{"text_format":"1","workbook":"xlsx","n_sheets":4,"sheets":["INTRO","MH algorithm","Data, model predictions, SSE","Model"],"not_rendered":{"charts":6,"images":1,"macros":1,"external_links":1,"defined_names":90,"comments":1,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"INTRO","index":0,"n_rows":59,"n_cols":4,"rows_kept":59,"cut":false,"hidden":false,"merged":[]}
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[null,"The Metropolis Hastings algorithm: an illustrative example"]
[]
[]
[null,"Aims"]
[null,"This spreadsheet is intended as an example to demonstrate how the Metropolis Hastings algorithm can be implemented within Excel to undertake Bayesian inference."]
[null,"If you are considering programming the Metropolis Hastings algorithm in another language/modelling package this example may be useful for you."]
[null,"This example uses a very simple state transition model (with 3 states) and uses data observations of persons in State B and persons moving to Stage C."]
[null,"This structure and approach can be extended to a larger more complex model and with more parameters and datasets."]
[null,"This spreadsheet may be a useful illustration of the process of the MH algorithm for those considering programming this algorithm in another package."]
[null,"Warning! This example is intended as a rough guide to the process only. For further details consult a statistics reference."]
[]
[null,"Referencing"]
[null,"The author has used a similar approach to calibrate a natural history model for colorectal cancer. The methods are published here:"]
[null,"Whyte S, Walsh C, Chilcott J. Bayesian Calibration of a Natural History Model with Application to a Population Model for Colorectal Cancer. Medical Decision Making 2011;31:625-641."]
[null,"http://www.ncbi.nlm.nih.gov/pubmed/21127321"]
[]
[null,"Please reference this webpage and the paper if they have been useful to you!"]
[null,"Please contact me with any feedback/suggestions: Sophie Whyte, Research Fellow, School of Health and Related Research (ScHARR), University of Sheffield, sophie.whyte@shef.ac.uk"]
[null,"Version 2, Feb 2012"]
[]
[null,"Instructions for use"]
[null,"(1)    Generate estimates for the variances:"]
[null,null,"Run the first macro to fit the model to one dataset at a time."]
[null,null,"A smaller number of runs will be sufficient for estimating variances."]
[null,null,"Adjust epsilon values and the proportion of parameters varied on each run to obtain a good jump rate."]
[null,null,"Adjust Max number of runs so that a good fit is obtained to each dataset."]
[null,null,"Check the resulting fit to each dataset."]
[null,"(2)    Run the MH algorithm:"]
[null,null,"Run the macro for the MH algorithm fitting to all datasets at once."]
[null,null,"The show progress graph feature can be used to check algorithm is working correctly."]
[null,null,"Examine the resulting plot of SSE for signs of convergence. "]
[null,"(3)    Repeat for several different initial sets:"]
[null,null,"Run the algorithm using a range of initial sets to form different chains."]
[null,null,"Different chains should all converge to the same region."]
[null,"(4)     A sample of parameter sets from after convergence can be used to represent parameter uncertainty (these could be taken from more than one chain)."]
[]
[null,"Notes:"]
[null,"If the number of parameters is high and the acceptance rate is low then it may help to just vary a proportion of the parameters on each run."]
[null,"Epsilon values for each parameter can be estimated by considering the relative variances of parameter sets from a set of runs."]
[null,"In this example parameter values are sampled from non-informative uniform priors (Beta(1,1)). However, if data is available on the values of model parameters this can be changed."]
[null,"The algorithm can be **animated** so you can see how close the model predictions are to the observed data for each run by using the 'show progress graph' option."]
[null,"Proposal sets are sampled from a uniform distribution around the current parameter value."]
[]
[null,"Sheet contents"]
[null,"Sheet 1: MH algorithm",null,"Contains macros for running MH, cells for adjusting how MH runs, and results from running MH algorithm included plots of parameter values and total SSE."]
