09 Analysis of Variance- Part1

Breakthrough Management GroupBMG

Analysis of Variance – Part I

ANOVA One-way –Hypothesis testing for multiple means

4μ1μ 3μ2μ

Pg 2© Breakthrough Management Group, Inc. Unpublished proprietary work available only under license. All rights reserved.

Module Objectives

To introduce the concepts of Analysis of VarianceSum of SquaresMean Square Error

To demonstrate and practice calculating the ANOVA tableManuallyMinitab

To practice ANOVA ExercisesHomeworkQuiz


Analyze Phase Deliverables

1) Describe ProjectObjective statementMetrics.xls chartsInitial validated forecast

2) FMEA3) ID Variation: Graphical Methods4) ID Variation: Statistical Methods

Correlation & RegressionMeans testingSigma testingProportions testingContingency tables

5) Planning for DOE6) Complete Phase Summary

Conclusions, Issues, & Next Steps

Failure Modes & Effects Analysis

ID Variation: Graphical Analysis

Plan for DOE

Analyze

ID Variation: Statistical Analysis


Mathematical Tools & Data Types

Discrete ContinuousD

iscr

ete

Analysis of Variance

Con

tinuo

us

Response VariableIn

depe

nden

t Var

iabl

e

A Menu of Six Sigma Tools.


Hypothesis Test Reference

Data Description Hypothesis Test Tool examples

Printer Model (X1) vs. Service Level (Y)

1 Discrete XContinuous Y

Ho: u = Target 1-Sample t test

2-sample t test; 2-sample paired t-testANOVA

Printer cartridge defect rate (p) vs. target


Ho: p = po 1 Proportion p-value

Comparison of quality levels of two products

2 Discrete XDiscrete Y

Ho: p1 – p2 = po 2 Proportions Difference, confidence interval, p-value

Observed Events vs. Expected Events

3+ Discrete X Discrete Y

Ho: p1=…=pk Chi-Square;Analysis of Means

Contingency table

Printer Models (X1, X2) vs. Fuser Life (Y)


Ho: u1 = u2

Time series charts; histogram; Capability analysisScatter plots; Dot Plots

Monday-Friday (X1, X2, X3, X4, X5) vs. Sales (Y)

3+ Discrete X Continuous Y

Ho: u1=…=uk Box Plots; Main Effects plots; Interaction plots; Pareto charts


Testing Means

Testing for one meanZ-test for large samples or when σ is knownt-test for smaller samples or when σ is unknown

Testing for two means2-sample t-testPaired t-test

Testing for three or more means1-way ANOVA

Ho: μ = μtargetHa: μ < μtarget

μ > μtargetμ ≠ μtarget

Ho: μ1 = μ2

Ha: μ1 < μ2μ1 > μ2μ1 ≠ μ2

Ho: μ1 = μ2= μ3= μ4

Ha: at least one μ is different from another

μtarget

μ

1μ 2μ

4μ1μ 3μ2μ


ANOVA Model

Mathematical Model for ANOVA

ijjijy ετμ ++=Where:yij = a single response from Treatment jμ = overall mean τj = the contribution from Treatment jεij = random error

000

≠=

ja one least@:Hs':H

ττ

different is one least atH

H

ja

j

μ

μμμ

:

...: 210 ===Mathematical Hypothesis Conventional Translation

ANOVA is a model for discrete inputs and continuous output features


ANOVA Introduction

Example:A fertilizer company wants to compare fields treated with four new, spring wheat fertilizers to fields with no fertilizer.Ten different fields were sampled for each treatment and the average bushels/acre was computed.

Problem:Were any of the fertilizers different from each other and the unfertilized control group?

How can we analyze this data?


