## APPENDIX K. Bias Factor Regression Equations (12 pages)

Non-Linear Equation (All Data)

Nonlinear Regression

Data Source: Copy of Data 1 in Notebook1 Equation: Exponential Decay, Double, 4 Parameter f=a*exp(-b*x)+c*exp(-d*x)

R Rsqr Adj Rsqr Standard Error of Estimate

0.5748 0.3304 0.2467 3.6214

Coefficient Std. Error t P VIF

a 470.8819 3.8303 122.9372 <0.0001 0.0000 b 0.0708 0.0008 93.0639 <0.0001 0.0000 c -458.8266 0.8733 -525.3738 <0.0001 0.0000 d 0.0708 3.8267E-0081849904.8463 <0.0001 0.0000

Analysis of Variance:

Uncorrected for the mean of the observations: DF SS MS Regression 4 602.6119 150.6530 Residual 24 314.7442 13.1143 Total 28 917.3561 32.7627

Corrected for the mean of the observations: DF SS MS F P Regression 3 155.3316 51.7772 3.9481 0.0202 Residual 24 314.7442 13.1143 Total 27 470.0758 17.4102

Statistical Tests:

PRESS 402.4287

Durbin-Watson Statistic 1.8406 Passed

Normality Test Passed (P = 0.1399)

K-S Statistic = 0.2123 Significance Level = 0.1399

Constant Variance Test Passed (P = 0.0539)

Power of performed test wi th alpha = 0.0500: 0.9055

Regression Diagnostics: Row Std. Res. Stud. Res. Stud. Del. Res.



### 11 0.2683 0.2713 0.2660 2 0.3842 0.3885 0.3815 3 0.3981 0.4025 0.3953 4 0.3870 0.3913 0.3843 5 0.5278 0.5337 0.5256 6 0.1744 0.1763 0.1727 7 0.1854 0.1875 0.1837 8 0.0186 0.0194 0.0190 9 0.4218 0.4390 0.4315 10 0.1898 0.1976 0.1936 11 0.1125 0.1171 0.1147 12 0.6399 0.6660 0.6581 13 -0.0090 -0.0094 -0.0092 14 -0.2050 -0.2134 -0.2091 15 -0.2655 -0.2740 -0.2687 16 -0.4119 -0.4250 -0.4177 17 -0.8040 -0.8296 -0.8241 18 -0.4837 -0.4991 -0.4912 19 0.5380 0.5552 0.5470 20 -0.6300 -0.6501 -0.6421 21 -0.7874 -0.8125 -0.8066 22 0.0724 0.0775 0.0759 23 -0.8223 -0.8797 -0.8754 24 0.0448 0.0479 0.0469 25 0.2353 0.2518 0.2468 26 4.0571< 4.3403< 9.1621< 27 -0.7091 -0.7586 -0.7516 28 -1.5237 -1.6300 -1.6921

Influence Diagnostics: Row Cook's Dist Leverage DFFITS 1 0.0004 0.0219 0.0398 2 0.0008 0.0219 0.0571 3 0.0009 0.0219 0.0592 4 0.0009 0.0219 0.0575 5 0.0016 0.0219 0.0787 6 0.0002 0.0219 0.0259 7 0.0002 0.0219 0.0275 8 7.8086E-006 0.0767 0.0055 9 0.0040 0.0767 0.1244 10 0.0008 0.0767 0.0558 11 0.0003 0.0767 0.0331 12 0.0092 0.0767 0.1897 13 1.8172E-006 0.0767 -0.0026 14 0.0009 0.0767 -0.0603 15 0.0012 0.0608 -0.0684 16 0.0029 0.0608 -0.1063 17 0.0111 0.0608 -0.2097 18 0.0040 0.0608 -0.1250 19 0.0050 0.0608 0.1392 20 0.0068 0.0608 -0.1634 21 0.0107 0.0608 -0.2053 22 0.0002 0.1263 0.0288 23 0.0280 0.1263 -0.3328 24 8.2997E-005 0.1263 0.0178 25 0.0023 0.1263 0.0938



