NG BB 34 Analysis of Variance (ANOVA)

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National GuardBlack Belt Training

Module 34

Analysis of Variance (ANOVA)

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CPI Roadmap – Analyze

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

TOOLS•Value Stream Analysis•Process Constraint ID •Takt Time Analysis•Cause and Effect Analysis •Brainstorming•5 Whys•Affinity Diagram•Pareto •Cause and Effect Matrix •FMEA•Hypothesis Tests•ANOVA•Chi Square •Simple and Multiple Regression

ACTIVITIES

• Identify Potential Root Causes

• Reduce List of Potential Root Causes

• Confirm Root Cause to Output Relationship

• Estimate Impact of Root Causes on Key Outputs

• Prioritize Root Causes

• Complete Analyze Tollgate

1.Validate the

Problem

4. Determine Root

Cause

3. Set Improvement

Targets

5. Develop Counter-

Measures

6. See Counter-MeasuresThrough

2. IdentifyPerformance

Gaps

7. Confirm Results

& Process

8. StandardizeSuccessfulProcesses

Define Measure Analyze ControlImprove

8-STEP PROCESS

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3Analysis of Variance (ANOVA)

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

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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

Continuous CategoricalC

ate

go

ric

al C

on

tin

uo

us

Independent Variable (X)D

ep

en

de

nt

Va

ria

ble

(Y

)

Regression ANOVA

Logistic

Regression

Chi-Square (2)

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

Facility

Pro

ce

ssin

g T

ime

Facility CFacility BFacility A

12

10

8

6

4

2

0

Boxplot of Processing Time by Facility

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One-Way ANOVA vs. Two-Sample t-Test

Old Method New Method

13.6 15.3

14.9 17.6

15.2 15.6

13.2 16.2

19.5 21.7

13.2 15.1

15.8 17.2

A two-sample t-test: What if we compare several methods?

Method 1 Method 2 Method 3 Method 4

16.3 17.2 19.4 20.5

15.2 17.3 17.9 18.8

14.9 16.0 18.1 21.3

19.2 20.5 22.8 25.0

20.1 22.6 24.7 26.4

13.2 14.3 17.3 18.5

15.8 17.6 19.7 23.2

Q: Is there a difference in the average for each method?

Q: Are there any statistically significant differences in the averages for the methods?

Q: If so, which are different from which others?

Let’s compare sets of data taken on different methods of processing invoices which vary a Factor A

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Is There a Difference?R

es

po

ns

e

5

10

15

20

25


x

Factor A

xx

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

x

30

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Is There a Difference Now?

Now that we have a bit more data, does factor A make a difference? Why or why

not?

x

Re

sp

on

se

5

10

15

20

25


Factor A

30

xx

x

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What About Now?

x

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

Re

sp

on

se

5

10

15

20

25


Factor A

30

xx

x

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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)

Total = Between + Within

“Between Subgroup Variation”

“Within Subgroup Variation”

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Questions Asked by ANOVA

Are any of the 4 population means different?

4321o :H

different is oneleast At :H ka

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4321

7

6

6

5

Factor/Level

Resp

on

se

0

5

0

5

yi,j = Individual measurement

y = Grand Mean of theexperiment

yj = Mean of Group

i =represents a data point within the jth group

j = represents the jth group

Sums of Squares

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Sums of Squares Formula

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)

SS(Error) SS(Factor) SS(Total)

1 1 1 1

2

1

22 )()()(

g

j

n

i

g

j

n

ijij

g

jjjij

j j

yyyynyy

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

different is oneleast At :Ha

:Ho

k

4321

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

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?

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

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

WITHIN SS(Error) SS(Error) / df error

TOTAL SS(Total) 11

a

j

jn

a

j

jn1

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)

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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

Let‟s look at some Boxplots

while we are here

Select > Boxplots of data

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ANOVA – Multiple Comparisons

Select Comparisons>Tukey’sWe will get into moredetail on these later

in this session

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Facility CFacility BFacility A

12

10

8

6

4

2

0

Facility

Pro

ce

ssin

g T

ime

Boxplot of Processing Time

Boxplots – What Do You Think?

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

How do we interpret these paired tests?

Pairwise Comparisons are simply confidence intervals for the difference between the tabulated pairs,with alpha being determined by the individual error rate

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

This is the output for the Tukey Test

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Tukey Pairwise Interpretation

Facility A is Statistically Significantly Different from Facilities B & C

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.

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

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Tukey Pairwise Interpretation (Cont.)

Facility B is Statistically Significantly Different from Facility C

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 Statistically Significant Difference between the pair. Here the interval Does Not contain zero so there is a Statistically Significant Difference between B and 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

Does the incentive plan seem to matter?

If you were themanager, what would you do?

EDCBA

1250

1200

1150

1100

1050

1000

plan

pro

du

cti

on

Boxplot of production

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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 Comparisonsanswer the question “Whichones 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 Tukeypairs 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 OKto go to Main Effects Plot

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What does the plot tell us?

