Dataplot Vol 1 Vol 2

# CALIBRATION

Name:
CALIBRATION
Type:
Analysis Command
Purpose:
Compute a linear or quadratic calibration using multiple methods.
Description:
The goal of calibration is to quantitatively convert measurements made on one of two measurement scales to the other measurement scale. There is also a model that describes the relationship between the two measurement scales.

The primary measurement scale is usually the scientifically relevant scale and measurements on this scale are typically more precise (relatively) than measurements on the secondary scale. However, the secondary scale is typically the easier measurement to obtain (i.e., it is typically cheaper or faster or more readily available).

So given a measurement on the secondary scale, we want to convert that to an estimate of the measurement on the primary scale. The steps involved are:

1. We start with a series of points that have been measured on both scales. The secondary measurement is treated as the response variable, Y, and the primary measurement is treated as the independent variable, X.

2. We perform a fit of Y and X. Currently, Dataplot supports calibration for the case where Y and X can be fit with either a linear fit

Y = A0 + A1*X

Y = A0 + Y = A0 + A2X2

This is typically referred to as the calibration curve.

Although these are the most common calibration models in practice, other calibration models are also used. For example, the fit could be multi-linear (i.e., more than one X variable), a higher order polynomial, or non-linear. These cases are not supported directly. However, you can use a bootstrap approach for many of these problems.

3. We then have one or more points measured on the secondary scale with no corresponding measurement on the primary scale.

We use the calibration curve to estimate the value of the measurement on the primary scale. In addition, we estimate a confidence interval for the estimated value on the primary scale.

The calibration problem has recieved significant attention and a number of different methods have been proposed for the calibration estimates. Most of these methods return the same value for the point estimate. However, the method for obtaining the confidence interval is typically different. We describe the "classical" method in some detail. For the other methods, we give references to the literature.

Given that in the calibration problem the primary measurement (the higher quality measurement) is assigned to the independent variable(s) (x axis) and the secondary measurement is assigned to the dependent (y axis) variable, a reasonable question is why don't we simply switch the axes and assign the secondary measurement to the independent variable? The reason is that least squares fitting assumes that the values for the indpendent variable are fixed (i.e., there is no error). In order to satisfy this assumption, we need to assign the higher quality measurement to the independent variable.

When Dataplot performs a calibration, it first prints out a summary of the initial fit. It then loops through each point being calibrated and prints the estimate for the primary scale and the corresponding confidence limits.

Calibration is discussed in the NIST/SEMATECH e-Handbook of Statistical Methods.

Description of Methods:
In this section, we only give the final computational formulas. A reference is given for most methods that discusses the derivation of the formula.

The following are some quantities that are used by several methods:

 $$\hat{y}$$ mean of the Y (secondary measurement) values $$\bar{x}$$ mean of the X (primary measurement) values A0: intercept value for the fit between Y and X A1: slope value for the fit between Y and X ssdx: $$\sum_{i=1}^{n}{(X_{i} - \hat{x})^2}$$ ssx: $$\sum_{i=1}^{n}{X_{i}^2}$$ ssdy: $$\sum_{i=1}^{n}{(Y_{i} - \hat{y})^2}$$ s: the residual standard deviation

For most of these methods, given a calibration point, Y0, the X0 is estimated from the original fit by

X0 = (Y0 - A0)/A1

with A0 and A1 denoting the coefficients from the original fit:

Y = A0 + A1 X

Dataplot generates the linear calibration using the following methods:

1. Inverse Prediction Limits (Eisenhart)

This method was originally recommended by Churchill Eisenhart and is based on inverting the prediction limits for Y given X0. The prediction interval is

$$Y_0 = \bar{Y} + A1 X_0 \pm t_{(1-\alpha/2,N-2)}s \sqrt{1 + \frac{1}{N} + \frac{X_0^2}{ssdx}}$$

The uncertainty is obtained from the linear regression prediction interval

$$\hat{Y} \pm t_{1 - \alpha/2,\nu} \hat{\sigma}_{p}$$

with $$\hat{\sigma}_{p}$$ denoting the standard deviation of the predicted value. The formula for $$\hat{\sigma}_{p}$$ is

$$\hat{\sigma}_{p} = \sqrt{\hat{\sigma}^2 + \hat{\sigma}_{f}^2}$$

with

$$\begin{array}{lcl} \hat{\sigma}^2 & = & \mbox{variance of the residuals} \\ & = & \sum_{i=1}^{N}{\frac{(Y - \hat{Y})^2}{N-1}} \end{array}$$

To find the confidence limits for X0 (X0L and X0U), we solve

$$\mbox{X0L} = (A0 + A1 \times X0) - t_{1 - \alpha/2,\nu} \hat{\sigma}_{p}$$

$$\mbox{X0U} = (A0 + A1 \times X0) + t_{1 - \alpha/2,\nu} \hat{\sigma}_{p}$$

2. Graybill-Iyer

This method is described on pages 427-431 of the Graybill and Iyer textbook (see the Reference section below).