### 226 0.6805 0.1263 3.4828< 27 0.0208 0.1263 -0.2857 28 0.0960 0.1263 -0.6432

95% Confidence: Row Predicted 95% Conf-L 95% Conf-U 95% Pred-L 95% Pred-U 1 0.4985 -0.6079 1.6049 -7.0571 8.0541 2 0.4985 -0.6079 1.6049 -7.0571 8.0541 3 0.4985 -0.6079 1.6049 -7.0571 8.0541 4 0.4985 -0.6079 1.6049 -7.0571 8.0541 5 0.4985 -0.6079 1.6049 -7.0571 8.0541 6 0.4985 -0.6079 1.6049 -7.0571 8.0541 7 0.4985 -0.6079 1.6049 -7.0571 8.0541 8 1.9525 -0.1178 4.0229 -5.8031 9.7081 9 1.9525 -0.1178 4.0229 -5.8031 9.7081 10 1.9525 -0.1178 4.0229 -5.8031 9.7081 11 1.9525 -0.1178 4.0229 -5.8031 9.7081 12 1.9525 -0.1178 4.0229 -5.8031 9.7081 13 1.9525 -0.1178 4.0229 -5.8031 9.7081 14 1.9525 -0.1178 4.0229 -5.8031 9.7081 15 4.8516 3.0084 6.6948 -2.8464 12.5497 16 4.8516 3.0084 6.6948 -2.8464 12.5497 17 4.8516 3.0084 6.6948 -2.8464 12.5497 18 4.8516 3.0084 6.6948 -2.8464 12.5497 19 4.8516 3.0084 6.6948 -2.8464 12.5497 20 4.8516 3.0084 6.6948 -2.8464 12.5497 21 4.8516 3.0084 6.6948 -2.8464 12.5497 22 7.6477 4.9920 10.3035 -0.2842 15.5797 23 7.6477 4.9920 10.3035 -0.2842 15.5797 24 7.6477 4.9920 10.3035 -0.2842 15.5797 25 7.6477 4.9920 10.3035 -0.2842 15.5797 26 7.6477 4.9920 10.3035 -0.2842 15.5797 27 7.6477 4.9920 10.3035 -0.2842 15.5797 28 7.6477 4.9920 10.3035 -0.2842 15.5797

Fit Equation Description: [Variables] x = col(1) y = col(2) reciprocal_y=1/abs(y) reciprocal_ysquare=1/y^2 'Automatic Initial Parameter Estimate Functions xnear0(q)=max(abs(q))-abs(q) yatxnear0(q,r)=xat ymax(q,xnear0(r)) [Parameters] a = yatxnear0(y,x)/2 ''Auto {{previous: 470.882}} b = if(x50(x,y)-min(x)=0, 1, -ln(.5)/(0.5*(x50(x,y)-min(x)))) ''Auto {{previous: 0.0707909}} c = yatxnear0(y,x)/2 ''Auto {{previous: -458.827}} d = if(x50(x,y)-min(x)=0, .5, -ln(.5)/(1.5*(x50(x,y)-min(x)))) ''Auto {{previous: 0.0707908}} [Equation] f=a*exp(-b*x)+c*exp(-d*x) fit f to y ''fit f to y with weight reciprocal_y



### 3''fit f to y with weight reciprocal_ysquare [Constraints] b>0 d>0 [Options] tolerance=1e-10 stepsize=0.1 iterations=200

Number of Iterations Performed = 36



### Upper 95% Confidence Interval for Non-Linear Equation (All Data)

Nonlinear Regression

Data Source: Copy of Copy of Da ta 1 in Non-linear Equations.JNB Equation: Exponential Decay, Double, 4 Parameter f=a*exp(-b*x)+c*exp(-d*x)

R Rsqr Adj Rsqr Standard Error of Estimate

0.9996 0.9993 0.9992 0.0932

Coefficient Std. Error t P VIF

a 1383.0681 128872.1352 0.0107 0.9915 14647896.9916< b 1.0676 14.5195 0.0735 0.9420 14698599.5014< c 11.6378 0.4017 28.9686 <0.0001 136.3957< d 0.0425 0.0012 36.6002 <0.0001 29.1386<

