Patent Application: US-201715607598-A

Abstract:
methods , systems , and computer program products are hereby claimed for including and distinguishing independent variables that denote lifespan in multivariable binary regression analyses of mortality and survivorship . these methods , systems , and computer program products are demonstrated here by including and distinguishing lifespan in multivariable binary regression analyses of humans &# 39 ; and medflies &# 39 ; mortality and survivorship , revealing advantages for the description , specification , measurement , analysis , explanation , and prediction of mortality and survivorship .

Description:
multivariable regression analyses of mortality or survivorship employ a computing environment that is schematically described in fig1 , in accordance with an aspect of the present invention . in the embodiments that are described herein , the computing environment includes computerized instructions and computer program products of the stata ° software from the stata corporation , college station , tex . ; this software is processed by a computer system that is employed in the embodiments that are presented here . to process the regression analysis the computing environment requires computerized memory and processors , as illustrated in the example of fig2 , in accordance with an aspect of the present invention . the computing environment of the present embodiments also includes respective regression instructions . in the embodiments that are presented here , each model is specified mathematically by an operator who translates this mathematically specified model into a computer program that is written as stata ° instructions for the regression analysis . the computing environment also includes data for the regression analysis . the instructions , model , and data are then obtained by a processor of the computing environment ; this processor performs the regression analysis by processing the instructions , model , and data , as illustrated in the example of fig1 , in accordance with an aspect of the present invention . it is , however , important to note that there are diverse kinds of computing environments for regression analyses as considered , for example , in u . s . pat . no . 8 , 645 , 966 b2 ( andrade et al . 2014 ). diverse kinds of computing environments can incorporate and use one or more aspects of the present invention . moreover , there are diverse kinds of multivariable binary regression models and binary link functions , as considered , for example , in u . s . pat . no . 8 , 417 , 541 b1 ( kramer 2013 ), epelbaum ( 2014 ), hilbe ( 2009 ), gündüz and fokoué ( 2015 ), and royston and sauerbrei ( 2008 ). these diverse kinds of computing environments and multivariable binary regression analysis can incorporate and use one or more aspects of the present invention . the following paragraphs present embodiments of the invention ; these embodiments include and distinguish age , lifespan , lifespan aggregate , and contemporary aggregate in multivariable binary regression analyses of humans &# 39 ; mortality and survivorship . in embodiments of the invention that are presented here , the data on mortality or survivorship of samples of humans are compiled from the deaths 1 × 1 and exposures 1 × 1 tables ( last modified on 14 jul . 2010 ) from the human mortality database , known as hivid . the hivid data are stored in the memory of a computer at the max planck institute for demographic research in germany , and — in the present embodiments of the invention — they are transmitted through the internet to the memory of a computer in the usa . in the present embodiments , a processor of a computer in the usa processes these hivid data and compiles them into pluralized data on age - sex - year - specific deaths and age - sex - year - specific exposures of males and females in ages 0 to 110 + in sweden 1751 - 2008 . the processor further processes these pluralized data and converts them to individualized data focusing on yearly events of each individual &# 39 ; s death or survival , where each individualized case is weighted by its corresponding number of age - lifespan - sex - specific identical individuals ( i . e ., the number of sex - specific individuals who are born in the year of birth of the criterion individual and who die in the year of death of the criterion individual ). computer intensive analyses impose restrictions on the size of the data file for the present analyses . therefore , the analytic individualized data file is restricted here to 188 , 087 weighted cases with 79 , 164 , 608 events of deaths or survivals of all individuals born in sweden in decennial years 1760 - 1930 , with deaths occurring between 1760 and 2008 . as depicted in fig3 , each row depicts one data record containing data on the following : one individual in a specific situation ( i denotes the individual and j denotes the situation , the situation refers here to the events occurring during a specific year in the life of the respective individual human ), this individual &# 39 ; s death or survival ( depicted in columns m ij and s ij , respectively denoting mortality or survivorship of individual i at situation j in fig3 ), this individual &# 39 ; s sex ( depicted in column g ij , wherein g ij = 1 denotes being female , and g ij = 0 denotes being male in fig3 ), this individual &# 39 ; s age during this situation ( during the mid - year , depicted in column a ij in fig3 ), this individual &# 39 ; s lifespan ( depicted in column l ij in fig3 ), this individual &# 39 ; s historic context ( depicted in column h ij , denoting a specific year , starting at year 1760 which is coded as year 0 . 5 in fig3 ), the individual &# 39 ; s lifespan aggregate size ( i . e ., number of corresponding age - lifespan - year - sex - specific - identical individuals , this is the number of age - sex - specific individuals with identical birth year and identical death year to the criterion individual , as depicted in column λ ij in fig3 ), the individuals &# 39 ; contemporary aggregate size ( i . e ., the number of age - year - sex - specific individuals that are exposed to the risk of death and prospect of survival during this situation , i . e ., during the specific year , as depicted in column c ij in fig3 ). the resultant data file contains 188 , 087 weighted cases corresponding to 79 , 164 , 608 events of deaths or survivals of all individuals born in sweden in decennial years 1760 - 1930 , wherein the individual &# 39 ; s lifespan aggregate size ( i . e ., as depicted in column λ ij in fig3 ) is the weighting variable in the analyses , and wherein j ( i . e ., the total number of situations j of each specific individual ) varies among individuals . based upon previous research , theoretical knowledge , and the available data , the analyst selected the following denoted variables : age , lifespan , lifespan aggregate size , contemporary aggregate size , historical time , and sex . these denoted variables are respectively denoted here with a ij , l ij , c ij , λ ij , h ij , and g ij for an individual i at situation j ; in these denotations a denotes age ( in years ), l denotes lifespan ( in years ), c denotes contemporary aggregate size , λ ( the greek capital letter lambda ) denotes lifespan aggregate size , h denotes historical time , and g denotes sex . numerical values for these denoted variables for individuals i at situations j are illustrated here in fig3 . transformations of these denoted variables are included as independent variables x vij in multivariable binary regression model n ij = β 0 + β v x vij + . . . + β w x wij for one of y ij = m ij and y ij = s ij of humans in sweden 1760 - 2008 . the analysis in the present embodiment employs forward selection methods in iterative multivariable binary regression analyses of humans &# 39 ; mortality or survivorship to select transformations of each of denoted variables a ij , l ij , c ij , λ ij , h ij , and g ij for an individual i at situation j . in these iterative analyses , the analyst tested power transformations of each of these denoted variables , selecting transformation that improved aic and bic . in these analyses , analyst conducted iterative multivariable binary regression analyses of humans &# 39 ; mortality or survivorship . the initial iterating multivariable binary regression analyses utilized diverse input models n ij β 0 + b 1 x 1ij + β 2 x 2ij with respective logit , probit , and complementary log - log binary link functions — as well as x 1 = g ij and diverse values of power coefficient p in x 2 =( a ij ) p wherein g ij denotes sex and a ij denotes age — for one of y ij = m ij and y ij = s ij ; these iterative analyses sought to optimize aic and bic values , selecting the power coefficient p beyond which aic and bic cease to improve , ensuring that all regression coefficients in the selected model are significant beyond the 0 . 