Case Studies

Case 1

Case 2Trial TmtA TmtB TmtC TmtD TmtCntrl11 6.6 7.0 6.1 7.5 5.52 6.4 7.0 5.9 7.5 5.73 6.5 7.0 6.0 7.6 5.54 6.6 6.8 6.1 7.4 5.45 6.5 7.1 5.9 7.4 5.56 6.6 7.0 5.8 7.4 5.57 6.5 6.9 6.0 7.6 5.48 6.6 6.9 5.9 7.7 5.59 6.7 6.9 6.1 7.5 5.5

10 6.5 6.9 6.1 7.6 5.7Mean 6.6 7.0 6.0 7.5 5.5

Trial Tmt1 Tmt2 Tmt3 Tmt4 TmtCntrl21 6.0 5.5 6.4 6.4 7.62 4.3 8.3 5.3 7.1 4.93 4.7 4.8 6.7 8.8 6.34 6.7 6.8 6.9 8.8 5.35 5.0 8.2 6.8 6.0 2.26 4.9 8.2 5.0 7.4 6.57 7.6 6.3 7.2 9.1 5.78 7.9 7.8 6.1 5.5 6.09 7.2 6.6 6.4 9.4 7.4

10 8.2 10.1 5.6 8.6 3.0Mean 6.3 7.3 6.2 7.7 5.5

Are the treatments different? Are the they different now?

IntroANOVA.mtw


Case 1 Graphs

Case 1 – Boxplot

Case 1 – Dotplot

Are the treatments different?

Dat

a

TmtCntrl1TmtDTmtCTmtBTmtA

8.0

7.5

7.0

6.5

6.0

5.5

Boxplot of TmtA, TmtB, TmtC, TmtD, TmtCntrl1

Dat

a

TmtCntrl1TmtDTmtCTmtBTmtA

8.0

7.5

7.0

6.5

6.0

5.5

Individual Value Plot of TmtA, TmtB, TmtC, TmtD, TmtCntrl1


Case 2 Graphs

Case 2 -- Boxplot

Case 2 -- Dotplot

How might one analyze these differences statistically?

Dat

a

TmtCntrl2Tmt4Tmt3Tmt2Tmt1

11

10

9

8

7

6

5

4

3

2

Boxplot of Tmt1, Tmt2, Tmt3, Tmt4, TmtCntrl2

Dat

a

TmtCntrl2Tmt4Tmt3Tmt2Tmt1

11

10

9

8

7

6

5

4

3

2

Individual Value Plot of Tmt1, Tmt2, Tmt3, Tmt4, TmtCntrl2


Multiple t-tests – Adequate?

An experimenter could run 2-sample t-tests against every combination of means

1 vs. 2, 1 vs. 3, 1 vs. 4, 1 vs. 52 vs. 3, 2 vs. 4, 2 vs. 53 vs. 4, 3 vs. 54 vs. 5

Why would this not be a good idea?Obviously, it is tedious and cumbersome

What about alpha risk?Each t-test has a risk of false rejection (α)

4.010^95.01

=−=totalα

What would be the total α risk?


Solution – Analysis of Variance

ANOVA is really an extension (generalization) of the 2-sample t-test

ANOVA is a method of detecting differences between multiple means of samples

Why is it called Analysis of Variance?ANOVA compares/analyzes variances

Variance within a groupVariance between groups

ANOVA is the mathematics behind the intuitive evaluation


ANOVA in Minitab – A Quick Demo

In Minitab select Stat>ANOVA>One-way (Unstacked)…

different is one least @:H

...:H

ja

j

μ

μμμ === 210


Minitab ANOVA Results – Case 1 One-way ANOVA: TmtA, TmtB, TmtC, TmtD, TmtCntrl1

Source DF SS MS F PFactor 4 24.65720 6.16430 643.60 0.000Error 45 0.43100 0.00958Total 49 25.08820

S = 0.09787 R-Sq = 98.28% R-Sq(adj) = 98.13%

Individual 95% CIs For Mean Based onPooled StDev

Level N Mean StDev --------+--------+--------+--------+TmtA 10 6.5500 0.0850 (*)TmtB 10 6.9500 0.0850 (*)TmtC 10 5.9900 0.1101 (*)TmtD 10 7.5200 0.1033 (*)TmtCntrl1 10 5.5200 0.1033 (*)

--------+--------+--------+--------+6.00 6.60 7.20 7.80

Pooled StDev = 0.0979

What do you think this means?

