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Gauge R&R

Measurement systems analysis after the AIAG MSA Reference Manual (4th edition): a crossed gauge repeatability and reproducibility study by the Average and Range method (equipment, appraiser and part variation from R̿, X̄diff and Rp with the K1, K2, K3 constants of the d2* table, %EV, %AV, %GRR, %PV, %Tolerance and the number of distinct categories, with the X̄ and R charts by appraiser) or by two-way ANOVA with the appraiser × part interaction (kept or pooled at the 0.25 rule, variance components and %Contribution), an attribute agreement analysis (within-appraiser, between-appraiser and versus-standard agreement with binomial intervals, Fleiss' and Cohen's κ, AIAG effectiveness, miss and false-alarm rates), a linearity and bias study (bias at each reference with its t test, the regression of bias on the reference value with confidence bands, the linearity as a share of the process variation, and the independent-sample bias study with the d2* repeatability estimate) and a Type 1 gauge study (Cg, Cgk, %Var). The default is the manual's own three-appraiser, ten-part, three-trial example, whose report (EV 0.2019, AV 0.2296, GRR 0.3058, %GRR 26.7 %, ndc 5) and ANOVA (%Study Variation 27.9 %, ndc 4) the page reproduces; the linearity, bias and attribute boxes carry the manual's linearity and bias examples and Fleiss' worked table.

  • Average and Range method with the d2* constants: EV, AV, PV, GRR, TV, %Study Variation, %Tolerance, %Process and ndc, with the AIAG verdict bands

  • ANOVA method with or without the appraiser × part interaction (0.25 rule), variance components, %Contribution and the X̄ / R / interaction charts

  • Attribute agreement: within, between and versus the standard with Wilson or exact intervals, Fleiss' and Cohen's κ, effectiveness, miss and false-alarm rates

  • Linearity and bias (regression of bias on the reference with confidence bands, t tests, %Linearity; independent-sample bias study) and the Type 1 study Cg / Cgk

Gauge R&R

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Gauge R&R: Average and Range, ANOVA, Attribute Agreement, Linearity and Bias, Type 1 | Method page
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Gauge R&R

Gauge R&R: Average and Range, ANOVA, Attribute Agreement, Linearity and Bias, Type 1

Why a measurement system is studied

Every recorded value is the part plus the measurement. A control chart or a capability index computed from such values carries the gauge's own variation inside it, and a gauge that spreads a part's reading over a third of the tolerance will pass bad parts and reject good ones whatever the process does. Measurement systems analysis (MSA) separates the variation of the measurement system from the variation of the parts and asks three questions of the system: is it stable (statistical control over time), is it accurate (bias and linearity against a reference), and is it precise (repeatability and reproducibility). The Automotive Industry Action Group's reference manual (AIAG MSA, 4th edition, 2010) is the common procedure in manufacturing; this page follows its Chapter III for variables gauges and attribute gauges, and adds the Type 1 study of the German automotive practice.

Repeatability, reproducibility and the crossed study

Repeatability (equipment variation, EV) is the spread of repeated readings of one part by one appraiser with one gauge: the noise of the instrument, its fixture and the method. Reproducibility (appraiser variation, AV) is the spread of the appraisers' averages: technique, reading of a scale, handling. Gauge R&R is their combination in quadrature, GRR² = EV² + AV², and it is compared with the part-to-part variation PV and the total variation TV² = GRR² + PV², or with the tolerance. The standard crossed study has every appraiser (two or three) measure every part (ten) several times (two or three) in random order, the parts chosen to span the process range; "crossed" because each appraiser sees each part, which lets the appraiser × part interaction be separated. The data box takes the AIAG data sheet layout (one line per appraiser and part with the trials across) or one measurement per line.

Average and Range method

The method of the AIAG report form (Table III-B 6b) estimates each standard deviation from a range through the constants d2 and d2*. For a single range of m values the expected range is d2(m) σ, so σ̂ = R/d2; when the estimate is a mean of g ranges the small-sample correction of Duncan's Table D3 (David 1951) is used, d2*(m, g) = √(d2² + d3²/g), which tends to d2 as g grows (the AIAG table switches to d2 for g > 15). The tool computes d2 = E[R]/σ and d3 = sd(R)/σ by numerical integration of the range distribution of normal samples and reproduces the Appendix C table to five decimals.