001 level . utilizing said selected power coefficient p in x 2ij =( a ij ) p , the analyst proceeded to conduct further iterative multivariable binary regression analyses with input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij and logit , probit , and complementary log - log binary link functions for one of y ij = m ij and y ij = s ij to find optimal aic and bic values for diverse values of power coefficient p in x 3ij =( l ij ) p . utilizing these forward selection methods per additional respective denoted variable , the analyst continued to select optimal power coefficients p in such iterative transformations of respective denoted variables in x 4ij =( c ij ) p , x 5ij =( λ ij ) p , and x 6ij =( h ij ) p in respective input models until reaching an optimal best - fitting first - degree polynomial input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij . utilizing said respective selected optimal first - degree polynomial power coefficients p , the analyst continued to test the optimality of second degree polynomial transformations for these respective selected power coefficients p , optimizing aic and bic criteria in such iterative transformations of respective denoted variables in x 7ij ={( a ij ) p } 2 , x 8ij ={( l ij ) p } 2 , x 9ij ={( c ij ) p } 2 , x 10ij ={( λ ij ) p } 2 , and x 11ij ={( h ij ) p } 2 in respective input models ; reaching an optimal second - degree powered polynomial input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij + β 11 x 11ij . utilizing said respective selected optimal first - degree and second - degree polynomials power coefficients p , the analyst continued to test the optimality of third - degree polynomial transformations for these respective selected power coefficients p , finding that respective x 12ij ={( a ij ) p } 3 , x 12ij ={( l ij ) p } 3 , x 12ij ={( c ij ) p } 3 , and x 12ij ={( λ ij ) p } 3 failed to improve aic and bic criteria in such iterative transformations , but finding that x 12ij ={( h ij ) p } 3 did improve aic and bic criteria in such iterative transformations , thus reaching a best - fitting input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij + β 11 x 11ij + β 12 x 12ij and a logit binary link function that incorporate an identity transformation of denoted binary variable g ij as well as first - degree and second - degree polynomial transformations of denoted powered variables a ij , l ij , c ij , λ ij , h ij , and third - degree polynomial transformation of denoted powered variable h ij . computer instructions for said input model of humans &# 39 ; mortality are depicted in fig4 . in all these iterative analyses , the analyst used non - negative values of power coefficientp ( in the interest of investigating power laws ), utilizing the natural logarithmic transformation when p = 0 ( e . g ., using x 1 = ln ( a ij ) instead of x 1 =( a ij ) 0 ), and allowing identity transformation ( i . e ., when p = 1 , e . g ., using x 1 =( a ij ) 1 = a ij ), ensuring that all regression coefficients in respective selected best - fitting models are significant beyond the 0 . 0001 level of significance . moreover , all analyses employed random effects input models . further information on these analyses is available in epelbaum ( 2014 ). in the best - fitting multivariable binary regression analysis of human &# 39 ; s mortality , the analyst created a data set consisting of one variable y ij and 12 variables x vij for 188 , 087 data records , wherein each record is weighted for depicting 79 , 164 , 608 situations j involving all individuals that were born in sweden in decennial years 1760 - 1930 and died between 1760 and 2008 . each data record contains yearly data on a respective individual i in a respective situation j , said data consisting of y ij = m ij , x 1ij = g ij , x 2ij =( a ij ) 0 . 