International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 1900

ISSN 2229-5518

On the estimation of variance components in

Gage R&R studies using Ranges & ANOVA

B. Ramesh, K.V.S. Sarma

—————————— ——————————

HE determination of an outcome of a quality characteristic is called measurement. It is a knowledge which evaluates the unknown quality in-terms of numerical values. These measurement systems are used every day in manufacturing, research, development, sales and marketing. Measurement System Analysis (MSA) is designed to help engineers, quality professionals in assessing, monitoring and reducing the variation that includes features of a measurement system like linearity, stability, repeatability, reproducibility (Gage R&R) and the calibration of measurement equipment. Thus it is a vital component for many quality improvement initiatives. The Automotive Industry Action Group (2006) has prepared a manual to work out the analysis of a measurement system. The basic idea is to estimate the variation in the measured values that can be attributed to several factors like operator, equipment etc. The method ranges is one approach to estimate the variance components while Analysis of Variance (ANOVA) with random

effects model, is another way.

The process of analyzing any measurement system involves three key dimensions viz., operator, equipment and material.

The causes of variation due to these three key dimensions should be kept at minimum so that the efficiency of the measurement system is maximized. In this paper we compare three existing methods of estimating the variance components viz., i) AIAG method ii) ANOVA method and iii) Modified AIAG method.

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• B. Ramesh is currently pursuing Ph. D., degree in Statistics in Sri Venkateswara University, Tirupati, India-517502, PH-09963833322. E- mail: buttaramesh7@gmail.com

• K.V.S. Sarma is currently a professor in Statistics in Sri Venkateswara University, Tirupati, India-517502, PH-09393621901. E-mail: sarma_kvs@rediffmail.com

The complete MSA is carried out by studying the measurement system variation to understand the components of variation. Let X denote the measurement made on a part using the given measurement system. It is common to assume that X ~ N(µ,σ2) and this assumption holds good in most cases of large scale production.

Variation in measurements occur over a period of time or when other assignable causes like material changes, operator changes and changes in machine settings take place. The variation can be classified into a) changes in the location (mean value) of the process and b) changes in width (variation) of the process. These two concepts are elaborated below.

The changes that occur in the process average accounts for deviations in the quality. The following are some indices of location variation.

a) Accuracy (or) Bias - It is a measure of the distance (closeness

of agreement) between the average value of a large number

of observed values of the characteristic and the true value or

reference value. The reference value is obtained by a

standardized procedure with properly calibrated

equipment.

b) Stability - It is the consistency of the performance over time and indicates the absence of assignable causes of variation, leaving only random variation. It measures the change in bias over time. Stability refers to the difference in the averages of at least two sets of measurements obtained with the same gage on the same characteristics taken at different times.

c) Linearity - It measures the consistency of accuracy over the range of measurements. When the true value is high, the observed value also should be high and vice versa. Linearity is expressed as, │Slope│* Process variation, where

││ indicates absolute value.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 1901

ISSN 2229-5518

The total variation found in the measurement system can be divided into the following components.

Where a = number of operators, b= number of parts, n= number of replications and B(A) denotes the parts within operator.

If 2

σ Y denotes the total variance in the experimental data then it

a) Precision (closeness) – Precision is the standard

deviation of the measurement system. The smaller the

spread of the distribution, the better is the precision.

can be split into components as follows.

2 2 2 2

Precision can be separated into two components, called repeatability and reproducibility.

(σ Y ) = (σ A ) + (σ B ( A) ) + (σ ε )

(2)

2 2

b) Repeatability: Repeatability is the inherent variation within the measuring instrument and is represented by σ2repeatability. It is the variation due to measurement equipment obtained with one instrument used several times by one appraiser while measuring the parts. This is also known as Equipment variation (EV).

c) Reproducibility: Reproducibility is the variation due to differences in appraisers denoted by σ2reproducibility. It is the variation in the average of measurements made by different operators using the same equipment when measuring the same characteristics on the same part. This is also known as Appraiser variation (AV).

d) Gage R&R - Gage Repeatability and Reproducibility

(R&R) is a measure which represents the variation due to the measurement system as a whole. It determines

σ ε is called σ repeatability or Equipment variation (EV).

We have considered three methods to estimate the variance components and are as follows.

