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[null,null,null,null,null,null,null,null,null,null,null,"Copyright © 2025 Society of Actuaries Research Institute"]
[null,"A Tool for Evaluating the Statistical Reliability of Mortality Improvement Estimates"]
[]
[null,"Disclaimer"]
[]
[null,"This software is made available free of charge with the understanding that neither the author nor the Society of Actuaries Research Institute "]
[null,"accepts any liability for its use. It may be used for any personal, research or commercial use provided that the user does not represent it as"]
[null,"his or her own work and does not distribute it to others for a fee."]
[null,""]

{"sheet":"Overview","index":1,"n_rows":7,"n_cols":2,"rows_kept":7,"cut":false,"hidden":false,"merged":[]}
[]
[null,"A Tool for Evaluating the Statistical Reliability of Mortality Improvement Estimates"]
[]
[null,"Estimating historical mortality trends—which this workbook refers to as “mortality improvement rates”—is statistically challenging. The frequently cited rule-of-thumb that 1,082 deaths are needed to achieve a credible mortality rate estimate is not applicable to improvement rate estimation. This rule-of-thumb assumes that a single rate, rather than the ratio of two rates, is being estimated. To produce a statistically meaningful estimate of a mortality improvement rate, a larger pool of data is required. This workbook uses stochastic simulation to evaluate how the number of persons (\"lives\") covered in a mortality study, and the length of the study period (in years), affects the statistical reliability of mortality improvement rate estimates. "]
[null,"If your operating system blocks this file, preventing the macros from operating, then it is necessary"]
[null,"to unblock the macros, as follows: download the file to your hard-drive; navigate to the file in"]
[null,"Windows Explorer; right-click on the file; click on \"Properties\", and click \"unblock\"."]

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[]
[null,"A Tool for Evaluating the Reliability of Mortality Improvement Estimates"]
[null,"This tool uses stochastic simulation to develop confidence intervals for mortality improvement rate estimates. 100K stochastic trials is sufficient to produce robust results, but you may select more trials if you wish."]
[null,"Below,  the term \"true\" describes an assumed mortality rate or MI rate. Each stochastic trial is based on the assumed \"true\" values, but the randomness of individual mortality risks causes simulation results to deviate from the assumptions."]
[null,"The gap between a simulated MI rate and the \"true\" MI rate can viewed as an estimation error. Using many stochastic trials, this tool creates distributions of errors, which, in turn, provide confidence intervals for MI estimates."]
[]
[]
[]
[]
[]
[null,null,null,null,null,"Enter Up to 25 Scenarios to Simulate in Sequence"]
[null,"Input Parameters",null,null,null,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]
[null,"Run this case? (\"Y\" = Yes)",null,null,null,"Y","Y","Y","Y"]
[null,"Exposure (Lives) in first year",null,null,null,25000,100000,400000,1600000]
[null,"Exposure (Lives) in final year",null,null,null,25000,100000,400000,1600000]
[null,"\"True\" mortality rate in first year",null,null,null,0.01,0.01,0.01,0.01]
[null,"\"True\" rate of mort improvement (\"MI\")",null,null,null,0.01,0.01,0.01,0.01]
[null,"Final year minus first year of data",null,null,null,1,1,1,1]
[null,"Use all years of data or only first & final?",null,null,"     ","First & Final","First & Final","First & Final","First & Final"]
[null,"Number of stochastic trials (1000s)",null,null,null,200,200,200,200]
[null,"Random number sequence",null,null,null,"Static","Static","Static","Static"]
[]
[null,"Outputs"]
[]
[null,"\"True\" mortality rate in first year",null,null,null,0.01,0.01,0.01,0.01]
[null,"Expected deaths in first year",null,null,null,250,1000,4000,16000]
[null,"Stdev of deaths",null,null,null,15.732132911682129,31.464265823364258,62.928531646728516,125.85706329345703]
[null,"Stdev of deaths as % of expected",null,null,null,6.2928535044193268E-2,3.1464267522096634E-2,1.5732133761048317E-2,7.8660668805241585E-3]
[]
[null,"\"True\" mortality rate in final year",null,null,null,9.8999999463558197E-3,9.8999999463558197E-3,9.8999999463558197E-3,9.8999999463558197E-3]
[null,"Expected deaths in final year",null,null,null,247.5,990,3960,15840]
[null,"Stdev of deaths",null,null,null,15.654065132141113,31.308130264282227,62.616260528564453,125.23252105712891]
[null,"Stdev of deaths as % of expected",null,null,null,6.324874609708786E-2,3.162437304854393E-2,1.5812186524271965E-2,7.9060932621359825E-3]
[]
[null,"\"True\" MI rate (copied from row 17)",null,null,null,0.01,0.01,0.01,0.01]
[]
[null,"MI Estimation Error: median",null,null,null,3.6512874066829681E-4,1.8215412274003029E-4,9.074946865439415E-5,4.5396387577056885E-5]
[null,"MI Estimation Error: average",null,null,null,-3.687137831579894E-3,-8.4864830140490084E-4,-1.7939757585525514E-4,-2.8696014836896212E-5]
[null,"MI Estimation Error: stdev",null,null,null,8.9117175734009957E-2,4.4192169421253334E-2,2.2051355519897779E-2,1.1020126385758479E-2]
[]
[null,"Percentiles of MI Error Terms"]