16 , x 3ij =( l ij ) 0 . 88 , x 4ij =( c ij ) 0 . 75 , x 5ij =( λ ij ) 0 . 30 , x 6ij =( h ij ) 1 . 41 , x 7ij ={( a ij ) 0 . 16 } 2 , x 8ij ={( l ij ) 0 . 88 } 2 , x 9ij ={( c ij ) 0 . 75 } 2 , x 10ij ={( λ ij ) 0 . 30 } 2 , x 11ij ={( h ij ) 1 . 41 } 2 , and x 12ij ={( h ij ) 1 . 41 } 3 , wherein , as noted and as illustrated in fig3 , i denotes an individual , j is a consecutive number of the year of life of this individual , m ij = 1 when the individual is dead , and m ij = 0 when the individual is not dead , a ij denotes the individual &# 39 ; s age ( in years ) at situation j , l ij denotes the individual &# 39 ; s lifespan ( in years ) at situation j , c ij denotes the individual &# 39 ; s contemporary aggregate size at situation j , λ ij denotes the individual &# 39 ; s lifespan aggregate size at situation j , h ij denotes the individual &# 39 ; s historical time at situation j ( where h denotes a calendar year transformed to a sequential number ), g ij = 1 when the individual is female , and g ij = 0 when the individual is male . in said best - fitting multivariable binary regression analysis of humans &# 39 ; mortality the analyst employed an input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij + β 11 x 11ij + β 12 x 12ij with a logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. computer instructions for said input model and said logit binary link function are shown in fig4 . a computer output for this estimated model is depicted in fig5 . utilizing corresponding best - fitting estimated regression coefficients that are depicted in fig5 , and utilizing respective denoting variables that correspond to variables x vij , the best - fitting estimated model is specified with n ij = 511 . 78 − 1074 . 55 ( a ij 0 . 16 )+ 546 . 12 ( a ij 0 . 16 ) 2 − 17 . 12 ( l ij 0 . 88 )+ 0 . 101 ( l ij 0 . 88 ) 2 + 0 . 006 ( c ij 0 . 75 )−( 4 . 39e − 7 )( c ij 0 . 75 ) 2 + 6 . 19 ( λ ij 0 . 30 )− 0 . 35 ( λ ij 0 . 30 ) 2 − 0 . 008 ( h ij 1 . 41 )+( 1 . 92e − 6 )( h ij 1 . 41 ) 2 −( 7 . 97e − 10 )( h ij 1 . 41 ) 3 − 1 . 13 ( g ij ) with a logit binary link function b ( n ij ) = ln { n ij /( 1 − n ij )}. as noted , an individualized probability of mortality refers to the probability of an individual &# 39 ; s mortality ; π ( m ij ) denotes here the probability of mortality of individual i at situation j , π ( m ij ) is calculated here with π ( m ij )= f ( n ij ). utilizing said best - fitting estimated model of humans &# 39 ; mortality in sweden 1760 - 2008 , the individualized probability of mortality π ( m ij ) is calculated with π ( m ij )= f ( n ij )= 1 /{ 1 + exp (− n ij )}={ exp ( n ij )}/{ 1 + exp ( n ij )} wherein cumulative distribution function f ( n ij ) corresponds to said best - fitting logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. utilizing said model and said cumulative distribution function f ( n ij ), fig6 shows selected n ij values and selected π ( m ij ) values that correspond to the data that are shown in fig3 , these values also correspond to the computer program that is shown in fig4 , these values further correspond to the results that are shown in fig5 . as noted , examples of a phenomenon - specific probability of mortality include an age - specific probability of mortality , a size - specific probability of mortality , and a sex - specific probability of mortality . as noted , an individualized z *- specific probability of mortality of individual i at situation j is denoted here by π ( m ijz * ), wherein z denotes a specific phenomenon , and z * denotes a specific selected value of this specific phenomeon . as further noted , π ( m ijz * ) is estimated here by utilizing n ijz * = β 0 + σβ v z * + σβ v x vij ˜ z and π ( m ijz * )= f ( h ijz * ). therefore , utilizing l = z wherein l denotes lifespan , π ( m ijl * ) denotes an individualized lifespan - specific probability of mortality of individual i at situation j and at a specific level l *, wherein l * denotes a specific level of lifespan l . therefore , based upon said best - fitting model of humans &# 39 ; mortality in sweden 1760 - 2008 , n ijl * = 511 . 