lmcal: Perform linear and nonlinear calibration of analytical method
In quantchem: Quantitative chemical analysis: calibration and evaluation of results
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
Fit given calibration data to several regression models.
Usage
1
2
Arguments
x
a vector of standard concentrations
y
a vector of corresponding respondes (peak areas etc.)
confint
confidence interval for graphing prediction intervals
gridratio
a part of x variable range, to extend plot range (default 5 percent)
Details
For linear 'lmcal' fitting, procedure is performed as follows. First, the
calibration data are fitted to OLS linear, quadratic, cubical, and
4th order polynomial. These models are called p1 - p4. Next, linear model is
reweighted using x and y raised to power gamma from range (-4,4) with 0.1 accuracy.
The optimal weights are detected by minimal mean relative error (MRE) according
to Almeida et al. (2002). The best weighting scheme is then chosen, and data
are fit to the same equations (called P1-P4, with uppercase).
Next, the optimal value of lambda for Box-Cox transform is estimated with accuracy
up to 0.001, for transformation of x and y. The transformed models are then
fitted (called bx and by).
Then, two next log-log transformed models, are fitted - linear called l1, and
quadratic (mentioned sometimes as Wagner transform), called l2.
Last, the same models as p1 - p4 and P1 - P4, are fitted using rlm robust
method, and called r1 - r4 and R1 - R4.
This function performs also computation of grid and corresponding predicted values
for easy graphing of fitted models.
For nonlinear 'nlscal' fitting, procedure is performed as above, but there are following
models fitted: asymptotic (a1), asymptotic through origin (a2), logistic (g1), four
parameter logistic (g2), Michaelis-Menten (m1) and nonparametric (loess) spline (s1).
There are no weighting nor transform when fitting by 'nlscal'.
Value
Returns object of class c("lmcal","cal") or c("nlscal","cal"), which is the list of following
components:
models
A list of fitted models (p1-p4,P1-P4,l1,l2,bx,by,r1-r4,R1-R4)
graph
A list used by plot() method to produce graphs. Stored permanently
to make custom graphing easier. Containing following elements: grid - a grid of x values,
fitted - corresponding fitted values, upperc,lowerc - upper and lower bound for
interval="confidence" prediction, upperp,lowerp - upper and lower bound for
interval="prediction" prediction
x
Concentration vector
y
Response vector
Linear calibration object contains also following elements:
weigh
A dataframe containing sequence of gamma values and corresponding
mean relative errors, estimated during weighting process
wx
Value of gamma for oprimal weighting on x
wy
Value of gamma for oprimal weighting on y
yw
Logical, if weighting on y gives better result than on x
px
Optimal Box-Cox power for transform of x variable
py
Optimal Box-Cox power for transform of y variable
Note
This function performs *no* decision which model should be chosen!
Such decision should be always made by analyst.
Author(s)
Lukasz Komsta
References
Almeida, A.M., Castel-Branco, M.M., Falcao, A.C. (2002) Linear regression for calibration lines
revisited: weighting schemes for bioanalytical methods. J. Chromatogr. B Biomed. Sci. Appl.
774, 215-222.
Nagaraja, N.V., Paliwal, J.K., Gupta, R.C. (1999) Choosing the calibration model in assay validation.
J. Pharm. Biomed. Anal. 20, 433-438.
Kimanani, E.K., Lavigne, J. (1998) Bioanalytical calibration curves: variability of optimal powers
between and within analytical methods. J. Pharm. Biomed. Anal. 16, 1107-1115.
Kirkup, L., Mulholland, M. (2004). Comparison of linear and non-linear equations
in univariate calibration. J. Chromatogr. A, 1029, 1-11.
Kimanani, E.K. (1998) Bioanalitical calibration curves: proposal for statistical criteria.
J. Pharm. Biomed. Anal. 16, 1117-1124.
Baumann, K., Waetzig, H. (1997) Regression and calibration for analytical separation techniques.
Part I. Design considerations. Process Control and Quality, 10, 59-73.
Baumann, K. (1997) Regression and calibration for analytical separation techniques.
Part II. Validation, weighted and robust regression. Process Control and Quality, 10, 75-112.
Coleman, D.E., Vanatta, L.E. (1999) Lack-of-fit testing of ion chromatographic calibration
curves with inexact replicates. J. Chromatogr. A 850, 43-51.
See Also
Examples
1
2
3
4
5
Example output
Loading required package: MASS
Loading required package: outliers
------------------------------------------
If you use this package, please cite the recent paper containing description of this software:
Komsta, L. Chemometric and statistical evaluation of calibration curves in pharmaceutical analysis
- a short review on trends and recommendations. J. AOAC Int. 2012, 95, 3, 669-672.