[null,"Sheet 2: Data, model predictions, SSE",null,"Contains the observed data, the model predictions, SSE calculations, plots of model predictions versus observed data, and the results of the variances macro."]
[null,"Sheet 3: Model",null,"Contains model parameters and formulae to calculate model predictions."]

{"sheet":"MH algorithm","index":1,"n_rows":5020,"n_cols":27,"rows_kept":2250,"cut":true,"hidden":false,"merged":[],"bounds":{"rows_first":1000,"rows_last":250,"rows_between":1000,"halvings":1}}
[]
[null,"Metropolis hastings algorithm implemented within Excel - an illustrative example"]
[]
[]
[null,"On this sheet cells in which the user should enter values are coloured white.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Summary statistics and plots"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Parameter values across all runs"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"L29","v":"SSE"},null,{"$f":"N29","v":"SSE/variance"},null,{"$f":"P29","v":"Total SSE"}]
[null,"User defined inputs",null,null,null,null,"Initial parameter values and generation of proposal set",null,null,null,null,null,null,null,null,null,null,null,{"$f":"H10","v":"Parameter 1"},{"$f":"I10","v":"Parameter 2"},{"$f":"J10","v":"Parameter 3"},{"$f":"K10","v":"Parameter 4"},{"$f":"L30","v":"Data set 1"},{"$f":"M30","v":"Data set 2"},{"$f":"N30","v":"Data set 1"},{"$f":"O30","v":"Data set 2"}]
[null,"Use initial set number",1,null,null,null,null,1,2,3,4,null,null,null,null,null,null,"Best fitting parameter set",{"$f":"VLOOKUP(1,$Q$20:$AA$5020,S17)","v":1.6087121906936853E-4},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,T17)","v":1E-8},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,U17)","v":6.6065485082928326E-3},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,V17)","v":4.5675620091659187E-2},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,W17)","v":118742.13803072379},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,X17)","v":2901507.5720745139},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,Y17)","v":3.8157076921804784},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,Z17)","v":20.236591521042769},{"$f":"VLOOKUP(1,$Q$20:$AA$5020,AA17)","v":24.052299213223247}]
[null,"Max number of runs",5000,null,null,null,null,{"$f":"\"Parameter \" & H9","v":"Parameter 1"},{"$f":"\"Parameter \" & I9","v":"Parameter 2"},{"$f":"\"Parameter \" & J9","v":"Parameter 3"},{"$f":"\"Parameter \" & K9","v":"Parameter 4"},null,null,null,null,null,null,"lower bound",{"$f":"MIN(S$20:S5020)","v":1E-8},{"$f":"MIN(T$20:T5020)","v":1E-8},{"$f":"MIN(U$20:U5020)","v":6.3400981537975829E-3},{"$f":"MIN(V$20:V5020)","v":4.0886738908333879E-2},{"$f":"MIN(W$20:W5020)","v":62540.415522669871},{"$f":"MIN(X$20:X5020)","v":2308016.8010528721},{"$f":"MIN(Y$20:Y5020)","v":2.0096989033519783},{"$f":"MIN(Z$20:Z5020)","v":16.097284624081393},{"$f":"MIN(AA$20:AA5020)","v":24.052299213223247}]
[null,"Store every ? runs",1,null,null,null,null,{"$f":"VLOOKUP(H9,Model!$B$7:$D$9,2)","v":"Trans AtoB age 20"},{"$f":"VLOOKUP(I9,Model!$B$7:$D$9,2)","v":"Trans BtoC age 20"},{"$f":"VLOOKUP(J9,Model!$B$7:$D$9,2)","v":"Trans AtoB  age 100"},{"$f":"VLOOKUP(K9,Model!$B$7:$D$9,2)","v":"Trans AtoB  age 100"},null,null,null,null,null,null,"upper bound",{"$f":"MAX(S$20:S5020)","v":2.8453963034077995E-4},{"$f":"MAX(T$20:T5020)","v":2.9388002377149355E-3},{"$f":"MAX(U$20:U5020)","v":7.2393859639904344E-3},{"$f":"MAX(V$20:V5020)","v":5.1890845492180142E-2},{"$f":"MAX(W$20:W5020)","v":656456.45144896442},{"$f":"MAX(X$20:X5020)","v":3975017.1874091751},{"$f":"MAX(Y$20:Y5020)","v":21.094836028025714},{"$f":"MAX(Z$20:Z5020)","v":27.723794307801992},{"$f":"MAX(AA$20:AA5020)","v":37.714781812800638}]