78 − 1074 . 55 ( a ij 0 . 16 )+ 546 . 12 ( a ij 0 . 16 ) 2 − 17 . 12 ( l * 0 . 88 ) + 0 . 101 ( l * 0 . 88 ) 2 + 0 . 006 ( c ij 0 . 75 )−( 4 . 39e − 7 )( c ij 0 . 75 ) 2 + 6 . 19 ( λ ij 0 . 30 )− 0 . 35 ( λ ij 0 . 30 ) 2 − 0 . 008 ( h ij 1 . 41 )+( 1 . 92e − 6 )( h ij 1 . 41 ) 2 −( 7 . 97e − 10 )( h ij 1 . 41 ) 3 − 1 . 13 ( g ij ), wherein all the independent variables — except variable l — apply to individual i at situation j . in said model for n ijl * , l denotes a specifically selected level of variable l , but l * applies to all individuals i at respective situations j . said best - fitting model for n ij has been estimated utilizing a logit binary link function ; therefore , π ( m ijl * )= f ( n ijl * )= 1 /{ 1 + exp (− n ijl * )}={ exp ( n ijl * )}/{ 1 + exp ( n ijl * )} for individual i at situation j and at a specific lifespan level l *. utilizing said best - fitting model for n ijl * and said π ( m ijl * )= f ( n ijl * ), and utilizing the data from fig6 , fig7 presents values of individualized lifespan - specific probabilities of mortality π ( m ijz * ) of selected individuals i at selected situations / and at the following specific levels of lifespan : l *= 0 . 5 year ( denoting less than 1 year ), l *= 40 years , l *= 60 years , and l *= 90 years . fig7 illustrates that if the lifespan is set at 0 . 5 year ( denoting less than 1 year ) then individualized lifespan - specific probability of mortality π ( m ijl * ) is estimated to be 1 for each of the about 6 , 519 males whose age was less than 1 year and whose contemporary aggregate size was about 27 , 866 in 1760 ; fig7 illustrates that if the lifespan is set at 40 , 60 , or 90 years then individualized lifespan - specific probability of mortality π ( m ijl * ) is estimated to be 0 for each of the about 6 , 519 males whose age was less than 1 year and whose contemporary aggregate size was about 27 , 866 in 1760 ; fig7 illustrates that if the lifespan is set at less than 1 year then individualized lifespan - specific probability of mortality π ( m ijl * ) is estimated to be 1 for each of the about 52 females whose age was 93 years and whose contemporary aggregate size is about 151 in 1760 ; fig7 illustrates that if the lifespan is set at 40 , 60 , or 90 years then individualized lifespan - specific probability of mortality π ( m ijl * ) is estimated to be 1 for each of the about 52 females whose age was 93 years and whose contemporary aggregate size was about 151 in 1760 ; fig7 illustrates that if the lifespan is set at less than 1 year then individualized lifespan - specific probability of mortality π ( m ijl * ) is estimated to be 1 for each of the about 576 females whose age was less than 1 year and whose contemporary aggregate size was about 60 , 364 in 1890 ; fig7 also illustrates that if the lifespan is set at 40 , 60 , or 90 years then individualized lifespan - specific probability of mortality π ( m ijl * ) is estimated to be 0 for each of the about 576 females whose age was less than 1 year and whose contemporary aggregate size was about 60 , 364 in 1890 . as noted , a plot of π ( m ijz ) is a plot of at least two individualized specific probabilities of mortality π ( m ijz * ), wherein π ( m ijz ) denotes at least two probabilities π ( m ijz * ) ( wherein said at least two probabilities π ( m ijz * ) denote respective individualized specific probabilities of mortality of individual i at situation j and at at least two specifically selected z * values of variable z ). based upon said best - fitting model of humans &# 39 ; mortality in sweden 1760 - 2008 , fig7 specifies π ( m ijl * ) values of individual i = 1 at situation j = 1 in fig6 and fig7 ; this individual was a less than 1 year old swedish male who died in 1760 ; fig6 and fig7 indicate that there were an estimated 6 , 519 swedish males who were less than 1 year old and that also died that year in sweden , further indicating that these males are included in the estimated 27 , 866 swedish males who were less than 1 year old at that year . the π ( m ijl * ) data about individual i = 1 at situation j = 1 and at lifespans 0 . 