------------------------------------------
There were 12 warnings (use warnings() to see them)
Optimal Box-Cox power on x: 1.0706
Optimal Box-Cox power on y: 0.9286
Optimal power of weighting on x: -1.1
Optimal power of weighting on y: -1.2
Better weighting on y: TRUE
Coefficients:
x0 x1 x2 x3 x4
p1 7.1855e+03 2.4367e+03 NA NA NA
p2 2.4477e+04 2.2503e+03 4.5050e-01 NA NA
p3 6.6297e+03 2.5568e+03 -1.1670e+00 2.6549e-03 NA
p4 8.4946e+04 7.8408e+02 1.3075e+01 -4.5838e-02 1e-04
P1 9.2852e+03 2.4262e+03 NA NA NA
P2 2.3786e+04 2.2577e+03 4.3268e-01 NA NA
P3 1.5885e+04 2.3970e+03 -3.2048e-01 1.2613e-03 NA
P4 9.9514e+04 4.5237e+02 1.5754e+01 -5.5003e-02 1e-04
l1 3.4461e+00 9.7722e-01 NA NA NA
l2 3.9466e+00 5.3114e-01 9.8918e-02 NA NA
bx 3.9029e+04 1.6761e+03 NA NA NA
by 1.6991e+04 9.5547e+02 NA NA NA
r1 6.2176e+03 2.4397e+03 NA NA NA
r2 6.5589e+04 1.9082e+03 1.1374e+00 NA NA
r3 -6.1566e+04 3.5305e+03 -5.6390e+00 9.2702e-03 NA
r4 1.8125e+05 -1.7845e+03 3.7062e+01 -1.3975e-01 2e-04
R1 6.9505e+03 2.4359e+03 NA NA NA
R2 6.3203e+04 1.9285e+03 1.0957e+00 NA NA
R3 -6.6491e+04 3.6044e+03 -5.9925e+00 9.8112e-03 NA
R4 1.7885e+05 -1.7263e+03 3.6555e+01 -1.3787e-01 2e-04
Goodness of fit:
R-Sq Adj-R-Sq AIC Sigma
p1 0.99981 0.99979 261.75205 2421.4910
p2 0.99994 0.99993 247.71781 1426.5197
p3 0.99995 0.99993 247.49803 1382.1149
p4 0.99996 0.99995 244.32234 1211.0032
P1 0.99980 0.99978 262.44697 0.9726
P2 0.99992 0.99991 250.87610 0.6256
P3 0.99993 0.99990 252.49472 0.6473
P4 0.99995 0.99992 249.93419 0.5798
l1 0.99968 0.99965 -120.27937 0.0029
l2 0.99990 0.99988 -134.86012 0.0017
bx 0.99993 0.99993 246.70642 1414.8727
by 0.99993 0.99993 220.63859 557.6887
r1 NA NA 262.21028 1641.7947
r2 NA NA 290.09308 1481.8118
r3 NA NA 276.52142 916.3124
r4 NA NA 291.36478 676.5453
R1 NA NA 264.54576 0.7086
R2 NA NA 297.47919 0.5314
R3 NA NA 287.70642 0.3199
R4 NA NA 284.47973 0.2587
Optimal Box-Cox power on x: 1.0706
Optimal Box-Cox power on y: 0.9286
Optimal power of weighting on x: -1.1
Optimal power of weighting on y: -1.2
Mean relative error:
0.003800207 on x
0.003778024 on y
0.003999273 unweighted
Better weighting on y: TRUE
Coefficients:
Estimate Std. Error t value Pr(>|t|)
x0 P1 9285.2032 1865.6239 4.9770 0.0003215 ***
x0 P2 23785.8581 3622.5547 6.5660 4.046e-05 ***
x0 P3 15884.8049 15495.0837 1.0252 0.3294508
x0 P4 99514.3329 47017.7766 2.1165 0.0633944 .
x0 R1 6950.5171 1654.4969 4.2010 2.658e-05 ***
x0 R2 63203.4992 1746.3796 36.1912 < 2.2e-16 ***
x0 R3 -66491.1832 5816.9866 -11.4305 < 2.2e-16 ***
x0 R4 178849.2739 12441.3020 14.3754 < 2.2e-16 ***
x0 bx 39029.0998 1150.3645 33.9276 2.733e-13 ***
x0 by 16991.1734 481.2495 35.3064 1.702e-13 ***
x0 l1 3.4461 0.0116 297.8942 < 2.2e-16 ***
x0 l2 3.9466 0.1004 39.2999 3.508e-13 ***
x0 p1 7185.5156 2089.5912 3.4387 0.0049061 **
x0 p2 24476.5774 3767.7857 6.4963 4.450e-05 ***
x0 p3 6629.6786 14096.2525 0.4703 0.6482197
x0 p4 84946.0666 40940.9648 2.0748 0.0678297 .