[null,"Epsilon Multiplier (changed automatically by macro)",0.38742048900000015,null,null,null,"Initial sets",null,null,null,null,null,null,null,null,null,null,"Mean",{"$f":"AVERAGE(S$20:S5020)","v":3.3299252974018673E-5},{"$f":"AVERAGE(T$20:T5020)","v":1.3018452340638118E-4},{"$f":"AVERAGE(U$20:U5020)","v":6.9228205038564371E-3},{"$f":"AVERAGE(V$20:V5020)","v":4.5958767762336629E-2},{"$f":"AVERAGE(W$20:W5020)","v":280611.59665204433},{"$f":"AVERAGE(X$20:X5020)","v":2645627.9827942317},{"$f":"AVERAGE(Y$20:Y5020)","v":9.0172860756744644},{"$f":"AVERAGE(Z$20:Z5020)","v":18.4519569480799},{"$f":"AVERAGE(AA$20:AA5020)","v":27.469243023754146}]
[null,"Current run number",5001,null,null,null,1,0.02,0.02,0.02,0.02,null,null,null,null,null,null,"2.5 percentile",{"$f":"PERCENTILE(S$20:S5020,0.025)","v":1E-8},{"$f":"PERCENTILE(T$20:T5020,0.025)","v":1E-8},{"$f":"PERCENTILE(U$20:U5020,0.025)","v":6.3400981537975829E-3},{"$f":"PERCENTILE(V$20:V5020,0.025)","v":4.1956860548041555E-2},{"$f":"PERCENTILE(W$20:W5020,0.025)","v":86579.703150816669},{"$f":"PERCENTILE(X$20:X5020,0.025)","v":2317801.8214901886},{"$f":"PERCENTILE(Y$20:Y5020,0.025)","v":2.7821870548919563},{"$f":"PERCENTILE(Z$20:Z5020,0.025)","v":16.165530340039822},{"$f":"PERCENTILE(AA$20:AA5020,0.025)","v":24.915171124554199}]
[null,"Proportion of parameters to vary on each run",0.5,null,null,null,2,4.0000000000000001E-3,0.5,0.5,0.5,null,null,null,null,null,null,"97.5 percentile",{"$f":"PERCENTILE(S$20:S5020,0.975)","v":2.8453963034077995E-4},{"$f":"PERCENTILE(T$20:T5020,0.975)","v":1.4270489410872621E-3},{"$f":"PERCENTILE(U$20:U5020,0.975)","v":7.1616774352419796E-3},{"$f":"PERCENTILE(V$20:V5020,0.975)","v":4.9516437992994038E-2},{"$f":"PERCENTILE(W$20:W5020,0.975)","v":423570.45071039506},{"$f":"PERCENTILE(X$20:X5020,0.975)","v":3515487.9104075497},{"$f":"PERCENTILE(Y$20:Y5020,0.975)","v":13.611183475050952},{"$f":"PERCENTILE(Z$20:Z5020,0.975)","v":24.518803095598052},{"$f":"PERCENTILE(AA$20:AA5020,0.975)","v":31.871366702471416}]
[null,"Show progress graph while running (slower)",0,null,null,null,3,3.0000000000000001E-3,0.7,0.7,0.7,null,null,null,null,null,null,"97.5 percentile",{"$f":"STDEV(S$501:S5021)","v":8.3873948338932598E-5},{"$f":"STDEV(T$501:T5021)","v":3.8423608942086322E-4},{"$f":"STDEV(U$501:U5021)","v":2.3153445841006307E-4},{"$f":"STDEV(V$501:V5021)","v":1.9707690651751568E-3}]
[null,null,null,null,null,null,{"$f":"CONCATENATE(\"Currently using initial set \", $C$9)","v":"Currently using initial set 1"},{"$f":"VLOOKUP(InitialSetNumber,$G$13:$L$15,H9+1)","v":0.02},{"$f":"VLOOKUP(InitialSetNumber,$G$13:$L$15,I9+1)","v":0.02},{"$f":"VLOOKUP(InitialSetNumber,$G$13:$L$15,J9+1)","v":0.02},{"$f":"VLOOKUP(InitialSetNumber,$G$13:$L$15,K9+1)","v":0.02}]
[null,"Buttons for running macros",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3,4,5,6,7,8,9,10,11]
[null,null,null,null,null,null,"epsilon max",0.01,0.01,0.01,0.01,null,null,null,null,null,null,null,null,null,null,null,{"$f":"W7","v":"SSE"},null,{"$f":"Y7","v":"SSE/variance"},null,{"$f":"AA7","v":"Total SSE"}]
[null,null,null,null,null,null,"Range for sampling parameters:",null,null,null,null,null,null,null,null,null,null,"Run Number",{"$f":"S8","v":"Parameter 1"},{"$f":"T8","v":"Parameter 2"},{"$f":"U8","v":"Parameter 3"},{"$f":"V8","v":"Parameter 4"},{"$f":"W8","v":"Data set 1"},{"$f":"X8","v":"Data set 2"},{"$f":"Y8","v":"Data set 1"},{"$f":"Z8","v":"Data set 2"}]
[null,null,null,null,null,null,"lower bound",1E-8,1E-8,1E-8,1E-8,null,null,null,null,null,{"$f":"IF(AA20=$AA$10,1,\"\")","v":""},1,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,null,null,null,null,null,"upper bound",0.99999999900000003,0.99999999900000003,0.99999999900000003,0.99999999900000003,null,null,null,null,null,"",2,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,null,null,null,null,null,"epsilon for current