Although this uses a different computational formula than the inverse prediction limits method, they appear to be equivalent.

3. Neter-Wasserman-Kutner

This method is described on pages 135-137 of the Neter, Wasserman, and Kutner textbook (see the Reference section below).

Although this method uses a different computational formula, it appears to be equivalent to the propogation of error as given in the e-Handbook (see the Reference section below).

4. Propogation of Error

The propogation error formulas here are different than those given in the e-Handbook. If you want to use a propogation of error method, we recommend the method used in the e-Handbook.

5. Propogation of Error (as given in the e-Handbook)

The propogation of error formula used here is from the NIST/SEMATECH e-Handbook of Statistical Methods

6. Inverse (Krutchkoff)

This method is described in the Krutchkoff paper (see the Reference Section below). Note that this method gives a different value for the estimate of the calibrated value than do the other methods. This method is not often used.

7. Maximum Likelihood

This method is from a private communication with Andrew Rukhin.

The basic algorithm used is

• Subtract out the mean from the X variable and perform the linear fit. Save the standard deviations of the A0 and A1 coefficients.

• Generate two sets of 10,000 standard normal random numbers.

• For each simulation, generate values for A0 and A1 by adding the normal random numbers (use one set for A0 and the other set for A1) multiplied by the standard deviations of A0 and A1 respectively. Then invert the linear fit to obtain the point estimate of the calibrated value. Note that the A1 coefficient is multiplied by the mean of the X values to restore the original scale.

• The confidence interval for the calibrated value is obtained from the appropriate percentiles of the 10,000 estimates of the calibrated value.

8. Bootstrap

For this method, the confidence limits are obtained by generating bootstrap samples, obtaining the point estimate for each bootstrap sample, and then computing a confidence interval based on the percentiles of these bootstrap point estimates. For example, a 95% confidence interval would be obtained from the 2.5 and 97.5 percentiles.

There are two methods for generating the bootstrap samples.

1. In the first approach, the least squares fit is computed from the original data. The residuals are then resampled. The residuals are added to the predicted values of the original fit to obtain a new Y vector. This new Y vector is then fit against the original X variable and the point estimate for the calibration is obtained from these Y and X. We call this approach residual resampling (or the Efron approach).

2. In the second approach, rows of the original data (both the Y vector and the corresponding rows of the X variables) are resampled. The resampled data are then fit. We call this approach data resampling (or the Wu approach).

Hamilton (see Reference below) gives some guidance on the contrasts between these approaches.

1. Residual resampling assumes fixed X values and independent and identically distributed residuals.

2. Data resampling does not assume independent and identically distributed residuals.

Given the above, if the assumption of fixed X is realistic (that is, we could readily collect new Y's with the same X values), then residual resampling is justified. For example, this would be the case in a designed experiment. However, if this assumption is not realistic (i.e., the X values vary randomly as well as the Y's), then data resampling is preferred.

The CALIBRATION command will generate estimates for both types of bootstrap sampling.

9. Fieller Method

The derivation and computational details for this method can be found in chapter 5 of Miller. Both a bias corrected (equation 5.32 in Miller) and the uncorrected interval (equation 5.35 in Miller) will be printed.

In addition, a simultaneous interval for the case when there more than one calibration points is given.

Dataplot generates the quadratic calibration using the following methods:

1. Inverse Prediction Limits (Eisenhart)

This uses the same idea as the linear calibration inverse prediction limits. That is, we invert the quadratic regression equation to obtain the point estimates and the confidence intervals are based on inverting the quadratic prediction limits. The algebraic details are not given here.

2. Propogation of Error

The propogation of error formula used here is from the NIST/SEMATECH e-Handbook of Statistical Methods

3. Bootstrap

See the comments above for the bootstrap using linear calibration. The same basic ideas apply except that we perform a quadratic rather than a linear fit.

Syntax 1:
LINEAR CALIBRATION <y> <x> <y0>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable (secondary measurements);
<x> is the independent variable (primary measurements);
<y0> is a number, parameter, or variable containing the secondary measurements where the calibration is to be performed;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes a linear calibration analysis.

Syntax 2:
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable (secondary measurements);
<x> is the independent variable (primary measurements);
<y0> is a number, parameter, or variable containing the secondary measurements where the calibration is to be performed;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes a quadratic calibration analysis.

Examples:
LINEAR CALIBRATION Y X Y0
LINEAR CALIBRATION Y X Y0SUBSET X > 2
Note:
To simplify the generation of additional plots and analysis, a number of results are written to external files.

The following variables are written to the file dpst1f.dat.

 Column 1 - method id Column 2 - Y0 (i.e., calibration point on secondary scale) Column 3 - X0 (i.e., for Y0, the estimate on the primary scale) Column 4 - lower confidence limit Column 5 - upper confidence limit

The following variables are written to the file dpst2f.dat.

 Column 1 - Y0 Columns 2 thru 9 - X0 for each of the 8 methods (only 3 methods for quadratic calibration)

The following variables are written to the file dpst3f.dat.