[null,"Percentile = 2.5 ",null,null,null,-0.19024936817586421,-9.0501298010349271E-2,-4.4253270141780376E-2,-2.1886718925088644E-2]
[null,"Percentile = 5.0",null,null,null,-0.15583266168832779,-7.4843408167362208E-2,-3.6734696663916114E-2,-1.8202625960111619E-2]
[null,"Percentile = 50 (median)",null,null,null,3.6512874066829681E-4,1.8215412274003029E-4,9.074946865439415E-5,4.5396387577056885E-5]
[null,"Percentile = 95",null,null,null,0.13578220605850216,7.0141414925456017E-2,3.5689901560544968E-2,1.8010526336729515E-2]
[null,"Percentile = 97.5",null,null,null,0.15982357822358609,8.3036412857472888E-2,4.2341453023254864E-2,2.1399865858256806E-2]
[null,"Percentile Delta: (95 - 5) / 2",null,null,null,0.14580743387341499,7.2492411546409119E-2,3.6212299112230541E-2,1.8106576148420567E-2]
[]
[null,"Confidence Intervals for MI Estimates"]
[null,"Prob of |MI Error| < 10%",null,null,null,0.74292000000000002,0.97579000000000005,0.99999499999999997,1]
[null,"Prob of |MI Error| < 5%",null,null,null,0.42934499999999998,0.74304499999999996,0.97631500000000004,0.99999499999999997]
[null,"Prob of |MI Error| < 1%",null,null,null,8.9995000000000006E-2,0.18002499999999999,0.35043999999999997,0.635625]
[null,"Prob of |MI Error| < 0.5%",null,null,null,4.5045000000000002E-2,8.9954999999999993E-2,0.18007500000000001,0.35050999999999999]
[null,"Prob of |MI Error| < 0.1%",null,null,null,8.9599999999999992E-3,1.8249999999999999E-2,3.6264999999999999E-2,7.195E-2]
[]
[null,"Percentiles of Mortality Rate Errors in First Year, as % of True Mortality Rate"]
[null,"Percentile = 2.5 ",null,null,null,-0.12323966845870019,-6.1619877815246582E-2,-3.0809938907623291E-2,-1.5404922887682915E-2]
[null,"Percentile = 5.0",null,null,null,-0.10321404226124287,-5.1606972701847553E-2,-2.5803530588746071E-2,-1.2901718728244305E-2]
[null,"Percentile = 50 (median)",null,null,null,2.8293579816818237E-4,1.4146789908409119E-4,7.0733949542045593E-5,3.5390257835388184E-5]
[null,"Percentile = 95",null,null,null,0.10305664502084255,5.1528322510421276E-2,2.5764205493032932E-2,1.288205850869417E-2]
[null,"Percentile = 97.5",null,null,null,0.12282834518700837,6.1414216179400682E-2,3.0707109253853559E-2,1.5353600028902292E-2]
[null,"Percentile Delta: (95 - 5) / 2",null,null,null,0.10313534364104271,5.1567647606134415E-2,2.5783868040889502E-2,1.2891888618469238E-2]
[]
[null,"Confidence Intervals for Mortality Rate Estimate in First Year of Study"]
[null,"Prob of |q Error| < 10% of q",null,null,null,0.88796315165709272,0.99851811901436172,0.9999999997934863,1]
[null,"Prob of |q Error| < 5% of q",null,null,null,0.57312593650238752,0.88796315165709272,0.99851811901436172,0.9999999997934863]
[null,"Prob of |q Error| < 1% of q",null,null,null,0.12626055172559086,0.24937916366099605,0.47499001265768426,0.79637215007880879]
[null,"Prob of |q Error| < 0.5% of q",null,null,null,6.3329445227349179E-2,0.12626055172559086,0.24937916366099605,0.47499001265768426]
[null,"Prob of |q Error| < 0.1% of q",null,null,null,1.2678684622269953E-2,2.5354168027954538E-2,5.0682740867783016E-2,0.10116118477617331]
[]
[]
[]
[]
[null,"Simulation Results for the 1000 Stochastic Trials of Case #1"]
[null,"The detail below is provided only for case #1"]
[]
[null,null,null,null,null,"True","Estimated",null,null,"Simulated Mortality Rates"]
[null,null,null,"Trial",null,"MI","MI","MI Error",null,"Year 1","Year 2"]
[]
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[null,null,null,2,null,9.9999997764825821E-3,-9.4797179102897644E-2,-0.10479717701673508,null,9.0350313112139702E-3,9.8915267735719681E-3]
[null,null,null,3,null,9.9999997764825821E-3,-7.0697449147701263E-2,-8.0697447061538696E-2,null,1.026234682649374E-2,1.0987868532538414E-2]
[null,null,null,4,null,9.9999997764825821E-3,-1.9610626623034477E-2,-2.9610626399517059E-2,null,9.2276819050312042E-3,9.4086425378918648E-3]
[null,null,null,5,null,9.9999997764825821E-3,-0.1019083634018898,-0.11190836131572723,null,9.3635562807321548E-3,1.0317781008780003E-2]
[null,null,null,6,null,9.9999997764825821E-3,-0.16514529287815094,-0.17514529824256897,null,9.4195064157247543E-3,1.0975093580782413E-2]
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[null,null,null,9,null,9.9999997764825821E-3,8.4824161604046822E-3,-1.5175836160778999E-3,null,9.5891216769814491E-3,9.5077827572822571E-3]
[null,null,null,10,null,9.9999997764825821E-3,-4.0110968053340912E-2,-5.0110965967178345E-2,null,1.0263317264616489E-2,1.0674988850951195E-2]
[null,null,null,11,null,9.9999997764825821E-3,0.16455842554569244,0.15455842018127441,null,1.0740387253463268E-2,8.9729661121964455E-3]
[null,null,null,12,null,9.9999997764825821E-3,0.15296277403831482,0.14296276867389679,null,9.9659999832510948E-3,8.4415730088949203E-3]
[null,null,null,13,null,9.9999997764825821E-3,9.1650709509849548E-3,-8.3492882549762726E-4,null,9.1021973639726639E-3,9.018775075674057E-3]
[null,null,null,14,null,9.9999997764825821E-3,6.0952074825763702E-2,5.095207691192627E-2,null,1.0799397714436054E-2,1.0141151957213879E-2]
[null,null,null,15,null,9.9999997764825821E-3,-4.5073870569467545E-4,-1.0450738482177258E-2,null,1.034967228770256E-2,1.0354337282478809E-2]
[null,null,null,16,null,9.9999997764825821E-3,4.4169023633003235E-2,3.4169025719165802E-2,null,9.7139915451407433E-3,9.2849340289831161E-3]
