Chapter 22 Variance Componentsweb.eng.fiu.edu/leet/TQM/chap22_2012.pdf · 3/18/2013 1 Chapter 22 Variance Components Introduction • The test attempts to quantify the variability

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Chapter 22

Variance Components

Introduction

• The test attempts to quantify the variability from factor

levels.

• Methodology described is a random effects (or components

of variance model), as opposed to a fixed-effects model.

• The statistical model for the random effects model is similar

to that of the fixed-effects model.

• The difference is that in the random effects model, the

levels (or treatments) could be a random sample from a

large population of levels, so the conclusion could be

extended to all population levels.

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22.1 S4/IEE Application Examples:

Variance Components

• Manufacturing 30,000-foot-level metric (KPOV): An S4/IEE

project was to improve the capability/ performance metric

for the diameter of a manufactured product (i.e., reduce the

number of parts beyond the specification limits). A cause-

and-effect matrix ranked the variability of diameter within a

part and between the four-cavity molds as important inputs

that could be affecting the overall 30,000-foot-level part

diameter metric. A variance component analysis was

conducted to test significance and estimate the

components.

22.1 S4/IEE Application Examples:

Variance Components

• Transactional 30,000-foot-level metric: DSO reduction was

chosen as an S4/IEE project. A cause-and-effect matrix

ranked company as an important input that could affect the

DSO response. The team wanted to estimate the variability

in DSO between and within companies. A variance

component analysis was conducted to test significance and

estimate the components.

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22.2 Description

• A fixed-effects model assesses how the level of key

process input variables (KPIVs) affects the mean response

of key process outputs, while a random effects model

assesses how the variability of key process input variables

(KPIVs) affect the variability of key process outputs.

• A key process output of a manufacturing process could be

the dimension or characteristic of a product.

• A key process output of a service or business process could

be time from initiation to delivery.

Process KPIVs KPOVs

22.2 Description

• The total affect of 𝑛 variance components on a key process

output can be expressed as the sum of the variances of

each of the components:

𝜎𝑡𝑜𝑡𝑎𝑙2 = 𝜎1

2 + 𝜎22 + 𝜎3

2 ⋯+ 𝜎𝑛2

• The components of variance within a manufacturing

process could be material, machines, operators, and the

measurement system.

• In service or business processes the variance components

can be the day of the month the request was initiated,

department-to-department variations when handling a

request, and the quality of the input request.

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22.2 Description

• An important use of variance components is the isolation of

different sources of variability that affect product or system

variability. This problem of product variability frequently

arises in quality assurance, where the isolation of the

sources for this variability can often be very difficult.

• A test to determine these variance components often has

the nesting structure as follows (Figure 22.1)

Batches

Samples

Tests

22.3 Example 22.1:

Variance Components of Pigment Paste

• Consider that numerous batches of a pigment paste are

sampled and tested once. We would like to understand the

variation of the resulting moisture content as a function of

process variation, sampling variation, and analytical variation.

𝜂 Process mean

Batch mean

Sample mean

𝜀𝑏

𝜀𝑠

𝜀𝑡

𝜀

𝜎𝐵

𝜎𝑆

𝜎𝑇

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22.3 Example 22.1:


• 𝜂 is the long-term process mean for moisture content.

• Process variation is the distribution of batch means about the

process mean, sampling variation is the distribution of samples

about the batch mean, and analytical variation is the

distribution of analytical test results about the sample mean.

• The overall error (𝜀 = 𝑦 − 𝜂) will contain the three separate

error components (𝜀 = 𝜀𝑏 + 𝜀𝑠 + 𝜀𝑡), where 𝜀𝑏 is the batch-to-

batch error, 𝜀𝑠 is the error made in taking the samples, and 𝜀𝑡 is the analytical test error.

• The mean of the errors (𝜀𝑏, 𝜀𝑠, and 𝜀𝑡) is zero.

• The assumption is made that the samples are random

(independent) from normal distributions with fixed variances,

𝜎𝑏2, 𝜎𝑠

2, 𝜎𝑡2.

𝜎 𝑇 = 0.96

Analytical test variation

Sample variation

Process variation

𝜎 𝑆 = 5.3

𝜎 𝐵 = 2.6

22.3 Example 22.1:


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22.3 Example 22.1:


Nested ANOVA: Moisture versus Batch, Sample

Analysis of Variance for Moisture

Source DF SS MS F P

Batch 14 1210.9333 86.4952 1.492 0.226

Sample 15 869.7500 57.9833 63.255 0.000

Error 30 27.5000 0.9167

Total 59 2108.1833

Variance Components

% of

Source Var Comp. Total StDev

Batch 7.128 19.49 2.670

Sample 28.533 78.01 5.342

Error 0.917 2.51 0.957

Total 36.578 6.048

Minitab:

Stat

ANOVA

Fully Nested ANOVA

Note: Enter the hierarchical

Order (Batch Sample)

22.4 Example 22.2:

Variance Components of a Manuf. Door including Measurement System Components

• When a door is closed, it needs to seal well with its mating

surface. Some twist of the door can be tolerated.

