Comparing two measurement devices
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Comparing Two Measurement Devices. Brian Novatny 2003. Each measurement made by an instrument or measuring device consists of the true, unknown level of the characteristic or item measured plus an error of measurement.

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Background

Each measurement made by an instrument or measuring device consists of the true, unknown level of the characteristic or item measured plus an error of measurement.

In practice it is important to know whether or not the variance in errors of measurement of an instrument, or the imprecision of measurement, is suitably small as compared to the variance of the characteristic or product measured, or the product variability.

Background


Background1

For efficiency of the measuring process, the variance in errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

High measurement error causes the power of most statistical tests to decrease unless compensated for by larger sample sizes

Power = 1 - Beta

Beta is the Type II error

Background


Measurements are indicated by
Measurements are indicated by errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Device 1 = B1 + Xi + Ei1

  • where B1 = bias for device 1

  • Xi = true value

  • Ei1 = random errors for device 1

  • Device 2 = B2 + Xi + Ei2


Regression approach
Regression Approach errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Y = b0 +b1*X

  • Device 1 = Intercept + Slope * Device 2

  • Intercept value should be zero, if not, it indicates bias of the two devices

  • Slope term should be around 1 indicating “Similarity”

  • Mean Square Error estimate “Precision”

  • R-Square term estimates some measure of strength


Problems with regression
Problems with Regression errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • The X variable (independent variable) should be measured without error

    • this is never the case, but errors in the X variable should be small, and it won’t be when comparing devices

    • they will be on equal footing

  • Asymmetry - specifically designate one device to predict the other

    • prefer symmetric approach where is doesn’t matter which variable is the input and which is the output

  • Inverse regression can pose several problems when trying to resolve the asymmetry problem


Problems with regression1
Problems with Regression errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • What should the R-square value be?

    • No objective justification

  • How to handle the case when there are multiple measuring devices?

    • Pairwise comparisons

    • multiple testing error problem


Some solutions
Some Solutions errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Grubbs model

    • uses sums and differences

  • Pittman and Morgan along with Maloney and Rastogi

    • provide proofs and refinements to Grubbs model

  • Blackwood and Bradley

    • multivariate test on bias and precision

  • Tan and Iglewicz modify the standard regression approach based on Mandel’s work of Errors in Variables


Grubbs model
Grubbs Model errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Involves calculating the Sums and Differences of the two devices

  • The differences will be used to estimate bias

    • standard paired t-test with n-1 degrees of freedom

  • Performing a correlation analysis analysis on the sums and differences is used to estimate precision of the two devices

    • follow student’s t distribution with n-2 degrees of freedom

  • Provides two independent tests

    • one for bias of the two measurement devices

    • one for precision equivalency


Simultaneous test for precision and bias
Simultaneous Test for Precision and Bias errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Multivariate approach instead of independent tests

  • Differences are regressed on the Sums

    • Difference = Intercept + Slope * Sums

  • Model F-test uses the UNCORRECTED SUMS OF SQUARES to get the correct number for the df

    • instead of the familiar corrected sums of squares

    • the overall Type I error (Alpha) rate is exact


Simultaneous test for precision and bias1
Simultaneous Test for Precision and Bias errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • If the Model F indicates significance, then tests for the Bias and Precision are just the individual F-tests

    • the overall test is generally more powerful

    • it can reject the equivalence assumption of the two devices even though each individual test does not


Simultaneous test for precision and bias2
Simultaneous Test for Precision and Bias errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • The Precision test is exact

  • The Bias test is exact only when the precision between the two devices is equal

    • use the paired difference t-test otherwise

    • more powerful

    • Uses UMVU (uniform minimum variance unbiased) estimate of the variance


Advantage of simultaneous test
Advantage of Simultaneous Test errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Type I Error Rate is exact

  • Overall test could reject even though individual test do not

    • power of test

  • Statistical modeling

    • usual array of diagnostics

      • residuals


Regression approach1
Regression approach errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Regression can still be used, but an adjustment to the model must be made

  • In Simple Linear Regression, the results (Beta Hat) are achieved by minimizing the sum of squared residuals in the direction of the dependent variable

  • The correction is to achieve Beta hat by minimizing the sum of squared residuals in the direction of -Lambda/Beta hat


Regression approach2
Regression approach errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Lambda is a calculated value to adjust the slope calculations

    • called the Precision Ratio

      • ratio of the machines repeatability

      • machine 1 / machine 2 or inverse

  • Lambda is determined by performing the standard Gauge RxR studies or taking repeated values


Regression approach3
Regression approach errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • The approach still uses regression and Lambda value helps solve the asymmetry problem

    • as lambda approaches infinity, implies X values approach 0, so Beta hat is Sxy/Sxx, which is where X is the independent variable

    • as lambda approaches 0, implies Y values approach 0, so Beta hat is Syy/Sxy, which is where Y is the independent variable


Regression approach4
Regression approach errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Approach handles precision but not bias