[null,null,null,17,null,9.9999997764825821E-3,-9.5040023326873779E-2,-0.10504002124071121,null,8.9125121012330055E-3,9.7595574334263802E-3]
[null,null,null,18,null,9.9999997764825821E-3,9.1746732592582703E-2,8.174673467874527E-2,null,1.037062332034111E-2,9.4191525131464005E-3]
[null,null,null,19,null,9.9999997764825821E-3,4.4389925897121429E-2,3.4389927983283997E-2,null,9.7851557657122612E-3,9.3507934361696243E-3]
[null,null,null,20,null,9.9999997764825821E-3,-1.306372694671154E-2,-2.3063726723194122E-2,null,9.8054688423871994E-3,9.9335648119449615E-3]
[null,null,null,21,null,9.9999997764825821E-3,8.639160543680191E-2,7.6391607522964478E-2,null,1.0831266641616821E-2,9.8955361172556877E-3]
[null,null,null,22,0.17573893070220947,9.9999997764825821E-3,5.5308930575847626E-2,4.5308932662010193E-2,null,1.0900826193392277E-2,1.0297913104295731E-2]
[null,null,null,23,0.82054263353347778,9.9999997764825821E-3,0.10099706798791885,9.0997070074081421E-2,null,1.1054622009396553E-2,9.9381376057863235E-3]
[null,null,null,24,6.2584161758422852E-2,9.9999997764825821E-3,-1.2191860005259514E-2,-2.2191859781742096E-2,null,1.0149179957807064E-2,1.0272917337715626E-2]
[null,null,null,25,0.49460166692733765,9.9999997764825821E-3,5.4590664803981781E-2,4.4590666890144348E-2,null,9.5827300101518631E-3,9.0596023947000504E-3]
[null,null,null,26,0.66162288188934326,9.9999997764825821E-3,-6.3831955194473267E-2,-7.3831953108310699E-2,null,1.0416748933494091E-2,1.108167041093111E-2]
[null,null,null,27,0.95883804559707642,9.9999997764825821E-3,-6.1651784926652908E-4,-1.0616517625749111E-2,null,1.0251051746308804E-2,1.0257371701300144E-2]
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[null,null,null,996,null,9.9999997764825821E-3,8.715907484292984E-2,7.7159076929092407E-2,null,9.8042720928788185E-3,8.9497407898306847E-3]
[null,null,null,997,null,9.9999997764825821E-3,0.10464826226234436,9.4648264348506927E-2,null,1.0249532759189606E-2,9.1769369319081306E-3]
[null,null,null,998,null,9.9999997764825821E-3,0.12440969049930573,0.11440969258546829,null,9.8979976028203964E-3,8.6665907874703407E-3]
[null,null,null,999,null,9.9999997764825821E-3,-5.2018947899341583E-2,-6.2018945813179016E-2,null,1.0201107710599899E-2,1.0731758549809456E-2]
[null,null,null,1000,null,9.9999997764825821E-3,-1.6479969024658203E-2,-2.6479968801140785E-2,null,9.5585072413086891E-3,9.7160311415791512E-3]

{"sheet":"Hidden","index":3,"n_rows":109,"n_cols":30,"rows_kept":109,"cut":false,"hidden":true,"merged":[]}
[]
[null,"A Tool for Evaluating the Reliability of Mortality Improvement Estimates"]
[null,"This tool uses stochastic simulation to develop confidence intervals for mortality improvement rate estimates. 100K stochastic trials is sufficient to produce robust results, but you may select more trials if you wish."]
[]
[]
[]
[]
[]
[null,null,null,null,null,"Enter Up to 20 Scenarios to Simulate in Sequence"]
[null,"Input Parameters",null,null,null,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]
[null,"Run this case? (\"Y\" = Yes)",null,null,null,"Y","Y","Y","Y","Y","Y",null,"Y","Y","Y","Y","Y","Y","Y",null,"Y","Y","Y","Y","Y",null,"Y","Y"]
[null,"Exposure (Lives) in first year",null,null,null,25000,100000,400000,12500,50000,200000,null,25000,100000,400000,1600000,6400000,25600000,102400000,null,6400000,6400000,6400000,6400000,6400000,null,100000,100000]
[null,"Exposure (Lives) in final year",null,null,null,25000,100000,400000,12500,50000,200000,null,25000,100000,400000,1600000,6400000,25600000,102400000,null,6400000,6400000,6400000,6400000,6400000,null,100000,100000]
[null,"True mortality rate in first year",null,null,null,0.01,0.01,0.01,0.02,0.02,0.02,null,0.01,0.01,0.01,0.01,0.01,0.01,0.01,null,0.01,0.01,0.01,0.01,0.01,null,0.01,0.01]
[null,"True rate of mortality improvement",null,null,null,0.02,0.02,0.02,0.02,0.02,0.02,null,0.02,0.02,0.02,0.02,0.02,0.02,0.02,null,0.02,0.02,0.02,0.02,0.02,null,0.02,0.02]
[null,"Final year minus first year of data",null,null,null,1,1,1,1,1,1,null,1,1,1,1,1,1,1,null,1,2,4,8,16,null,10,10]
[null,"Use all years of data or only first & final?",null,null,null,"First & Final","First & Final","First & Final","First & Final","First & Final","First & Final",null,"First & Final","First & Final","First & Final","First & Final","First & Final","First & Final","First & Final",null,"First & Final","First & Final","First & Final","First & Final","First & Final",null,"First & Final","All Years"]
[null,"Number of stochastic trials (1000s)",null,null,null,200,200,200,200,200,200,null,200,200,200,200,200,200,200,null,200,200,200,200,200,null,200,200]
[null,"Random number sequence",null,null,null,"Static","Static","Static","Static","Static","Static",null,"Static","Static","Static","Static","Static","Static","Static",null,"Static","Static","Static","Static","Static",null,"Static","Static"]
[]
[null,"Outputs"]
[]
[null,"Expected death rate in first year",null,null,null,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,1.9999999552965164E-2,1.9999999552965164E-2,1.9999999552965164E-2,null,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,null,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,null,9.9999997764825821E-3,9.9999997764825821E-3]