run",{"$f":"H18*EpsilonMultiplier","v":3.8742048900000017E-3},{"$f":"I18*EpsilonMultiplier","v":3.8742048900000017E-3},{"$f":"J18*EpsilonMultiplier","v":3.8742048900000017E-3},{"$f":"K18*EpsilonMultiplier","v":3.8742048900000017E-3},null,null,null,null,null,"",3,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,null,null,null,null,null,"U[-e,e]",{"$f":"(-H22+RAND()*2*H22)","v":3.4879706669498287E-3},{"$f":"(-I22+RAND()*2*I22)","v":-3.2762001819305127E-3},{"$f":"(-J22+RAND()*2*J22)","v":-3.0347140830026555E-3},{"$f":"(-K22+RAND()*2*K22)","v":2.7385095424697949E-3},null,null,null,null,null,"",4,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Estimating Variances",null,null,null,null,"Random number",{"$f":"RAND()","v":3.6051268629041511E-2},{"$f":"RAND()","v":0.59036301034707195},{"$f":"RAND()","v":0.57034303020321742},{"$f":"RAND()","v":0.40099012639434417},null,null,null,null,null,"",5,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"For each data set the MH algorithm is run to just fit to that one data set.",null,null,null,null,"Vary parameter on current run?",{"$f":"IF(H24<$C$14,1,0)","v":1},{"$f":"IF(I24<$C$14,1,0)","v":0},{"$f":"IF(J24<$C$14,1,0)","v":0},{"$f":"IF(K24<$C$14,1,0)","v":1},null,null,null,null,null,"",6,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"The resulting minimum SSE is used to generate an estimate for the variance of a data set ",null,null,null,null,"Values for proposal set",{"$f":"MIN(H21,MAX(H20,H33+H23*H25))","v":3.4879806669498286E-3},{"$f":"MIN(I21,MAX(I20,I33+I23*I25))","v":1E-8},{"$f":"MIN(J21,MAX(J20,J33+J23*J25))","v":7.0444744926159431E-3},{"$f":"MIN(K21,MAX(K20,K33+K23*K25))","v":4.9022811503721009E-2},null,null,null,null,null,"",7,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Number of runs to estimate variances",1000,null,null,null,null,null,null,null,null,null,null,null,null,null,"",8,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Estimate variances",0,null,null,null,"Comparison of current set and proposal set",null,null,null,null,null,null,null,null,null,"",9,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Current data set number",0,null,null,null,null,null,null,null,null,"SSE",null,"SSE/variance",null,"Total SSE","",10,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Total SSE for current data set",{"$f":"OFFSET(C37,0,EstimateVariancesNumber-1)","v":"Current value of SSE"},null,null,null,null,null,null,null,null,"Data set 1","Data set 2","Data set 1","Data set 2",null,"",11,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Min total SSE for current data set",{"$f":"OFFSET(L32,0,EstimateVariancesNumber-1)","v":4.6284301961251217E-2},null,null,null,null,{"$f":"H10","v":"Parameter 1"},{"$f":"I10","v":"Parameter 2"},{"$f":"J10","v":"Parameter 3"},{"$f":"K10","v":"Parameter 4"},null,null,null,null,null,"",12,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,null,1,2,null,null,"Proposal Set",1E-8,1E-8,7.0444744926159431E-3,4.6284301961251217E-2,{"$f":"'Data, model predictions, SSE'!M21","v":280107.48452345969},{"$f":"'Data, model predictions, SSE'!N21","v":2443942.494207581},{"$f":"L32/C36","v":9.0010867334807632},{"$f":"M32/D36","v":17.045299633954269},{"$f":"IF(EstimateVariances=0,SUM(N32:O32),C30)","v":26.046386367435034},"",13,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,null,"Data set 1","Data set 2",null,null,"Current Set",1E-8,1E-8,7.0444744926159431E-3,4.6284301961251217E-2,280107.48452345969,2443942.494207581,9.0010867334807632,17.045299633954269,26.046386367435034,"",14,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Minimum SSE when model is just fit to this data set",155596.4811901889,1577171.8898230263,null,null,null,null,null,null,null,null,null,null,null,null,"",15,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Number of points in data set",{"$f":"COUNT('Data, model predictions, SSE'!C10:C20)","v":5},{"$f":"COUNT('Data, model predictions, SSE'!G10:G20)","v":11},null,null,"Contribtuions from priors",null,null,null,null,null,null,null,null,null,"",16,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Estimate of variance",{"$f":"C34/C35","v":31119.296238037779},{"$f":"D34/D35","v":143379.26271118422},null,null,null,{"$f":"H10","v":"Parameter 