Summary

8.1

Updated April-09

Lecture Notes

Chapter 8

Analyze and Improve

Effectiveness

ENTERPRISE EXCELLENCE

8.2

Updated April-09

Learning Objectives

• Analysis of Variance (ANOVA)

• One-Way Analysis of Variance (ANOVA)

8.3

Updated April-09

Analysis of valiance (ANOVA)

• Analysis of valiance (ANOVA) is a statistical technique by which the source of variability within a process is identified.

• ANOVA is widely used in industry to help identify the source of potential problems in a production process, and identify whether variation in measured output values is due to variability between various manufacturing processes or within them.

• By varying the factors in a predetermined pattern and then analyzing the output, an accurate assessment

• ANOVA tells us whether the variance we see in our data is significant and quantifies that significance.

• ANOVA accomplishes this by measuring the variance of different treatments and treatment levels.

8.4

Updated April-09

Hypothesis Testing

• When performing an ANOVA, the assumption is always made that the population is homogeneous. This is done by creating a statement of "equality" or "no change,“ which we call the null hypothesis. It is designated by H0. The null hypothesis is a statement that the means (µ) of the population for all treatments are equal:

• The opposite of the null hypothesis is the alternative hypothesis. It is designated by Ha. The alternative hypothesis is that at least one of the population means is not equal:

8.5

Updated April-09

Fisher's F Statistic

• ANOVA determines whether the mean values for several treatments are equal by examining population variances using a value known as the F statistic.

• The F statistic is based on the evaluation of the variance (s2) of the data

• ANOVA compares two estimates of this variance, one estimate attributable to the variance within treatments (Swithin

2), which is also called error, and one estimate from between treatment means (Streatments

2) • We calculate the first estimate from the variance within

all the data from several distinct treatments or different levels of one treatment

• The second estimate of the population variance is calculated from the variance between the individual treatments means (Streatments)

8.6

Updated April-09

ONE-WAY ANOVA

• One-way ANOVA provides for the analysis of two populations with a single treatment. This assists in determining whether there are differences in such things as:

1. The quality of materials coming from two suppliers

2. The warranty returns from two different areas

3. The differences between two processes producing the same product, or a other combination of inputs to a single treatment

8.7

Updated April-09

ONE-WAY ANOVA

• There are six steps in our approach to the ANOVA: 1. Calculate the sum of the squares

2. Determine the degrees of freedom

3. Calculate the mean squares

4. Calculate the F ratio

5. Look up the critical value of F

6. Calculate the percent contribution

• In performing any ANOVA, the sum of the squares must be calculated for the following:

1. Total variation (SStotal)

2. Variation attributed to the treatment (SStreatments)

3. Variation attributed to error within (SSwithn)

8.8

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ONE-WAY ANOVA- EXAMPLE

Two suppliers have delivered to us six lots of goods. Each lot contains

100 items (which are supposed to be exactly the same). However,

defective product is found in each of the lots. The data represents the

acceptance yield of the lots received from the two suppliers.

8.9

Updated April-09

ONE-WAY ANOVA- EXAMPLE

Sum for the squares for TOTAL VARIATION:

Sum for the squares for the TREATMENT VARIATION. Summarize the squared values for each treatment (Al, A2, etc.) divided by the number of data points in each level (n=6), and subtract the squared value of the summation of (y)2 divided by the total number of data points (N=12), as indicated in the equation:

The sum of the squares for WITHIN VARIATION is the total sum of the squares minus the sum of the squares for the treatment.:

8.10

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ONE-WAY ANOVA- EXAMPLE

• Degrees of freedom (df) are the number of independent comparisons available to evaluate the data.