There are three methods in use, for estimating the variance components in a measurement system. In addition to these, there are three ratios used to evaluate the performance of the measurement system. We call them vital ratios discussed in detail below.

how much of the observed process variation has

occurred due to the measurement system variation. It is

the combined estimate of R&R and denoted by

2 2 2

normally distributed data, the ratio R is an estimate of the

*d *2

process standard deviation σ, where R is the mean of ranges in the subgroups. For different sizes of the sub groups the values

σ Gauge + σ Re peatability + σ Re producibility

of d are tabulated. Another related constant is d *

which is a

The problem is to estimate the variance components either by generating data through an experiment or by collecting data from the production line using the measurement system.

function of the levels of the factor (like operator) in the measurement system. The value of both d and d * for different sample sizes were provided by Duncan (1955). For n ≥ 25 , the

*

constant d2 shall be used and for n < 25 the constant d 2

shall be

*

Any item from the process on which a measurement is made by the operator with given equipment is called a part. The total variation in the measurement could be due to part, due to equipment or due to operator. These three components shall be estimated by observing the process over a period of time or by conducting an experiment. Factorial experiments with random effects model are used to estimate the variance components. Let

the system contains two factors i) Operator (A) and ii) Part (B)

used. The constant for the factor equipment is denoted by d 2,e

and for parts it is d * .

We consider a measurement system with a operators

(appraisers) and b parts and n replications (trials).

The R method makes use of means of sample ranges R to

a

R __ __ ∑ Ri

which contribute to variation, apart from random variation. Then the variation due to A and variation due to B within A,

estimate σe as

σˆ e =

d 2

where*R *= i =1 and

a

denoted by B(A) represents σ2reproducibility and the variation

due to the experimental error (residual) denotes σ2repeatability.

R = Max(X

)− Min(X )

for the ith sample for j=1,2,...,b.

*

Let Yijk denote the kth value obtained on the jth part by the ith operator. Then the design for this experiment can be expressed by the two way random effects model given by,

From the tables of d2 values we get d 2,a

and a = 3 respectively.

= 1.41 and 1.91 for a = 2

Yijk = µ + Ai +

B(A) j(i)

*i *= 1,2,..., *a*

+ ε *j *= 1,2,..., *b*

*k *= 1,2,..., *n*

(1)

The sample lay out with a = 3, b = 5 and n = 2 appears as shown in Table-1 where X denotes the actual measurement made on each part, in suitable units.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 1902

ISSN 2229-5518

Since each part is measured twice, the ranges within each part are calculated for each operator and the average of these ranges

is found to give R = 4.267. Since all the 30 data values are treated as a single group we have n > 25 and hence d2 = 1.128.

Wheeler (2009) has provided the following estimators.

Wheeler (2009) observed that these VRs are not ratios in a proper sense as they would not add to unity and proposed the following modified Vital Ratios which can be interpreted in a nice way.

a) VRrepeat = EV2/TV2 in place of EV/TV b) VRreprod = AV2/TV2 in place of AV/TV c) VRpart = PV2/TV2 in place of PV/TV d) VRGRR = GRR2/TV2 in place of GRR/TV

Ermer (2006) observed that the use of ratio of variances in place

of the ratio of standard deviations is justified due to the fact that

[σ2a+b = σ2a + σ2b ]but [ σa+b ≠ σa + σb ] and the same can be established by the Pythagorean theorem. Wheeler (2006) also showed that the VR method based on the standard deviations leads to trigonometrically derived quantities which do not add to unity. However it can be seen that (VRrepeat +VRreprod +VRpart

) = (VRGRR + VRpart ) = 1. For the data given in illustration-1 we get the results as shown in Table-2.

1. Equipment Variation (EV) =

R

σˆ Re peatability = σˆ e =

d 2

2. Appraiser Variation (AV) =

2

σ σ *R*a

*a * σ 2

ˆ Re producibility =

ˆ a =

*d **

− *a*.*b*.*n *[ ˆ e ]

It can be seen that the ratios in the fourth row of Table-2 do not

2,a

add up to unity while the modified ratios in the fifth row add up to unity.