[null,"Expected deaths in first year",null,null,null,250,1000,4000,250,1000,4000,null,250,1000,4000,16000,64000,256000,1024000,null,64000,64000,64000,64000,64000,null,1000,1000]
[null,"Stdev of deaths",null,null,null,15.732132911682129,31.464265823364258,62.928531646728516,15.652475357055664,31.304950714111328,62.609901428222656,null,15.732132911682129,31.464265823364258,62.928531646728516,125.85706329345703,251.71412658691406,503.42825317382813,1006.8565063476563,null,251.71412658691406,251.71412658691406,251.71412658691406,251.71412658691406,251.71412658691406,null,31.464265823364258,31.464265823364258]
[null,"Stdev of deaths as % of expected",null,null,null,6.2928535044193268E-2,3.1464267522096634E-2,1.5732133761048317E-2,6.2609903514385223E-2,3.1304951757192612E-2,1.5652475878596306E-2,null,6.2928535044193268E-2,3.1464267522096634E-2,1.5732133761048317E-2,7.8660668805241585E-3,3.9330334402620792E-3,1.9665167201310396E-3,9.8325836006551981E-4,null,3.9330334402620792E-3,3.9330334402620792E-3,3.9330334402620792E-3,3.9330334402620792E-3,3.9330334402620792E-3,null,3.1464267522096634E-2,3.1464267522096634E-2]
[]
[null,"Expected death rate in final year",null,null,null,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,1.9999999552965164E-2,1.9999999552965164E-2,1.9999999552965164E-2,null,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,null,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,9.9999997764825821E-3,null,8.1707276403903961E-3,8.1707276403903961E-3]
[null,"Expected deaths in final year",null,null,null,250,1000,4000,250,1000,4000,null,250,1000,4000,16000,64000,256000,1024000,null,64000,64000,64000,64000,64000,null,817.07275390625,817.07275390625]
[null,"Stdev of deaths",null,null,null,15.732132911682129,31.464265823364258,62.928531646728516,15.652475357055664,31.304950714111328,62.609901428222656,null,15.732132911682129,31.464265823364258,62.928531646728516,125.85706329345703,251.71412658691406,503.42825317382813,1006.8565063476563,null,251.71412658691406,251.71412658691406,251.71412658691406,251.71412658691406,251.71412658691406,null,28.467466354370117,28.467466354370117]
[null,"Stdev of deaths as % of expected",null,null,null,6.2928535044193268E-2,3.1464267522096634E-2,1.5732133761048317E-2,6.2609903514385223E-2,3.1304951757192612E-2,1.5652475878596306E-2,null,6.2928535044193268E-2,3.1464267522096634E-2,1.5732133761048317E-2,7.8660668805241585E-3,3.9330334402620792E-3,1.9665167201310396E-3,9.8325836006551981E-4,null,3.9330334402620792E-3,3.9330334402620792E-3,3.9330334402620792E-3,3.9330334402620792E-3,3.9330334402620792E-3,null,3.4840796142816544E-2,3.4840796142816544E-2]
[]
[null,"\"True\" MI rate",null,null,null,0,0,0,0,0,0,null,0,0,0,0,0,0,0,null,0,0,0,0,0,null,1.9999999552965164E-2,1.9999999552965164E-2]
[]
[null,"MI Estimation Error: median",null,null,null,4.9905027844943106E-4,-3.192124713677913E-4,9.8265377346251626E-6,-4.8130111827049404E-4,-2.1202326752245426E-4,-8.1007110566133633E-5,null,-3.9538103737868369E-4,2.3007687559584156E-4,1.1221159365959466E-4,2.0871836568403523E-5,1.5733186046418268E-5,3.1608835342922248E-5,5.5848634019639576E-6,null,1.6827441868372262E-5,-1.4418088767342851E-6,-3.5868009717887617E-6,-4.1158386920869816E-6,1.3231599496066337E-6,null,-8.1714242696762085E-6,1.7704442143440247E-5]
[null,"MI Estimation Error: average",null,null,null,-3.5926800053359214E-3,-1.1217411570256064E-3,-2.1070032409211932E-4,-4.1379689628081818E-3,-1.1452217525014503E-3,-2.8709722032510711E-4,null,-4.2959004512531261E-3,-7.6506995358869342E-4,-1.8606821122009863E-4,-4.9957273945906449E-5,-8.8400445764202122E-6,1.4591651563890693E-5,4.6003766077850373E-6,null,-1.2538096182336177E-5,-6.137565222646728E-6,-4.6913506781844916E-7,-3.6527093093161156E-6,1.0584497950137094E-6,null,5.1985584944486616E-6,9.9538238439708951E-6]
[null,"MI Estimation Error: stdev",null,null,null,9.0117150550540884E-2,4.4407300777422359E-2,2.2372076224018149E-2,8.9444336097037266E-2,4.4452507879584491E-2,2.2229511374494668E-2,null,9.0006636031016607E-2,4.4545310067415467E-2,2.2254543436714729E-2,1.1124963551768384E-2,5.5709071493808271E-3,2.776003748617255E-3,1.3872426454897314E-3,null,5.5729921140616531E-3,2.7822123242281335E-3,1.3926613834525667E-3,6.9644227565663166E-4,3.4854554879047012E-4,null,4.6072016463921698E-3,3.1020329887331945E-3]
[]
[null,"Percentiles of MI Error Terms"]
[null,"Percentile = 2.5 ",null,null,null,-0.19282271414995195,-9.1151366569101805E-2,-4.4789937976747747E-2,-0.19009605832397938,-9.1215725988149643E-2,-4.4578058365732427E-2,null,-0.19200980849564075,-9.0773209184408191E-2,-4.4788850098848344E-2,-2.2117571951821447E-2,-1.0973534942604602E-2,-5.4700531880371274E-3,-2.7016278647352008E-3,null,-1.09290418215096E-2,-5.4535393835976722E-3,-2.740312973037362E-3,-1.3690933032194153E-3,-6.8088826956227424E-4,null,-9.0300226351246234E-3,-6.0848666820675131E-3]
[null,"Percentile = 5.0",null,null,null,-0.15830465853214262,-7.5584995001554481E-2,-3.7534624524414542E-2,-0.15627572461962699,-7.5440501049160946E-2,-3.7491559609770776E-2,null,-0.15911468565464021,-7.541262693703174E-2,-3.716508317738771E-2,-1.8445656448602674E-2,-9.1531630139797919E-3,-4.5775617007166149E-3,-2.2772461757995187E-3,null,-9.185539931058884E-3,-4.5769899152219296E-3,-2.2925233002752065E-3,-1.1493730824440717E-3,-5.7266164221800862E-4,null,-7.5747655238956211E-3,-5.09361126460135E-3]