• It is necessary to determine the degrees of freedom for TREATEMENT, WITHIN, and TOTAL:

8.11

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ONE-WAY ANOVA- EXAMPLE

12 – 1 = 11

2 – 1 = 1

11 – 1 = 10

8.12

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ONE-WAY ANOVA- EXAMPLE

• The mean squares (MS) element of the ANOVA table is the quotient of the sum of the squares of the treatment and within and the degrees of freedom (df), as indicated in the following equations:

8.13

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ONE-WAY ANOVA- EXAMPLE

• The F ratio is the quotient of the MStreatment and MSwithin

• It is calculated using the following equation:

8.14

Updated April-09

ONE-WAY ANOVA- EXAMPLE

• The F Critical: We compare the Fratio to the critical value of F' (Fcritical) to determine whether the variance is significant or not. The critical value df is determined by referring to an F table for the applicable degrees of freedom and significance level selected for your evaluation.

• The level of significance applied to the analysis can be a very subjective choice where no specific standards exist.

• It is important that some standard for selection of the significance level be implemented and applied to analysis uniformly throughout.

• The selection of the level of significance often reflects the consequences of the decision that will result from the analysis. A typical decision table for level of significance is provided in Table 8.1.

8.15

Updated April-09

ONE-WAY ANOVA- EXAMPLE

TABLE 8.1

Decision Example Level Critical systems parameters System reliability .01>

System safety requirements System performance Systems competitive capability

Process efficiency or effectiveness Process improvement options .05> Process selection Process differentiation Equipment selection

Administrative/business decisions Payment of bonuses .10> Return on investment Marketing decisions

8.16

Updated April-09

ONE-WAY ANOVA- EXAMPLE

Select a significance level of .05, based on process

efficiency:

F' (Fcritical): df1, df10, .05 = 4.96 (See Table V)

F ratio > Fcritical = Significant F ratio < Fcritical = Not Significant 1.91 < 4.96 = Not Significant

8.17

Updated April-09

ONE-WAY ANOVA- EXAMPLE

8.18

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ONE-WAY ANOVA- EXAMPLE

8.19

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ONE-WAY ANOVA- EXAMPLE

• Percent Contribution (% contribution) element of the ANOVA table is the quotient of the sum of the squares for the treatment and within and the sum of the squares for total, as follows:

8.20

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ONE-WAY ANOVA- EXAMPLE

8.21

Updated April-09

ONE-WAY ANOVA- EXAMPLE

A few of the important facts that we can extract from the table are:

• The product variance caused by the treatment (supplier) was not significant.

• The supplier treatment is contributing 16% to the overall product variability.

• In this analysis, 84 percent of the product variability is not accounted for.

8.22

Updated April-09

• One-way ANOVA deals only with one treatment (Supplier)

• It is often necessary to determine whether two different treatments are

affecting a process or product and whether their effect is significant.

• Two-way ANOVA provides a tool to make that assessment of two

treatments. Treatment A (supplier A and Supplier A2), and treatment B

(Testing Set).

• Cause-and-effect analysis or engineering data may have suggested

that suppliers and test set are contributors to the variation

• The data now represent the acceptance yield of six lots of 100

received from two different suppliers and accepted on two different

acceptance test sets.

• Two-way ANOVA can be accomplished in the same six steps as one-

way ANOVA. The difference is the addition of the second treatment to

the ANOVA computation and decision table.

• Objective again is to determine if suppliers are significant or if acceptance testing methodology is significant as well as percent contrinbution of each.

Two Way ANOVA

8.23

Updated April-09

• Multivariate ANOVA (a.k.a. MANOVA) can be used for the evaluation of

process data and in designed experiments.

• It extends analysis of variance methods to handle cases where there

is more than one dependent variable and where the dependent

variables cannot simply be combined:

• Supplier A1, Supplier A2

• Test Set B1, B2, B3, B4

• Vacuum C1, C, C3, C4

• It identifies whether changes in the independent variables have a

significant effect on the dependent variables.

• It also seeks to identify the interactions among the independent

variables and the association between dependent variables. MANOVA

involves six steps:

• 1) Calculate the sum of the squares, 2) Determine the degrees of

freedom. 3) Calculate the mean squares. 4) Calculate the F ratio

• Statistical software can be used for the calculations (Minitab, etc.)

Multi-Variate ANOVA

8.24

Updated April-09

• In this chapter, we have learned the following:

• Analysis of Variance (ANOVA)

• One-Way Analysis of Variance (ANOVA)

Wrap-up