Graphical Analysis – Main Effects Plots

EDCBA

1200

1175

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1125

1100

1075

1050

plan

Me

an

Main Effects Plot for productionData Means

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Graphical Analysis – Interval Plots

What do the Interval plots tell us about our experiment?

Select Stat>ANOVA>Interval Plot

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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 variableAnd 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 !

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

EDCBA

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Interval Plot of production95% CI for the Mean

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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?”

statisticsincurrencyfreedomofdegrees

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”?

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

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

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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.

The variance is assumed constant for all factor levels.

We can investigate these assumptions with residual plots.

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 factorPress OK

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Constant Variance Assumption (Cont.).

Both Bartlett’s Testand Levene’s Testare run on the data and are reported at the same time.

pla

n

95% Bonferroni Confidence Intervals for StDevs

E

D

C

B

A

120100806040200

Bartlett's Test

0.764

Test Statistic 4.95

P-Value 0.292

Levene's Test

Test Statistic 0.46

P-Value

Test for Equal Variances for production

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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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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

At a high level, a Two-Way ANOVA (two factor) can be viewed as a two-factor experiment

The factors can take on many levels; you are not limited to two levels for each

Low High

Low

High

A

B

69

65 82

80

63

59

44

42

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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:

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

Factor B

1 2 . . . b

1

Factor A 2

.

a

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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

1

a

SSMS A

AE

A

MS

MSF 0

Factor B SSB b - 1

1

b

SSMS B

BE

B

MS

MSF 0

Interaction SSAB (a - 1)(b - 1))1)(1(

ba

SSMS AB

ABE

AB

MS

MSF 0

Error SSE ab(n - 1))1(

nab

SSMS E

E

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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Check both boxes for Display means

Enter the Response andFactors – the choice of row vs. column for factors is unimportant

Click on OK to get the analysis in your Session Window

Marketing Example Data (Cont.)

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

Marketing Example – ANOVA Table

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Generating a Main Effects Plot

Let‟s look at a Main Effects Plot

Select Stat>ANOVA>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

Which factor has the stronger effect?

321

800

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point of saleplaincolor

region

Me

an

packaging

Main Effects Plot for salesData Means

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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

How do we interpret Interaction Plots?

point of saleplaincolor

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400

packaging

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an

1

2

3

region

Interaction Plot for salesData Means

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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

What are we looking for?

What are theassumptions we want to verify?

2001000-100-200

99.9

99

90

50

10

1

0.1

Residual

Pe

rce

nt

N 270

AD 0.409

P-Value 0.343

12001000800600400

200

100

0

-100

-200

Fitted ValueR

esid

ua

l

120600-60-120

40

30

20

10

0

Residual

Fre

qu

en

cy

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200

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0

-100

-200

Observation Order

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ua

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Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for sales

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Here is another option for checking for Equal Variances

Select Stat>Basic Statistics>2 Variances

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We can only check one factor at a time in this dialog box. First do Salesby Region. Then click on OK to get this comparison of variances.

Now go back and repeat the analysis. This time do Sales by Packaging. Then click on OK to get this comparison of variances.

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3

2

1

300250200150100

reg

ion


Test Statistic 42.13

P-Value 0.000


P-Value 0.000

Bartlett's Test

Levene's Test

Test for Equal Variances for sales


Do the factor levels have equal variances?

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What about the Variances for these factor levels?

point of sale

plain

color

350300250200150100

pa

cka

gin

g



P-Value 0.000


P-Value 0.000

Bartlett's Test

Levene's Test

Test for Equal Variances for sales

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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

Sums of Squares

ANOVA Table

ANOVA Boxplots, Multiple Comparisons

Tukey Pairwise Comparisons

Main Effects Plots

Interactions Plot

Two-Factor ANOVA

Two-Factor Model

Residual Analysis


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What other comments or questions

do you have?

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APPENDIX

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Another look at the ANOVA table

Analysis of Variance Using Minitab

1093 = 33.1

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(*Only if subgroup sizes are equal)

Reading the ANOVA Table

One-Way Analysis of Variance

Analysis of Variance on Response

Source DF SS MS F p

Factor 3 4,626 1542.0 13.76 0.000

Error 20 2,242 112.1

Total 23 6,868

4

2

4

2

3

2

2

2

12

Pooled

If P is small (say, less than 5%), then we

conclude that at least one subgroup mean is different. In this case,

we reject the hypothesis that all the subgroup

means are equal

The F-test is close to 1.00when subgroup means are similar. In this case, This

F-test ratio is much greater than 1.00, hence subgroup

means are NOT similar.

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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‟

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14121086420

0 .7

0 .6

0 .5

0 .4

0 .3

0 .2

0 .1

0 .0

F- V a lu e

Pro

b

F - D is t r ib 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

10% Point

5% Point

1% Point

Observed Point

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.

What the F-Distribution Explains

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F Distribution:

Probability Distribution Function (PDF) Plots

1, 1 d.f.

3, 3 d.f.

5, 8 d.f.

8, 8 d.f

0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

F

pdf(

F)

* N1 refers to the d.f. in the numerator

N1* N2

NG BB 34 Analysis of Variance (ANOVA)

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