 Column 1 - Y0 Columns 2 thru 9 - lower limit for X0 for each of the 8 methods (only 3 methods for quadratic calibration)

The following variables are written to the file dpst4f.dat.

 Column 1 - Y0 Columns 2 thru 9 - upper limit for X0 for each of the 8 methods (only 3 methods for quadratic calibration)
Note:
The default confidence limits are for a 95% confidence interval (i.e., $$\alpha$$ = 0.05). To use a different alpha, enter the command (before entering the CALIBRATION command):

LET ALPHA = <value>

For example, to generate 90% confidence intervals, enter

LET ALPHA = 0.10
Note:
If you want the fit to be generated without a constant term, enter the command

Note:
Although linear and quadratic calibrations are often sufficient, for some applications more complicated calibration curves may be required. In these cases, inverting the prediction limits or computing the propogation of error formulas may become difficult. The bootstrap can be used for these applications. This is demonstrated in the Program 3 example.
Default:
None
Synonyms:
None
Related Commands:
 FIT = Perform a fit. BOOTSTRAP FIT = Generate a bootstrap fit. BOOTSTRAP PLOT = Generate a bootstrap plot.
References:
Churchill Eisenhart (1939). "The Interpretation of Certain Regression Methods and Their Use in Biological and Industrial Research," Annals of Mathematical Statistics, Vol. 10, pp. 162-182.

F. Graybill and H. Iyer. "Regression Anaysis," First Edition, Duxbury Press, pp. 427-431.

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 2012.

Neter, Wasserman, and Kuttner. "Applied Linear Statistical Models," Third Edition, Irwin, pp. 173-175.

Miller (1997). "Beyond ANOVA: Basics of Applied Statistics", Chapman & Hall/CRC, pp. 177-181.

Fieler (1954). "Some Problems in Interval Estimation", Journal of the Royal Statistical Society, Series B, Vol.16, pp. 175-185.

R. G. Krutchkoff (1967). "Classical and Inverse Methods of Calibration," Technometrics, Vol. 9, pp. 425-439.

B. Hoadley (1970). "A Bayesian Look at Inverse Linear Regresssion," Journal of the American Statistical Association, Vol. 65, pp. 356-369.

H. Scheffe (1973). "A Statistical Theory of Calibration," Annals of Statistics, Vol. 1, pp. 1-37.

P. J. Brown (1982). "Multivariate Calibration," (with discussion), JRSBB, Vol. 44, pp. 287-321.

A. Racine-Poon (1988). "A Bayesian Approach to Nonlinear Calibration Problems," Journal of the American Statistical Association, Vol. 83, pp. 650-656.

C. Osborne (1991). "Statistical Calibration: A Review," International Statistical Review, Vol. 59, pp. 309-336.

Hamilton (1992). "Regression with Graphics: A Second Course in Applied Statistics," Duxbury Press.