• Let’s consider this situation from the point of view of the

supplier of the door. The burden of how well a door latches

does not completely lie with the supplier of the door. (The

doorframe could be twisted, but can’t be checked.)

• The customer of the door supplier often rejects doors. The

supplier can only manufacture to the specification.

• The question arises of how to measure the door twist.

• Drawing specifications indicate that the area has a 0.031”

tolerance. Currently, this dimension is measured in the fixture

that manufactures the door.

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22.4 Example 22.2:

Variance Components of a Manuf. Door including Measurement System Components

• It has been noticed that the door tends to spring into a different

position after leaving the fixture.

• It was concluded that there needed to build a measurement

fixture for checking the door which simulated where the door

would be mounted on hinges in taking the measurements.

• A nested experiment was planned. There was only one

manufacturing jig and one inspection jig.

• Sources of variability considered:

• Week-to-week, shift-to-shift, operator-to-operator,

• Within-part variability, inspector measurement repeatability,

and inspector measurement reproducibility.

22.5 Example 22.3: Determining Process

Capability/Performance Metrics using

Variance Components

• Data set was presented as Exercise 3 in Chapter 10 on control

charts. Example 11.2 described a procedure used to calculate

process capability/performance metrics.

• This chapter gives an additional calculation procedure for

determining standard deviations from the process.

• Additional procedures for process capability/performance

metrics calculations are treated with single-factor analysis of

variance (Chapter 24).

• This control chart data has samples nested within the

subgroups.

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11.12 Example 11.2:

Process Capability/Performance Indices Study

# 𝑥 R

1 0.65 0.70 0.65 0.65 0.85 0.70 0.20

2 0.75 0.85 0.75 0.85 0.65 0.77 0.20

3 0.75 0.80 0.80 0.70 0.75 0.76 0.10

4 0.60 0.70 0.70 0.75 0.65 0.68 0.15

5 0.70 0.75 0.65 0.85 0.80 0.75 0.20

6 0.60 0.75 0.75 0.85 0.70 0.73 0.25

7 0.75 0.80 0.65 0.75 0.70 0.73 0.15

8 0.60 0.70 0.80 0.75 0.75 0.72 0.20

9 0.65 0.80 0.85 0.85 0.75 0.78 0.20

10 0.60 0.70 0.60 0.80 0.65 0.67 0.20

11 0.80 0.75 0.90 0.50 0.80 0.75 0.40

12 0.85 0.75 0.85 0.65 0.70 0.76 0.20

13 0.70 0.70 0.75 0.75 0.70 0.72 0.05

14 0.65 0.70 0.85 0.75 0.60 0.71 0.25

15 0.90 0.80 0.80 0.75 0.85 0.82 0.15

16 0.75 0.80 0.75 0.80 0.65 0.75 0.15

LSL=0.500

USL=0.900

𝑥 = 0.7375

𝑅 = 0.1906

11.12 Example 11.2:


Method 2: Short-term Estimate of 𝜎: Using 𝑅

𝜎 =𝑅

𝑑2=

0.1906

2.326= 0.0819

𝑍𝑈𝑆𝐿 =𝑈𝑆𝐿 − 𝜇

𝜎 =

0.900 − 0.7375

0.0819= 1.9831

𝑍𝐿𝑆𝐿 =𝜇 − 𝐿𝑆𝐿

𝜎 =

0.7375 − 0.500

0.0819= 2.8983

𝐶𝑝𝑘 =𝑍𝑚𝑖𝑛

3=

1.9831

3= 0.6610

𝐶𝑝 =𝑈𝑆𝐿 − 𝐿𝑆𝐿

6𝜎=

0.900 − 0.500

6(0.0819)= 0.8136

𝑝𝑝𝑚𝑈𝑆𝐿 = Φ 𝑍𝑈𝑆𝐿 × 106

= Φ 1.98 × 106

= 23,679

𝑝𝑝𝑚𝐿𝑆𝐿 = Φ 𝑍𝐿𝑆𝐿 × 106

= Φ 2.90 × 106

= 1,876

𝑝𝑝𝑚𝑡𝑜𝑡𝑎𝑙 = 𝑝𝑝𝑚𝑈𝑆𝐿 + 𝑝𝑝𝑚𝐿𝑆𝐿

= 23,679 + 1,876= 25,555

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11.12 Example 11.2:


Method 1: Long-term Estimate of 𝜎: Using Individual Data

𝜎 = (𝑥𝑖 − 𝑥 )2

(𝑛 − 1)

𝑛

𝑖=1

= (𝑥𝑖 − 0.7375)2

(80 − 1)

80

𝑖=1

= 0.0817

𝑃𝑝 =𝑈𝑆𝐿 − 𝐿𝑆𝐿

6𝜎=

0.900 − 0.500

6(0.0817)= 0.8159

𝑃𝑝𝑘 = 𝑚𝑖𝑛𝑈𝑆𝐿 − 𝜇

3𝜎,𝜇 − 𝐿𝑆𝐿

3𝜎= min 0.6629, 0.9688 = 0.6629

11.12 Example 11.2:


• Process capability and process performance metrics are

noted to be almost identical.