Reject Ho or Fail to Reject Ho?


Minitab ANOVA Results – Case 2One-way ANOVA: Tmt1, Tmt2, Tmt3, Tmt4, TmtCntrl2

Source DF SS MS F PFactor 4 31.51 7.88 3.89 0.009Error 45 91.25 2.03Total 49 122.77

S = 1.424 R-Sq = 25.67% R-Sq(adj) = 19.06%

Individual 95% CIs For Mean Based onPooled StDev

Level N Mean StDev --+---------+---------+---------+-----Tmt1 10 6.250 1.457 (------*-------)Tmt2 10 7.260 1.561 (-------*------)Tmt3 10 6.240 0.729 (-------*-------)Tmt4 10 7.710 1.412 (------*-------)TmtCntrl2 10 5.490 1.748 (-------*------)

--+---------+---------+---------+-----4.8 6.0 7.2 8.4

Pooled StDev = 1.424

What do you think this means?



Analysis of Variance – General Recipe

1. State the practical problem

2. State the null hypothesis

3. State the alternate hypothesis

4. Do the model assumptions hold?

5. Construct the Analysis of Variance Table

6. Do the assumptions for the errors hold (residual analysis)?

7. Interpret the p-value for the factor effect (p < α)

8. Calculate %SS for the factor and error terms

9. Translate the conclusion into practical terms

A general recipe for all types of hypothesis tests.


Control 66 67 74 73 75 64Herbicide A 85 85 76 82 79 86Herbicide B 91 93 88 87 90 86

An ANOVA Calculation Example

Peaches & HerbsTwo herbicides were tested to determine if treatment of the surrounding weeds improved the growth of peach tree seedlings. A third group was left untreated as a control.Eighteen seedlings were selected for the test, six assigned randomly to each of the three groups. At the end of the study period, the height, in cm, was recorded for each seedling.Use ANOVA and the following data to detect differences among the different seedling heights. Use α = 0.05

What is the first thing you should do in any analysis?

Peaches.mtw


Graphical Exploration – Minitab

What conclusions can you draw?

Group

Hei

ght

HerbBHerbACntrl

95

90

85

80

75

70

65

60

Boxplot of Height by Group

Group

Hei

ght

HerbBHerbACntrl

95

90

85

80

75

70

65

60

Individual Value Plot of Height vs Group


Group

Hei

ght

HerbBHerbACntrl

95

90

85

80

75

70

65

60

Grand Mean = 80.39

89.1667

82.1667

69.8333


ANOVA Visually


ANOVA Assumptions

Each sample is an independent, random sampleIndependent

The selection of any sample is not dependent on any other sample being selected or not selected

RandomAll members of the population have an equal chance of being selected

The measurements within each group are normally distributed and have equal variances

This only applies for the within group variation, not between group variationThe variances for each group (treatment) are equal

ANOVA Assumptions: Normality and Equal Variances.


Group

Hei

ght

HerbBHerbACntrl

95

90

85

80

75

70

65

60

89.17

82.17

69.83


ANOVA Assumptions – Visually

2Cntrlσ

2HerbAσ

2HerbBσ

222HerbBHerbACntrl σσσ ==


Normality Testing for ANOVA

How many normality tests should we run?

Height_HerbB

Perc

ent

9694929088868482

99

95

90

80

70

60504030

20

10

5

1

Mean

0.859

89.17StDev 2.639N 6AD 0.178P-Value

Probability Plot of Height_HerbBNormal

Height_HerbA

Perc

ent

90858075

99

95

90

80

70

60504030

20

10

5

1

Mean

0.304


Probability Plot of Height_HerbANormal

Height_Cntrl

Perc

ent

8075706560

99

95

90

80

70

60504030

20

10

5

1

Mean

0.230


Probability Plot of Height_CntrlNormal


Equal Variances – Another ANOVA Assumption

Select Stat>ANOVA>Test for Equal Variances…

Have we proven the variances are equal? What if the variances were unequal?