  • EV = R̿ K1, where R̿ is the mean of the appraisers' average ranges and K1 = 1/d2 for the number of trials (0.8862 for two, 0.5908 for three, the report form's fixed constants; the option "Appendix C" uses d2*(trials, appraisers × parts) instead, which differs only in small studies).
  • AV = √((X̄diff K2)² − EV²/(n r)), where X̄diff is the largest minus the smallest appraiser average, K2 = 1/d2*(appraisers, 1), n the parts and r the trials. The subtraction removes the repeatability that the appraiser averages carry; a negative radicand means the appraisers differ less than the noise predicts and AV is set to zero.
  • PV = Rp K3 with Rp the range of the part averages and K3 = 1/d2*(parts, 1). When a historical process standard deviation is entered, TV is taken from it and PV = √(TV² − GRR²), as the form allows.
  • %EV, %AV, %GRR and %PV are ratios of standard deviations to TV (they do not add to 100). The study variation is 6σ (99.73 % of a normal spread; the 3rd edition used 5.15σ, 99 %), and %Tolerance compares 6σ with USL − LSL.
  • ndc = 1.41 PV/GRR, truncated: the number of non-overlapping 97 % confidence groups the gauge can resolve across the part range (√2 PV/GRR). Five or more is wanted for process control.

The X̄ chart by appraiser plots the part averages with the limits X̿ ± A2 R̿; because those limits are the repeatability band, a usable gauge has most averages outside them (the parts differ by more than the measurement noise), the reverse of a process chart. The range chart with UCL = D4 R̿ exposes a trial that differs from its mates (reread, retrain); the interaction plot of appraiser averages against the part shows parallel lines when no interaction exists.

ANOVA method

The two-factor random-effects model y = μ + appraiser + part + (appraiser × part) + error gives, from the expected mean squares, the components σ²rep = MSE, σ²int = (MSAP − MSE)/r, σ²app = (MSA − MSAP)/(n r) and σ²part = (MSP − MSAP)/(o r). The AIAG rule tests the interaction against the error at α = 0.25; when it is not significant the interaction sum of squares is pooled into the error and the components are taken from the pooled mean square. Reproducibility is appraiser plus interaction; the percentages follow as in the range method, with %Contribution (ratio of variances, adding to 100) added. The ANOVA method separates the interaction, uses all the data rather than ranges, and is the one most software reports; on the AIAG example the two methods give %GRR of 26.7 % (range) and 27.9 % (ANOVA) and ndc of 5 and 4.

Acceptance bands

AIAG (Chapter II Section D): %GRR under 10 % of the study variation (or of the tolerance) is acceptable; 10 to 30 % may be acceptable depending on the importance of the characteristic, the cost of the gauge and of repairs; over 30 % is unacceptable and the system should be improved. The percentages compared with the total variation judge the gauge for process control; those compared with the tolerance judge it for inspection against the specification. When repeatability dominates, the instrument or the fixture is the problem; when reproducibility dominates, training and the method instruction are.

Attribute agreement analysis

When the gauge is a judgement (pass / fail, a grade of 1 to 5), AIAG Chapter III Section C compares appraisers with each other and with a known standard. A part counts as agreement only when every rating of the set is the same: within an appraiser (all trials agree, the repeatability of the judgement), between appraisers (all ratings of all appraisers agree), each appraiser versus the standard (every trial correct) and all appraisers versus the standard. Each proportion carries a 95 % interval, Wilson's score interval or the exact Clopper–Pearson one that AIAG and Minitab print. Agreement beyond chance is measured by kappa: Fleiss' κ = (P̄ − P̄e)/(1 − P̄e) over all ratings of a set, where P̄ is the mean pairwise agreement over the parts and P̄e = Σ pj² the agreement expected from the category shares; Cohen's κ for one appraiser against the standard or against another appraiser, with the Fleiss–Cohen–Everitt standard error. Landis and Koch's labels (slight, fair, moderate, substantial, almost perfect) are shown; AIAG calls κ above 0.75 good and below 0.40 poor. For a binary rating with a reject code the AIAG rates are added: effectiveness (parts rated correctly on every trial), miss rate (nonconforming parts accepted, over the nonconforming decisions) and false-alarm rate (conforming parts rejected, over the conforming decisions), with the bands 90 / 80 %, 2 / 5 % and 5 / 10 %.

Linearity and bias

Bias is the difference between the mean reading and the reference value; linearity is the change of bias over the operating range. The AIAG linearity study measures five or more reference parts across the range ten or more times each, computes the bias of every reading, and fits bias = a + b × reference by least squares over all readings. The slope b is the linearity (as a share of the process variation, %Linearity = 100 |b|; as a value, |b| × process variation); t tests of the slope and the intercept against zero use the residual standard deviation s with n − 2 degrees of freedom, and the gauge is linear when the line bias = 0 lies inside the 95 % confidence band of the fit, ŷ ± t s √(1/n + (x − x̄)²/Sxx), over the whole range. The independent-sample bias study takes ten or more readings of one reference: σr = R/d2* from the range (d2* for n values in one range), σb = σr/√n, t = bias/σb, and the bias is acceptable when zero lies in bias ± (d2/d2*) σb tν,0.975, ν being the degrees of freedom of the range estimate from Appendix C (Patnaik's approximation). The same test with the sample standard deviation and n − 1 degrees of freedom is shown beside it.