[null,"Percentile = 50 (median)",null,null,null,4.9905027844943106E-4,-3.192124713677913E-4,9.8265377346251626E-6,-4.8130111827049404E-4,-2.1202326752245426E-4,-8.1007110566133633E-5,null,-3.9538103737868369E-4,2.3007687559584156E-4,1.1221159365959466E-4,2.0871836568403523E-5,1.5733186046418268E-5,3.1608835342922248E-5,5.5848634019639576E-6,null,1.6827441868372262E-5,-1.4418088767342851E-6,-3.5868009717887617E-6,-4.1158386920869816E-6,1.3231599496066337E-6,null,-8.1714242696762085E-6,1.7704442143440247E-5]
[null,"Percentile = 95",null,null,null,0.1367523260414599,7.0034912228584281E-2,3.6006712540984136E-2,0.13676982671022395,7.0522525161504687E-2,3.5910842753946765E-2,null,0.13694295436143858,7.0454421639442336E-2,3.5864679329097073E-2,1.8079492822289461E-2,9.1113319620490067E-3,4.5557474251836519E-3,2.2936290362849821E-3,null,9.0878428891301086E-3,4.5693465741351207E-3,2.28486027335748E-3,1.1378584720660001E-3,5.7371469447389226E-4,null,7.6045528985559897E-3,5.1084453240036872E-3]
[null,"Percentile = 97.5",null,null,null,0.16124454922974091,8.3313227072358123E-2,4.2629857920110176E-2,0.16098977662622907,8.3390125073492444E-2,4.273871257901192E-2,null,0.16081914976239201,8.3320135623216607E-2,4.2753005307167767E-2,2.1529449569061391E-2,1.0894998093135651E-2,5.4090430028736558E-3,2.7231400832533827E-3,null,1.0831547994166611E-2,5.4148778552189445E-3,2.7263129304628802E-3,1.3522614462999625E-3,6.8293647636892226E-4,null,9.0661269612610212E-3,6.0739700682461211E-3]
[null,"Percentile Delta: (95 - 5) / 2",null,null,null,0.14752849228680126,7.2809953615069381E-2,3.6770668532699335E-2,0.14652277566492547,7.2981513105332824E-2,3.6701201181858767E-2,null,0.14802882000803941,7.2933524288237045E-2,3.6514881253242391E-2,1.8262574635446067E-2,9.1322474880143993E-3,4.5666545629501334E-3,2.2854376060422506E-3,null,9.1366914100944954E-3,4.5731682446785247E-3,2.2886917868163432E-3,1.1436157772550359E-3,5.7318816834595044E-4,null,7.5896592112258054E-3,5.1010282943025186E-3]
[]
[null,"Confidence Intervals for MI Estimates"]
[null,"Prob of |MI Error| < 10%",null,null,null,0.73770000000000002,0.97502999999999995,0.99999000000000005,0.74250000000000005,0.97475999999999996,0.99997999999999998,null,0.73756999999999995,0.97536,1,1,1,1,1,null,1,1,1,1,1,null,1,1]
[null,"Prob of |MI Error| < 5%",null,null,null,0.42320999999999998,0.74156999999999995,0.97480999999999995,0.42845,0.74063000000000001,0.97507999999999995,null,0.42675999999999997,0.73953999999999998,0.97491000000000005,0.99997999999999998,1,1,1,null,1,1,1,1,1,null,1,1]
[null,"Prob of |MI Error| < 1%",null,null,null,8.8279999999999997E-2,0.17893999999999999,0.34615000000000001,8.9419999999999999E-2,0.17674000000000001,0.34866000000000003,null,9.1120000000000007E-2,0.17727000000000001,0.34909000000000001,0.63241000000000003,0.92767999999999995,0.99978,1,null,0.92820000000000003,0.99970999999999999,1,1,1,null,0.96963999999999995,0.99865999999999999]
[null,"Prob of |MI Error| < 0.5%",null,null,null,4.4040000000000003E-2,9.017E-2,0.1772,4.3729999999999998E-2,8.8700000000000001E-2,0.17918999999999999,null,4.6080000000000003E-2,9.0329999999999994E-2,0.17957000000000001,0.34909000000000001,0.62983999999999996,0.92835000000000001,0.99966999999999995,null,0.62902999999999998,0.92776000000000003,0.99968999999999997,1,1,null,0.72302,0.8931]
[null,"Prob of |MI Error| < 0.1%",null,null,null,8.9999999999999993E-3,1.804E-2,3.6229999999999998E-2,8.9800000000000001E-3,1.7760000000000001E-2,3.5299999999999998E-2,null,9.4699999999999993E-3,1.772E-2,3.5709999999999999E-2,7.0580000000000004E-2,0.14376,0.28153,0.52834000000000003,null,0.14172999999999999,0.28156999999999999,0.52690999999999999,0.84940000000000004,0.99595999999999996,null,0.17136000000000001,0.25245000000000001]
[]
[null,"Percentiles of Mort Rate Errors in First Year, as % of True Mort Rate"]
[null,"Percentile = 2.5 ",null,null,null,-0.12410016730427742,-6.1668038833886385E-2,-3.0922275036573407E-2,-0.12215357515960931,-6.1732497997581959E-2,-3.0731691513210535E-2,null,-0.12320163901895284,-6.1724768020212643E-2,-3.0884405132383108E-2,-1.5517615247517824E-2,-7.711756043136119E-3,-3.8244761526584625E-3,-1.9294256344437599E-3,null,-7.6677906326949596E-3,-7.6527846977114677E-3,-7.6913367956876755E-3,-7.7842758037149906E-3,-7.7408738434314728E-3,null,-6.1938504222780466E-2,-6.1647605616599321E-2]
[null,"Percentile = 5.0",null,null,null,-0.10344677045941353,-5.1957275718450539E-2,-2.5968197733163834E-2,-0.10308734513819216,-5.1638497970998273E-2,-2.5699157267808911E-2,null,-0.10350249260663985,-5.1738331094384193E-2,-2.5820834562182426E-2,-1.2932252138853073E-2,-6.4729992300271988E-3,-3.209342248737812E-3,-1.6169669106602669E-3,null,-6.4421584829688072E-3,-6.4296741038560867E-3,-6.4325658604502678E-3,-6.5348204225301743E-3,-6.5066013485193253E-3,null,-5.1688789390027523E-2,-5.1665473729372025E-2]
[null,"Percentile = 50 (median)",null,null,null,5.798880010843277E-4,2.0866282284259796E-4,1.8021091818809509E-5,-1.8361024558544159E-4,-1.2312084436416626E-4,1.8160790205001831E-5,null,-1.9175931811332703E-4,1.7941929399967194E-4,6.1979517340660095E-5,1.4388933777809143E-5,8.149072527885437E-6,8.149072527885437E-6,6.5192580223083496E-7,null,4.3306499719619751E-6,-8.9406967163085938E-6,8.2887709140777588E-6,-7.0780515670776367E-6,2.0954757928848267E-6,null,4.9592927098274231E-5,8.1816688179969788E-5]