5 , 40 , 60 , and 90 in fig7 is specified in greater detail in fig8 . fig8 also adds π ( m ijl * ) data at lifespans 1 , 3 , and 6 of this individual . based upon the π ( m ijl * ) data about individual i = 1 at situation j = 1 in fig8 , panel a of fig9 shows a scatterplot and a corresponding supersmoothed friedman line of π ( m ijl ) by lifespan l , depicting the trajectory of the individualized lifespan - specific probabilities of mortality π ( m ijl ) of the less than 1 year old male who died in sweden in 1760 , wherein π ( m ijl ) denotes individualized lifespan - specific probabilities of mortality . as noted , an averaged z *- specific probability of mortality of individuals i at respective situations j is denoted here by π ( m z * ), wherein z denotes a specific phenomenon , z * denotes a specific level of this specific phenomeon , and “ average ” refers to one of the statistical measures of location or central tendency ( e . g ., mean , median , mode ). as also noted , estimation of π ( y z * ) of one of π ( m z * )= π ( y z * ) and π ( s z * ) = π ( y z * ) utilizes n ijz * = β 0 + σβ v z *+ σβ v x vij ˜ z and at least one of π ( y z * )= f { average ( n ijz * )} and π ( y z * )= average { f ( n ijz * )}. fig8 illustrates π ( y l * ) of each of π ( m l * )= π ( y l * ) and π ( s l * ) at selected specific levels of lifespan l *; these π ( y l * ) were calculated utilizing arithmetic means of the 19 , 394 events of death or survival in sweden 1760 - 2008 that are shown in fig7 . as noted , a plot of π ( m z ) is a plot of at least two averaged specific probabilities of mortality π ( m z * ), wherein π ( m z ) denotes at least two probabilities π ( m z * ) ( wherein said at least two probabilities π ( m z * ) denote respective averaged specific probabilities of mortality at at least two specifically selected z * values of variable z ). utilizing the respective π ( m l * )= average { f ( n ijl * )} data in fig8 , panel b of fig9 illustrates a scatterplot and a supersmoothed friedman lineplot of averaged lifespan - specific probabilities of mortality that are denoted by π ( m l ), wherein π ( m l ) denotes more than one selected averaged lifespan - specific probabilities of mortality π ( m l * ). similarly , utilizing the respective π ( m l * )= f { average ( n ijl * )} data in fig8 , panel c of fig9 illustrates a scatterplot and a supersmoothed friedman lineplot of averaged lifespan - specific probabilities of mortality that are denoted by π ( m l ), wherein π ( m l ) denotes more than one selected averaged lifespan - specific probabilities of mortality π ( m l * ). individualized age - specific probability of mortality of individual i at situation j and at a specific level of age is estimated here utilizing n ija * β 0 + σβ v a *+ σβ v x vij ˜ a , wherein a denotes age and wherein a * denotes the specific level of age , and further utilizing π ( m ija * )= f ( h ija * ), wherein π ( m ija * ) denotes the individualized age - specific probability of mortality of an individual i at a respective situation j at said specific level of age a *. utilizing the best - fitting model of humans &# 39 ; mortality in sweden 1760 - 2008 , n ija * = 511 . 78 − 1074 . 55 ( a * 0 . 16 )+ 546 . 12 ( a * 0 . 16 ) 2 − 17 . 12 ( l ij 0 . 88 )+ 0 . 101 ( l ij 0 . 88 ) 2 + 0 . 006 ( c ij 0 . 75 )−( 4 . 39e − 7 )( c ij 0 . 75 ) 2 + 6 . 19 ( λ ij 0 . 30 )− 0 . 35 ( λ ij 0 . 30 ) 2 − 0 . 008 ( h ij 1 . 41 )+( 1 . 92e − 6 )( h ij 1 . 41 ) 2 −( 7 . 97e − 10 )( h ij 1 . 41 ) 3 − 1 . 13 ( g ij ) and π ( m ija * )= f ( n ija * )= 1 /{ 1 + exp (− n ija * )}={ exp ( n ija * )}/{ 1 + exp ( n ija )} for individual i at situation j and at a specific age a *, fig1 presents values of individualized age - specific probabilities of mortality π ( m ija * ) of selected individuals i at selected situations j and at the following specific levels of age : a *= 0 . 5 year ( denoting less than 1 year ), a *= 40 years , a *= 60 years , and a *= 90 years . fig1 illustrates that if age is set at 0 . 