MLE3_Dose_Response-Ver10

# MLE3_Dose_Response-Ver10 - Instructions for Running the MLE...

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Unformatted text preview: Instructions for Running the MLE Dose­Response Model 1. Enter Dose­Response data into the yellow cells of the sheet entitled "Dose­Response". [Cells C15:E19] 2. Enter Guess of Parameters in the blue cells [F7:F9]. 3. Use Solver to maximize the likelihood function and determine corresponding optimal parameters.** Note: Solver is set to go. DO NOT change any of the options within solver. The likelihood function [Cell J24] is to be maximized by changing the parameter guesses in the blue cells. 4. To produce 3D plots, reasonable parameter values must be manually copied into the orange cells. [Cells G7:G9] Note: Manual transfer is necessary as the program is too complex to handle any other method. 5. Further instructions for manipulation of 3D plots are included on the sheets governing each plot. Hopefully this is not necessary. **If you do not have solver as an add­in with your Excel package, please e­mail CEE 597 TA. Interpretating the Graphs In the example that is currently implemented, you can observe how very different solutions can provide a reasonable fit. The solution shown initially shows a reasonable multi­stage model . However, it is not the MLE. Can you compute the MLE with solver? The correct answer is given on the last sheet marked sample. Look at the 1­2 contour sheet, and Chart 1­2. You should see the tradeoff between q1 and q2. A whole range of values provide nearly the same fit (likelihood value) to the data. As a result, the data cannot resolve which of these pairs is most reasonable. So what should EPA do? Use this spreadsheet to solve the problem specified in the homework assignment. Jery Stedinger he blue cells. Cells G7:G9] Dose Response Model 1.000 0.800 0.600 0.400 0.200 0.000 Probability Cancer 0 100 200 300 400 500 Dose P(d) 600 700 800 900 1000 1000 Demonstration of Dose­Response Modelling using Maximum Likelihood Instructions Enter data in Yellow Cells Enter Guess of parameters in Blue Cells F7:F9 Enter reasonable parameters for 3D plots in Orange cells G7:G9 F7:F9for 3 parameters F7:F8 for 2 parameters & set F9 = 0 Jery Stedinger and Bryan Tolson Cornell University April 12, 2001; revised 2004. For MLEs select Solver under TOOLS. Guess for Enter in Orange Optimization Values for 3D Plotting q­zero 0.34000 0.25000 q­one 0.00200 0.00150 q­two 0.00150 0.00250 Model: Pr{Cancer} = 1 ­ exp(­qzero ­ qone*dose ­ [qtwo*dose]2 ) Dose 0 125 250 500 750 Animals 70 70 72 60 50 with Cancer 18 30 50 55 48 < ­­­ (Note definition of of q­two!) logarithms of individual terms in Observed Guess/Opt Likelihood function = k ln(p) + (n ­ k) ln(1 ­ p) P(d) Model P(d) Guess/Opt Full Model 0.257 0.288 ­40.07 ­39.90 0.429 0.465 ­47.99 ­47.80 0.694 0.625 ­45.08 ­44.32 0.917 0.851 ­18.40 ­17.21 0.960 0.955 ­8.41 ­8.40 Dose Response Model 1.000 ANALYSIS FOR BLUE DATA Log-Likelihood Value: log­plot_factor based on Full Model ­157.630 log[ likelihood ] ­ log[ plot_factor ] ­2.321 For the initial data set, observe how the blue line with a qtwo =0.0015 bends up very little only at d = 0, whereas the orange line with qtwo = 0.0025 has a noticable upward twist for d up to 300. The shape of the function depends upon the Probability Cancer 0.800 0.600 0.400 0.200 0.000 -159.951 0 100 200 300 400 500 600 700 800 900 1000 Dose 0.000 0 100 200 300 400 500 600 700 800 900 1000 Dose P(d) absolute and relative values of qone & qtwo. qzero determines Pr{Cancer} at d = 0. qone determines the incremental risk from sm Play with the parameters and watch the changes. See if you can maximize the likelihood value by adjusting the blue parameters. MODEL PREDICTIONS FOR PLOTTING Prob of Model using Guess/ Prob of fitted Optimize orange model 0.2882 0.2212 0.3025 0.2333 0.3168 0.2461 0.3310 0.2596 0.3453 0.2739 0.3596 0.2887 0.3738 0.3041 0.3880 0.3200 0.4021 0.3363 0.4162 0.3531 0.4302 0.3703 0.4441 0.3878 0.4580 0.4055 0.4717 0.4234 0.4853 0.4415 0.4987 0.4597 0.5121 0.4780 0.5253 0.4962 0.5383 0.5145 0.5512 0.5326 0.5640 0.5507 Dose 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 0.5765 0.5889 0.6011 0.6131 0.6249 0.6365 0.6480 0.6592 0.6702 0.6810 0.6916 0.7019 0.7121 0.7220 0.7317 0.7412 0.7504 0.7595 0.7683 0.7769 0.7852 0.7934 0.8013 0.8090 0.8165 0.8238 0.8309 0.8377 0.8444 0.8508 0.8570 0.8631 0.8689 0.8746 0.8800 0.8853 0.5686 0.5863 0.6037 0.6209 0.6378 0.6544 0.6706 0.6865 0.7020 0.7171 0.7317 0.7459 0.7596 0.7729 0.7858 0.7981 0.8100 0.8214 0.8323 0.8428 0.8528 0.8623 0.8713 0.8800 0.8882 0.8959 0.9033 0.9102 0.9167 0.9229 0.9287 0.9341 0.9392 0.9440 0.9485 0.9526 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 0.8904 0.8953 0.9001 0.9046 0.9090 0.9133 0.9173 0.9213 0.9250 0.9286 0.9321 0.9355 0.9387 0.9417 0.9447 0.9475 0.9502 0.9527 0.9552 0.9576 0.9598 0.9620 0.9640 0.9660 0.9678 0.9696 0.9713 0.9729 0.9744 0.9759 0.9772 0.9786 0.9798 0.9810 0.9821 0.9832 0.9565 0.9601 0.9635 0.9666 