Where

Ra is the range of operator averages. For some data

We have got VRpart = 0.944 which means that 94.4% of the

values, the content inside the square root turns out to be

negative in which case it is reset to zero.

variation in the measurements can be attributed to parts (production) while the equipment variation accounts for 2.5% and the appraiser variation consumes 3.1%. The sum of these

3. Gage R&R (GR&R) = σˆ

R&R =

(σˆ e

)2 + (σˆ )2

2

ratios adds up to unity.

Now we consider another method called WR method to

4. Part Variation (PV) = σˆ p =

*R *p

*d **

where R p is

estimate the variance components by using within sample ranges.

the range of the parts.

2, p

5. Total Variation (TV) = σˆT =

(σˆ e )

+ (σˆ a )

+ (σˆ )2

= (σˆ

R&R

)2 + (σˆ )2

In this method σˆ Re peatability is estimated by taking the average of the ranges within parts instead of taking the average range from

After estimating these variance components, we express each of them as a percentage of TV. We get three Vital Ratios (VR).

the entire data. For each operator, the average of the ranges

within parts is calculated. The average of these averages is used

*

Each VR represents the portion of the total variation attributable to the factor under consideration. These ratios can be expressed as percentages. The AIAG group has used the ratios as EV/TV, AV/TV, PV/TV and GRR/TV for the equipment, operator, product and R&R respectively.

to find the EV. It also makes use of the constant

values are available in Duncan (1955).

The variance components in this case are as follows.

d 2 whose

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International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 1903

ISSN 2229-5518

1. σˆ = *WR*

2,*e*

where WR is the average of the averages of

range within parts.

2

*R*a [σ__ˆ __e ]

2. σˆ a =

*d*

2,*a *

2

*bn *

It can be seen that the sum of modified VR’s is unity. Closer this

3. σˆ

R&R =

(σˆ e

)2 + (σˆ )2

2

sum to unity better is the method estimating the variance components.

p [σˆ e ]

In a normally distributed data the range method, though

4. σˆ p =

*R *

*d *

2

*an *

unbiased, is not the only way of estimating the variance

5. σˆT =

(σˆ

*

2, *p *

)2 + (σˆ

)2 + (σˆ

)2 = (σˆ

R&R

)2 + (σˆ )2

components. The Analysis of Variance (ANOVA) with random effects model is a known procedure. In the following section, we consider Nested ANOVA method and compare the estimate by the three methods.

For some data values, the content inside the square root turns

out to be negative in which case AV and PV are set to zero.

For the case of EV, the numerator is based on m = 5 x 3 = 15

According to Montgomery (2002) , in an experiment if the levels

ranges (< 30) and with n = 2 replications we get

*

2,*e*

= 1.15.

of one factor are similar but not identical for different levels of

another factor then such an arrangement is said to be nested

Similarly we get d *

(b = 5).

= 1.91 for AV (a = 3) and d *

= 2.48 for PV

design. Here we have the levels of factor B nested under the levels of factor A. There are b parts of raw materials available from each operator (a), and n trials are to be taken for each part. This is a two-stage nested design, with parts nested within operators.

Reconsider the data in Table-1 of illustration-1. The method of using within ranges of parts produces the following intermediate calculations.

We use Two-way Nested ANOVA with random effects model to estimate the variance components.

Define

Then

yi.. = mean of the ith operator y.j.. = mean of the jth part

y… = overall mean of all observations.

yij. = mean of observations at the ith operator and jth

part.

1. Sum of squares (SS) due to operator (SSA) =

a *y *2

*y *2

∑ i.. − ...

i=1 *bn *

*abn *

From the Table-3 we get the following intermediate calculations.

2. SS due to parts within operator within A (SSB(A) )=

a b 2 a 2

y

__= __ ij .

*y*i..

a) The average of WR gives WR = 4.267

∑ ∑

*n * − ∑ *bn *

i=1

j =1

i=1

b) The range of operator averages becomes 8.5

c) Each part appears twice with each operator and hence

3. SS due to Residual (SSЄ ) =

a b n

a

b 2

the average of X is based on 6 observations and hence we get five

(y 2 ) −

y

__=__ij .

values of part averages as {158.0, 206.2, 182.0, 184.8, 148.0} so that

∑ ∑ ∑

ijk

∑ ∑ *n *

the range of these averages is 58.167.

i=1

j =1 k =1

i=1

a b

j =1

n

*y *2

The summary of results including the modified VRs is shown in

4. Total SS (SST ) =

∑ ∑ ∑ (*y*

2

ijk

) − ...