1"},{"$f":"I10","v":"Parameter 2"},{"$f":"J10","v":"Parameter 3"},{"$f":"K10","v":"Parameter 4"},null,null,null,null,null,"",17,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Current value of SSE",{"$f":"L32","v":280107.48452345969},{"$f":"M32","v":2443942.494207581},null,null,"Prior (beta dist)                 a",1,1,1,1,null,null,null,null,null,"",18,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Minimum value of SSE",{"$f":"W10","v":62540.415522669871},{"$f":"X10","v":2308016.8010528721},null,null,"b",1,1,1,1,null,null,null,null,null,"",19,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,null,null,null,null,null,null,null,null,null,null,"Total contributions from priors",null,null,"Total SSE+prior contrs.",null,"",20,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Data on jump rate from last set of runs",null,null,null,null,"Proposal Set",{"$f":"-((H$37-1)*LOG(H32)+(H$38-1)*LOG(1-H32))","v":0},{"$f":"-((I$37-1)*LOG(I32)+(I$38-1)*LOG(1-I32))","v":0},0,0,{"$f":"SUM(H40:K40)","v":0},null,null,null,{"$f":"P32+L40","v":26.046386367435034},"",21,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Tuning - Epsilon values should be selected to obtain a jump rate of between 0.2 and 0.3.",null,null,null,null,"Current Set",{"$f":"-((H$37-1)*LOG(H33)+(H$38-1)*LOG(1-H33))","v":0},0,0,0,{"$f":"SUM(H41:K41)","v":0},null,null,null,{"$f":"P33+L41","v":26.046386367435034},"",22,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,"Run Number","Jump rate","Episilon Multiplier",null,null,null,null,null,null,null,null,null,null,null,null,"",23,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,100,0.13,0.9,null,null,null,null,null,null,null,null,null,null,null,null,"",24,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,200,0.21,0.9,null,null,"Accept/Reject proposal set",{"$f":"IF((P40-P41)<0,1,IF(RAND()<EXP(-0.5*(P40-P41)),1,0))","v":1},null,null,null,null,null,null,null,null,"",25,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,300,0.16,0.9,null,null,null,null,null,null,null,null,null,null,null,null,"",26,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,400,0.17,0.9,null,null,null,null,null,null,null,null,null,null,null,null,"",27,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,500,0.22,0.9,null,null,null,null,null,null,null,null,null,null,null,null,"",28,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,600,0.15,0.9,null,null,null,null,null,null,null,null,null,null,null,null,"",29,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,700,0.11,0.81,null,null,null,null,null,null,null,null,null,null,null,null,"",30,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,800,0.08,0.72900000000000009,null,null,null,null,null,null,null,null,null,null,null,null,"",31,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,900,0.13,0.65610000000000013,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1000,0.13,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",32,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1100,0.15,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",33,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1200,0.17,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",34,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1300,0.25,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",35,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1400,0.15,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",36,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1500,0.24,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",37,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1600,0.22,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",38,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1700,0.16,0.59049000000000018,null,null,null,null,null,null,null,null,null,null,null,null,"",39,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1800,0.12,0.53144100000