Applications:
Calibration
Implementation Date:
2003/07
2011/07: Support for Fieler methods
2016/11: Support fit with no intercept term
2016/11: Support for propogation of error methods given in
NIST/SEMATECH Engineering Statistics Handbook
2017/11: Add coverage factor and expanded error columns to output
 Program 1:  SKIP 25 READ NATR533.DAT Y X LET Y0 = DATA 150 200 250 300 . LINEAR CALIBRATION Y X X0  The following output is generated:  Linear Calibration Analysis Summary of Linear Fit Between Y and X Number of Observations: 16 Estimate of Intercept: 13.5059 SD(Intercept): 21.0476 t(Intercept): 0.6417 Estimate of Slope: 0.7902 SD(Slope): 0.0710 t(Slope): 11.1237 CV(Slope): 0.0899 Residual Standard Deviation: 26.2078 Linear Calibration Summary Y0 = 150.0000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 172.7310 90.6910 246.3665 34.3323 2.1448 73.6355 2. Graybill-Iyer: 172.7310 90.6910 246.3665 34.3323 2.1448 73.6355 3. Neter-Wasserman-Kutner: 172.7310 96.4652 248.9967 35.5587 2.1448 76.2658 4. Propogation of Error: 172.7310 83.3506 262.1114 41.6734 2.1448 89.3804 5. Propogation of Error (e-Handbook): 172.7310 96.4652 248.9967 35.5587 2.1448 76.2658 6. Inverse (Krutchkoff): 183.7930 106.8874 260.6986 35.8570 2.1448 76.9056 7. Maximum Likelihood: 172.7310 142.7454 194.8587 13.2553 2.2622 29.9855 8. Bootstrap (Residuals): 172.7310 145.7334 192.7618 12.2189 2.2095 26.9976 9. Bootstrap (Data): 172.7310 146.6036 193.8465 12.1874 2.1438 26.1273 10. Fieller (Bias Corrected): 172.7310 140.1836 196.8739 13.2158 2.4628 32.5474 11. Fieller (No Bias Correction): 172.7310 145.2251 200.2369 12.8245 2.1448 27.5059 12. Fieller (Simultaneous): 172.7310 128.8803 202.5816 13.4759 3.2540 43.8507 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 200.0000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 236.0051 158.9669 309.5252 34.2785 2.1448 73.5201 2. Graybill-Iyer: 236.0051 158.9669 309.5252 34.2785 2.1448 73.5201 3. Neter-Wasserman-Kutner: 236.0051 162.1587 309.8515 34.4307 2.1448 73.8464 4. Propogation of Error: 236.0051 134.6395 337.3707 47.2614 2.1448 101.3656 5. Propogation of Error (e-Handbook): 236.0051 162.1587 309.8515 34.4307 2.1448 73.8464 6. Inverse (Krutchkoff): 240.6357 168.0773 313.1941 33.8301 2.1448 72.5584 7. Maximum Likelihood: 236.0051 216.1080 253.3034 9.4703 2.1010 19.8971 8. Bootstrap (Residuals): 236.0051 217.8168 251.9631 8.7004 2.0905 18.1883 9. Bootstrap (Data): 236.0051 220.3723 254.0829 8.5301 2.1193 18.0778 10. Fieller (Bias Corrected): 236.0051 213.9560 254.5361 9.4602 2.3307 22.0492 11. Fieller (No Bias Correction): 236.0051 216.1707 255.8395 9.2477 2.1448 19.8344 12. Fieller (Simultaneous): 236.0051 206.8218 259.3279 9.6005 3.0398 29.1833 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 250.0000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 299.2793 225.1549 374.7718 35.1982 2.1448 75.4925 2. Graybill-Iyer: 299.2793 225.1549 374.7718 35.1982 2.1448 75.4925 3. Neter-Wasserman-Kutner: 299.2793 225.8776 372.6809 34.2233 2.1448 73.4017 4. Propogation of Error: 299.2793 185.8826 412.6759 52.8709 2.1448 113.3967 5. Propogation of Error (e-Handbook): 299.2793 225.8776 372.6809 34.2233 2.1448 73.4017 6. Inverse (Krutchkoff): 297.4785 225.7040 369.2529 33.4646 2.1448 71.7745 7. Maximum Likelihood: 299.2793 283.0275 316.7070 8.5216 2.0451 17.4278 8. Bootstrap (Residuals): 299.2793 283.5742 316.0866 8.1504 2.0622 16.8074 9. Bootstrap (Data): 299.2793 283.3301 318.4212 8.8429 2.1647 19.1420 10. Fieller (Bias Corrected): 299.2793 281.4960 318.4307 8.6103 2.2242 19.1514 11. Fieller (No Bias Correction): 299.2793 281.1709 317.3876 8.4430 2.1448 18.1084 12. Fieller (Simultaneous): 299.2793 276.5729 324.2647 8.7202 2.8652 24.9855 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 300.0000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 362.5534 289.2164 442.1448 37.1093 2.1448 79.5914 2. Graybill-Iyer: 362.5534 289.2164 442.1448 37.1093 2.1448 79.5914 3. Neter-Wasserman-Kutner: 362.5534 287.5868 437.5200 34.9530 2.1448 74.9666 4. Propogation of Error: 362.5534 237.0931 488.0137 58.4955 2.1448 125.4603 5. Propogation of Error (e-Handbook): 362.5534 287.5868 437.5200 34.9530 2.1448 74.9666 6. Inverse (Krutchkoff): 354.3212 279.7674 428.8750 34.7605 2.1448 74.5538 7. Maximum Likelihood: 362.5534 343.2002 387.2146 11.1659 2.2086 24.6612 8. Bootstrap (Residuals): 362.5534 345.3757 384.6494 10.0983 2.1881 22.0960 9. Bootstrap (Data): 362.5534 342.0918 388.3067 11.9157 2.1613 25.7533 10. Fieller (Bias Corrected): 362.5534 341.3596 390.0016 11.3396 2.4206 27.4482 11. Fieller (No Bias Correction): 362.5534 338.8869 386.2199 11.0345 2.1448 23.6665 12. Fieller (Simultaneous): 362.5534 336.2003 399.3252 11.5421 3.1859 