Analysis of Variance:

Uncorrected for the mean of the observations: DF SS MS Regression 4 1187.9849 296.9962 Residual 24 0.2083 0.0087 Total 28 1188.1932 42.4355

Corrected for the mean of the observations: DF SS MS F P Regression 3 292.0892 97.3631 11217.2314 <0.0001 Residual 24 0.2083 0.0087 Total 27 292.2975 10.8258

Statistical Tests:

PRESS 0.2835

Durbin-Watson Statistic 0.4108 Failed

Normality Test Passed (P = 0.0737)

K-S Statistic = 0.2365 Significance Level = 0.0737



### 4Constant Variance Test Failed (P = <0.0001)

Power of performed test wi th alpha = 0.0500: 1.0000

Regression Diagnostics: Row Std. Res. Stud. Res. Stud. Del. Res. 2 -1.2049 -1.3014 -1.3215 3 -1.2049 -1.3014 -1.3215 4 -1.2049 -1.3014 -1.3215 5 -1.2049 -1.3014 -1.3215 6 -1.2049 -1.3014 -1.3215 7 -1.2049 -1.3014 -1.3215 8 -1.2049 -1.3014 -1.3215 9 1.3274 1.4338 1.4679 10 1.3274 1.4338 1.4679 11 1.3274 1.4338 1.4679 12 1.3274 1.4338 1.4679 13 1.3274 1.4338 1.4679 14 1.3274 1.4338 1.4679 15 1.3274 1.4338 1.4679 16 -0.4622 -0.4993 -0.4913 17 -0.4622 -0.4993 -0.4913 18 -0.4622 -0.4993 -0.4913 19 -0.4622 -0.4993 -0.4913 20 -0.4622 -0.4993 -0.4913 21 -0.4622 -0.4993 -0.4913 22 -0.4622 -0.4993 -0.4913 23 0.0318 0.0344 0.0337 24 0.0318 0.0344 0.0337 25 0.0318 0.0344 0.0337 26 0.0318 0.0344 0.0337 27 0.0318 0.0344 0.0337 28 0.0318 0.0344 0.0337 29 0.0318 0.0344 0.0337

Influence Diagnostics: Row Cook's Dist Leverage DFFITS 2 0.0706 0.1429 -0.5395 3 0.0706 0.1429 -0.5395 4 0.0706 0.1429 -0.5395 5 0.0706 0.1429 -0.5395 6 0.0706 0.1429 -0.5395 7 0.0706 0.1429 -0.5395 8 0.0706 0.1429 -0.5395 9 0.0857 0.1429 0.5993 10 0.0857 0.1429 0.5993 11 0.0857 0.1429 0.5993 12 0.0857 0.1429 0.5993 13 0.0857 0.1429 0.5993 14 0.0857 0.1429 0.5993 15 0.0857 0.1429 0.5993 16 0.0104 0.1429 -0.2006 17 0.0104 0.1429 -0.2006 18 0.0104 0.1429 -0.2006 19 0.0104 0.1429 -0.2006 20 0.0104 0.1429 -0.2006



### 521 0.0104 0.1429 -0.2006 22 0.0104 0.1429 -0.2006 23 4.9303E-005 0.1429 0.0137 24 4.9303E-005 0.1429 0.0137 25 4.9303E-005 0.1429 0.0137 26 4.9303E-005 0.1429 0.0137 27 4.9303E-005 0.1429 0.0137 28 4.9303E-005 0.1429 0.0137 29 4.9303E-005 0.1429 0.0137