5 year ( denoting less than 1 year ) or 40 , 60 , or 90 years then individualized age - specific probability of mortality π ( m ija * ) is estimated to be 1 for each of the about 6 , 519 males whose lifespan was less than 1 year and whose contemporary aggregate size was about 27 , 866 in 1760 ; fig1 illustrates that if age is set at less than 1 year or 40 , 60 , or 90 years then individualized age - specific probability of mortality π ( m ija * ) is estimated to be 0 for each of the about 52 females whose lifespan was 93 . 5 years and whose contemporary aggregate size is about 151 in 1760 ; fig1 illustrates that if age is set at less than 1 year then individualized age - specific probability of mortality π ( m ija * ) is estimated to be 0 for each of the about 576 females whose lifespan was 93 . 5 years and whose contemporary aggregate size was about 60 , 364 in 1890 ; fig1 also illustrates that if age is set at 40 , 60 , or 90 years then individualized age - specific probability of mortality π ( m ija * ) is estimated to be 1 for each of the about 576 females whose lifespan was 93 . 5 years and whose contemporary aggregate size was about 60 , 364 in 1890 . the π ( m ija * ) data about individual i = 1 at situation j = 1 and at ages 0 . 5 , 40 , 60 , and 90 in fig1 is specified in greater detail in fig1 . fig1 also adds π ( m ija * ) data at ages 1 , 3 , and 6 of this individual . based upon the π ( m ija * ) data about individual i = 1 at situation j = 1 in fig1 , panel a in fig1 shows a scatterplot and a corresponding supersmoothed friedman line of π ( m ija ) by age a , depicting the trajectory of the individualized age - specific probabilities of mortality π ( m ija ) of the less than 1 year old male who died in sweden in 1760 , wherein π ( m ija ) denotes individualized age - specific probabilities of mortality . fig1 illustrates π ( y a * ) of each of π ( m a * )= π ( y a * ) and π ( s a * )= π ( y a * ) at selected specific levels of age a *; these π ( y a * ) were calculated utilizing arithmetic means of the 19 , 394 events of death or survival in sweden 1760 - 2008 that are shown in fig1 . utilizing the respective π ( m a * )= average { f ( n ija * )} data in fig1 , panel b in fig1 illustrates a scatterplot and a supersmoothed friedman lineplot of averaged age - specific probabilities of mortality that are denoted by π ( m a ), wherein π ( m a ) denotes more than one especially selected averaged age - specific probabilities of mortality π ( m a * ). similarly , utilizing the respective π ( m a * )= f { average ( n ija * )} data in fig1 , panel c in fig1 illustrates a scatterplot and a supersmoothed friedman lineplot of averaged age - specific probabilities of mortality that are denoted by π ( m a ), wherein π ( m a ) denotes more than one especially selected averaged age - specific probabilities of mortality π ( m a * ). the invention is also applied here in multivariable binary regression analysis of humans &# 39 ; survivorship . in the best - fitting multivariable binary regression analysis of human &# 39 ; s survivorship , the analyst created a data set consisting of the same data and denotations as the corresponding mortality data set except for using y ij = s ij instead of y ij = m ij , so that s ij = 1 when the individual is alive , and s ij = 0 when the individual is not alive . in said best - fitting multivariable binary regression analysis of humans &# 39 ; survivorship the analyst also employed an input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij + β 11 x 11ij + β 12 x 12ij with a logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. the best - fitting multivariable binary regression analyses of humans &# 39 ; survivorship also yielded a best - fitting estimated model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij + β 11 x 11ij + β 12 x 12ij with an f ( n ij )= 1 /{ 1 + exp (− n ij )}={ exp ( n ij )}/{ 1 + exp ( n ij )} cumulative distribution function corresponding to said logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. utilizing corresponding best fitting estimated regression coefficients , and utilizing respective denoting variables that correspond to variables x vij , the best - fitting estimated model for humans &# 39 ; survivorship is specified with n ij =− 511 . 78 + 1074 . 55 ( a ij 0 . 16 )− 546 . 12 ( a ij 0 . 16 ) 2 + 17 . 12 ( l ij 0 . 88 )− 0 . 101 ( l ij 0 . 88 ) 2 − 0 . 006 ( c ij 0 . 75 )+( 4 . 39e − 7 )( c ij 0 . 75 ) 2 − 6 . 19 ( λ ij 0 . 