36.7718 ----------------------------------------------------------------------------------------------------------------------------------  Program 2:  skip 25 read loadcell.dat x y . let y0 = sequence 2 0.1 5 set write decimals -7 quadratic calibration y x y0  The following output is generated:  Quadratic Calibration Analysis Summary of Quadratic Fit Between Y and X Number of Observations: 33 Estimate of Intercept: -0.1839805E-04 SD(Intercept): 0.2450722E-04 t(Intercept): -0.7507195E+00 Estimate of Linear Term: 0.1001025E+00 SD(Linear Term): 0.4838699E-05 t(Linear Term): 0.2068789E+05 Estimate of Quadratic Term: 0.7031865E-05 SD(Quadratic Term): 0.2013613E-06 t(Quadratic Term): 0.3492164E+02 Residual Standard Deviation: 0.3764029E-04 Quadratic Calibration Summary Y0 = 2.000000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.1995174E+02 0.1995098E+02 0.1995251E+02 0.3752013E-03 0.2042272E+01 0.7662633E-03 2. Propogation of Error (Handbook): 0.1995174E+02 0.1995094E+02 0.1995255E+02 0.3943881E-03 0.2042272E+01 0.8054479E-03 7. Bootstrap (Residuals): 0.1995174E+02 0.1995152E+02 0.1995199E+02 0.1175987E-03 0.2138490E+01 0.2514836E-03 8. Bootstrap (Data): 0.1995174E+02 0.1995162E+02 0.1995188E+02 0.6466544E-04 0.2166359E+01 0.1400886E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.100000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2094786E+02 0.2094709E+02 0.2094862E+02 0.3751334E-03 0.2042272E+01 0.7661247E-03 2. Propogation of Error (Handbook): 0.2094786E+02 0.2094703E+02 0.2094868E+02 0.4033869E-03 0.2042272E+01 0.8238259E-03 7. Bootstrap (Residuals): 0.2094786E+02 0.2094759E+02 0.2094816E+02 0.1421178E-03 0.2136924E+01 0.3036949E-03 8. Bootstrap (Data): 0.2094786E+02 0.2094771E+02 0.2094804E+02 0.8543484E-04 0.2208017E+01 0.1886416E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.200000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2194383E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2194383E+02 0.2194298E+02 0.2194468E+02 0.4166365E-03 0.2042272E+01 0.8508852E-03 7. Bootstrap (Residuals): 0.2194383E+02 0.2194353E+02 0.2194421E+02 0.1793797E-03 0.2118225E+01 0.3799665E-03 8. Bootstrap (Data): 0.2194383E+02 0.2194362E+02 0.2194411E+02 0.1166699E-03 0.2350679E+01 0.2742535E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.300000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2293967E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2293967E+02 0.2293878E+02 0.2294056E+02 0.4348091E-03 0.2042272E+01 0.8879986E-03 7. Bootstrap (Residuals): 0.2293967E+02 0.2293929E+02 0.2294009E+02 0.2061083E-03 0.2071231E+01 0.4268977E-03 8. Bootstrap (Data): 0.2293967E+02 0.2293938E+02 0.2293998E+02 0.1514034E-03 0.2059494E+01 0.3118145E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.400000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2393536E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2393536E+02 0.2393443E+02 0.2393630E+02 0.4584265E-03 0.2042272E+01 0.9362318E-03 7. Bootstrap (Residuals): 0.2393536E+02 0.2393490E+02 0.2393588E+02 0.2590127E-03 0.2003900E+01 0.5190355E-03 8. Bootstrap (Data): 0.2393536E+02 0.2393500E+02 0.2393577E+02 0.1995916E-03 0.2042275E+01 0.4076209E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.500000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2493092E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2493092E+02 0.2492993E+02 0.2493192E+02 0.4878336E-03 0.2042272E+01 0.9962891E-03 7. Bootstrap (Residuals): 0.2493092E+02 0.2493035E+02 0.2493154E+02 0.2923929E-03 0.2096521E+01 0.6130077E-03 8. Bootstrap (Data): 0.2493092E+02 0.2493047E+02 0.2493139E+02 0.2363052E-03 0.1989333E+01 0.4700896E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.600000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2592634E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2592634E+02 0.2592527E+02 0.2592741E+02 0.5232012E-03 0.2042272E+01 0.1068519E-02 7. Bootstrap (Residuals): 0.2592634E+02 0.2592570E+02 0.2592708E+02 0.3545057E-03 0.2070759E+01 0.7340960E-03 8. Bootstrap (Data): 0.2592634E+02 0.2592580E+02 0.2592697E+02 0.2864968E-03 0.2185835E+01 0.6262349E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.700000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2692162E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2692162E+02 0.2692047E+02 0.2692278E+02 0.5645540E-03 0.2042272E+01 0.1152973E-02 7. Bootstrap (Residuals): 0.2692162E+02 0.2692086E+02 0.2692244E+02 0.4017102E-03 0.2022701E+01 0.8125395E-03 8. Bootstrap (Data): 0.2692162E+02 0.2692093E+02 0.2692229E+02 0.3491369E-03 0.1982133E+01 0.6920357E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.800000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2791677E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2791677E+02 