chapter 22 variance componentsweb.eng.fiu.edu/leet/tqm/chap22_2012.pdf · 3/18/2013 1 chapter 22...
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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:
Variance Components of Pigment Paste
• 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:
Variance Components of Pigment Paste
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22.3 Example 22.1:
Variance Components of Pigment Paste
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:
Process Capability/Performance Indices Study
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:
Process Capability/Performance Indices Study
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/Performance Indices Study
• 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
22.5 Example 22.3: Determining Process
Capability/Performance Metrics using
Variance Components
22.5 Example 22.3: Determining Process
Capability/Performance Metrics using
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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15.3 Example 15.1: Multi-Vari Chart of Injection Molding Data
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
15.3 Example 15.1: Multi-Vari Chart of Injection Molding Data
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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:
Variance Components Analysis of
Injection-Molding Data
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:
Variance Components Analysis of
Injection-Molding Data
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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:
Variance Components Analysis of
Injection-Molding Data