x0 r1 6217.5759 2299.2674 2.7042 0.0068478 **
x0 r2 65588.9586 2189.9951 29.9494 < 2.2e-16 ***
x0 r3 -61566.2330 7037.2999 -8.7486 < 2.2e-16 ***
x0 r4 181251.9827 13538.4495 13.3879 < 2.2e-16 ***
x1 P1 2426.2468 9.9272 244.4036 < 2.2e-16 ***
x1 P2 2257.7038 40.2376 56.1093 7.112e-15 ***
x1 P3 2397.0219 268.3580 8.9322 4.429e-06 ***
x1 P4 452.3725 1071.8938 0.4220 0.6829048
x1 R1 2435.8586 8.8038 276.6832 < 2.2e-16 ***
x1 R2 1928.5087 19.3980 99.4182 < 2.2e-16 ***
x1 R3 3604.3647 100.7439 35.7775 < 2.2e-16 ***
x1 R4 -1726.2560 283.6322 -6.0862 1.156e-09 ***
x1 bx 1676.1184 3.9161 428.0083 < 2.2e-16 ***
x1 by 955.4713 2.2442 425.7567 < 2.2e-16 ***
x1 l1 0.9772 0.0051 193.3863 < 2.2e-16 ***
x1 l2 0.5311 0.0893 5.9447 9.666e-05 ***
x1 p1 2436.7245 9.7442 250.0686 < 2.2e-16 ***
x1 p2 2250.3211 38.8158 57.9744 4.969e-15 ***
x1 p3 2556.7962 236.8145 10.7966 7.840e-07 ***
x1 p4 784.0799 907.5734 0.8639 0.4100597
x1 r1 2439.7193 10.7220 227.5435 < 2.2e-16 ***
x1 r2 1908.1636 22.5614 84.5766 < 2.2e-16 ***
x1 r3 3530.5491 118.2254 29.8629 < 2.2e-16 ***
x1 r4 -1784.4548 300.1184 -5.9458 2.750e-09 ***
x2 P2 0.4327 0.1020 4.2425 0.0013831 **
x2 P3 -0.3205 1.4371 -0.2230 0.8280176
x2 P4 15.7540 8.7301 1.8046 0.1046309
x2 R2 1.0957 0.0492 22.2859 < 2.2e-16 ***
x2 R3 -5.9925 0.5395 -11.1077 < 2.2e-16 ***
x2 R4 36.5554 2.3100 15.8245 < 2.2e-16 ***
x2 l2 0.0989 0.0198 4.9954 0.0004055 ***
x2 p2 0.4505 0.0928 4.8556 0.0005062 ***
x2 p3 -1.1670 1.2373 -0.9432 0.3677970
x2 p4 13.0751 7.1807 1.8209 0.1019624
x2 r2 1.1374 0.0539 21.0926 < 2.2e-16 ***
x2 r3 -5.6390 0.6177 -9.1293 < 2.2e-16 ***
x2 r4 37.0616 2.3745 15.6080 < 2.2e-16 ***
x3 P3 0.0013 0.0024 0.5255 0.6106812
x3 P4 -0.0550 0.0303 -1.8153 0.1028663
x3 R3 0.0098 0.0009 10.8887 < 2.2e-16 ***
x3 R4 -0.1379 0.0080 -17.1958 < 2.2e-16 ***
x3 p3 0.0027 0.0020 1.3108 0.2192314
x3 p4 -0.0458 0.0242 -1.8915 0.0911248 .
x3 r3 0.0093 0.0010 9.1678 < 2.2e-16 ***
x3 r4 -0.1397 0.0080 -17.4378 < 2.2e-16 ***
x4 P4 0.0001 0.0000 1.8616 0.0955657 .
x4 R4 0.0002 0.0000 18.7017 < 2.2e-16 ***
x4 p4 0.0001 0.0000 2.0064 0.0757681 .
x4 r4 0.0002 0.0000 19.4460 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residuals:
0% 25% 50% 75% 100% W
p1 -3411.2350 -956.2609 -633.8371 675.2061 4691.8080 0.8997
p2 -1791.1304 -1110.4795 -185.7831 1364.0765 1904.8696 0.8774
p3 -1653.4307 -953.4773 -287.4505 904.7724 2392.5513 0.9501
p4 -1709.4648 -668.0869 192.0693 673.1431 1986.5352 0.9720
P1 -3206.8646 -1398.9152 -530.8360 -19.7872 4901.1638 0.8633
P2 -1675.1390 -1096.8205 -140.9642 1303.0759 2020.8610 0.8798
P3 -1533.4564 -1038.8358 -260.8895 1144.0320 2162.5436 0.8972
P4 -1766.8812 -702.0159 89.2569 710.3186 1929.1188 0.9638
l1 -0.0028 -0.0021 -0.0010 0.0005 0.0066 0.8638
l2 -0.0027 -0.0010 -0.0002 0.0011 0.0034 0.9562
bx -1876.4174 -1214.4929 -38.1910 1368.0099 1819.5826 0.8834
by -756.9503 -463.0558 -23.7449 533.4825 759.1336 0.8932
r1 -3101.8541 -795.0184 -174.4162 1274.9338 5348.5869 0.9228
r2 -14769.0697 -5811.0169 -363.8833 -38.3051 1271.6388 0.7286
r3 -1084.6571 -157.4836 186.7870 420.2634 10272.1227 0.5976
r4 -14008.3794 -199.7239 -69.2792 133.9334 3682.5651 0.5809
R1 -2985.8187 -567.9099 -251.8039 1016.6706 5016.7765 0.8930
R2 -14047.1060 -5558.1056 -349.6547 -96.1537 1255.7213 0.7334
R3 -1052.9120 -135.2842 121.7895 517.0114 10736.4510 0.5918
R4 -13802.8794 -191.1081 -73.5248 125.5459 3689.5663 0.5829
Pr(<W)
p1 0.1115569
p2 0.0533595 .
p3 0.5617375
p4 0.9018410
P1 0.0337856 *
P2 0.0576933 .
