals should be fairly normally distributed.

Practical Note: Moderate departures from normality of
the residuals are of little concern. We always want to
check the residuals, though, because they are an
opportunity to learn more about the data.


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Constant Variance Assumption
 There are two tests we can use to test the assumption of
constant (equal) variance:
 Bartlett's Test is frequently used to test this hypothesis for
data that is normally distributed.
 Levene's Test can be used when the data is not normally
distributed.

Note: Minitab will perform this analysis for us with the procedure
called „Test for Equal Variances’

Practical Note: Balanced designs (consistent sample size for all
factor levels) are very robust to the constant variance assumption.
Still, make a habit of checking for constant variances. It is an
opportunity to learn if factor levels have different amounts of
variability, which is useful information.

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Test for Equal Variances
Select: Stat>ANOVA>Test for Equal Variances

Then place production in response
&
Plan in factor
Press OK


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Constant Variance Assumption (Cont.).

Test for Equal Variances for production

Bartlett's Test
A Both Bartlett’s Test
Test Statistic 4.95
P-Value 0.292
and Levene’s Test
Levene's Test are run on the data
B
Test Statistic 0.46 and are reported
P-Value 0.764 at the same time.
plan

C

D

E

0 20 40 60 80 100 120
95% Bonferroni Confidence Intervals for StDevs


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Model Adequacy – More Good News
 By selecting an adequate sample size and randomly
conducting the trials, your experiment should be robust to
the normality assumption (remember the Central Limit
Theorem)
 Although there are certain assumptions that need to be
verified, there are precautions you can take when
designing and conducting your experiment to safeguard
against some common mistakes
 Protect the integrity of your experiment right from the start
 Often, problems can be easily corrected by collecting a
larger sample size of data


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One-Way ANOVA Wrap-Up
 We will formally address the checking of model assumptions
during the Two-Way ANOVA analysis.
 Re-capping One-Way ANOVA methodology:
1. Select a sound sample size and factor levels
2. Randomly conduct your trials and collect the data
3. Conduct your ANOVA analysis
4. Follow up with pairwise comparisons, if indicated
5. Examine the residuals, variance and normality assumptions
6. Generate main effects plots, interval plots, etc.
7. Draw conclusions

 This short procedure is not meant to be an exhaustive
methodology.
 What other items would you add?

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Individual Exercise
 A market research firm for the Defense Commissary
Agency (DECA) believed that the sales of a given
product in units was dependent upon its placement
 Items placed at eye level tended to have higher sales
than items placed near the floor
 Using the data in the Minitab file Sales vs Product
Placement.mtw, draw some conclusions about the
relationship between sales and product placement
 You will have 10 minutes to complete this exercise


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National Guard
Black Belt Training
Two-Way ANOVA

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One-Way vs. Two-Way ANOVA
 In One-Way ANOVA, we looked at how different
levels of a single factor impacted a response variable.
 In Two-Way ANOVA, we will examine how different
levels of two factors and their interaction impact a
response variable.


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Now We Can Consider Two Factors

A
Low High
 At a high level, a Two-
69 80 Way ANOVA (two
Low 65 82 factor) can be viewed
as a two-factor
B experiment
59  The factors can take
42
High on many levels; you
44 63
are not limited to two
levels for each


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Two-Way ANOVA
 Experiments often involve the study of more than one
factor.
 Factorial designs are very efficient methods to
investigate various combinations of levels of the
factors.
 These designs evaluate the effect on the response
caused by different levels of factors and their
interaction.
 As in the case of One-Way ANOVA, we will be building
a model and verifying some assumptions.