[null,"Percentile = 95",null,null,null,0.10339909233152855,5.1425909623503671E-2,2.6153624057769768E-2,0.10306395478546611,5.1143155433237518E-2,2.5718752294778824E-2,null,0.10250091552734372,5.1706340163946138E-2,2.5920919142663462E-2,1.2934161350131024E-2,6.486335769295682E-3,3.2349536195397368E-3,1.6228295862674713E-3,null,6.5012881532311431E-3,6.5077142789959899E-3,6.4927246421575529E-3,6.4800074324011681E-3,6.5180566161870939E-3,null,5.1921005360782028E-2,5.17087150365114E-2]
[null,"Percentile = 97.5",null,null,null,0.12373242899775497,6.1262280214577851E-2,3.1064029317349185E-2,0.12312181536108241,6.0684289783239351E-2,3.0687020625919052E-2,null,0.12206961400806902,6.1820610426366246E-2,3.0945155303925276E-2,1.552990637719631E-2,7.746948394924397E-3,3.848690539598465E-3,1.9278423860669125E-3,null,7.7808531932532727E-3,7.7538238838315001E-3,7.7276560477912409E-3,7.7281729318201403E-3,7.797251455485813E-3,null,6.1925710178911679E-2,6.1455627437680946E-2]
[null,"Percentile Delta: (95 - 5) / 2",null,null,null,0.10342293139547104,5.1691592670977102E-2,2.6060910895466801E-2,0.10307564996182914,5.1390826702117892E-2,2.5708954781293869E-2,null,0.10300170406699179,5.1722335629165166E-2,2.5870876852422946E-2,1.2933206744492049E-2,6.4796674996614404E-3,3.2221479341387744E-3,1.6198982484638691E-3,null,6.4717233180999756E-3,6.4686941914260387E-3,6.4626452513039103E-3,6.5074139274656712E-3,6.5123289823532096E-3,null,5.1804897375404779E-2,5.1687094382941709E-2]
[]
[null,"Confidence Intervals for Mortality Rate Estimate in First Year of Study"]
[null,"Prob of |q Error| < 10% of q",null,null,null,0.88796315165709272,0.99851811901436172,0.9999999997934863,0.88977696266573747,0.99859869881206853,0.99999999983276999,null,0.88796315165709272,0.99851811901436172,0.9999999997934863,1,1,1,1,null,1,1,1,1,1,null,0.99851811901436172,0.99834450660525365]
[null,"Prob of |q Error| < 5% of q",null,null,null,0.57312593650238752,0.88796315165709272,0.99851811901436172,0.57547513356872759,0.88977696266573747,0.99859869881206853,null,0.57312593650238752,0.88796315165709272,0.99851811901436172,0.9999999997934863,1,1,1,null,1,1,1,1,1,null,0.88796315165709272,0.88427518121213144]
[null,"Prob of |q Error| < 1% of q",null,null,null,0.12626055172559086,0.24937916366099605,0.47499001265768426,0.12689767655669071,0.25060581213740796,0.47709676051044519,null,0.12626055172559086,0.24937916366099605,0.47499001265768426,0.79637215007880879,0.98899584278058561,0.99999963263252556,1,null,0.98899584278058561,0.98899584278058561,0.98899584278058561,0.98899584278058561,0.98899584278058561,null,0.24937916366099605,0.24693069876637619]
[null,"Prob of |q Error| < 0.5% of q",null,null,null,6.3329445227349179E-2,0.12626055172559086,0.24937916366099605,6.3651054185748401E-2,0.12689767655669071,0.25060581213740796,null,6.3329445227349179E-2,0.12626055172559086,0.24937916366099605,0.47499001265768426,0.79637215007880879,0.98899584278058561,0.99999963263252556,null,0.79637215007880879,0.79637215007880879,0.79637215007880879,0.79637215007880879,0.79637215007880879,null,0.12626055172559086,0.12498954455600386]
[null,"Prob of |q Error| < 0.1% of q",null,null,null,1.2678684622269953E-2,2.5354168027954538E-2,5.0682740867783016E-2,1.274320172912935E-2,2.5483153125267999E-2,5.0940318504947124E-2,null,1.2678684622269953E-2,2.5354168027954538E-2,5.0682740867783016E-2,0.10116118477617331,0.20070274737230726,0.3889066495677771,0.69085931103699294,null,0.20070274737230726,0.20070274737230726,0.20070274737230726,0.20070274737230726,0.20070274737230726,null,2.5354168027954538E-2,2.5096894004634329E-2]
[]
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[null,0,-4.5774984359741211]
[null,1,-4.6047525405883789]
[null,2,-4.6539559364318848]
[null,3,-4.620734691619873]
[null,4,-4.6568608283996582]
[null,5,-4.7486171722412109]

{"sheet":"Help","index":4,"n_rows":62,"n_cols":8,"rows_kept":62,"cut":false,"hidden":false,"merged":[]}
[]
[null,"A Description of the Inputs and Outputs"]
[]
[]
[null,"Input Parameters",null,null,null,null,null,"In this example, a time interval of 5 years is analyzed, corresponding to 6 consecutive years of data. For example, the data might run from 2014 to 2019."]
[null,"Run this case? (\"Y\" = Yes)",null,null,null,"Y",null,"Exposure in the first year is 150K, and in the final year is 160K. This tool assumes that exposure increases linearly across time; therefore, in this example,"]
[null,"Exposure (Lives) in first year",null,null,null,150000,null,"exposure would be 150K in the first year, 152K in the second year, etc. The initial mortality rate is assumed to be 1%, and improvement is assumed to be "]
[null,"Exposure (Lives) in final year",null,null,null,160000,null,"2% per year. Therefore, in the 2nd year, the \"true\" mortality rate would be 1% * (100% - 2%) = 0.98%. In the final year, the true mortality rate would be"]
[null,"True mortality rate in first year",null,null,null,0.01,null,"1% * (100% - 2%) ^ 5 = 0.904%. This scenario uses all 6 years of data to estimate the improvement rate, as opposed to using solely the first and final years."]