P3 0.1025857
P4 0.7850142
l1 0.0343363 *
l2 0.6603242
bx 0.0649205 .
by 0.0898782 .
r1 0.2412125
r2 0.0007420 ***
r3 3.864e-05 ***
r4 2.753e-05 ***
R1 0.0893096 .
R2 0.0008371 ***
R3 3.431e-05 ***
R4 2.866e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Variances:
Min Median Max K Pr(>K)
Pure 2380.50 186660.50 6830208.00 12.297 0.055653 .
Log 0.00 0.00 0.00 17.741 0.006914 **
Box-Cox 371.52 28344.99 1149255.19 12.650 0.048946 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Goodness of fit:
R-Sq Adj-R-Sq AIC Sigma SSR SSPE F
p1 0.99981 0.99979 261.75205 2.4215e+03 7.0363e+07 7.9093e+06 11.0548
p2 0.99994 0.99993 247.71781 1.4265e+03 2.2385e+07 7.9093e+06 3.2028
p3 0.99995 0.99993 247.49803 1.3821e+03 1.9102e+07 7.9093e+06 3.3021
p4 0.99996 0.99995 244.32234 1.2110e+03 1.3199e+07 7.9093e+06 2.3407
P1 0.99980 0.99978 262.44697 9.7260e-01 1.1351e+01 7.9093e+06 NA
P2 0.99992 0.99991 250.87610 6.2560e-01 4.3057e+00 7.9093e+06 NA
P3 0.99993 0.99990 252.49472 6.4730e-01 4.1900e+00 7.9093e+06 NA
P4 0.99995 0.99992 249.93419 5.7980e-01 3.0251e+00 7.9093e+06 NA
l1 0.99968 0.99965 -120.27937 2.9000e-03 1.0000e-04 0.0000e+00 5.8256
l2 0.99990 0.99988 -134.86012 1.7000e-03 0.0000e+00 0.0000e+00 1.0133
bx 0.99993 0.99993 246.70642 1.4149e+03 2.4022e+07 7.9093e+06 2.8521
by 0.99993 0.99993 220.63859 5.5769e+02 3.7322e+06 1.3106e+06 2.5867
r1 NA NA 262.21028 1.6418e+03 7.2705e+07 7.9093e+06 NA
r2 NA NA 290.09308 1.4818e+03 4.6182e+08 7.9093e+06 NA
r3 NA NA 276.52142 9.1631e+02 1.5185e+08 7.9093e+06 NA
r4 NA NA 291.36478 6.7655e+02 3.8005e+08 7.9093e+06 NA
R1 NA NA 264.54576 7.0860e-01 1.3187e+01 7.9093e+06 NA
R2 NA NA 297.47919 5.3140e-01 1.2015e+02 7.9093e+06 NA
R3 NA NA 287.70642 3.1990e-01 5.1822e+01 7.9093e+06 NA
R4 NA NA 284.47973 2.5870e-01 3.5676e+01 7.9093e+06 NA
Pr(>F)
p1 0.003222 **
p2 0.085570 .
p3 0.087435 .
p4 0.166576
P1 NA
P2 NA
P3 NA
P4 NA
l1 0.019460 *
l2 0.461544
bx 0.102212
by 0.123754
r1 NA
r2 NA
r3 NA
r4 NA
R1 NA
R2 NA
R3 NA
R4 NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Sensitivity and autocorrelation:
Sens LOD LOQ Auto DW Pr(<DW)
p1 0.9937 3.2794 9.9375 0.0939 1.38728 0.077872 .
p2 0.6339 2.0919 6.3392 -0.1521 2.13482 0.445225
p3 0.5406 1.7839 5.4057 -0.2232 2.14595 0.450685
p4 1.5445 5.0968 15.4449 -0.1887 2.04487 0.420956
P1 0.0004 0.0013 0.0040 0.0418 1.54238 0.134268
P2 0.0003 0.0009 0.0028 -0.1701 2.14728 0.454901
P3 0.0003 0.0009 0.0027 -0.1870 2.13902 0.445513
P4 0.0013 0.0042 0.0128 -0.2006 2.07907 0.444676
l1 NA NA NA 0.0226 1.41554 0.087374 .
l2 NA NA NA -0.1287 1.88321 0.271245
bx NA NA NA -0.1422 2.13281 0.533920
by NA NA NA -0.1441 2.12070 0.524305
r1 0.6729 2.2207 6.7294 0.1021 1.28980 0.053014 .
r2 0.7766 2.5627 7.7656 0.4207 0.89283 0.005037 **
r3 0.2595 0.8565 2.5954 0.0000 1.30219 0.043064 *
r4 -0.3791 -1.2511 -3.7913 0.0018 1.44432 0.104051
R1 0.0003 0.0010 0.0029 0.0724 1.35904 0.069900 .