  • Uses Polar coordinates for confidence intervals

    • Slope = TAN (Theta)

    • Intercept = Tau/COS (Theta)


Example
Example errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

Device1 Device2 Sum Diff

5.00 4.73 9.73 0.27

5.17 4.83 10.00 0.33

5.17 4.63 9.80 0.53

5.00 4.37 9.37 0.63

8.50 7.03 15.53 1.47

5.67 4.50 10.17 1.17

8.00 7.03 15.03 0.97

9.00 7.93 16.93 1.07

8.50 7.50 16.00 1.00

11.17 9.57 20.73 1.60

9.00 7.70 16.70 1.30


Standard regression analysis
Standard Regression Analysis errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • R-Square = 98.24%

  • SQRT MSE = 0.31

  • Device 2 = -0.28 + 1.2 * Device 1

  • 95% Confidence Intervals

    • Intercept (-1.07,0.52)

    • Slope (1.07,1.31)

  • Conclusion

    • No Bias, but not similar


Simultaneous test for bias and precision
Simultaneous Test for Bias and Precision errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Regress Diff on Sums (Uncorrected SS)

  • Model F = 73.22 ==> p-value = 0.0000027

  • Therefore, devices are different relative to their bias and precision

  • Individual Precision Test

    • F = 17.5 ==> p-value 0.0024

  • Individual Bias Test (not exact)

    • F = 128.93 ==> p-value = 0.0000012

  • Paired t-test has a T value = 6.98 and p-value less then 0.00001


Correct conclusion
Correct Conclusion errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • The two devices measure differently

  • Strong Bias

  • Strong lack of precision (repeatability)


Potential problems
Potential Problems errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Methods do not account for a difference in Gain, or slope of devices

  • Devices might measure equally well or poor at the low and high ends of the scale, but the relationship is not constant

    • collect data at one end of the data range

    • power of the test could be compromised


Multiple measuring devices
Multiple Measuring Devices errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Grubbs and others propose technique for three measuring devices

    • comparisons when one device is a “Standard”

    • with three devices, get a more powerful test

  • Multivariate methods lead to fuller choice of sub-hypothesis and can be used regardless of the number of measurement devices


Multiple measuring devices1
Multiple Measuring Devices errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • One method involves performing a multivariate regression on q-1 measurement devices

    • independent variable = mean of each part

    • dependent variable = deviations from that mean

  • Independent variable is averaged each part across all the measurement devices

  • Dependent variable is calculated by the differences of each value from that mean


Multiple measuring devices2
Multiple Measuring Devices errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Generally have to fit a Full model and a Reduced model (intercepts only)

  • Then compare the two models

    • usually through some matrix manipulation

  • Technique can be performed by most software packages that can perform MANOVA techniques


Authors opinion
Authors Opinion errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • As the title says, this is just my opinion and not based on any concrete proof

    • such as simulation studies

  • My preferred method of analysis would be the Multivariate approach using Blackwood and Bradley’s Regression with the Uncorrected Sums of Squares

    • this procedure seems to have a more powerful test in finding differences

    • eliminates the possibility of getting a negative variance, which Grubbs method could get


Authors opinion1
Authors Opinion errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • With the Multivariate approach, there is a natural extension to testing more than two measuring devices

  • Of course, there is no reason to try both the multivariate approach and Grubbs approach since they are easily computed using standard data analysis techniques


Final comments
Final Comments errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • These methods are not to replace Gage RxR studies, but to evaluate two devices against each other

  • Each device should be tested for bias and repeatability and linearity as desired

    • corrective action should be taken as needed


Final comments1
Final Comments errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • The test for Bias is only a test for agreement between the two devices, not a bias against a standard

    • both devices could be grossly off from the standard (but in the same direction and amount)

  • If there is a claim that one device is “superior” to another (better precision), these methods could prove the validity of the claim and provide the precision estimates


References for two device comparison
References for Two Device Comparison errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Grubbs, F.E. (1973). “Errors of Measurement, Precision, Accuracy and the Statistical Comparison of Measuring Instruments” , Technometrics Vol. 15 pp. 53-66

  • Bradley, E.L. and Blackwood, L.G (1991). “An Omnibus Test for Comparing Two Measuring Devices”, Journal of Quality Technology, Vol. 23 pp. 12-16

  • Tan, C.Y. and Iglewicz, B. (1999). “Measurement-Methods Comparisons and Linear Statistical Relationship”, Technometrics, Vol. 41 pp. 192-201


References for multiple device comparison
References for Multiple Device Comparison errors of measurement should be several or many times smaller than the variability of the characteristic measured or product variance.

  • Christensen, R. and Blackwood, L.G (1993). “Tests for Precision and Accuracy of Multiple Measuring Devices”, Technometrics, Vol. 35 pp. 411-420

  • Bedrick, E.J. (2001). “An Efficient Scores Test for Comparing Several Measuring Devices”, Journal of Quality Technology, Vol. 33 pp. 96-102



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