0.9695 0.9722 0.9747 0.9769 0.9791 0.9810 0.9828 0.9844 0.9859 0.9873 0.9885 0.9896 0.9907 0.9916 0.9925 0.9933 0.9940 0.9946 0.9952 0.9957 0.9962 0.9966 0.9970 0.9973 0.9976 0.9979 0.9981 0.9984 0.9985 0.9987 0.9989 0.9990 930 940 950 960 970 980 990 1000 0.9842 0.9851 0.9860 0.9869 0.9877 0.9884 0.9892 0.9898 0.9991 0.9992 0.9993 0.9994 0.9995 0.9996 0.9996 0.9997 r and Bryan Tolson ll University 001; revised 2004. f of q­two!) vidual terms in n = k ln(p) + (n ­ k) ln(1 ­ p) FOR BLUE DATA ood Value: sed on Full Model log[ plot_factor ] data set, observe ine with a qtwo =0.0015 little only at d = 0, whereas with qtwo = 0.0025 has ward twist for d up to 300. he function depends upon the elative values of qone & qtwo. nes Pr{Cancer} at d = 0. es the incremental risk from small doses. arameters and watch the changes. aximize the likelihood value blue parameters. ONLY CHANGE THE YELLOW CELLS ON THIS SHEET ­all other cells referenced to MLE for Dose­Response sheet Dose 0 125 250 500 750 Animals 70 70 72 60 50 with Cancer 18 30 50 55 48 5 q­zero q­one q­two Observed Pr{ C } 0.257 0.429 0.694 0.917 0.960 times their MLE values Values for q­one and q­two are plotted to endpoints equal to Adjust these values to eliminate poor cel q­one 2 q­two ­2.56E+00 0.00 2.17E­04 4.35E­04 6.52E­04 8.70E­04 1.09E­03 1.30E­03 1.52E­03 1.74E­03 1.96E­03 2.17E­03 2.39E­03 2.61E­03 2.83E­03 3.04E­03 3.26E­03 3.48E­03 3.70E­03 3.91E­03 4.13E­03 4.35E­03 4.57E­03 4.78E­03 0.00E+00 ­1.75867E+02 ­1.69010E+02 ­1.51487E+02 ­1.29065E+02 ­1.06083E+02 ­8.48247E+01 ­6.62479E+01 ­5.06E+01 ­3.79E+01 ­2.78E+01 ­2.00E+01 ­1.44E+01 ­1.05E+01 ­8.18E+00 ­7.20E+00 ­7.40E+00 ­8.67E+00 ­1.09E+01 ­1.40E+01 ­1.79E+01 ­2.25E+01 ­2.79E+01 ­3.39E+01 6.25E­04 ­8.04E+01 ­7.82E+01 ­7.22E+01 ­6.34E+01 ­5.34E+01 ­4.33E+01 ­3.38E+01 ­2.55E+01 ­1.86E+01 ­1.32E+01 ­9.16E+00 ­6.49E+00 ­5.03E+00 ­4.68E+00 ­5.33E+00 ­6.89E+00 ­9.30E+00 ­1.25E+01 ­1.64E+01 ­2.10E+01 ­2.63E+01 ­3.22E+01 ­3.87E+01 5.00E­03 ­4.06E+01 ­4.58E+01 Input q*s 2.50E­01 1.50E­03 2.50E­03 Fitted Model 0.2212 0.4144 0.6378 0.9229 0.9925 their MLE values 2.50E­001 MLE estimates of parameters 1.50E­003 2.500E­03 Likelihood Term in Likelihood ­40.16 ­47.83 ­44.83 ­17.23 ­10.14 ­160.19 would plot as: ­2.56 st these values to eliminate poor cells 1.25E­03 ­4.01E+01 ­3.91E+01 ­3.61E+01 ­3.18E+01 ­2.66E+01 ­2.12E+01 ­1.61E+01 ­1.16E+01 ­8.03E+00 ­5.36E+00 ­3.66E+00 ­2.90E+00 ­3.05E+00 ­4.04E+00 ­5.82E+00 ­8.35E+00 ­1.16E+01 ­1.55E+01 ­2.00E+01 ­2.52E+01 ­3.10E+01 ­3.73E+01 ­4.42E+01 1.88E­03 ­1.93E+01 ­1.87E+01 ­1.72E+01 ­1.48E+01 ­1.21E+01 ­9.24E+00 ­6.60E+00 ­4.40E+00 ­2.80E+00 ­1.88E+00 ­1.69E+00 ­2.23E+00 ­3.48E+00 ­5.42E+00 ­8.02E+00 ­1.13E+01 ­1.51E+01 ­1.96E+01 ­2.46E+01 ­3.02E+01 ­3.63E+01 ­4.30E+01 ­5.02E+01 2.50E­03 ­8.29E+00 ­8.00E+00 ­7.20E+00 ­6.02E+00 ­4.65E+00 ­3.30E+00 ­2.16E+00 ­1.38E+00 ­1.09E+00 ­1.34E+00 ­2.18E+00 ­3.62E+00 ­5.66E+00 ­8.28E+00 ­1.15E+01 ­1.53E+01 ­1.96E+01 ­2.45E+01 ­2.99E+01 ­3.58E+01 ­4.22E+01 ­4.92E+01 ­5.66E+01 3.13E­03 ­3.12E+00 ­3.00E+00 ­2.65E+00 ­2.15E+00 ­1.64E+00 ­1.23E+00 ­1.06E+00 ­1.22E+00 ­1.81E+00 ­2.87E+00 ­4.43E+00 ­6.52E+00 ­9.13E+00 ­1.23E+01 ­1.59E+01 ­2.01E+01 ­2.48E+01 ­3.00E+01 ­3.57E+01 ­4.19E+01 ­4.86E+01 ­5.57E+01 ­6.33E+01 3.75E­03 ­1.68E+00 ­1.65E+00 ­1.59E+00 ­1.54E+00 ­1.58E+00 ­1.79E+00 ­2.27E+00 ­3.07E+00 ­4.27E+00 ­5.90E+00 ­7.99E+00 ­1.06E+01 ­1.36E+01 ­1.71E+01 ­2.11E+01 ­2.56E+01 ­3.06E+01 ­3.61E+01 ­4.20E+01 ­4.84E+01 ­5.53E+01 ­6.27E+01 ­7.04E+01 4.38E­03 ­2.76E+00 ­2.79E+00 ­2.92E+00 ­3.16E+00 ­3.57E+00 ­4.21E+00 ­5.13E+00 ­6.39E+00 ­8.03E+00 ­1.01E+01 ­1.25E+01 ­1.55E+01 ­1.88E+01 ­2.27E+01 ­2.70E+01 ­3.17E+01 ­3.69E+01 ­4.26E+01 ­4.87E+01 ­5.53E+01 ­6.24E+01 ­6.98E+01 ­7.76E+01 ­5.16E+01 ­5.78E+01 ­6.44E+01 ­7.13E+01 ­7.84E+01 ­8.56E+01 5.00E­03 ­5.60E+00 ­5.68E+00 ­5.93E+00 ­6.38E+00 ­7.05E+00 ­7.98E+00 ­9.22E+00 ­1.08E+01 ­1.28E+01 ­1.51E+01 ­1.79E+01 ­2.11E+01 ­2.47E+01 ­2.87E+01 ­3.33E+01 ­3.82E+01 ­4.36E+01 ­4.95E+01 ­5.57E+01 ­6.25E+01 ­6.97E+01 ­7.72E+01 ­8.51E+01 5.63E­03 ­9.73E+00 ­9.84E+00 ­1.02E+01 ­1.08E+01 ­1.16E+01 ­1.28E+01 ­1.42E+01 ­1.61E+01 ­1.82E+01 ­2.08E+01 ­2.38E+01 ­2.72E+01 ­3.10E+01 ­3.53E+01 ­4.00E+01 ­4.51E+01 ­5.06E+01 ­5.66E+01 ­6.30E+01 ­6.99E+01 ­7.72E+01 ­8.48E+01 ­9.26E+01 6.25E­03 ­1.48E+01 ­1.49E+01 ­1.53E+01 ­1.60E+01 ­1.70E+01 ­1.83E+01 ­2.00E+01 ­2.19E+01 ­2.43E+01 ­2.71E+01 ­3.02E+01 ­3.38E+01 ­3.78E+01 ­4.22E+01 ­4.70E+01 ­5.22E+01 ­5.79E+01 ­6.40E+01 ­7.05E+01 ­7.75E+01 ­8.48E+01 ­9.25E+01 ­1.00E+02 6.88E­03 ­2.06E+01 ­2.07E+01 ­2.12E+01 ­2.20E+01 ­2.30E+01 ­2.45E+01 ­2.62E+01 ­2.84E+01 ­3.09E+01 ­3.37E+01 ­3.70E+01 ­4.07E+01 ­4.48E+01 ­4.93E+01 ­5.43E+01 ­5.96E+01 ­6.54E+01 ­7.16E+01 ­7.82E+01 ­8.53E+01 ­9.26E+01 ­1.00E+02 ­1.08E+02 7.50E­03 ­2.70E+01 ­2.71E+01 ­2.76E+01 ­2.84E+01 ­2.96E+01 ­3.11E+01 ­3.29E+01 ­3.52E+01 ­3.78E+01 ­4.08E+01 ­4.42E+01 ­4.80E+01 ­5.22E+01 ­5.68E+01 ­6.18E+01 ­6.72E+01 ­7.31E+01 ­7.94E+01 ­8.61E+01 ­9.32E+01 ­1.01E+02 ­1.08E+02 ­1.16E+02 ­9.29E+01 ­1.00E+02 ­1.08E+02 ­1.15E+02 ­1.23E+02 Value of