Table-4.

i=1

j =1 k =1

*abn *

The corresponding mean sum of squares (MSS) are obtained by dividing each SS by the corresponding degrees of freedom.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 1904

ISSN 2229-5518

The variance components are given by

1. σˆ e = [*MSS*ε ]

*MSS *A − *MSS *B A

standard deviations is not a correct method, though the percentages individually appear to be correct.

2. σˆ 2 = ( )

*na *

Minitab provides a module for Gage R & R studies using nested

ANOVA. We have developed a template in Excel to work out

*MSS *B A

− *MSS*ε

3. σˆ 2 = ( )

the calculations for a = 3, b = 5 and n = 2 as shown in Figure-1.

4. R&R

= (σˆ e

n

)2 + (σˆ )2

In the Minitab output the Gage R&R section produces variance

components and % contribution of each component to total

variance. These are nothing but the VRs defined in section 4.1.

5. σˆ 2 = (σˆ

)2 + (σˆ

)2 + (σˆ

)2 = (σˆ

R&R

)2 + (σˆ )2

A single index of measurement system performance is the

Reconsider the data given in illustration-1. By using Nested– ANOVA method the following output is found.

r - coefficient given by,

( σˆ 2 + σˆ 2 )

*r * e a

( σˆ 2 + σˆ 2 + σˆ 2 )

e a p

The VRs are found in the usual way for which the ratio property

σˆ 2 + σˆ 2 + σˆ 2

Table-7 shows the convention used to classify a process into one of the three categories.

holds well. It can be seen that e

a

σˆ 2

p = 1 which

means that the relative utilities add up to unity which describes that how strongly the units in the same group resemble each other and is used to study the performance of a measurement system.

Remark: It is possible that for certain data sets a vital ratio turns

out to be more than unity, in which case it is reset to unity.

We considered a = 3, b = 5 and n = 2 and generated a random sample from N(µ,σ2) by using Data Analysis Pak of Excel with given mean (µ) & Standard Deviation (σ). For µ = 175 & σ = 2.5 the variance components and the vital ratios are shown in Table-

6.

It follows that both ANOVA and WR methods produce meaningful percentage contributions of different components to the measurement system variation. The method of using ratio of

The value of r is 0.24 by the R – method and WR – method but it is only 0.15 by ANOVA. However by all the three methods the process is classified as marginal variation in the measurement system.

The computation of above mentioned three methods is shown in

Figure-1.

From the above study we observe the following.

1) More variation is due to EV when compared with AV &

PV.

2) The Equipment Variation needed improvement.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 1905

ISSN 2229-5518

3) ANOVA gives the exact ratio than R and WR method.

4) By posting the experimental data into the column-D of the Excel template, the calculations are automatically changed.

The first author acknowledges the financial support by UGC- BSR.

[1] D.S. Ermer, “Improved Gage R&R measurement studies,” part one, Quality Progress, 2006.

[2] Easy GR&RTM for windows, Gage Repeatability & Reproducibility Analysis Software, User’s Manual Version 3.0, Distributed by Math Options Inc for Easy Software, 888 – 764 - 3958, 1994 – 2001 Easy Software, www.MathOptions.com.

[3] D.S.Ermer, “Appraiser variation in Gage R&R

measurement,” part two, Quality Progress, 2006.

[4] D.C. Montgomery, “Design and analysis of experiments,” third edition, John Wiley & sons, pp. 440-

444, 1991.

[5] D.J. Wheeler, “An honest Gage R&R study,” SPC

press, 2009.

[6] D.C. Montgomery, “Introduction to Statistical Quality

Control,” second edition, John Wiley & sons, pp. 369-

396, 2007.

[7] Measurement System Capability manual, “Measurement System Analysis,” Southfield, MI, Automotive Industry Action Group (AIAG) manual, http://www.6sigma.us/MeasurementSystemsMSA/me asurement-systems-analysis-MSA-p1.html.

[8] A.J. Duncan, “Quality Control and Industrial Statistics,” A.J. Duncan, 1969.

[9] H.J, Mittag, and H. Rinne, “Statistical methods of

Quality Assurance,” Chapman & Hall, 1993.

[10] K.V.S. Sarma, “Statistics made simple – Basic concepts,”

2007.

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