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 with alpha = 0.0500: 0.9055

Regression Diagnostics:

Row Std. Res. Stud. Res. Stud. Del. Res. 

1
1 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 

2
26 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)=xatymax(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 Data 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 

4
Constant Variance Test Failed (P = <0.0001) 

Power of performed test with 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 

5
21 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)=xatymax(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}} 

6
c = 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) 

7
Total 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 with 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)=xatymax(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 with alpha = 0.0500: 0.9858

Regression Diagnostics:

Row Std. Res. Stud. Res. Stud. Del. Res. 

9
1 -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 

10
26 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