log­likelihood function for various values of qone and qtwo at the MLE estimate of qzero 0.00000E+00 ­2.00000E+01 ­4.00000E+01 ­6.00000E+01 ­8.00000E+01 ­1.00000E+02 ­1.20000E+02 ­1.40000E+02 ­1.60000E+02 ­1.80000E+02 ­2.00000E+02 0.00 8.70E­04 1.74E­03 5.63E­03 2.61E­03 3.48E­03 4.13E­03 qtwo 2.50E­03 4.78E­03 0.00E+00 qone imate of qzero 3 qone Contours of log­likelihood function for various values of qone and qtwo at the MLE estimate of qzero 0.00 8.70E­041.52E­03 5.63E­03 2.17E­03 2.83E­03 3.48E­03 4.13E­03 4.78E­03 qtwo 2.50E­03 0.00E+00 qone stimate of qzero qone ONLY CHANGE THE YELLOW CELLS ON THIS SHEET ­all other cells referenced to MLE for Dose­Response sheet Dose 0 125 250 500 750 Animals 70 70 72 60 50 with Cancer 18 30 50 55 48 4 q­zero q­one q­two Observed Pr{ C } 0.257 0.429 0.694 0.917 0.960 times their MLE values Values for q­zero and q­two are plotted to endpoints equal to Adjust these values to eliminate poor cell q­zero 3 q­two ­2.56E+00 0.00E+00 3.26E­04 6.52E­04 9.78E­04 1.30E­03 1.63E­03 1.96E­03 2.28E­03 2.61E­03 2.93E­03 3.26E­03 3.59E­03 3.91E­03 4.24E­03 4.57E­03 4.89E­03 5.22E­03 5.54E­03 5.87E­03 6.20E­03 6.52E­03 6.85E­03 7.17E­03 0.00E+00 ­3.61E+02 ­3.58E+02 ­3.49E+02 ­3.39E+02 ­3.27E+02 ­3.17E+02 ­3.10E+02 ­3.05E+02 ­3.02E+02 ­3.01E+02 ­3.02E+02 ­3.05E+02 ­3.10E+02 ­3.17E+02 ­3.25E+02 ­3.34E+02 ­3.45E+02 ­3.56E+02 ­3.66E+02 ­3.76E+02 ­3.88E+02 ­4.00E+02 ­4.13E+02 8.33E­02 ­6.13E+01 ­5.87E+01 ­5.20E+01 ­4.31E+01 ­3.41E+01 ­2.64E+01 ­2.06E+01 ­1.71E+01 ­1.57E+01 ­1.63E+01 ­1.88E+01 ­2.29E+01 ­2.86E+01 ­3.58E+01 ­4.44E+01 ­5.43E+01 ­6.53E+01 ­7.67E+01 ­8.73E+01 ­9.81E+01 ­1.10E+02 ­1.22E+02 ­1.35E+02 7.50E­03 ­4.27E+02 ­1.49E+02 Input q*s 2.50E­01 1.50E­03 2.50E­03 Fitted Model 0.2212 0.4144 0.6378 0.9229 0.9925 their MLE values 2.50E­001 MLE estimates of parameters 1.50E­003 2.500E­03 Likelihood Term in Likelihood ­40.16 ­47.83 ­44.83 ­17.23 ­10.14 ­160.19 would plot as: ­2.56 st these values to eliminate poor cells 1.67E­01 ­4.14E+01 ­3.93E+01 ­3.37E+01 ­2.64E+01 ­1.90E+01 ­1.29E+01 ­8.57E+00 ­6.32E+00 ­6.07E+00 ­7.67E+00 ­1.10E+01 ­1.59E+01 ­2.23E+01 ­3.00E+01 ­3.91E+01 ­4.95E+01 ­6.09E+01 ­7.25E+01 ­8.34E+01 ­9.43E+01 ­1.06E+02 ­1.19E+02 ­1.32E+02 2.50E­01 ­3.02E+01 ­2.84E+01 ­2.37E+01 ­1.76E+01 ­1.15E+01 ­6.67E+00 ­3.54E+00 ­2.32E+00 ­2.97E+00 ­5.36E+00 ­9.36E+00 ­1.49E+01 ­2.18E+01 ­3.00E+01 ­3.95E+01 ­5.02E+01 ­6.19E+01 ­7.37E+01 ­8.48E+01 ­9.60E+01 ­1.08E+02 ­1.21E+02 ­1.34E+02 3.33E­01 ­2.35E+01 ­2.20E+01 ­1.81E+01 ­1.29E+01 ­7.99E+00 ­4.16E+00 ­1.99E+00 ­1.61E+00 ­3.00E+00 ­6.04E+00 ­1.06E+01 ­1.66E+01 ­2.39E+01 ­3.26E+01 ­4.24E+01 ­5.34E+01 ­6.54E+01 ­7.73E+01 ­8.86E+01 ­9.99E+01 ­1.12E+02 ­1.25E+02 ­1.39E+02 4.17E­01 ­1.99E+01 ­1.86E+01 ­1.53E+01 ­1.10E+01 ­6.94E+00 ­4.00E+00 ­2.63E+00 ­2.96E+00 ­4.97E+00 ­8.55E+00 ­1.36E+01 ­2.00E+01 ­2.77E+01 ­3.67E+01 ­4.68E+01 ­5.81E+01 ­7.02E+01 ­8.23E+01 ­9.37E+01 ­1.05E+02 ­1.17E+02 ­1.30E+02 ­1.44E+02 5.00E­01 ­1.84E+01 ­1.74E+01 ­1.45E+01 ­1.10E+01 ­7.69E+00 ­5.51E+00 ­4.82E+00 ­5.76E+00 ­8.29E+00 ­1.23E+01 ­1.78E+01 ­2.46E+01 ­3.26E+01 ­4.18E+01 ­5.22E+01 ­6.37E+01 ­7.60E+01 ­8.83E+01 ­9.97E+01 ­1.11E+02 ­1.24E+02 ­1.37E+02 ­1.51E+02 5.83E­01 ­1.86E+01 ­1.77E+01 ­1.54E+01 ­1.24E+01 ­9.81E+00 ­8.27E+00 ­8.18E+00 ­9.64E+00 ­1.26E+01 ­1.71E+01 ­2.29E+01 ­3.00E+01 ­3.83E+01 ­4.77E+01 ­5.83E+01 ­7.00E+01 ­8.25E+01 ­9.48E+01 ­1.06E+02 ­1.18E+02 ­1.31E+02 ­1.44E+02 ­1.58E+02 ­1.46E+02 ­1.48E+02 ­1.53E+02 ­1.59E+02 ­1.65E+02 ­1.72E+02 6.67E­01 ­2.02E+01 ­1.94E+01 ­1.74E+01 ­1.50E+01 ­1.30E+01 ­1.20E+01 ­1.25E+01 ­1.44E+01 ­1.78E+01 ­2.26E+01 ­2.87E+01 ­3.60E+01 ­4.46E+01 ­5.42E+01 ­6.50E+01 ­7.69E+01 ­8.95E+01 ­1.02E+02 ­1.14E+02 ­1.25E+02 ­1.38E+02 ­1.51E+02 ­1.65E+02 7.50E­01 ­2.28E+01 ­2.21E+01 ­2.05E+01 ­1.86E+01 ­1.71E+01 ­1.66E+01 ­1.75E+01 ­1.98E+01 ­2.35E+01 ­2.86E+01 ­3.50E+01 ­4.26E+01 ­5.14E+01 ­6.12E+01 ­7.22E+01 ­8.42E+01 ­9.69E+01 ­1.09E+02 ­1.21E+02 ­1.33E+02 ­1.46E+02 ­1.59E+02 ­1.73E+02 8.33E­01 ­2.63E+01 ­2.58E+01 ­2.44E+01 ­2.29E+01 ­2.19E+01 ­2.18E+01 ­2.31E+01 ­2.58E+01 ­2.98E+01 ­3.52E+01 ­4.18E+01 ­4.96E+01 ­5.86E+01 ­6.86E+01 ­7.97E+01 ­9.18E+01 ­1.05E+02 ­1.17E+02 ­1.29E+02 ­1.41E+02 ­1.54E+02 ­1.67E+02 ­1.81E+02 9.17E­01 ­3.06E+01 ­3.01E+01 ­2.91E+01 ­2.79E+01 ­2.73E+01 ­2.76E+01 ­2.93E+01 ­3.23E+01 ­3.66E+01 ­4.22E+01 ­4.90E+01 ­5.70E+01 ­6.61E+01 ­7.63E+01 ­8.75E+01 ­9.98E+01 ­1.13E+02 ­1.25E+02 ­1.37E+02 ­1.49E+02 ­1.62E+02 ­1.75E+02 ­1.90E+02 1.00E+00 ­3.55E+01 ­3.51E+01 ­3.43E+01 ­3.34E+01 ­3.32E+01 ­3.39E+01 ­3.58E+01 ­3.91E+01 ­4.37E+01 ­4.95E+01 ­5.65E+01 ­6.47E+01 ­7.39E+01 ­8.42E+01 ­9.56E+01 ­1.08E+02 ­1.21E+02 ­1.33E+02 ­1.45E+02 ­1.57E+02 ­1.70E+02 ­1.84E+02 ­1.98E+02 ­1.80E+02 ­1.88E+02 ­1.96E+02 ­2.04E+02 ­2.13E+02 Value of likelihood function for various values of qzero and qtwo at the MLE estimate of qone 0.00E+00 ­2.00E+01 ­4.00E+01 ­6.00E+01 ­8.00E+01 ­1.00E+02 ­1.20E+02 ­1.40E+02 ­1.60E+02 ­1.80E+02 ­2.00E+02 0.00E+00 1.30E­03 2.28E­03 3.26E­03 4.24E­03 5.22E­03 6.20E­03 7.17E­03 qtwo 7.50E­01 3.33E­01 0.00E+00 qzero of qone