95% Confidence: Row Predicted 95% Conf-L 95% Conf-U 95% Pred-L 95% Pred-U 2 1.7172 1.6445 1.7898 1.5116 1.9227 3 1.7172 1.6445 1.7898 1.5116 1.9227 4 1.7172 1.6445 1.7898 1.5116 1.9227 5 1.7172 1.6445 1.7898 1.5116 1.9227 6 1.7172 1.6445 1.7898 1.5116 1.9227 7 1.7172 1.6445 1.7898 1.5116 1.9227 8 1.7172 1.6445 1.7898 1.5116 1.9227 9 3.8992 3.8266 3.9719 3.6937 4.1048 10 3.8992 3.8266 3.9719 3.6937 4.1048 11 3.8992 3.8266 3.9719 3.6937 4.1048 12 3.8992 3.8266 3.9719 3.6937 4.1048 13 3.8992 3.8266 3.9719 3.6937 4.1048 14 3.8992 3.8266 3.9719 3.6937 4.1048 15 3.8992 3.8266 3.9719 3.6937 4.1048 16 6.7379 6.6652 6.8105 6.5323 6.9434 17 6.7379 6.6652 6.8105 6.5323 6.9434 18 6.7379 6.6652 6.8105 6.5323 6.9434 19 6.7379 6.6652 6.8105 6.5323 6.9434 20 6.7379 6.6652 6.8105 6.5323 6.9434 21 6.7379 6.6652 6.8105 6.5323 6.9434 22 6.7379 6.6652 6.8105 6.5323 6.9434 23 10.3005 10.2279 10.3732 10.0950 10.5061 24 10.3005 10.2279 10.3732 10.0950 10.5061 25 10.3005 10.2279 10.3732 10.0950 10.5061 26 10.3005 10.2279 10.3732 10.0950 10.5061 27 10.3005 10.2279 10.3732 10.0950 10.5061 28 10.3005 10.2279 10.3732 10.0950 10.5061 29 10.3005 10.2279 10.3732 10.0950 10.5061

Fit Equation Description: [Variables] x = col(1) y = col(5) reciprocal_y=1/abs(y) reciprocal_ysquare=1/y^2 'Automatic Initial Parameter Estimate Functions xnear0(q)=max(abs(q))-abs(q) yatxnear0(q,r)=xat ymax(q,xnear0(r)) [Parameters] a = yatxnear0(y,x)/2 ''Auto {{previous: 1383.07}} b = if(x50(x,y)-min(x)=0, 1, -ln(.5)/(0.5*(x50(x,y)-min(x)))) ''Auto {{previous: 1.06758}}



### 6c = yatxnear0(y,x)/2 ''Auto {{previous: 11.6378}} d = if(x50(x,y)-min(x)=0, .5, -ln(.5)/(1.5*(x50(x,y)-min(x)))) ''Auto {{previous: 0.0425242}} [Equation] f=a*exp(-b*x)+c*exp(-d*x) fit f to y ''fit f to y with weight reciprocal_y ''fit f to y with weight reciprocal_ysquare [Constraints] b>0 d>0 [Options] tolerance=1e-5 ''tolerance=1e-10 stepsize=0.1 iterations=10000 ''iterations=1000 ''iterations=200 ''iterations=600

Number of Iterations Performed = 102



### Non-Linear Equation (Average Data)

Nonlinear Regression

Data Source: Copy of Copy of Copy of Data 1 in BIASFACTOR MODEL.JNB Equation: Exponential Decay, Double, 4 Parameter f=a*exp(-b*x)+c*exp(-d*x)

R Rsqr Adj Rsqr Standard Error of Estimate

1.0000 1.0000 (+inf) (+inf)

Coefficient

a 170.5297 b 0.5580 c 4.1246 d 0.0186

Analysis of Variance:

Uncorrected for the mean of the observations: DF SS MS Regression 4 91.4065 22.8516 Residual 0 0.0000 (+inf) Total 4 91.4065 22.8516

Corrected for the mean of the observations: DF SS MS F P Regression 3 26.5235 8.8412 0.0000 (NAN) Residual 0 0.0000 (+inf)



### 7Total 3 26.5235 8.8412

Statistical Tests:

PRESS 0.0000

Durbin-Watson Statistic (+inf) Failed

Normality Test Failed (P = <0.0001)