Method 1

LT

Method 2

ST

𝑠 𝑅 𝑑2

𝜎 0.0817 𝜎 0.0819

𝑃𝑝 0.8159 𝐶𝑝 0.8136

𝑃𝑝𝑘 0.6629 𝐶𝑝𝑘 0.6610

𝑍𝑈𝑆𝐿 𝑍𝑈𝑆𝐿 1.98

𝑍𝐿𝑆𝐿 𝑍𝐿𝑆𝐿 2.90

ppm ppm 25555

• Calculation for ST variability

were slightly larger which is

not reasonable. Using 𝑠 , the

𝜎 = 0.0811.

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Minitab:

Stat

ANOVA

Fully Nested ANOVA

Nested ANOVA: Data versus Subgroup

Analysis of Variance for Data

Source DF SS MS F P

Subgroup 15 0.1095 0.0073 1.118 0.360

Error 64 0.4180 0.0065

Total 79 0.5275

Variance Components

% of

Source Var Comp. Total StDev

Subgroup 0.000 2.30 0.012

Error 0.007 97.70 0.081

Total 0.007 0.082



Variance Components



Variance Components

• An interpretation of this output is that the long-term standard

deviation would be the total component of 0.082, while the

short-term standard deviation component would be the error

component of 0.081.

• Variance components technique can be useful for determining

process capability/performance metrics when a hierarchy of

sources affects process variability.

• The technique will also indicate where process improvement

focus should be given to reduce the magnitude of component

variabilities.

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• Data set was presented as Example 15.1 for multi-vari analysis

of the injection-molding data.

• It was thought that differences between cavities affected the

diameter of parts.

22.6 Example 22.4:

Variance Components Analysis of

Injection-Molding Data

15.3 Example 15.1: Multi-Vari Chart of Injection Molding Data

Time 1 Time 2 Time 3

Cavity 1 2 3 4 1 2 3 4 1 2 3 4

Location

Part1 Top 0.2522 0.2501 0.251 0.2489 0.2518 0.2498 0.2516 0.2494 0.2524 0.2488 0.2511 0.249

Part1 Middle 0.2523 0.2497 0.2507 0.2481 0.2512 0.2484 0.2496 0.2485 0.2518 0.2486 0.2504 0.2479

Part1 Bottom 0.2518 0.2501 0.2516 0.2485 0.2501 0.2492 0.2507 0.2492 0.2512 0.2497 0.2503 0.2488

Part2 Top 0.2514 0.2501 0.2508 0.2485 0.252 0.2499 0.2503 0.2483 0.2517 0.2496 0.2503 0.2485

Part2 Middle 0.2513 0.2494 0.2495 0.2478 0.2514 0.2495 0.2501 0.2482 0.2509 0.2487 0.2497 0.2483

Part2 Bottom 0.2505 0.2495 0.2507 0.2484 0.2513 0.2501 0.2504 0.2491 0.2513 0.25 0.2492 0.2495

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Minitab

Stat

Quality Tools

Multi-vari

Factor1:pos

Factor2:cav

Factor3:time

Factor4:part

Minitab

Stat

Quality Tools

Multi-vari

Factor1:cav

Factor2:pos

Factor3:time

Factor4:part


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• A variance components analysis of the factors yielded the

following results (the raw data were multiplied by 10,000 so that

the magnitude of the variance components would be large

enough to be quantified.

22.6 Example 22.4:



Minitab:

Stat

ANOVA

Fully Nested ANOVA

Nested ANOVA: Diameter1 versus Time1, Cavity1, Part1, Position

Analysis of Variance for Diameter1

Source DF SS MS F P

Time1 2 56.4444 28.2222 0.030 0.970

Cavity1 9 8437.3750 937.4861 17.957 0.000

Part1 12 626.5000 52.2083 1.772 0.081

Position 48 1414.0000 29.4583

Total 71 10534.3194

Variance Components

Source Var Comp. % Total StDev

Time1 -37.886* 0.00 0.000

Cavity1 147.546 79.93 12.147

Part1 7.583 4.11 2.754

Position 29.458 15.96 5.428

Total 184.588 13.586

* Value is negative, and is estimated by zero.

22.6 Example 22.4:



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• In this analysis, variability between position was used to

estimate error.

• Using position measurements to estimate error, the p-value for

cavity is the only factor less than 0.05. We estimate that the

variability between cavities is the largest contributor, at most

80% of total variability.

• We also note that the percentage value for position has a fairly

high percentage value relative to time. This could indicate that

there are statistically significant differences in measurements

across the parts, which is consistent with our observation from

the multi-vari chart.

22.6 Example 22.4:



Documents

Chapter 22 Variance Componentsweb.eng.fiu.edu/leet/TQM/chap22_2012.pdf · 3/18/2013 1 Chapter 22 Variance Components Introduction • The test attempts to quantify the variability