R2 0.0003 0.0009 0.0028 0.4162 0.90904 0.005519 **
R3 0.0001 0.0003 0.0009 0.0006 1.30923 0.044293 *
R4 -0.0001 -0.0005 -0.0015 0.0014 1.44412 0.103986
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
quantchem documentation built on May 30, 2017, 5:28 a.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
Description
Fit given calibration data to several regression models.
Usage
1 2 |
Arguments
x |
a vector of standard concentrations |
y |
a vector of corresponding respondes (peak areas etc.) |
confint |
confidence interval for graphing prediction intervals |
gridratio |
a part of x variable range, to extend plot range (default 5 percent) |
Details
For linear 'lmcal' fitting, procedure is performed as follows. First, the calibration data are fitted to OLS linear, quadratic, cubical, and 4th order polynomial. These models are called p1 - p4. Next, linear model is reweighted using x and y raised to power gamma from range (-4,4) with 0.1 accuracy. The optimal weights are detected by minimal mean relative error (MRE) according to Almeida et al. (2002). The best weighting scheme is then chosen, and data are fit to the same equations (called P1-P4, with uppercase).
Next, the optimal value of lambda for Box-Cox transform is estimated with accuracy up to 0.001, for transformation of x and y. The transformed models are then fitted (called bx and by).
Then, two next log-log transformed models, are fitted - linear called l1, and quadratic (mentioned sometimes as Wagner transform), called l2.
Last, the same models as p1 - p4 and P1 - P4, are fitted using rlm robust
method, and called r1 - r4 and R1 - R4.
This function performs also computation of grid and corresponding predicted values for easy graphing of fitted models.
For nonlinear 'nlscal' fitting, procedure is performed as above, but there are following models fitted: asymptotic (a1), asymptotic through origin (a2), logistic (g1), four parameter logistic (g2), Michaelis-Menten (m1) and nonparametric (loess) spline (s1). There are no weighting nor transform when fitting by 'nlscal'.
Value
Returns object of class c("lmcal","cal") or c("nlscal","cal"), which is the list of following
components:
models |
A list of fitted models (p1-p4,P1-P4,l1,l2,bx,by,r1-r4,R1-R4) |
graph |
A list used by |
x |
Concentration vector |
y |
Response vector |
Linear calibration object contains also following elements:
weigh |
A dataframe containing sequence of gamma values and corresponding mean relative errors, estimated during weighting process |
wx |
Value of gamma for oprimal weighting on x |
wy |
Value of gamma for oprimal weighting on y |
yw |
Logical, if weighting on y gives better result than on x |
px |
Optimal Box-Cox power for transform of x variable |
py |
Optimal Box-Cox power for transform of y variable |
Note
This function performs *no* decision which model should be chosen! Such decision should be always made by analyst.
Author(s)
Lukasz Komsta
References
Almeida, A.M., Castel-Branco, M.M., Falcao, A.C. (2002) Linear regression for calibration lines revisited: weighting schemes for bioanalytical methods. J. Chromatogr. B Biomed. Sci. Appl. 774, 215-222.
Nagaraja, N.V., Paliwal, J.K., Gupta, R.C. (1999) Choosing the calibration model in assay validation. J. Pharm. Biomed. Anal. 20, 433-438.
Kimanani, E.K., Lavigne, J. (1998) Bioanalytical calibration curves: variability of optimal powers between and within analytical methods. J. Pharm. Biomed. Anal. 16, 1107-1115.
Kirkup, L., Mulholland, M. (2004). Comparison of linear and non-linear equations in univariate calibration. J. Chromatogr. A, 1029, 1-11.
Kimanani, E.K. (1998) Bioanalitical calibration curves: proposal for statistical criteria. J. Pharm. Biomed. Anal. 16, 1117-1124.
Baumann, K., Waetzig, H. (1997) Regression and calibration for analytical separation techniques. Part I. Design considerations. Process Control and Quality, 10, 59-73.
Baumann, K. (1997) Regression and calibration for analytical separation techniques. Part II. Validation, weighted and robust regression. Process Control and Quality, 10, 75-112.
Coleman, D.E., Vanatta, L.E. (1999) Lack-of-fit testing of ion chromatographic calibration curves with inexact replicates. J. Chromatogr. A 850, 43-51.
See Also
Examples
1 2 3 4 5 |
Example output
Loading required package: MASS
Loading required package: outliers
------------------------------------------
If you use this package, please cite the recent paper containing description of this software:
Komsta, L. Chemometric and statistical evaluation of calibration curves in pharmaceutical analysis
- a short review on trends and recommendations. J. AOAC Int. 2012, 95, 3, 669-672.