0.2791552E+02 0.2791802E+02 0.6118100E-03 0.2042272E+01 0.1249483E-02 7. Bootstrap (Residuals): 0.2791677E+02 0.2791594E+02 0.2791770E+02 0.4586082E-03 0.2026136E+01 0.9292027E-03 8. Bootstrap (Data): 0.2791677E+02 0.2791602E+02 0.2791756E+02 0.3961834E-03 0.2002283E+01 0.7932712E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 2.900000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2891177E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2891177E+02 0.2891041E+02 0.2891313E+02 0.6648206E-03 0.2042272E+01 0.1357745E-02 7. Bootstrap (Residuals): 0.2891177E+02 0.2891082E+02 0.2891281E+02 0.4999159E-03 0.2088879E+01 0.1044264E-02 8. Bootstrap (Data): 0.2891177E+02 0.2891090E+02 0.2891268E+02 0.4514173E-03 0.2011385E+01 0.9079739E-03 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.000000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.2990664E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.2990664E+02 0.2990516E+02 0.2990811E+02 0.7234045E-03 0.2042272E+01 0.1477389E-02 7. Bootstrap (Residuals): 0.2990664E+02 0.2990550E+02 0.2990790E+02 0.6065934E-03 0.2079718E+01 0.1261543E-02 8. Bootstrap (Data): 0.2990664E+02 0.2990557E+02 0.2990766E+02 0.5163847E-03 0.2066543E+01 0.1067131E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.100000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3090136E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3090136E+02 0.3089976E+02 0.3090297E+02 0.7873713E-03 0.2042272E+01 0.1608027E-02 7. Bootstrap (Residuals): 0.3090136E+02 0.3090013E+02 0.3090268E+02 0.6579362E-03 0.2000254E+01 0.1316039E-02 8. Bootstrap (Data): 0.3090136E+02 0.3090022E+02 0.3090250E+02 0.5874540E-03 0.1944684E+01 0.1142412E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.200000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3189595E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3189595E+02 0.3189420E+02 0.3189770E+02 0.8565366E-03 0.2042272E+01 0.1749281E-02 7. Bootstrap (Residuals): 0.3189595E+02 0.3189454E+02 0.3189735E+02 0.7346464E-03 0.1922963E+01 0.1412698E-02 8. Bootstrap (Data): 0.3189595E+02 0.3189466E+02 0.3189731E+02 0.6618454E-03 0.2050840E+01 0.1357339E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.300000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3289040E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3289040E+02 0.3288850E+02 0.3289230E+02 0.9307304E-03 0.2042272E+01 0.1900805E-02 7. Bootstrap (Residuals): 0.3289040E+02 0.3288882E+02 0.3289209E+02 0.8318344E-03 0.2023262E+01 0.1683019E-02 8. Bootstrap (Data): 0.3289040E+02 0.3288899E+02 0.3289190E+02 0.7421759E-03 0.2012738E+01 0.1493805E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.400000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3388471E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3388471E+02 0.3388265E+02 0.3388678E+02 0.1009801E-02 0.2042272E+01 0.2062288E-02 7. Bootstrap (Residuals): 0.3388471E+02 0.3388295E+02 0.3388664E+02 0.9214684E-03 0.2086196E+01 0.1922364E-02 8. Bootstrap (Data): 0.3388471E+02 0.3388318E+02 0.3388635E+02 0.7975925E-03 0.2053929E+01 0.1638198E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.500000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3487889E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3487889E+02 0.3487665E+02 0.3488112E+02 0.1093614E-02 0.2042272E+01 0.2233457E-02 7. Bootstrap (Residuals): 0.3487889E+02 0.3487697E+02 0.3488093E+02 0.9611747E-03 0.2122007E+01 0.2039619E-02 8. Bootstrap (Data): 0.3487889E+02 0.3487723E+02 0.3488068E+02 0.8849014E-03 0.2020613E+01 0.1788043E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.600000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3587292E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3587292E+02 0.3587051E+02 0.3587534E+02 0.1182054E-02 0.2042272E+01 0.2414076E-02 7. Bootstrap (Residuals): 0.3587292E+02 0.3587084E+02 0.3587520E+02 0.1125396E-02 0.2023944E+01 0.2277738E-02 8. Bootstrap (Data): 0.3587292E+02 0.3587106E+02 0.3587492E+02 0.9760089E-03 0.2050670E+01 0.2001472E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.700000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3686682E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3686682E+02 0.3686422E+02 0.3686942E+02 0.1275022E-02 0.2042272E+01 0.2603942E-02 7. Bootstrap (Residuals): 0.3686682E+02 0.3686462E+02 0.3686914E+02 0.1164792E-02 0.1988408E+01 0.2316082E-02 8. Bootstrap (Data): 0.3686682E+02 0.3686474E+02 0.3686887E+02 0.1036396E-02 0.2008434E+01 0.2081532E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.800000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3786058E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3786058E+02 0.3785778E+02 