[null,"True rate of mortality improvement",null,null,null,0.02,null,"Lastly, this scenario uses 100K stochastic trials. For each of these trials, a mortality rate is randomly generated using a normal distribution. For example,"]
[null,"Final year minus first year of data",null,null,null,5,null,"for the first year of the period, the normal distribution would be have a mean of 150K * 1% = 1500 deaths, and a variance equal to 150K * 1% * (100% - 1%)."]
[null,"Use all years of data or only first & final?",null,null,null,"All Years",null,"For each trial, a time series of mortality rates is simulated, the rates are logged, and a trend line is fit to the data. The improvement rate is equal to the "]
[null,"Number of stochastic trials (1000s)",null,null,null,100,null,"slope of the trend line multiplied by negative 1, as explained on the Methodology worksheet."]
[null,"Random number sequence",null,null,null,"Static"]
[]
[null,"Outputs"]
[]
[null,"Expected death rate in first year",null,null,null,9.9999997764825821E-3,null,"This value echoes the parameter entered above in cell F9."]
[null,"Expected deaths in first year",null,null,null,1500,null,"Exposure in the first year multiplied by the expected mortality rate in the first year."]
[null,"Stdev of deaths",null,null,null,38.535697937011719,null,"Standard deviation of deaths in the first year, equal to SQR[ Exposure * q * (1-q) ]. "]
[null,"Stdev of deaths as % of expected",null,null,null,2.5690466165542603E-2,null,"The smaller this value, the tighter will be the confidence interval for the estimated mortality rate."]
[]
[null,"Expected death rate in final year",null,null,null,9.039207361638546E-3,null,"The \"true\" mortality rate in the final year. This is equal to q_initial * (1 - improvement rate) ^ N, where N = final year minus initial year."]
[null,"Expected deaths in final year",null,null,null,1446.273193359375,null,"The final \"true\" mortality rate multiplied by the final year's exposure."]
[null,"Stdev of deaths",null,null,null,37.857627868652344,null,"Standard deviation of deaths in the first year, equal to SQR[ Exposure * q * (1-q) ]. "]
[null,"Stdev of deaths as % of expected",null,null,null,2.6175986975431442E-2,null,"The smaller this value, the tighter will be the confidence interval for the estimated mortality rate."]
[]
[null,"\"True\" MI rate",null,null,null,1.9999999552965164E-2,null,"The expected rate of mortality improvement. This echoes parameter F10."]
[]
[null,"MI Estimation Error: median",null,null,null,-1.7539598047733307E-5,null,"The MI error is equal to the estimated MI rate minus the “true” MI rate. For example, if the true MI rate is 2%, and a particular stochastic trial"]
[null,"MI Estimation Error: average",null,null,null,-2.9656118778511883E-5,null,"generates a result of 2.5%, then the error is 2.5% minus 2% = 0.5%. Across a large number of stochastic trials, the median MI error is likely to"]
[null,"MI Estimation Error: stdev",null,null,null,6.0785991624561402E-3,null,"be close to zero. The lower the stdev of the MI errors, the more reliable is the MI estimator."]
[]
[null,"Percentiles of MI Error Terms"]
[null,"Percentile = 2.5 ",null,null,null,-1.202594437636435E-2,null,"Various percentiles extracted from the distribution of MI error terms. In this example, the 5th percentile is negative 1%, the median is very close to 0%,"]
[null,"Percentile = 5.0",null,null,null,-1.0087987873703241E-2,null,"and the 95th percentile is a little greater than positive 1%. Therefore, there is about a 90% chance of an MI estimate falling within 1% of the true value."]
[null,"Percentile = 50 (median)",null,null,null,-1.7539598047733307E-5,null,"For example, if this analysis produced an MI estimate of 1.9%, we could conclude that there is a 90% chance that the \"true\" MI rate falls within 1% of "]
[null,"Percentile = 95",null,null,null,9.9598846398293965E-3,null,"the estimate, resulting in a 90% confidence interval running from 0.9% to 2.9%."]
[null,"Percentile = 97.5",null,null,null,1.1843475978821492E-2]
[null,"Percentile Delta: (95 - 5) / 2",null,null,null,1.0023936256766319E-2]
[]
[null,"Confidence Intervals for MI Estimates"]
[null,"Prob of |MI Error| < 10%",null,null,null,1,null,"These results are another way of expressing the reliability of the MI estimate. For example, if the MI estimate is 1.9%, we could conclude"]
[null,"Prob of |MI Error| < 5%",null,null,null,1,null,"that there is a 59% probability that the true MI rate falls within 0.5% of this value, resulting in a confidence interval running from "]
[null,"Prob of |MI Error| < 1%",null,null,null,0.89932000000000001,null,"1.4% to 2.4%."]
[null,"Prob of |MI Error| < 0.5%",null,null,null,0.58957000000000004]
[null,"Prob of |MI Error| < 0.1%",null,null,null,0.13037000000000001]
[]
[null,"Percentiles of Mort Rate Errors in First Year, as % of True Mort Rate"]
[null,"Percentile = 2.5 ",null,null,null,-5.0203031860291958E-2,null,"These results are focused on mortality rates as opposed to mortality improvement rates. "]
[null,"Percentile = 5.0",null,null,null,-4.2340452782809734E-2,null,"The results show the distribution of mortality rate errors for the first year of the study."]
[null,"Percentile = 50 (median)",null,null,null,-1.3611279428005219E-4,null,"If the true mortality rate is 1.00%, and the estimated mortality rate is 1.02%,"]
[null,"Percentile = 95",null,null,null,4.208873026072972E-2,null,"then the error is 2%. Roughly, the results indicate that there is a 90% chance that the"]
[null,"Percentile = 97.5",null,null,null,5.0219865515828077E-2,null,"error is less than 4.2%."]
[null,"Percentile Delta: (95 - 5) / 2",null,null,null,4.2214591521769727E-2]
[]
[null,"Confidence Intervals for Mortality Rate Estimate in First Year of Study"]
[null,"Prob of |q Error| < 10% of q",null,null,null,0.99989493433022325,null,"These are confidence intervals for the mortality rate estimate in the first year of the study."]