K-S Statistic = <0.0001 Significance Level = <0.0001

Constant Variance Test Failed (P = <0.0001)

Power of performed test wi th alpha = 0.0500: 1.0000

Fit Equation Description: [Variables] x = col(1) y = col(2) reciprocal_y=1/abs(y) reciprocal_ysquare=1/y^2 'Automatic Initial Parameter Estimate Functions xnear0(q)=max(abs(q))-abs(q) yatxnear0(q,r)=xat ymax(q,xnear0(r)) [Parameters] a = yatxnear0(y,x)/2 ''Auto {{previous: 170.53}} b = if(x50(x,y)-min(x)=0, 1, -ln(.5)/(0.5*(x50(x,y)-min(x)))) ''Auto {{previous: 0.558022}} c = yatxnear0(y,x)/2 ''Auto {{previous: 4.12461}} d = if(x50(x,y)-min(x)=0, .5, -ln(.5)/(1.5*(x50(x,y)-min(x)))) ''Auto {{previous: 0.0185501}} [Equation] f=a*exp(-b*x)+c*exp(-d*x) fit f to y ''fit f to y with weight reciprocal_y ''fit f to y with weight reciprocal_ysquare [Constraints] b>0 d>0 [Options] tolerance=1e-10 stepsize=0.1 iterations=200

Number of Iterations Performed = 22

The regression produces a perfect fit



### 8

### Log-Linear Equation (All Data)

Nonlinear Regression

Data Source: Data 1 in Notebook1 Equation: Polynomial, Linear f=y0+a*x

R Rsqr Adj Rsqr Standard Error of Estimate

0.6807 0.4634 0.4428 0.2258

Coefficient Std. Error t P VIF

y0 0.7862 0.0779 10.0900 <0.0001 3.3333 a -0.0137 0.0029 -4.7386 <0.0001 3.3333

Analysis of Variance:

Uncorrected for the mean of the observations: DF SS MS Regression 2 7.5229 3.7615 Residual 26 1.3259 0.0510 Total 28 8.8488 0.3160

Corrected for the mean of the observations: DF SS MS F P Regression 1 1.1451 1.1451 22.4542 <0.0001 Residual 26 1.3259 0.0510 Total 27 2.4709 0.0915

Statistical Tests:

PRESS 1.5307

Durbin-Watson Statistic 1.5424 Passed

Normality Test Passed (P = 0.8502)

K-S Statistic = 0.1124 Significance Level = 0.8502

Constant Variance Test Passed (P = 0.0823)

Power of performed test wi th alpha = 0.0500: 0.9858

Regression Diagnostics: Row Std. Res. Stud. Res. Stud. Del. Res.



### 91 -0.0046 -0.0049 -0.0048 2 0.4787 0.5100 0.5026 3 0.5289 0.5635 0.5560 4 0.4889 0.5208 0.5134 5 0.9461 1.0080 1.0084 6 -0.5105 -0.5439 -0.5364 7 -0.4436 -0.4726 -0.4654 8 -0.5659 -0.5768 -0.5692 9 0.4802 0.4895 0.4822 10 -0.0511 -0.0520 -0.0510 11 -0.2667 -0.2718 -0.2669 12 0.8737 0.8905 0.8868 13 -0.6635 -0.6763 -0.6691 14 -1.5514 -1.5813 -1.6310 15 -0.0872 -0.0896 -0.0878 16 -0.3689 -0.3787 -0.3724 17 -1.4252 -1.4631 -1.4976 18 -0.5238 -0.5377 -0.5302 19 0.9869 1.0131 1.0136 20 -0.8844 -0.9079 -0.9047 21 -1.3667 -1.4029 -1.4309 22 0.8868 0.9237 0.9210 23 -0.1266 -0.1319 -0.1294 24 0.8624 0.8982 0.8948 25 1.0252 1.0678 1.0708 26 2.8836< 3.0035< 3.6444< 27 0.0352 0.0367 0.0360 28 -1.6364 -1.7044 -1.7733