Type 1 gauge study

Before an R&R study, one appraiser reads one reference part 25 to 50 times. With K the share of the tolerance granted to the gauge (20 % by convention) and L the number of standard deviations taken as its spread (6), Cg = (K/100) T/(L s) compares the gauge spread with that share, and Cgk = ((K/200) T − |x̄ − xm|)/(L s/2) also charges the bias. Both should reach 1.33 (Dietrich and Schulze; Bosch Booklet 10; VDA 5). %Var is the share of the tolerance the gauge consumes, L s/T, and K/Cgk the share consumed with bias. The t test of the bias uses the sample standard deviation.

Reading the results

Read the range chart first: a trial beyond the limit is a reading to check before anything is concluded. Then %GRR against the bands, and ndc; a system that is marginal for process control may still be acceptable for inspection when its %Tolerance is small, and the reverse. The components say where to work: repeatability on the instrument and fixture, reproducibility on the appraisers, an interaction on part features that some appraisers read differently. The attribute study is read the same way: within-appraiser agreement is repeatability, agreement with the standard is accuracy, and the miss rate is the risk the customer carries. Linearity and bias are calibration questions and are corrected before the R&R numbers are trusted.

Limits

The variables study must be balanced (every appraiser measures every part the same number of times, at least twice). Confidence intervals on the R&R components (Burdick, Borror and Montgomery) and the EMP method of Wheeler and Lyday are not computed. The attribute study treats ratings as nominal (no weighted kappa, no Kendall coefficients for ordered grades). The Type 1 study covers one appraiser and one part only. Stability of the gauge over time (a control chart of a reference) is the SPC and Capability tool's business.



References

Automotive Industry Action Group (2010). Measurement Systems Analysis Reference Manual, 4th ed. AIAG, Southfield, MI.

Duncan, A. J. (1974). Quality Control and Industrial Statistics, 4th ed. Richard D. Irwin, Homewood, IL.

David, H. A. (1951). "Further Applications of Range to the Analysis of Variance." Biometrika, Vol. 38, No. 3/4, pp. 393–409.

Patnaik, P. B. (1950). "The Use of Mean Range as an Estimator of Variance in Statistical Tests." Biometrika, Vol. 37, No. 1/2, pp. 78–87.

Montgomery, D. C. (2013). Introduction to Statistical Quality Control, 7th ed. Wiley, Hoboken, NJ.

Burdick, R. K., Borror, C. M., and Montgomery, D. C. (2005). Design and Analysis of Gauge R&R Studies: Making Decisions with Confidence Intervals in Random and Mixed ANOVA Models. SIAM, Philadelphia, PA.

Wheeler, D. J., and Lyday, R. W. (1989). Evaluating the Measurement Process, 2nd ed. SPC Press, Knoxville, TN.

Fleiss, J. L. (1971). "Measuring Nominal Scale Agreement among Many Raters." Psychological Bulletin, Vol. 76, No. 5, pp. 378–382.

Fleiss, J. L., Nee, J. C. M., and Landis, J. R. (1979). "Large Sample Variance of Kappa in the Case of Different Sets of Raters." Psychological Bulletin, Vol. 86, No. 5, pp. 974–977.

Cohen, J. (1960). "A Coefficient of Agreement for Nominal Scales." Educational and Psychological Measurement, Vol. 20, No. 1, pp. 37–46.

Fleiss, J. L., Cohen, J., and Everitt, B. S. (1969). "Large Sample Standard Errors of Kappa and Weighted Kappa." Psychological Bulletin, Vol. 72, No. 5, pp. 323–327.

Landis, J. R., and Koch, G. G. (1977). "The Measurement of Observer Agreement for Categorical Data." Biometrics, Vol. 33, No. 1, pp. 159–174.

Wilson, E. B. (1927). "Probable Inference, the Law of Succession, and Statistical Inference." Journal of the American Statistical Association, Vol. 22, No. 158, pp. 209–212.

Clopper, C. J., and Pearson, E. S. (1934). "The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial." Biometrika, Vol. 26, No. 4, pp. 404–413.

Newcombe, R. G. (1998). "Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods." Statistics in Medicine, Vol. 17, No. 8, pp. 857–872.

Dietrich, E., and Schulze, A. (2011). Measurement Process Qualification: Gage Acceptance and Measurement Uncertainty According to Current Standards, 3rd ed. Hanser, Munich.

NIST/SEMATECH (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology, Gaithersburg, MD.

Marsaglia, G. (2004). "Evaluating the Normal Distribution." Journal of Statistical Software, Vol. 11, No. 4, pp. 1–11.

Worked Examples

Published examples this tool reproduces; each page lists the inputs to enter and the printed answers.

All worked examples

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