Calibration StudiesRegression Analysis and CalibrationIE 361 Module 7Calibration Studies and Inference Based on Simple Linear RegressionReading: Section 2.5 of Revised SQAMEProf. Steve Vardeman and Prof. Max MorrisIow a State Univer sityVardeman and Morris (I owa State Uni versity) IE 361 Module 7 1 / 13Calibration StudiesCalibration is an essential activity in the quali…cation and maintenance ofmeasurement devices or systems. The basic idea is that one uses ameasurement device to produce measurements on "standard" specimenswith (relatively well-) "known" values of a measurand, and sees how themeasurements compare to the measurand. In the event that there aresystematic discrepancies between what is known to be true and what thedevice reads, the plan is then to invent some conversion scheme to (infuture use of the device) adjust what is read to something that is hopefullycloser to the (future) truth. A slight extension of "regression" analysis(curve …tting) from an elementary statistics course is the relevantstatistical methodology in making this conversion.Vardeman and Morris (I owa State University) IE 361 Module 7 2 / 13Regression Analysis and CalibrationCalibration studies produce "true"/gold-standard-measurement values xand "local" measurements y and seek a "conversion" method from y to x.(Strictly speaking, y need not even be in the same units as x.) Regressionanalysis can provide both "point conversions" and measures of uncertainty(the latter through inversion of "prediction limits").The simplest version of this is the case wherey  β0+ β1xThis is "linear calibration." The standard statistical model for such acircumstance isy = β0+ β1x + efor a normal error e with mean 0 and standard deviation σ . (σ describeshow much y’s vary for a …xed x, and in the present context amounts to a"repeatability" standard deviation.) This can be pictured as follows.Vardeman and Morris (I owa State University) IE 361 Module 7 3 / 13Regression Analysis and CalibrationFigure: Normal Simple Linear Regression ModelVardeman and Morris (I owa State University) IE 361 Module 7 4 / 13Regression Analysis and CalibrationFor n data pairs(xi, yi), simple linear regression methodology allows oneto make con…dence intervals and tests associated with the model, andwhat is more important for our present purposes, prediction limits for anew y asso ciated with a new x. These are of the form(b0+ b1x)tsLFs1 +1n+(x  ¯x)2∑(x  ¯x)2where the least squares line is ˆy = b0+ b1x and sLFis an estimate of σderived from the …t of the line to the data. Any good stati stical packagewill compute and plot these limits along with a least squares line throughthe data set.Vardeman and Morris (I owa State University) IE 361 Module 7 5 / 13Regression Analysis and CalibrationExample 7-1 (Mandel NBS/NIST)"Gold-standard" and "local" measurements on n = 14 specimens (unitsnot given) are as below.Figure: JMP Data Table for Mandel’s Calibration ExampleVardeman and Morris (I owa State University) IE 361 Module 7 6 / 13Regression Analysis and CalibrationExample 7-1 (Mandel NBS/NIST)A JMP report for simple linear regression including prediction limits for anadditional value of y (that, of course, change with x) plotted is below.Figure: JMP Report for Simple Linear Regression With Mandel’s DataVardeman and Morris (I owa State University) IE 361 Module 7 7 / 13Regression Analysis and CalibrationWhat is of most interest here is (of course) what regression technologyindicates about measurement and calibration. In particular:From a simple linear regression output,sLF=pMSE = "root mean square error"is a kind of estimated repeatability standard deviation. One maymake con…dence intervals for σrepeatabilitybased on this "samplestandard deviation" using ν = n 2 degrees of freedom and limitssLFsn 2χ2u pperand sLFsn 2χ2lowerThe least squares equation ˆy = b0+ b1x can be solved for x, givingˆx =y  b0b1as a way of estimating a "gold-standard" value of a measurand xfrom a measured local value y.Vardeman and Morris (I owa State University) IE 361 Module 7 8 / 13Regression Analysis and CalibrationIt turns out that one can take the prediction limits for y and "turnthem around" to get con…dence limits for the x corresponding to ameasured local y. This provides a defensible way to set "errorbounds" on what y indicates about x.Vardeman and Morris (I owa State University) IE 361 Module 7 9 / 13Regression Analysis and CalibrationExample 7-1 (Mandel NBS/NIST)Since from the JMP reporty = 42.216299 + 0.881819x with sLF= 25.32578we might expect a local (y) repeatability standard deviation of around 25(in the y units). In fact, 95% con…dence limits for σ can be made (usingn 2 = 12 degrees of freedom) limits25.3r1223.337and 25.3r124.4004i.e.18.1 and 41.8Making use of the slope and intercept of the least squares line, a"conversion formula" for going from y to x isˆx =y  42.2162990.881819Vardeman and Morris (I owa State University) IE 361 Module 7 10 / 13Regression Analysis and CalibrationExample 7-1 (Mandel NBS/NIST)The following …gure shows how one can set 95% con…dence limits on x ify = 1500 is observed, using a plot of 95% prediction limits for y given x.Figure: 95% Con…dence Limits for x When y = 1500 is Measured, Derived FromTraces of 95% Prediction Limits for y at Given x ValuesVardeman and Morris (I owa State University) IE 361 Module 7 11 / 13Regression Analysis and CalibrationAs one …nal consideration for Example 7-1, it is worthwhile to note what astandard simple linear regression analysis has to say about the "linearity"of the local measurement device assuming that the (unavailable) units of xand y are meant to be the same. While a scatterplot of the n = 14 datapairs(xi, yi)in the example is reasonably straight-line, that is not reallythe issue to be discussed, but rather something stronger. As wementioned in Module 2, the term "linearity" as typically employed inmetrology contexts concerns the matter of constant bias. In regressionterms, this requires a straight-line relationship between measurand andaverage measurement with slope of 1.The methods of elementary regression analysis say that con…dence limitsfor the slope β1of the simple linear regression model areb1tsLFq∑(xi ¯x)2Vardeman and Morris (I owa State University) IE 361 Module 7 12 / 13Regression Analysis and CalibrationExample 7-1 (Mandel NBS/NIST)The report on panel 7 shows thatb1= .882 andsLFq∑(xi