Grou

p

95% Bonferroni Confidence Intervals for StDevs

HerbB

HerbA

Cntrl

1614121086420

Bartlett's Test

0.158

Test Statistic 1.47P-Value 0.480

Levene's Test

Test Statistic 2.09P-Value

Test for Equal Variances for Height


Group

Hei

ght

HerbBHerbACntrl

95

90

85

80

75

70

65

60


ANOVA Visually

Data points can be characterized by distances

80.4

89.2

82.2

69.8

The distance of the group mean from the grand

mean

The distance of the data

point from the group mean


A Variation Estimate – Sum of Squares

The variability of n sample measurements about their mean can be measured using the sum of squared deviations from the grand mean

Likewise, the sum of square deviations within each group is:

The variability of each group as compared to the grand mean is:

According to the model:

2)(∑∑ −=

i jijTotal yySS

2)(∑∑ −=

i jjijWithin yySS

2)(∑∑ −=

i jjBetween yySS

WithinBetweenTotal SSSSSS +=

Variances are Sums of Squares divided by degrees of freedom.


60

70

80

90

100

0 5 10 15 20 25 30

The Sum of Squares Model

Putting it all together:

( ) ( )2 2 2

1 1 1 1( )

k m k m

ij ijj jj i i j j i

y y y y y y= = = =

− = − + −∑∑ ∑∑ ∑∑

TotalSS WithinSSBetweenSS

1Y

2Y

3Y

Y


k is the number of groups (or factors)d.f. (between) =

n is the total number of samplesd.f. (within) =d.f. (total) =

The ANOVA table: Degrees of Freedom

SourceDegrees ofFreedom

Sum of Squares

MeanSquare F-Statistic

Between(or Factor) k-1 SS Between s 2

Between = SS Between /k-1 s 2Between /s 2

Within

Within(or Error) n-k SS Within s 2

Within = SS Within /n-k

Totals n-1 SS Total

2

1517


The ANOVA table: Sum of Squares


Sum of Squares


Between(or Factor) 2 SS Between s 2


Within

Within(or Error) 15 SS Within s 2


Totals 17 SS Total

2)(∑∑ −=

i jjijWithin yySS

2)(∑∑ −=

i jijTotal yySS

2)(∑∑ −=

i jjBetween yySS

How else can we calculate SSTotal?


Calculating the SSQs – the Grand Mean

Select Stat>Basic Stat>Store Descriptive Statistics…


Calculating the SSQ’s – the Group Means

Select Stat>Basic Stat>Store Descriptive Statistics…


Calculating the Squared Differences

Select Calc>Calculator…


Calculating the Squared Differences

Results of the Sum of Squares Calculations


Summing the Squared Differences

Select Calc>Column Statistics…Repeat for SSB and SSW

Minitab Results:Sum of SSB = 1149.8Sum of SSW = 224.5

Sum of TSS = 1374.3

The SSTotal equals the sum of SSBetween and SSWithin.

TSS = TOTAL SUMS OF SQUARES

SSB = BETWEEN GROUP SUMS OF SQUARES

SSW = WITHIN GROUP SUMS OF SQUARES



Sum of Squares


Between(or Factor) 2 1150 s 2


Within

Within(or Error) 15 224 s 2


Totals 17 1374

224 1150 - 1374 ==− WithinBetweenTotal SSSSSS

WithinBetweenTotal SSSSSS +=

The ANOVA table: SSWithin

Remember:

Notice that:



Sum of Squares


Between(or Factor) 2 1150 s 2


Within

Within(or Error) 15 224 s 2


Totals 17 1374


Sum of Squares


Between(or Factor) 2 1150 575 38.33

Within(or Error) 15 224 15

Totals 17 1374

===

2BetweenBetweenBetween dfSSs

===

2WithinWithinWithin dfSSs

===

22WithinBetweenCalc ssF

5752 1150

2

=== BetweenBetweenBetween dfSSs

1515 224

2

=== WithinWithinWithin dfSSs

33.3815 575

22

=== WithinBetweenCalc ssF

The ANOVA Table: Mean Square & Fcalc

What is an F test? An F distribution?