------------------------------------------
There were 12 warnings (use warnings() to see them)
Optimal Box-Cox power on x: 1.0706
Optimal Box-Cox power on y: 0.9286
Optimal power of weighting on x: -1.1
Optimal power of weighting on y: -1.2
Better weighting on y: TRUE
Coefficients:
x0 x1 x2 x3 x4
p1 7.1855e+03 2.4367e+03 NA NA NA
p2 2.4477e+04 2.2503e+03 4.5050e-01 NA NA
p3 6.6297e+03 2.5568e+03 -1.1670e+00 2.6549e-03 NA
p4 8.4946e+04 7.8408e+02 1.3075e+01 -4.5838e-02 1e-04
P1 9.2852e+03 2.4262e+03 NA NA NA
P2 2.3786e+04 2.2577e+03 4.3268e-01 NA NA
P3 1.5885e+04 2.3970e+03 -3.2048e-01 1.2613e-03 NA
P4 9.9514e+04 4.5237e+02 1.5754e+01 -5.5003e-02 1e-04
l1 3.4461e+00 9.7722e-01 NA NA NA
l2 3.9466e+00 5.3114e-01 9.8918e-02 NA NA
bx 3.9029e+04 1.6761e+03 NA NA NA
by 1.6991e+04 9.5547e+02 NA NA NA
r1 6.2176e+03 2.4397e+03 NA NA NA
r2 6.5589e+04 1.9082e+03 1.1374e+00 NA NA
r3 -6.1566e+04 3.5305e+03 -5.6390e+00 9.2702e-03 NA
r4 1.8125e+05 -1.7845e+03 3.7062e+01 -1.3975e-01 2e-04
R1 6.9505e+03 2.4359e+03 NA NA NA
R2 6.3203e+04 1.9285e+03 1.0957e+00 NA NA
R3 -6.6491e+04 3.6044e+03 -5.9925e+00 9.8112e-03 NA
R4 1.7885e+05 -1.7263e+03 3.6555e+01 -1.3787e-01 2e-04
Goodness of fit:
R-Sq Adj-R-Sq AIC Sigma
p1 0.99981 0.99979 261.75205 2421.4910
p2 0.99994 0.99993 247.71781 1426.5197
p3 0.99995 0.99993 247.49803 1382.1149
p4 0.99996 0.99995 244.32234 1211.0032
P1 0.99980 0.99978 262.44697 0.9726
P2 0.99992 0.99991 250.87610 0.6256
P3 0.99993 0.99990 252.49472 0.6473
P4 0.99995 0.99992 249.93419 0.5798
l1 0.99968 0.99965 -120.27937 0.0029
l2 0.99990 0.99988 -134.86012 0.0017
bx 0.99993 0.99993 246.70642 1414.8727
by 0.99993 0.99993 220.63859 557.6887
r1 NA NA 262.21028 1641.7947
r2 NA NA 290.09308 1481.8118
r3 NA NA 276.52142 916.3124
r4 NA NA 291.36478 676.5453
R1 NA NA 264.54576 0.7086
R2 NA NA 297.47919 0.5314
R3 NA NA 287.70642 0.3199
R4 NA NA 284.47973 0.2587
Optimal Box-Cox power on x: 1.0706
Optimal Box-Cox power on y: 0.9286
Optimal power of weighting on x: -1.1
Optimal power of weighting on y: -1.2
Mean relative error:
0.003800207 on x
0.003778024 on y
0.003999273 unweighted
Better weighting on y: TRUE
Coefficients:
Estimate Std. Error t value Pr(>|t|)
x0 P1 9285.2032 1865.6239 4.9770 0.0003215 ***
x0 P2 23785.8581 3622.5547 6.5660 4.046e-05 ***
x0 P3 15884.8049 15495.0837 1.0252 0.3294508
x0 P4 99514.3329 47017.7766 2.1165 0.0633944 .
x0 R1 6950.5171 1654.4969 4.2010 2.658e-05 ***
x0 R2 63203.4992 1746.3796 36.1912 < 2.2e-16 ***
x0 R3 -66491.1832 5816.9866 -11.4305 < 2.2e-16 ***
x0 R4 178849.2739 12441.3020 14.3754 < 2.2e-16 ***
x0 bx 39029.0998 1150.3645 33.9276 2.733e-13 ***
x0 by 16991.1734 481.2495 35.3064 1.702e-13 ***
x0 l1 3.4461 0.0116 297.8942 < 2.2e-16 ***
x0 l2 3.9466 0.1004 39.2999 3.508e-13 ***
x0 p1 7185.5156 2089.5912 3.4387 0.0049061 **
x0 p2 24476.5774 3767.7857 6.4963 4.450e-05 ***
x0 p3 6629.6786 14096.2525 0.4703 0.6482197
x0 p4 84946.0666 40940.9648 2.0748 0.0678297 .