Influence Diagnostics: Row Cook's Dist Leverage DFFITS 1 1.6345E-006 0.1190 -0.0018 2 0.0176 0.1190 0.1848 3 0.0215 0.1190 0.2044 4 0.0183 0.1190 0.1887 5 0.0687 0.1190 0.3707 6 0.0200 0.1190 -0.1972 7 0.0151 0.1190 -0.1711 8 0.0065 0.0374 -0.1122 9 0.0047 0.0374 0.0951 10 5.2642E-005 0.0374 -0.0101 11 0.0014 0.0374 -0.0526 12 0.0154 0.0374 0.1748 13 0.0089 0.0374 -0.1319 14 0.0486 0.0374 -0.3216 15 0.0002 0.0510 -0.0204 16 0.0039 0.0510 -0.0863 17 0.0575 0.0510 -0.3473 18 0.0078 0.0510 -0.1229 19 0.0276 0.0510 0.2350 20 0.0222 0.0510 -0.2098 21 0.0529 0.0510 -0.3318 22 0.0362 0.0782 0.2683 23 0.0007 0.0782 -0.0377 24 0.0342 0.0782 0.2607 25 0.0484 0.0782 0.3120



### 1026 0.3828 0.0782 1.0617 27 5.7113E-005 0.0782 0.0105 28 0.1233 0.0782 -0.5166

95% Confidence: Row Predicted 95% Conf-L 95% Conf-U 95% Pred-L 95% Pred-U 1 0.1684 0.0082 0.3285 -0.3227 0.6594 2 0.1684 0.0082 0.3285 -0.3227 0.6594 3 0.1684 0.0082 0.3285 -0.3227 0.6594 4 0.1684 0.0082 0.3285 -0.3227 0.6594 5 0.1684 0.0082 0.3285 -0.3227 0.6594 6 0.1684 0.0082 0.3285 -0.3227 0.6594 7 0.1684 0.0082 0.3285 -0.3227 0.6594 8 0.4331 0.3433 0.5229 -0.0397 0.9059 9 0.4331 0.3433 0.5229 -0.0397 0.9059 10 0.4331 0.3433 0.5229 -0.0397 0.9059 11 0.4331 0.3433 0.5229 -0.0397 0.9059 12 0.4331 0.3433 0.5229 -0.0397 0.9059 13 0.4331 0.3433 0.5229 -0.0397 0.9059 14 0.4331 0.3433 0.5229 -0.0397 0.9059 15 0.6097 0.5048 0.7145 0.1338 1.0855 16 0.6097 0.5048 0.7145 0.1338 1.0855 17 0.6097 0.5048 0.7145 0.1338 1.0855 18 0.6097 0.5048 0.7145 0.1338 1.0855 19 0.6097 0.5048 0.7145 0.1338 1.0855 20 0.6097 0.5048 0.7145 0.1338 1.0855 21 0.6097 0.5048 0.7145 0.1338 1.0855 22 0.6979 0.5681 0.8277 0.2159 1.1799 23 0.6979 0.5681 0.8277 0.2159 1.1799 24 0.6979 0.5681 0.8277 0.2159 1.1799 25 0.6979 0.5681 0.8277 0.2159 1.1799 26 0.6979 0.5681 0.8277 0.2159 1.1799 27 0.6979 0.5681 0.8277 0.2159 1.1799 28 0.6979 0.5681 0.8277 0.2159 1.1799

Fit Equation Description: [Variables] x = col(1) y = col(2) reciprocal_y = 1/abs(y) reciprocal_ysquare = 1/y^2 'Automatic Initial Parameter Estimate Functions F(q)=ape(x,y,1,0,1) [Parameters] y0 = F(0)[1] ''Auto {{previous: 0.786168}} a = F(0)[2] ''Auto {{previous: -0.0137291}} [Equation] f=y0+a*x fit f to y ''fit f to y with weight reciprocal_y ''fit f to y with weight reciprocal_ysquare [Constraints] [Options] tolerance=1e-10 stepsize=1 iterations=200



### 11

### 12 Number of Iterations Performed = 1