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Two-Factor Factorial Design
 The general two-factor factorial experiment takes the following form. As
in the case of a one-factor ANOVA, randomizing the experiment is
important:
Factor B
1 2 ... b
1
Factor A 2
.
a

 In this experiment, Factor A has levels ranging from 1 to a, Factor B
has levels ranging from 1 to b, while the replications have replicates 1
to n
A balanced design is always preferred (same number of observations
for each treatment) because it buffers against any inequality of
variances Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 60

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Two-Factor Factorial Design
 Just as in the One-Factor ANOVA, the total variability can
be segmented into its component sum of squares:
SST= SSA+ SSB + SSAB + SSe
Given:
 SST is the total sum of squares,
 SSA is the sum of squares from factor A,
 SSB is the sum of squares from factor B,
 SSAB is the sum of squares due to the interaction between
A&B
 SSe is the sum of squares from error


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Degrees of Freedom – Two Factor ANOVA
 Each Sum of Squares has associated degrees of
freedom:
Source Sum of Squares Degrees of Freedom Mean Square F0
Factor A SSA a-1 SS A MS A
MS A  F0 
a 1 MS E

Factor B SSB b-1 SS MS B
MS B  B F0 
b 1 MS E

Interaction SSAB (a - 1)(b - 1) SS AB MS AB
MS AB  F0 
(a  1)(b  1) MS E

SS E
Error SSE ab(n - 1) MS E 
ab(n  1)

Total SST abn - 1


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Marketing Example
 AAFES is trying to introduce their own brand of candy and
wants to find out which product packaging or regions will
yield the highest sales.
 They sold their candy in either a plain brown bag, a colorful
bag or a clear plastic bag at the cash register (point of sale).
 AAFES had stores in regions which varied economically and
the information was captured to see if different regions
affect sales.
 The data set is: Two Way ANOVA Marketing.mtw.
 As a class we will analyze the data.


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Marketing Example Data
The team collected sales
data for three different
packaging styles in three
geographic regions.
They are interested in
knowing if the packaging
affects sales in any of the
regions.


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Marketing Example Data (Cont.)
Selection Stat>ANOVA>Two-Way


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Marketing Example Data (Cont.)
Check both boxes for
Display means
Enter the Response and
Factors – the choice of
row vs. column for
factors is unimportant
Click on OK to get the
analysis in your
Session Window


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Marketing Example – ANOVA Table

What is significant?
Who wants to give it a try?


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Generating a Main Effects Plot
Select Stat>ANOVA>Main Effects Plot

Let‟s look at a Main Effects Plot

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Selecting Main Effects
Fill in Responses and Factors, then click on OK to get Plots


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Main Effects Plot
Main Effects Plot for sales
Data Means
region packaging
800

750

700
Mean

650

600

550

1 2 3 color plain point of sale

Which factor has the stronger effect?

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Generating an Interaction Plot
Select Stat>ANOVA>Interactions plot


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Selecting Interactions
Enter the Responses and Factors, then click on OK to get Plot


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Interactions Plot
Interaction Plot for sales
Data Means
region
1100
1
2
1000 3

900

800
Mean

700

600

500

400

color plain point of sale
packaging

How do we interpret Interaction Plots?

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Residual Analysis
Select Store residuals and Store fits, then select Graphs
and select Four in one (under Residual Plots) and click on OK and OK
again so we can do some model confirmation


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Residual Four Pack
Residual Plots for sales
Normal Probability Plot Versus Fits
99.9
N 270
200
99 AD 0.409
90
P-Value 0.343 100

Residual
Percent

What are we
50 0

10

1
-100
looking for?
0.1 -200
-200 -100 0 100 200 400 600 800 1000 1200
Residual Fitted Value
What are
Histogram Versus Order
200
the
40
100
assumptions
Frequency

we want to
Residual

30
0
20
10 -100 verify?
0 -200
-120 -60 0 60 120 1 20 40 60 80 00 20 40 60 80 00 20 40 60
1 1 1 1 1 2 2 2 2
Residual
Observation Order


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Select Stat>Basic Statistics>2 Variances

Here is another option for checking for Equal Variances

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We can only check one factor at a Now go back and repeat the analysis.
time in this dialog box. First do Sales This time do Sales by Packaging.
by Region. Then click on OK to get Then click on OK to get this comparison
this comparison of variances. of variances.


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Test for Equal Variances for sales

Bartlett's Test
Test Statistic 42.13
1 P-Value 0.000
Lev ene's Test
P-Value 0.000
region

2

3

100 150 200 250 300

Do the factor levels have equal variances?