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",40,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,1900,0.15,0.53144100000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",41,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2000,0.19,0.53144100000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",42,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2100,0.22,0.53144100000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",43,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2200,0.3,0.53144100000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",44,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2300,0.17,0.53144100000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",45,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2400,0.14000000000000001,0.47829690000000014,null,null,null,null,null,null,null,null,null,null,null,null,"",46,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2500,0.16,0.47829690000000014,null,null,null,null,null,null,null,null,null,null,null,null,"",47,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2600,0.22,0.47829690000000014,null,null,null,null,null,null,null,null,null,null,null,null,"",48,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2700,7.0000000000000007E-2,0.43046721000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",49,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2800,0.16,0.43046721000000016,null,null,null,null,null,null,null,null,null,null,null,null,"",50,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,2900,0.14000000000000001,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",51,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,3000,0.26,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",52,1E-8,1E-8,7.0259070664834174E-3,4.6428457920918299E-2,266818.49794360017,2448725.3757387251,8.574053085990494,17.078657885633788,25.65271097162428]
[null,3100,0.2,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",53,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3200,0.25,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",54,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3300,0.23,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",55,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3400,0.26,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",56,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3500,0.27,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",57,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3600,0.19,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",58,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3700,0.22,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",59,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3800,0.24,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",60,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,3900,0.17,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",61,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4000,0.2,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",62,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4100,0.21,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",63,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4200,0.26,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",64,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4300,0.22,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(AA85=$AA$10,1,\"\")","v":""},65,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4400,0.17,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",66,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4500,0.21,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",67,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4600,0.21,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",68,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4700,0.31,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",69,