x0 r1 6217.5759 2299.2674 2.7042 0.0068478 **
x0 r2 65588.9586 2189.9951 29.9494 < 2.2e-16 ***
x0 r3 -61566.2330 7037.2999 -8.7486 < 2.2e-16 ***
x0 r4 181251.9827 13538.4495 13.3879 < 2.2e-16 ***
x1 P1 2426.2468 9.9272 244.4036 < 2.2e-16 ***
x1 P2 2257.7038 40.2376 56.1093 7.112e-15 ***
x1 P3 2397.0219 268.3580 8.9322 4.429e-06 ***
x1 P4 452.3725 1071.8938 0.4220 0.6829048
x1 R1 2435.8586 8.8038 276.6832 < 2.2e-16 ***
x1 R2 1928.5087 19.3980 99.4182 < 2.2e-16 ***
x1 R3 3604.3647 100.7439 35.7775 < 2.2e-16 ***
x1 R4 -1726.2560 283.6322 -6.0862 1.156e-09 ***
x1 bx 1676.1184 3.9161 428.0083 < 2.2e-16 ***
x1 by 955.4713 2.2442 425.7567 < 2.2e-16 ***
x1 l1 0.9772 0.0051 193.3863 < 2.2e-16 ***
x1 l2 0.5311 0.0893 5.9447 9.666e-05 ***
x1 p1 2436.7245 9.7442 250.0686 < 2.2e-16 ***
x1 p2 2250.3211 38.8158 57.9744 4.969e-15 ***
x1 p3 2556.7962 236.8145 10.7966 7.840e-07 ***
x1 p4 784.0799 907.5734 0.8639 0.4100597
x1 r1 2439.7193 10.7220 227.5435 < 2.2e-16 ***
x1 r2 1908.1636 22.5614 84.5766 < 2.2e-16 ***
x1 r3 3530.5491 118.2254 29.8629 < 2.2e-16 ***
x1 r4 -1784.4548 300.1184 -5.9458 2.750e-09 ***
x2 P2 0.4327 0.1020 4.2425 0.0013831 **
x2 P3 -0.3205 1.4371 -0.2230 0.8280176
x2 P4 15.7540 8.7301 1.8046 0.1046309
x2 R2 1.0957 0.0492 22.2859 < 2.2e-16 ***
x2 R3 -5.9925 0.5395 -11.1077 < 2.2e-16 ***
x2 R4 36.5554 2.3100 15.8245 < 2.2e-16 ***
x2 l2 0.0989 0.0198 4.9954 0.0004055 ***
x2 p2 0.4505 0.0928 4.8556 0.0005062 ***
x2 p3 -1.1670 1.2373 -0.9432 0.3677970
x2 p4 13.0751 7.1807 1.8209 0.1019624
x2 r2 1.1374 0.0539 21.0926 < 2.2e-16 ***
x2 r3 -5.6390 0.6177 -9.1293 < 2.2e-16 ***
x2 r4 37.0616 2.3745 15.6080 < 2.2e-16 ***
x3 P3 0.0013 0.0024 0.5255 0.6106812
x3 P4 -0.0550 0.0303 -1.8153 0.1028663
x3 R3 0.0098 0.0009 10.8887 < 2.2e-16 ***
x3 R4 -0.1379 0.0080 -17.1958 < 2.2e-16 ***
x3 p3 0.0027 0.0020 1.3108 0.2192314
x3 p4 -0.0458 0.0242 -1.8915 0.0911248 .
x3 r3 0.0093 0.0010 9.1678 < 2.2e-16 ***
x3 r4 -0.1397 0.0080 -17.4378 < 2.2e-16 ***
x4 P4 0.0001 0.0000 1.8616 0.0955657 .
x4 R4 0.0002 0.0000 18.7017 < 2.2e-16 ***
x4 p4 0.0001 0.0000 2.0064 0.0757681 .
x4 r4 0.0002 0.0000 19.4460 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residuals:
0% 25% 50% 75% 100% W
p1 -3411.2350 -956.2609 -633.8371 675.2061 4691.8080 0.8997
p2 -1791.1304 -1110.4795 -185.7831 1364.0765 1904.8696 0.8774
p3 -1653.4307 -953.4773 -287.4505 904.7724 2392.5513 0.9501
p4 -1709.4648 -668.0869 192.0693 673.1431 1986.5352 0.9720
P1 -3206.8646 -1398.9152 -530.8360 -19.7872 4901.1638 0.8633
P2 -1675.1390 -1096.8205 -140.9642 1303.0759 2020.8610 0.8798
P3 -1533.4564 -1038.8358 -260.8895 1144.0320 2162.5436 0.8972
P4 -1766.8812 -702.0159 89.2569 710.3186 1929.1188 0.9638
l1 -0.0028 -0.0021 -0.0010 0.0005 0.0066 0.8638
l2 -0.0027 -0.0010 -0.0002 0.0011 0.0034 0.9562
bx -1876.4174 -1214.4929 -38.1910 1368.0099 1819.5826 0.8834
by -756.9503 -463.0558 -23.7449 533.4825 759.1336 0.8932
r1 -3101.8541 -795.0184 -174.4162 1274.9338 5348.5869 0.9228
r2 -14769.0697 -5811.0169 -363.8833 -38.3051 1271.6388 0.7286
r3 -1084.6571 -157.4836 186.7870 420.2634 10272.1227 0.5976
r4 -14008.3794 -199.7239 -69.2792 133.9334 3682.5651 0.5809
R1 -2985.8187 -567.9099 -251.8039 1016.6706 5016.7765 0.8930
R2 -14047.1060 -5558.1056 -349.6547 -96.1537 1255.7213 0.7334
R3 -1052.9120 -135.2842 121.7895 517.0114 10736.4510 0.5918
R4 -13802.8794 -191.1081 -73.5248 125.5459 3689.5663 0.5829
Pr(<W)
p1 0.1115569
p2 0.0533595 .
p3 0.5617375
p4 0.9018410
P1 0.0337856 *
P2 0.0576933 .