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Test for Equal Variances for sales

Bartlett's Test
color P-Value 0.000
Lev ene's Test
P-Value 0.000
packaging

plain

point of sale

100 150 200 250 300 350

What about the Variances for these factor levels?

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ANOVA Conclusions
 Did our model assumptions hold up?
 How comfortable are we with the conclusions drawn?
 Questions?


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Individual Exercise - Employee Productivity
A manager wanted to increase productivity due to the
organization‟s slim margins.

 The hope was to increase productivity by 8%-10%
and reduce payroll through attrition.

 The manager piloted a program across three
departments that involved 99 employees.

 The manager was evaluating the effect on productivity
of a four day work week, flextime, and the status quo.

 Using the data collected in Two Way ANOVA.mtw,
help the manager interpret the results.

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Takeaways

 Conceptual ANOVA  Main Effects Plots

 Sums of Squares  Interactions Plot

 ANOVA Table  Two-Factor ANOVA

 ANOVA Boxplots,  Two-Factor Model
Multiple Comparisons
 Residual Analysis
 Tukey Pairwise
Comparisons  Test for Equal
Variances


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What other comments or questions
do you have?

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National Guard
Black Belt Training
APPENDIX

UNCLASSIFIED / FOUO

UNCLASSIFIED / FOUO

UNCLASSIFIED / FOUO

Analysis of Variance Using Minitab
Another look at the ANOVA table

1093 = 33.1

UNCLASSIFIED / FOUO

Reading the ANOVA Table

One-Way Analysis of Variance
If P is small (say, less
Analysis of Variance on Response than 5%), then we
Source DF SS MS F p conclude that at least
one subgroup mean is
Factor 3 4,626 1542.0 13.76 0.000
different. In this case,
Error 20 2,242 112.1 we reject the hypothesis
Total 23 6,868 that all the subgroup
means are equal

 12   22   32   42 The F-test is close to 1.00
 2
Pooled  when subgroup means are
4
(*Only if subgroup sizes are equal)
similar. In this case, This
F-test ratio is much greater
than 1.00, hence subgroup
means are NOT similar.


UNCLASSIFIED / FOUO

Multiple Comparisons
 Tukey’s – Family error rate controlled
 Fisher’s – Individual error rate controlled
 Dunnett’s – Compares all results to a control group
 Hsu’s MCB – Compares all results to a known best group
 Which one do you use? In general, Tukey’s is recommended
because it‟s „tighter‟. In other words, you will be less likely to
find a difference between means (less statistical power), but you
will be protected against a “false positive”, especially when there
are a lot of groups.
 Tukey‟s makes each test at a higher level of significance (a‟ > .05) and
holds the family error rate to a = .05
 Fisher‟s makes all tests at the specified significance level (usually a = .05)
and reports the “family” error rate, a‟


UNCLASSIFIED / FOUO

What the F-Distribution Explains
Here we see the F-Distribution and the F-test dynamics illustrated. This is the distribution of F-ratios
that would occur if all methods produced the same results. Notice that the F-ratio we observed from
the experiment is way out in the tail of the distribution. For this distribution, 3 is the d.f. for the
numerator and 20 is the d.f. for the denominator.

F - D is t r i b u t io n f o r 3 a n d 2 0 d e g r e e s o f F r e e d o m

0 .7
10% Point
0 .6

0 .5
5% Point Observed Point
0 .4
Prob

0 .3

0 .2
1% Point
0 .1

0 .0

0 2 4 6 8 10 12 14

F - V a lu e


UNCLASSIFIED / FOUO

F Distribution:
Probability Distribution Function (PDF) Plots

1.0
N1* N2
0.9
1, 1 d.f.
0.8 3, 3 d.f.
0.7 5, 8 d.f.
0.6 8, 8 d.f
0.5
* N1 refers to the d.f. in
0.4 the numerator
0.3
0.2
0.1
0.0
0 1 2 3
F


NG BB 34 Analysis of Variance (ANOVA)

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