ImmunoAssay Example Data Comparison of two fitting procedures Data input file name ImmunoAssayData.txt This file, ImmunoAssayFourOutput.txt, was created at 18:34:01 on 09-Feb-2011 Equations compared: Equation One: Four parameter logistic function Equation Two: Five parameter logistic function Eqation Eqation One Two Sum of squares 0.1132 0.0486 Degrees of freedom 11 10 Extra sum of squares F-ratio = 13.3059 F-ratio probabilty = 0.0045 F value at the 0.05 significance level = 4.9646 Statitically Equation Two is the preferred fit. The additional parameter has, given a 0.05 significance level, significantly improved the fit. However, a choice of model should include, along with this comparison, observation of the displayed graphs and of the detailed analyses listed below Details of the two compared fitting exercises Five parameter logistic fitting: r = top + (bottom - top)/((1 + (a/C50)^HillSlope)^asymm) r = assay response; a = analyte concentration Non-linear regression (Nelder and Mead simplex procedure) Estimated parameters The statistics are obtained assuming that the model behaves as a linear model about the minimum. The Hessian matrix is calculated as the numerically derived second derivatives of chi square with respect to all pairs of parameters. Consequentlty treat the statistics with great caution. Best Estimate of Coefficient t-value p-value estimate the error of t P > |t| variation (%) top -0.0394 0.7757 1968.8138 -0.0508 0.9605 bottom 4.0111 0.9387 23.4018 4.2732 0.0016 C50 4.1981 3.2346 77.0481 1.2979 0.2235 HillSlope -5.4265 12.8645 237.0678 -0.4218 0.6821 Asymm 0.3436 1.2428 361.6797 0.2765 0.7878 Best Pre-minimum Post-minimum Initial Fractional estimate gradient gradient estimate step top -0.0394 -3.5563E-4 2.5299E-4 0.0060 6.0E-4 bottom 4.0111 -0.0026 0.0028 3.95 0.395 C50 4.1981 -0.0015 0.0013 2.75 0.275 HillSlope -5.4265 -1.6286E-4 1.3308E-4 1.004 0.1004 Asymm 0.3436 -0.0038 0.0026 1.0 0.1 analyte observed calculated weight unweighted weighted estimated analyte concn response response residual residual concn error * 0.0 0.0060 -0.0394 1.0 -0.0454 -0.0454 0.3765 0.5 0.049 0.0372 1.0 -0.0118 -0.0118 0.2512 1.0 0.103 0.2396 1.0 0.1366 0.1366 0.1885 1.5 0.61 0.5542 1.0 -0.0558 -0.0558 0.0912 2.0 0.993 0.9708 1.0 -0.0222 -0.0222 0.0747 2.5 1.507 1.4708 1.0 -0.0362 -0.0362 0.0652 3.0 2.009 2.0168 1.0 0.0078 0.0078 0.0633 3.5 2.498 2.5485 1.0 0.0505 0.0505 0.0684 4.0 3.01 3.003 1.0 -0.0070 -0.0070 0.0875 4.5 3.407 3.3456 1.0 -0.0614 -0.0614 0.134 5.0 3.496 3.5802 1.0 0.0842 0.0842 0.1561 5.5 3.708 3.7321 1.0 0.0241 0.0241 0.2716 6.0 3.888 3.8281 1.0 -0.0599 -0.0599 0.8405 6.5 3.941 3.8889 1.0 -0.0521 -0.0521 2.0075 10.0 3.95 3.9987 1.0 0.0487 0.0487 2.1111 * the estimated error in the estimated concentration on entering this response via getSampleConcn(response) Degrees of freedom 10 Number of data points 15 Number of estimated paramaters 5 Sum of squares of the unweighted residuals 0.0486 Chi Square 0.0486 Reduced Chi Square 0.0049 Correlation: analyte concentration and responses Weighted Linear Correlation Coefficient (R) 0.9108 Weighted Linear Correlation Coefficient Probability 1.189E-6 Correlation: observed responses and calculated responses Weighted Linear Correlation Coefficient 0.9993 Weighted Linear Correlation Coefficient Probability 3.8784E-20 Parameter - parameter correlation coefficients top bottom C50 HillSlope Asymm top 1.0 0.1369 -0.461 0.2936 0.4668 bottom 0.1369 1.0 -0.3728 0.7454 0.6476 C50 -0.461 -0.3728 1.0 -0.8289 -0.9301 HillSlope 0.2936 0.7454 -0.8289 1.0 0.9598 Asymm 0.4668 0.6476 -0.9301 0.9598 1.0 Coefficient of determination, R = 0.9986 Adjusted Coefficient of determination, R' = 0.9984 Coefficient of determination, F-ratio = 8940.1753 Coefficient of determination, F-ratio probability = 7.7567E-20 Total (weighted) sum of squares = 33.4358 Regression (weighted) sum of squares = 33.3873 Error (weighted) sum of squares = 0.0486 Number of iterations taken 1113 Maximum number of iterations allowed 3000 Number of restarts taken 3 Maximum number of restarts allowed 3 Standard deviation of the simplex at the minimum 6.6069E-10 Convergence tolerance 1.0E-9 simplex sd < the tolerance times the mean of the absolute values of the y values Step used in numerical differentiation to obtain Hessian matrix d(parameter) = parameter*1.0E-4 Four parameter logistic fitting: r = top + (bottom - top)/(1 + (a/C50)^HillSlope) r = assay response; a = analyte concentration Non-linear regression (Nelder and Mead simplex procedure) Estimated parameters The statistics are obtained assuming that the model behaves as a linear model about the minimum. The Hessian matrix is calculated as the numerically derived second derivatives of chi square with respect to all pairs of parameters. Consequentlty treat the statistics with great caution. Best Estimate of Coefficient t-value p-value estimate the error of t P > |t| variation (%) top 4.2044 0.9477 22.5415 4.4363 0.0010 bottom 0.0522 0.6355 1218.1974 0.0821 0.9361 C50 3.0189 0.7322 24.2529 4.1232 0.0017 Hill Slope 3.1946 2.2668 70.9588 1.4093 0.1864 Best Pre-minimum Post-minimum Initial Fractional estimate gradient gradient estimate step top 4.2044 -0.0026 0.0024 3.95 0.395 bottom 0.0522 4.886E-5 1.0507E-4 0.0060 6.0E-4 C50 3.0189 -0.0019 0.0019 2.75 0.275 Hill Slope 3.1946 -2.4677E-4 1.643E-4 1.004 0.1004 analyte observed calculated weight unweighted weighted estimated analyte concn response response residual residual concn error * 0.0 0.0060 0.0522 1.0 0.0462 0.0462 Outside the working range 0.5 0.049 0.0654 1.0 0.0164 0.0164 Outside the working range 1.0 0.103 0.1704 1.0 0.0674 0.0674 0.7229 1.5 0.61 0.4537 1.0 -0.1563 -0.1563 0.1114 2.0 0.993 0.9307 1.0 -0.0623 -0.0623 0.0899 2.5 1.507 1.5211 1.0 0.0141 0.0141 0.0837 3.0 2.009 2.1074 1.0 0.0984 0.0984 0.0894 3.5 2.498 2.6097 1.0 0.1117 0.1117 0.1069 4.0 3.01 3.0032 1.0 -0.0068 -0.0068 0.1501 4.5 3.407 3.2976 1.0 -0.1094 -0.1094 0.235 5.0 3.496 3.5137 1.0 0.0177 0.0177 0.2702 5.5 3.708 3.6717 1.0 -0.0363 -0.0363 0.418 6.0 3.888 3.788 1.0 -0.1 -0.1 0.7546 6.5 3.941 3.8745 1.0 -0.0665 -0.0665 0.9756 10.0 3.95 4.1158 1.0 0.1658 0.1658 1.0259 * the estimated error in the estimated concentration on entering this response via getSampleConcn(response) Degrees of freedom 11 Number of data points 15 Number of estimated paramaters 4 Sum of squares of the unweighted residuals 0.1132 Chi Square 0.1132 Reduced Chi Square 0.0103 Correlation: analyte concentration and responses Weighted Linear Correlation Coefficient (R) 0.9108 Weighted Linear Correlation Coefficient Probability 1.189E-6 Correlation: observed responses and calculated responses Weighted Linear Correlation Coefficient 0.9983 Weighted Linear Correlation Coefficient Probability 9.4961E-18 Parameter - parameter correlation coefficients top bottom C50 Hill Slope top 1.0 -0.2712 0.601 -0.7362 bottom -0.2712 1.0 0.3698 0.5343 C50 0.601 0.3698 1.0 -0.1426 Hill Slope -0.7362 0.5343 -0.1426 1.0 Coefficient of determination, R = 0.9966 Adjusted Coefficient of determination, R' = 0.9964 Coefficient of determination, F-ratio = 3828.5861 Coefficient of determination, F-ratio probability = 1.8992E-17 Total (weighted) sum of squares = 33.4358 Regression (weighted) sum of squares = 33.3227 Error (weighted) sum of squares = 0.1132 Number of iterations taken 580 Maximum number of iterations allowed 3000 Number of restarts taken 3 Maximum number of restarts allowed 3 Standard deviation of the simplex at the minimum 7.7212E-10 Convergence tolerance 1.0E-9 simplex sd < the tolerance times the mean of the absolute values of the y values Step used in numerical differentiation to obtain Hessian matrix d(parameter) = parameter*1.0E-4 End of file