[null,"Prob of |q Error| < 5% of q",null,null,null,0.94753428898029401,null,"In this example, there is a 94.75% chance that the estimated mortality rate falls within"]
[null,"Prob of |q Error| < 1% of q",null,null,null,0.30187927862137759,null,"5% of the true mortality rate. Another way of looking at this is there is a 94.75% chance"]
[null,"Prob of |q Error| < 0.5% of q",null,null,null,0.15376873532360852,null,"that the true mortality rate falls between 95% and 105% of the estimated mortality rate."]
[null,"Prob of |q Error| < 0.1% of q",null,null,null,3.0938875576266511E-2]

{"sheet":"Methodology","index":5,"n_rows":49,"n_cols":13,"rows_kept":49,"cut":false,"hidden":false,"merged":[]}
[]
[null,"Methodology"]
[]
[null,"Below is an illustration of how mortality data is simulated as a function of user-specified assumptions, and "]
[null,"then used to produce a mortality improvement rate estimate. This example uses 6 consecutive years of "]
[null,"simulated data. The data below is for one stochastic trial. Press the \"Generate\" button to produce a fresh"]
[null,"set of random numbers, and a fresh set of simulated data."]
[]
[null,"Assumptions"]
[null,100000,"Exposure"]
[null,0.01,"True mortality rate in first year"]
[null,0.02,"True rate of mortality improvement"]
[null,null,null,null,null,null,null,null,null,"Logged"]
[null,"Simulation Results",null,"\"True\"",null,null,"Random",null,"Simulated","Simulated"]
[null,null,null,"Mortality","Expected","Stdev","Number","Simulated","Mortality","Mortality"]
[null,null,"Year","Rate","Deaths","of Deaths","0 to 1","Deaths","Rate","Rate"]
[]
[null,null,1,{"$f":"B11","v":0.01},{"$f":"D18*$B$10","v":1000},{"$f":"SQRT($B$10*D18*(1-D18))","v":31.464265445104548},{"$f":"RAND()","v":0.81955818307313433},{"$f":"_xlfn.NORM.INV(G18,E18,F18)","v":1028.7483527978491},{"$f":"H18/$B$10","v":1.0287483527978491E-2},{"$f":"LN(I18)","v":-4.5768273141487859},null,{"$f":"-SLOPE(J18:J23,C18:C23)","v":3.614968024859664E-2}," Estimated Instantaneous Geometric Rate of Mortality Improvement"]
[null,null,2,{"$f":"D18*(1-$B$12)","v":9.7999999999999997E-3},980,31.151179752940337,{"$f":"RAND()","v":0.70815273641671284},{"$f":"_xlfn.NORM.INV(G19,E19,F19)","v":997.07072744542143},9.9707072744542151E-3,{"$f":"LN(I19)","v":-4.6081037572583119}]
[null,null,3,{"$f":"D19*(1-$B$12)","v":9.6039999999999997E-3},960.4,30.841146515653403,0.88585670602054756,997.55692867328275,9.9755692867328268E-3,-4.6076162464830555,null,{"$f":"1-EXP(-L18)","v":3.5504083310365142E-2}," Estimated Instantaneous Geometric Rate of Mortality Improvement"]
[null,null,4,{"$f":"D20*(1-$B$12)","v":9.411919999999999E-3},941.19199999999989,30.534137881907846,0.28092276255222193,923.47907150834578,9.2347907150834574E-3,-4.6847773277335323]
[null,null,5,{"$f":"D21*(1-$B$12)","v":9.2236815999999985E-3},922.36815999999988,30.23012619514153,0.32347040759649481,908.52227300504967,9.0852227300504962E-3,-4.7011060611934097,null,{"$f":"L18-B12","v":1.6149680248596639E-2}," Estimation Error (equal to the estimated MI rate minus the \"true\" MI rate)"]
[null,null,6,{"$f":"D22*(1-$B$12)","v":9.0392079679999984E-3},903.92079679999983,29.929083994187401,6.1355289927018775E-2,{"$f":"_xlfn.NORM.INV(G23,E23,F23)","v":857.72538803503414},8.5772538803503412E-3,-4.7586414772778083]
[]
[]
[null,"A statistically meaningful MI estimate is one that has a high probability of falling close to the \"true\" MI value."]
[null,"By design, the model knows the \"true\" MI value. The reliability of an estimation process is assessed by calculating"]
[null,"the distribution of potential MI estimates, and translating each estimate into an MI error term. The error is "]
[null,"the difference between an MI estimate and the true MI rate. Ideally, the distribution of error terms should have "]
[null,"a low standard deviation -- the lower the standard deviation of the estimator, the greater its reliability."]
[]
[null,"This tool uses simulation to calculate the distribution of MI error terms. The user can choose 100K, 200K, 500K, "]
[null,"or 1M stochastic trials. For each trial, mortality rates are simulated, and the mortality improvement rate is estimated"]
[null,"from the simulated mortality rates. The simulated mortality rates are drawn from a normal distribution that"]
[null,"reflects the assumed \"true\" or \"expected\" mortality rate, and the assumed exposure. For example, if the "]
[null,"expected mortality rate in the initial year is 1% and the exposure is 100K lives, then the simulated mortality rate"]
[null,"will be drawn from a normal distribution with a mean of 1% and a variance equal to 1% * 99% / 100K. This is "]
[null,"accomplished by generating a random number between 0 and 1, and plugging this value into the inverse cumulative"]
[null,"normal distribution. Random numbers are independently generated to produce a mortality rate estimate for"]
[null,"each year in the assumed period of analysis. If the period is 5 years in length, then a time series of 5 mortality"]
[null,"rate estimates is generated. The normal distribution is modified across time to reflect the assumed rate of "]
[null,"mortality improvement, as well as any changes in exposure."]
[]
[null,"To produce improvement rate estimates, the user has the option of using only the first and final years of simulated"]
[null,"data, or using all years of simulated data. If only the first and final years are used, each MI estimate is calculated "]
[null,"as follows: MI rate = 1 - (q_final_simulated / q_initial _simulated) ^ (1 / N), where N = the final year minus the"]
[null,"first year. If the user selects the option to use all years of simulated data, then the MI rate is estimated by first"]
[null,"taking the log of each mortality rate in the simulated time series, and then fitting a regression to this data. The"]
[null,"slope of the regression line, multiplied by negative 1, is equal to the geometric rate of mortality improvement."]