   Six Sigma Glossary from MiC Quality
Analysis of Variance (ANOVA)
 Analysis of Variance

See ANOVA

 ANOVA

A statistical technique that partitions the variation in a dataset into the variation caused by the each of the factors, and the common cause variation. The purpose is to find out how much of the variation can be explained by each factor, and the statistical significance of each component of variation. See One Way ANOVA and Two Way ANOVA for examples.

The ANOVA method requires the process conforms to a normal distribution. For nonparametric tests:

Kruskal Wallace Test
Mood's Median Test
Friedman's Test (equivalent to two factor ANOVA)

 F Test

The F Test is used to compare the variances of two samples to test the hypothesis that they may come from the same distribution. The hypothesis is:

H0 the samples are drawn from populations with equal variances
H1 the samples are drawn from populations with different variances

The F test is based on the ratio of the variances of two samples. It depends on the number of degrees of freedom of both samples:

 1 2 3 4 5 Variance A 8 .7 9.8 12.2 12.9 9.5 3.327 B 23.8 21.5 18.3 24.3 22.9 5.788

This gives an F statistic: This can be compared to the critical F value from tables, or the p-value can be obtained using the Excel function:

=FDIST(Fo, DOFn, DOFd)

 Hypothesis Test for Variance

See F Test

 One Way ANOVA

One Way ANOVA is used to determine whether varying levels or values of a single factor affect the process. The following data shows the strength of a fibre with varying percentages of a synthetic material:

 1 2 3 4 5 15% 8 .7 9.8 12.2 12.9 9.5 20% 14.5 17.3 14.9 15.5 12.7 25% 16.0 13.8 18.3 17.2 18 30% 23.8 21.5 18.3 24.3 22.9 35% 9.0 14.3 10 14.3 7.1

An ANOVA analysis tests the hypothesis:

H0 The fibres are all of equal strength
H1 At least one of the fibres is of different strength

Use a level of significance of 0.05.

The analysis is:

 Source of Variation Sum of Squares Degrees of Freedom Mean Square F0 p-value Factor 448.32 4 112.08 21.78 0 Error 102.91 20 5.15 Total 551.23 24

The alternative hypothesis is accepted.

 Single Factor Experiment

See One Way ANOVA

 Two Way ANOVA

In the ANOVA example the five replications within each row were all taken under the same test conditions. Consider the example where three treatments are evaluated on four different patients:

 Therapy Andrew Belinda Chris Dave Relaxed 110 140 100 130 Normal 115 150 105 135 High Intensity 117 155 100 135

The hypothesis is:

H0 The therapies all give the same result
H1 At least one of the therapies gives a different response

Use a level of significance of 0.05.

The patients are all different, and a 'One Way ANOVA' would not cause the null hypothesis to be rejected. However a two way ANOVA separates the variation due to the therapy from that due to different patient characteristics:

 Source of Variation Sum of Squares Degrees of Freedom Mean Square F0 p-value Therapy 113.17 2 56.58 5.40 0.046 Patient 3832.67 3 1277.56 121.99 0 Error 62.83 6 10.47 Total 4008.67 11

The null hypothesis is rejected, the therapies give different results at the 0.05 level of significance.

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