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Using the ANOVA Test

June 2007

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ANOVA Purpose and Use

ANOVA Purpose

Compares >2 buckets of data

Determines if means are the same or different

Different: at least one pair of means are different

ANOVA Assumptions

Attribute data as the input (X)

Continuous data as the output (Y)

Variance (std deviation) is equal for each bucket

Normally distributed output data

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ANOVA Applications

Process

How does supplier impact yield?

Is there a purity difference from different catalysts?

Analytical

Are the results the same for 3 different columns?

Is there a difference between 4 laboratories?

Troubleshooting Did the different TMBS lots affect the impurity level?

Does day of the week impact yield?

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Data

Lot 3Lot 2Lot 1

79

78

77

76

75

74

73

72

71

70

Boxplot of Lot 1, Lot 2, Lot 3

6/13/2007;BKD

Data

Lot 3Lot 2Lot 1

120

110

100

90

80

70

60

50

40

30

Boxplot of Lot 1, Lot 2, Lot 3

6/13/2007;BKD

Mean Yields

Lot 1 = 75Lot 2 = 73Lot 3 = 77

Mean YieldsLot 1 = 75

Lot 2 = 73Lot 3 = 77

ANOVA ResultsMeans for eachlot are different

ANOVA Results

Means for eachlot are the same

Means Dont Always Tell the Story

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Example Data

Ten samples are analyzed on three different instrumentsAre the results independent of the instrument?

Instr 1 Instr 2 Instr 3

98.2991 97.6095 98.0251

98.0422 97.9073 97.9751

98.1325 97.7797 98.3714

98.3308 97.5003 98.1120

98.4422 97.5736 98.0919

98.4103 98.0124 98.2839

98.1786 97.6929 97.6889

98.5282 97.7577 98.1245

98.6183 98.0015 98.1470

98.2532 97.7338 98.3623

Summary Statistics

Mean Std Dev

Instr 1 98.324 0.180

Instr 2 97.757 0.174

Instr 3 98.118 0.202

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Step 1 Check for Normality andEqual Variance

Anderson-Darling Normality Test

Variance 0.033

Skewness 0 .091780

Kurtosis -0.701420

N 10

A-Squared

0.10

P-Value 0.993

Mean 98.324

StDev 0.180

Anderson-Darling Normality Test

Variance 0.041

Skewness -0.80461

Kurtosis 1. 31358

N 10

A-Squared

0.34

P-Value 0.409

Mean 98.118

StDev 0.202

Anderson-Darling Normality Test

Variance 0.030

Skewness 0 .214828

Kurt osis -0.971612

N 10

A-Squared

0.23

P-Value 0.745

Mean 97.757

StDev 0.174

Instr

95% Bonferroni Confidence Intervals for StDevs

Instr 3

Instr 2

Instr 1

0.450.400.350.300.250.200.150.10

Bartlett's Test

0.989

Test Sta tist ic 0.20

P-Value 0.904

Levene's Test

Test Sta tist ic 0.01

P-Value

Test for Equal Variances for Result

6/13/2007;BKD

p-value >0.05Normal distributions

p-value >0.05Variances are the same

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MinitabTesting for Equal Variances

Stat ANOVATest for Equal Variances

Note:

Data must be instacked format whenusing the test for equalvariances

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Step 2 ANOVA Test

Stat ANOVA One-Way or Stat ANOVA One Way (Unstacked)(If youre using the unstacked method, how did you test for equal variances?????)

Comparisons: Select Fishers to compare differences between each bucket

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ANOVA ResultsOne-way ANOVA: Instr 1, Instr 2, Instr 3

Source DF SS MS F P

Factor 2 1.6462 0.8231 23.82 0.000

Error 27 0.9328 0.0345Total 29 2.5790

S = 0.1859 R-Sq = 63.83% R-Sq(adj) = 61.15%

Individual 95% CIs For Mean Based on

Pooled StDev

Level N Mean StDev -----+---------+---------+---------+----

Instr 1 10 98.324 0.180 (----*----)

Instr 2 10 97.757 0.174 (----*----)

Instr 3 10 98.118 0.202 (----*----)-----+---------+---------+---------+----

97.75 98.00 98.25 98.50

Pooled StDev = 0.186

Fisher 95% Individual Confidence Intervals

All Pairwise Comparisons

Simultaneous confidence level = 88.07%

Instr 1 subtracted from:

Lower Center Upper -+---------+---------+---------+--------

Instr 2 -0.7372 -0.5667 -0.3961 (----*----)

Instr 3 -0.3759 -0.2053 -0.0348 (----*----)

-+---------+---------+---------+--------

-0.70 -0.35 0.00 0.35

Instr 2 subtracted from:

Lower Center Upper -+---------+---------+---------+--------

Instr 3 0.1908 0.3614 0.5319 (----*----)

-+---------+---------+---------+--------

-0.70 -0.35 0.00 0.35

p-value

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ANOVA Example Conclusions

Results are not independent of the instrument

Mean are statistically different

From Fishers comparisons Instr 1 mean is different than Instr 2 & 3 mean

Instr 2 mean is different than Instr 3 mean

61% of the variability is due to the instrument

39% of the variability has yet to be identified

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Planning for ANOVA

Identify the buckets

Make sure the sample size is adequate

Use the power & sample size calculator Know what level of difference you want to detect

Fill the bucket with meaningful data

Independent samples

Samples representative of the process

Follow the steps!!!