0.3786338E+02 0.1372431E-02 0.2042272E+01 0.2802879E-02 7. Bootstrap (Residuals): 0.3786058E+02 0.3785816E+02 0.3786313E+02 0.1258081E-02 0.2029352E+01 0.2553089E-02 8. Bootstrap (Data): 0.3786058E+02 0.3785829E+02 0.3786280E+02 0.1175100E-02 0.1946960E+01 0.2287874E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 3.900000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3885420E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3885420E+02 0.3885119E+02 0.3885721E+02 0.1474210E-02 0.2042272E+01 0.3010737E-02 7. Bootstrap (Residuals): 0.3885420E+02 0.3885147E+02 0.3885687E+02 0.1366383E-02 0.1998703E+01 0.2730994E-02 8. Bootstrap (Data): 0.3885420E+02 0.3885169E+02 0.3885685E+02 0.1292122E-02 0.2046363E+01 0.2644150E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.000000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.3984769E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.3984769E+02 0.3984446E+02 0.3985091E+02 0.1580293E-02 0.2042272E+01 0.3227389E-02 7. Bootstrap (Residuals): 0.3984769E+02 0.3984473E+02 0.3985071E+02 0.1505074E-02 0.2007024E+01 0.3020721E-02 8. Bootstrap (Data): 0.3984769E+02 0.3984509E+02 0.3985036E+02 0.1356550E-02 0.1974391E+01 0.2678361E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.100000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4084103E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4084103E+02 0.4083758E+02 0.4084448E+02 0.1690629E-02 0.2042272E+01 0.3452724E-02 7. Bootstrap (Residuals): 0.4084103E+02 0.4083810E+02 0.4084419E+02 0.1555796E-02 0.2030616E+01 0.3159225E-02 8. Bootstrap (Data): 0.4084103E+02 0.4083801E+02 0.4084394E+02 0.1484237E-02 0.2034256E+01 0.3019318E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.200000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4183424E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4183424E+02 0.4183055E+02 0.4183793E+02 0.1805169E-02 0.2042272E+01 0.3686646E-02 7. Bootstrap (Residuals): 0.4183424E+02 0.4183089E+02 0.4183789E+02 0.1708715E-02 0.2135974E+01 0.3649772E-02 8. Bootstrap (Data): 0.4183424E+02 0.4183119E+02 0.4183748E+02 0.1630422E-02 0.1984928E+01 0.3236270E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.300000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4282731E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4282731E+02 0.4282338E+02 0.4283124E+02 0.1923873E-02 0.2042272E+01 0.3929073E-02 7. Bootstrap (Residuals): 0.4282731E+02 0.4282390E+02 0.4283097E+02 0.1755553E-02 0.2083943E+01 0.3658473E-02 8. Bootstrap (Data): 0.4282731E+02 0.4282408E+02 0.4283086E+02 0.1716201E-02 0.2070531E+01 0.3553447E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.400000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4382024E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4382024E+02 0.4381606E+02 0.4382442E+02 0.2046707E-02 0.2042272E+01 0.4179932E-02 7. Bootstrap (Residuals): 0.4382024E+02 0.4381683E+02 0.4382402E+02 0.1842181E-02 0.2051369E+01 0.3778992E-02 8. Bootstrap (Data): 0.4382024E+02 0.4381680E+02 0.4382379E+02 0.1829786E-02 0.1937297E+01 0.3544839E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.500000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4481304E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4481304E+02 0.4480860E+02 0.4481748E+02 0.2173639E-02 0.2042272E+01 0.4439162E-02 7. Bootstrap (Residuals): 0.4481304E+02 0.4480906E+02 0.4481689E+02 0.1999264E-02 0.1989872E+01 0.3978281E-02 8. Bootstrap (Data): 0.4481304E+02 0.4480921E+02 0.4481729E+02 0.1974569E-02 0.2155767E+01 0.4256711E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.600000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4580569E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4580569E+02 0.4580099E+02 0.4581040E+02 0.2304642E-02 0.2042272E+01 0.4706707E-02 7. Bootstrap (Residuals): 0.4580569E+02 0.4580128E+02 0.4580996E+02 0.2184767E-02 0.2020918E+01 0.4415235E-02 8. Bootstrap (Data): 0.4580569E+02 0.4580159E+02 0.4580971E+02 0.2069881E-02 0.1984320E+01 0.4107307E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.700000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4679821E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4679821E+02 0.4679323E+02 0.4680320E+02 0.2439693E-02 0.2042272E+01 0.4982519E-02 7. Bootstrap (Residuals): 0.4679821E+02 0.4679385E+02 0.4680299E+02 0.2278381E-02 0.2094687E+01 0.4772494E-02 8. Bootstrap (Data): 0.4679821E+02 0.4679404E+02 0.4680261E+02 0.2209621E-02 0.1988902E+01 0.4394720E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.800000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4779059E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4779059E+02 0.4778533E+02 0.4779586E+02 0.2578772E-02 0.2042272E+01 0.5266554E-02 