The ANOVA Table – Complete


Sum of Squares


Between(or Factor) 2 1150 575 38.33

Within(or Error) 15 224 15

Totals 17 1374


P (F >= f) = 1.000 – 1.0000 or p-value = 0.0000

Calculating a P-value from an F Statistic

In Minitab select Calc > Probability Distributions > F…

Minitab Outputx P( X <= x )38.3300 1.0000

dfBetween

dfWithin

F



In Summary

ANOVA is a powerful tool for testing equality of meansANOVA compares the variability between group means to the average variability within a group

ANOVA p-values are significant when the group-to-group differences are too large to be explained by the within group variability

The initial assumptions for ANOVA:Each subgroup is normally distributedThe variances of all the subgroups are equal

The ANOVA table is the standard way of reporting ANOVA analysis


The (ω)Rap!

Key TakeawaysANOVA is a powerful tool for determining the differences between multiple meansANOVA compares the between-group variability to the within-group variabilityThe assumptions of ANOVA are subgroup normality and equal variancesOthers:__________________________________________________________________

How can I use this in my project?____________________________________________________________________________________________________________________________________________________________________________________________________________________________


10-QQ! (Choose the best answers)1) ANOVA is used to test hypotheses of means.

a) Trueb) False

2) If the group means are different, the p-value will be large.a) Trueb) False

3) Excel and Minitab calculate p-values from F-distributions in the same way.a) Trueb) False

4) Which statements are not true?a) Manual calculations are less accurate than

Minitab calculationsb) ANOVA is just an easier way to conduct multiple

t testsc) ANOVA is a test for only one input at multiple

levelsd) All of the Above

5) The most important step in ANOVA is:a) Checking the initial assumptionsb) Making practical conclusionsc) Gathering large amounts of datad) Asking a practical questione) Letting Minitab run the analysis automatically

6) The null hypothesis of Analysis of Variance is:a) All of the means are differentb) All of the variances are differentc) At least one of the variances is different from

anotherd) At least on of the means is different from anothere) None of the above

7) ANOVA is a way of analyzing:a) Continuous independent variables and continuous

dependent variablesb) Continuous dependent variables and discrete

independent variablesc) Discrete independent variables and continuous

independent variablesd) Discrete dependent variables and discrete

dependent variables

8) The initial assumptions of ANOVA are:a) Normality of the datasetb) All groups have equal variancesc) All groups are normally distributedd) All samples are measured perfectlye) All of the groups have the same mean

9) Which statement is not true?a) ANOVA is a hypothesis test for meansb) The father of modern statistics was R. A. Fisherc) Pooled standard deviation is the square root of an

average varianced) ANOVA requires relatively few samples per groupe) R.A. Fisher did farm research in Iowa


New and Important Terms

ANOVA TableContinuous VariableDiscrete (Categorical) VariableEmpirical ModelF-testOne-way Analysis of Variance


Glossary

ANOVA TableThe standard way of displaying the results of the calculations from ANOVA

Continuous VariableVariables (data) whose possible of values form a whole interval, range, or continuum (e.g., Temperature)

Discrete (Categorical) VariableVariables (data) whose possible values are distinct or separate (0, 1, 2, etc.)

Empirical ModelAn equation derived from the data the expresses a relationship between the inputs and an output (Y=f(x))

F-testA hypothesis test for comparing variances

One-way Analysis of Variance One-way analysis of variance tests the equality of population means when classification is by one variable. The classification variable, or factor, usually has three or more levels.

09 Analysis of Variance- Part1

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