ero ONLY CHANGE THE YELLOW CELLS ON THIS SHEET ­all other cells referenced to MLE for Dose­Response sheet Dose 0 125 250 500 750 Animals 70 70 72 60 50 with Cancer 18 30 50 55 48 3 q­zero q­one q­two Observed Pr{ C } 0.257 0.429 0.694 0.917 0.960 times their MLE values Values for q­zero and q­one are plotted to endpoints equal to Adjust these values to eliminate poor cell q­zero 5 q­one ­2.56E+00 0.00E+00 3.26E­04 6.52E­04 9.78E­04 1.30E­03 1.63E­03 1.96E­03 2.28E­03 2.61E­03 2.93E­03 3.26E­03 3.59E­03 3.91E­03 4.24E­03 4.57E­03 4.89E­03 5.22E­03 5.54E­03 5.87E­03 6.20E­03 6.52E­03 6.85E­03 7.17E­03 0.00E+00 ­3.43E+02 ­3.28E+02 ­3.18E+02 ­3.10E+02 ­3.05E+02 ­3.01E+02 ­2.98E+02 ­2.97E+02 ­2.96E+02 ­2.96E+02 ­2.97E+02 ­2.97E+02 ­2.99E+02 ­3.00E+02 ­3.02E+02 ­3.05E+02 ­3.07E+02 ­3.10E+02 ­3.13E+02 ­3.16E+02 ­3.19E+02 ­3.22E+02 ­3.25E+02 6.25E­02 ­4.90E+01 ­3.88E+01 ­3.14E+01 ­2.61E+01 ­2.24E+01 ­1.99E+01 ­1.83E+01 ­1.76E+01 ­1.75E+01 ­1.80E+01 ­1.89E+01 ­2.03E+01 ­2.19E+01 ­2.39E+01 ­2.61E+01 ­2.86E+01 ­3.13E+01 ­3.41E+01 ­3.72E+01 ­4.03E+01 ­4.36E+01 ­4.70E+01 ­5.05E+01 7.50E­03 ­3.29E+02 ­5.41E+01 Input q*s 2.50E­01 1.50E­03 2.50E­03 Fitted Model 0.2212 0.4144 0.6378 0.9229 0.9925 their MLE values 2.50E­001 MLE estimates of parameters 1.50E­003 2.500E­03 Likelihood Term in Likelihood ­40.16 ­47.83 ­44.83 ­17.23 ­10.14 ­160.19 would plot as: ­2.56 st these values to eliminate poor cells 1.25E­01 ­2.97E+01 ­2.24E+01 ­1.70E+01 ­1.33E+01 ­1.07E+01 ­9.14E+00 ­8.37E+00 ­8.26E+00 ­8.71E+00 ­9.63E+00 ­1.10E+01 ­1.26E+01 ­1.46E+01 ­1.68E+01 ­1.93E+01 ­2.20E+01 ­2.48E+01 ­2.78E+01 ­3.10E+01 ­3.43E+01 ­3.77E+01 ­4.12E+01 ­4.48E+01 1.88E­01 ­1.89E+01 ­1.34E+01 ­9.55E+00 ­6.91E+00 ­5.27E+00 ­4.44E+00 ­4.29E+00 ­4.70E+00 ­5.59E+00 ­6.90E+00 ­8.56E+00 ­1.05E+01 ­1.27E+01 ­1.52E+01 ­1.79E+01 ­2.07E+01 ­2.37E+01 ­2.69E+01 ­3.02E+01 ­3.36E+01 ­3.71E+01 ­4.07E+01 ­4.44E+01 2.50E­01 ­1.22E+01 ­8.19E+00 ­5.42E+00 ­3.68E+00 ­2.76E+00 ­2.54E+00 ­2.91E+00 ­3.76E+00 ­5.03E+00 ­6.66E+00 ­8.60E+00 ­1.08E+01 ­1.33E+01 ­1.59E+01 ­1.88E+01 ­2.18E+01 ­2.49E+01 ­2.82E+01 ­3.16E+01 ­3.51E+01 ­3.87E+01 ­4.24E+01 ­4.62E+01 3.13E­01 ­8.26E+00 ­5.30E+00 ­3.41E+00 ­2.39E+00 ­2.08E+00 ­2.36E+00 ­3.16E+00 ­4.38E+00 ­5.98E+00 ­7.89E+00 ­1.01E+01 ­1.25E+01 ­1.52E+01 ­1.80E+01 ­2.10E+01 ­2.42E+01 ­2.74E+01 ­3.08E+01 ­3.43E+01 ­3.79E+01 ­4.16E+01 ­4.54E+01 ­4.92E+01 3.75E­01 ­6.13E+00 ­4.06E+00 ­2.88E+00 ­2.46E+00 ­2.66E+00 ­3.38E+00 ­4.55E+00 ­6.09E+00 ­7.97E+00 ­1.01E+01 ­1.25E+01 ­1.52E+01 ­1.80E+01 ­2.10E+01 ­2.41E+01 ­2.74E+01 ­3.08E+01 ­3.43E+01 ­3.79E+01 ­4.16E+01 ­4.53E+01 ­4.92E+01 ­5.31E+01 4.38E­01 ­5.39E+00 ­4.04E+00 ­3.47E+00 ­3.56E+00 ­4.19E+00 ­5.28E+00 ­6.77E+00 ­8.60E+00 ­1.07E+01 ­1.31E+01 ­1.57E+01 ­1.85E+01 ­2.15E+01 ­2.46E+01 ­2.79E+01 ­3.13E+01 ­3.48E+01 ­3.84E+01 ­4.21E+01 ­4.58E+01 ­4.97E+01 ­5.36E+01 ­5.75E+01 ­4.85E+01 ­4.81E+01 ­5.00E+01 ­5.31E+01 ­5.70E+01 ­6.15E+01 5.00E­01 ­5.73E+00 ­4.99E+00 ­4.94E+00 ­5.46E+00 ­6.47E+00 ­7.89E+00 ­9.66E+00 ­1.17E+01 ­1.41E+01 ­1.67E+01 ­1.94E+01 ­2.24E+01 ­2.55E+01 ­2.88E+01 ­3.21E+01 ­3.56E+01 ­3.92E+01 ­4.29E+01 ­4.67E+01 ­5.05E+01 ­5.44E+01 ­5.84E+01 ­6.24E+01 5.63E­01 ­6.94E+00 ­6.73E+00 ­7.12E+00 ­8.02E+00 ­9.36E+00 ­1.11E+01 ­1.31E+01 ­1.54E+01 ­1.79E+01 ­2.07E+01 ­2.36E+01 ­2.67E+01 ­2.99E+01 ­3.33E+01 ­3.68E+01 ­4.04E+01 ­4.40E+01 ­4.78E+01 ­5.16E+01 ­5.55E+01 ­5.95E+01 ­6.35E+01 ­6.76E+01 6.25E­01 ­8.87E+00 ­9.11E+00 ­9.89E+00 ­1.11E+01 ­1.27E+01 ­1.47E+01 ­1.70E+01 ­1.94E+01 ­2.22E+01 ­2.51E+01 ­2.81E+01 ­3.14E+01 ­3.47E+01 ­3.82E+01 ­4.18E+01 ­4.54E+01 ­4.92E+01 ­5.30E+01 ­5.69E+01 ­6.09E+01 ­6.49E+01 ­6.89E+01 ­7.31E+01 6.88E­01 ­1.14E+01 ­1.20E+01 ­1.31E+01 ­1.47E+01 ­1.66E+01 ­1.87E+01 ­2.12E+01 ­2.39E+01 ­2.67E+01 ­2.98E+01 ­3.30E+01 ­3.63E+01 ­3.98E+01 ­4.33E+01 ­4.70E+01 ­5.07E+01 ­5.46E+01 ­5.84E+01 ­6.24E+01 ­6.64E+01 ­7.05E+01 ­7.46E+01 ­7.88E+01 7.50E­01 ­1.44E+01 ­1.54E+01 ­1.68E+01 ­1.86E+01 ­2.07E+01 ­2.31E+01 ­2.58E+01 ­2.86E+01 ­3.16E+01 ­3.48E+01 ­3.81E+01 ­4.15E+01 ­4.51E+01 ­4.87E+01 ­5.24E+01 ­5.63E+01 ­6.02E+01 ­6.41E+01 ­6.81E+01 ­7.22E+01 ­7.63E+01 ­8.04E+01 ­8.46E+01 ­6.64E+01 ­7.17E+01 ­7.72E+01 ­8.30E+01 ­8.89E+01 Value of log­likelihood function for various values of qzero and qone at the estimate of qtwo 0.00E+00 ­2.00E+01 ­4.00E+01 ­6.00E+01 ­8.00E+01 ­1.00E+02 ­1.20E+02 ­1.40E+02 ­1.60E+02 ­1.80E+02 ­2.00E+02 0.00E+00 1.30E­03 2.61E­03 5.63E­01 3.91E­03 4.89E­03 qone 5.87E­03 6.85E­03 2.50E­01 0.00E+00 qzero ate of qtwo qzero Sample Problem 3 par. MLEs 0.28091 0.00262 0 Dose 0 125 250 500 750 2 par. MLEs 0.25815 0.00373 0 Animals 70 70 72 60 50 with Cancer 18 30 50 55 48 3 par MLEs 2 par MLEs Observed Guess/Opt Guess/Opt Pr{ C } Model Prob Model Prob 0.2449 0.2275 0.257 0.4776 0.5154 0.429 0.6669 0.6960 0.694 0.8941 0.8804 0.917 0.9757 0.9529 0.960 ...
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## This note was uploaded on 09/28/2008 for the course CEE 5970 taught by Professor Stedinger during the Spring '07 term at Cornell.

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