7. Bootstrap (Residuals): 0.4779059E+02 0.4778571E+02 0.4779524E+02 0.2508210E-02 0.1945844E+01 0.4880587E-02 8. Bootstrap (Data): 0.4779059E+02 0.4778613E+02 0.4779502E+02 0.2294179E-02 0.1946596E+01 0.4465839E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 4.900000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4878284E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4878284E+02 0.4877728E+02 0.4878840E+02 0.2721858E-02 0.2042272E+01 0.5558775E-02 7. Bootstrap (Residuals): 0.4878284E+02 0.4877841E+02 0.4878759E+02 0.2464032E-02 0.1929710E+01 0.4754867E-02 8. Bootstrap (Data): 0.4878284E+02 0.4877807E+02 0.4878797E+02 0.2483865E-02 0.2063787E+01 0.5126169E-02 ---------------------------------------------------------------------------------------------------------------------------------- Y0 = 5.000000 ---------------------------------------------------------------------------------------------------------------------------------- 95% 95% Standard Expanded Method X0 Lower Limit Upper Limit Error Coverage Error ---------------------------------------------------------------------------------------------------------------------------------- 1. Inverse Prediction Limits: 0.4977495E+02 ** ** ** ** ** 2. Propogation of Error (Handbook): 0.4977495E+02 0.4976909E+02 0.4978081E+02 0.2868935E-02 0.2042272E+01 0.5859146E-02 7. Bootstrap (Residuals): 0.4977495E+02 0.4976989E+02 0.4978051E+02 0.2686516E-02 0.2071153E+01 0.5564186E-02 8. Bootstrap (Data): 0.4977495E+02 0.4976933E+02 0.4978023E+02 0.2674777E-02 0.2099327E+01 0.5615231E-02 ----------------------------------------------------------------------------------------------------------------------------------  Program 3:  . Step 1: Read the data . dimension 20 columns . x = Dose . y = Response . read x y 1 0.4148 1 0.4173 1 0.4141 1 0.4156 2 0.8321 2 0.8315 2 0.8317 2 0.8285 3 1.2396 3 1.2367 3 1.2356 3 1.2343 5 2.0127 5 2.0281 5 2.0321 5 2.0260 7 2.7950 7 2.7948 7 2.7962 7 2.8022 10 3.8656 10 3.8625 10 3.8695 10 3.8668 20 7.1166 20 7.1236 20 7.1132 20 7.1079 30 9.8641 30 9.8512 30 9.8558 30 9.8498 50 14.0724 50 14.0646 50 14.0644 50 14.0801 70 17.0616 70 17.0760 70 17.0488 70 17.1096 100 20.3078 100 20.2990 100 20.3642 100 20.3210 end of data . let n = number y . . Step 2: Perform initial fit. Save parameter estimates . to use as starting values for bootstrap fits. . fit y = a*(1-exp((-x)/b)) let astart = a let bstart = b . . Step 3: Save predicted and residual values from initial fit. . . The "y0" identifies the distinct points in predicted . values. These are the calibration points. . let res2 = res let pred2 = pred let y0 = distinct pred2 let ndist = size y0 let xtag = sequence 1 1 ndist . . Step 4: Perform first bootstrap sample . . Note that bootstrap sample is drawn from . the residuals of the fit, not the original . observations. Add the bootstrapped residuals . to the predicted values to obtain the new . set of observations. . . The "x0" points are the inverse calibration points. . let a = astart let b = bstart let ind = bootstrap index for i = 1 1 n let res3 = bootstrap sample res2 ind let y3 = pred2+res3 fit y3 = a*(1-exp((-x)/b)) let x0 = (-b)*log(1-(y0/a)) let tag = xtag . . Step 5: Now generate the remaining bootstrap . samples. . let numboot = 500 loop for k = 2 1 numboot let a = astart let b = bstart let ind = bootstrap index for i = 1 1 n let res3 = bootstrap sample res2 ind let y3 = pred2+res3 fit y3 = a*(1-exp((-x)/b)) let xtemp = (-b)*log(1-(y0/a)) extend tag xtag extend x0 xtemp end of loop . . Step 6: Now compute the mean and standard deviation . of each of the calibration points. . skip 1 tabulate standard deviation x0 tag read dpst1f.dat junk sb tabulate mean x0 tag read dpst1f.dat junk mb let rb = (sb/mb)*100 . write y0 mb sb rb  The following output is generated: ------------------------------------------------------------ Y0 MB SB RB ------------------------------------------------------------ 0.4127830E+00 0.9977350E+00 0.2384049E-02 0.2389461E+00 0.8187348E+00 0.1995482E+01 0.4729158E-02 0.2369933E+00 0.1217969E+01 0.2993241E+01 0.7034910E-02 0.2350265E+00 0.1996725E+01 0.4988796E+01 0.1152672E-01 0.2310521E+00 0.2749919E+01 0.6984403E+01 0.1585634E-01 0.2270249E+00 0.3833610E+01 0.9977912E+01 0.2204026E-01 0.2208905E+00 0.7078018E+01 0.1995722E+02 0.3987126E-01 0.1997837E+00 0.9823781E+01 0.2993818E+02 0.5337987E-01 0.1783003E+00 0.1411415E+02 0.4990638E+02 0.7102655E-01 0.1423196E+00 0.1718705E+02 0.6988584E+02 0.1026442E+00 0.1468741E+00 0.2024178E+02 0.9988721E+02 0.2825655E+00 0.2828846E+00  NIST is an agency of the U.S. Commerce Department. Date created: 09/09/2010 Last updated: 11/29/2017 Please email comments on this WWW page to alan.heckert@nist.gov.