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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

      or a quadratic fit

        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

      http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3671.htm

    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:
    QUADRATIC 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 quadratic calibration analysis.

Examples:
    LINEAR CALIBRATION Y X Y0
    QUADRATIC 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

      SET FIT ADDITIVE CONSTANT OFF
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: 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
          

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Date created: 09/09/2010
Last updated: 11/29/2017

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