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4800,0.15,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",70,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,4900,0.23,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",71,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,5000,0.24,0.38742048900000015,null,null,null,null,null,null,null,null,null,null,null,null,"",72,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",73,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",74,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",75,1E-8,1E-8,6.9417515300134475E-3,4.6428457920918299E-2,339086.15771215188,2412663.3586280257,10.896331174021848,16.82714301222186,27.723474186243706]
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{"sheet":"Data, model predictions, SSE","index":2,"n_rows":21,"n_cols":19,"rows_kept":21,"cut":false,"hidden":false,"merged":[[7,2,7,4],[7,5,7,7],[6,12,6,13]]}
[]
["Observed data, model predictions, and SSE",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Results of variance estimation run"]
[]
[]
[]
[null,null,"Observed data",null,null,null,null,null,"Model predictions",null,null,null,"Sum squared error (SSE)"]
[null,null,"Observed counts",null,null,null,null,null,null,null,null,null,"Difference between observed counts and model predctions, squared"]
[null,"Age","Data set 1 (persons in State B)",null,null,"Data set 2 (persons moving to State C)",null,null,"Predicted rates",null,"Predicted counts - calculated using denominators from observed counts data",null,"Data set 1","Data set 2",null,null,"Predicted rates from variance estimation run"]
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{"sheet":"Model","index":3,"n_rows":103,"n_cols":10,"rows_kept":103,"cut":false,"hidden":false,"merged":[[11,1,13,3],[14,1,15,3]]}
[]
[null,"Model inputs",null,null,null,"Model"]
[]
[null,null,null,null,null,"A very simple state transition model example with 3 states and 4 parameters."]
[]
[null,"Parameter Number","Parameter Name","Parameter value",null,"State transition matrices"]
[null,1,"Trans AtoB age 20",{"$f":"TRANSPOSE('MH algorithm'!H32:L32)","v":1E-8}]
[null,2,"Trans BtoC age 20",1E-8,null,"age 20","State A","State B","State C"]
[null,3,"Trans AtoB  age 100",7.0444744926159431E-3,null,"State A",{"$f":"1-H9","v":0.99999998999999995},{"$f":"Parameter1","v":1E-8},0]
[null,4,"Trans BtoC age 100",4.6284301961251217E-2,null,"State B",0,{"$f":"1-I10","v":0.99999998999999995},{"$f":"Parameter2","v":1E-8}]
[null,null,null,null,null,"State C",0,0,1]
[null,"The transition probabilities for age 20-100 are assumed to vary linearly (i.e. the rate for age 60 will be midway between the rates for ages 20 and 100)"]
[null,null,null,null,null,"age 100","State A","State B","State C"]
[null,null,null,null,null,"State A",{"$f":"1-H14","v":0.99295552550738408},{"$f":"Parameter3","v":7.0444744926159431E-3},0]
[null,"The transition matrix for age i is hence: ((matrix100-matrix20)*i/80)-(1/4)*matrix100+(5/4)*matrix20",null,null,null,"State B",0,{"$f":"1-I15","v":0.95371569803874878},{"$f":"Parameter4","v":4.6284301961251217E-2}]
[null,null,null,null,null,"State C",0,0,1]
[]
[]
[null,null,null,null,null,"Model Predictions"]
[null,null,null,null,null,"Model starts at age 20 with initial state distribution (1,0,0) i.e. all persons in State A."]
[]
[null,null,null,null,null,"Time","State A","State B","State C","Moving to state C"]
[null,null,null,null,null,20,1,0,0]
[null,null,null,null,null,21,{"$f":"MMULT(G23:I23,((matrix100-matrix20)*F23/80)-(1/4)*matrix100+(5/4)*matrix20)","v":0.99999998999999984},1.0000000000014846E-8,0,{"$f":"I24-I23","v":0}]
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[null,null,null,null,null,24,{"$f":"MMULT(G26:I26,((matrix100-matrix20)*F26/80)-(1/4)*matrix100+(5/4)*matrix20)","v":0.99947171046688055},5.2772925734272764E-4,5.6027577616568003E-7,4.5835595931166994E-7]
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