P3 0.1025857
P4 0.7850142
l1 0.0343363 *
l2 0.6603242
bx 0.0649205 .
by 0.0898782 .
r1 0.2412125
r2 0.0007420 ***
r3 3.864e-05 ***
r4 2.753e-05 ***
R1 0.0893096 .
R2 0.0008371 ***
R3 3.431e-05 ***
R4 2.866e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Variances:
Min Median Max K Pr(>K)
Pure 2380.50 186660.50 6830208.00 12.297 0.055653 .
Log 0.00 0.00 0.00 17.741 0.006914 **
Box-Cox 371.52 28344.99 1149255.19 12.650 0.048946 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Goodness of fit:
R-Sq Adj-R-Sq AIC Sigma SSR SSPE F
p1 0.99981 0.99979 261.75205 2.4215e+03 7.0363e+07 7.9093e+06 11.0548
p2 0.99994 0.99993 247.71781 1.4265e+03 2.2385e+07 7.9093e+06 3.2028
p3 0.99995 0.99993 247.49803 1.3821e+03 1.9102e+07 7.9093e+06 3.3021
p4 0.99996 0.99995 244.32234 1.2110e+03 1.3199e+07 7.9093e+06 2.3407
P1 0.99980 0.99978 262.44697 9.7260e-01 1.1351e+01 7.9093e+06 NA
P2 0.99992 0.99991 250.87610 6.2560e-01 4.3057e+00 7.9093e+06 NA
P3 0.99993 0.99990 252.49472 6.4730e-01 4.1900e+00 7.9093e+06 NA
P4 0.99995 0.99992 249.93419 5.7980e-01 3.0251e+00 7.9093e+06 NA
l1 0.99968 0.99965 -120.27937 2.9000e-03 1.0000e-04 0.0000e+00 5.8256
l2 0.99990 0.99988 -134.86012 1.7000e-03 0.0000e+00 0.0000e+00 1.0133
bx 0.99993 0.99993 246.70642 1.4149e+03 2.4022e+07 7.9093e+06 2.8521
by 0.99993 0.99993 220.63859 5.5769e+02 3.7322e+06 1.3106e+06 2.5867
r1 NA NA 262.21028 1.6418e+03 7.2705e+07 7.9093e+06 NA
r2 NA NA 290.09308 1.4818e+03 4.6182e+08 7.9093e+06 NA
r3 NA NA 276.52142 9.1631e+02 1.5185e+08 7.9093e+06 NA
r4 NA NA 291.36478 6.7655e+02 3.8005e+08 7.9093e+06 NA
R1 NA NA 264.54576 7.0860e-01 1.3187e+01 7.9093e+06 NA
R2 NA NA 297.47919 5.3140e-01 1.2015e+02 7.9093e+06 NA
R3 NA NA 287.70642 3.1990e-01 5.1822e+01 7.9093e+06 NA
R4 NA NA 284.47973 2.5870e-01 3.5676e+01 7.9093e+06 NA
Pr(>F)
p1 0.003222 **
p2 0.085570 .
p3 0.087435 .
p4 0.166576
P1 NA
P2 NA
P3 NA
P4 NA
l1 0.019460 *
l2 0.461544
bx 0.102212
by 0.123754
r1 NA
r2 NA
r3 NA
r4 NA
R1 NA
R2 NA
R3 NA
R4 NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Sensitivity and autocorrelation:
Sens LOD LOQ Auto DW Pr(<DW)
p1 0.9937 3.2794 9.9375 0.0939 1.38728 0.077872 .
p2 0.6339 2.0919 6.3392 -0.1521 2.13482 0.445225
p3 0.5406 1.7839 5.4057 -0.2232 2.14595 0.450685
p4 1.5445 5.0968 15.4449 -0.1887 2.04487 0.420956
P1 0.0004 0.0013 0.0040 0.0418 1.54238 0.134268
P2 0.0003 0.0009 0.0028 -0.1701 2.14728 0.454901
P3 0.0003 0.0009 0.0027 -0.1870 2.13902 0.445513
P4 0.0013 0.0042 0.0128 -0.2006 2.07907 0.444676
l1 NA NA NA 0.0226 1.41554 0.087374 .
l2 NA NA NA -0.1287 1.88321 0.271245
bx NA NA NA -0.1422 2.13281 0.533920
by NA NA NA -0.1441 2.12070 0.524305
r1 0.6729 2.2207 6.7294 0.1021 1.28980 0.053014 .
r2 0.7766 2.5627 7.7656 0.4207 0.89283 0.005037 **
r3 0.2595 0.8565 2.5954 0.0000 1.30219 0.043064 *
r4 -0.3791 -1.2511 -3.7913 0.0018 1.44432 0.104051
R1 0.0003 0.0010 0.0029 0.0724 1.35904 0.069900 .
R2 0.0003 0.0009 0.0028 0.4162 0.90904 0.005519 **
R3 0.0001 0.0003 0.0009 0.0006 1.30923 0.044293 *
R4 -0.0001 -0.0005 -0.0015 0.0014 1.44412 0.103986
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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