30 )+ 0 . 35 ( λ ij 0 . 30 ) 2 + 0 . 008 ( h ij 1 . 41 )−( 1 . 92e − 6 )( h ij 1 . 41 ) 2 +( 7 . 97e − 10 )( h ij 1 . 41 )+ 1 . 13 ( g ij ) for which the probability of survivorship π ( s ij ) is calculated with π ( s ij )= f ( n ij )= 1 /{ 1 + exp (− n ij )}={ exp ( n ij )}/{ 1 + exp ( n ij )} corresponding to said logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. as noted , π ( ms ijz ) denotes the combination of corresponding π ( m ijz ) and π ( s ijz ) at at least two specifically selected z * values of variable z . utilizing respective supersmoothed friedman lineplots for respective π ( m ijl * ) and π ( s ijl * ) data from fig8 , fig1 shows π ( ms ijl ) plots of the combination of corresponding π ( m ijl ) and π ( s ijl ) trajectories . utilizing respective supersmoothed friedman lineplots for respective π ( m ija * ) and π ( s ija * ) data from fig1 , fig1 shows π ( ms ija ) plots of the combination of corresponding π ( m ija ) and π ( s ija ) trajectories . as noted , π ( ms z ) denotes the combination of corresponding π ( m z ) and π ( s z ) at at least two specifically selected z * values of variable z . utilizing respective supersmoothed friedman lineplots for respective π ( m l * )= f { average ( n ijl * )} and π ( s l * )= f { average ( n ijl * )} data from fig8 , fig1 shows π ( ms l ) plots of the combination of corresponding π ( m l ) and π ( s l ) trajectories . utilizing respective supersmoothed friedman lineplots for respective π ( m z * )= f { average ( n ija * )} mortality data and π ( s a * )= f { average ( n ija * )} survivorship data from fig1 , fig1 shows π ( ms a ) plots of the combination of corresponding π ( m a ) and π ( s a ) trajectories . utilizing respective supersmoothed friedman lineplots for respective π ( m l * )= f { average ( n ijl * )} and π ( s l * )= f { average ( n ijl * )} data from fig8 , fig1 shows π ( ms l ) plots of the combination of corresponding π ( m l ) and π ( s l ) trajectories . utilizing respective supersmoothed friedman lineplots for respective π ( m a * )= average { f ( n ija * )} mortality data and π ( s a * )= average { f ( n ija * )} survivorship data from fig1 , fig1 shows π ( ms a ) plots of the combination of corresponding π ( m a ) and π ( s a ) trajectories . as noted , π ( ms ijz + ) denotes the combination of corresponding π ( ms ijz ) at at least two specifically selected z * values of at least two variables z ). utilizing respective supersmoothed friedman lineplots for respective π ( m ijl * ) and π ( s ijl * ) data from fig8 and fig1 , and utilizing respective supersmoothed friedman lineplots for respective π ( m ija * ) and π ( s ija * ) data from fig1 and fig1 , fig1 shows π ( ms ijz + ) plots of the combination of corresponding π ( m ijl ), π ( s ijl ), π ( m ija ), and π ( s ija ) trajectories . as noted , π ( ms z + ) denotes the combination of at least two corresponding π ( ms z ) of at least two variables z . utilizing respective supersmoothed friedman lineplots for respective π ( m l * )= f { average ( n ijl * )} and π ( s l * )= f { average ( n ijl * )} data from fig8 and fig1 , and utilizing respective supersmoothed friedman lineplots for respective π ( m z * )= f { average ( n ija * )} mortality data and π ( s a * )= f { average ( n ija * )} survivorship data from fig1 and fig1 , fig2 shows π ( ms z + ) plots of the combination of corresponding π ( m l ), π ( s l ), π ( m a ), and π ( s a ) trajectories . utilizing respective supersmoothed friedman lineplots for respective π ( m l * )= f { average ( n ijl * )} and π ( s l * )= average { f ( n ijl * )} data from fig8 and fig1 , and utilizing respective supersmoothed friedman lineplots for respective π ( m a * )= average { f ( n ija * )} mortality data and π ( s a * )= average { f ( n ija * )} survivorship data from fig1 and fig1 , fig2 shows π ( ms z + ) plots of the combination of corresponding π ( m l ), π ( s l ), π ( m a ), and π ( s a ) trajectories . as noted , an “ average ” refers here to a statistical measure of location or central tendency , such as a mean ( e . g ., arithmetic , geometric , harmonic , or other mean ), median , or mode . whereas fig1 presents respective π ( m a * )= average { f ( n ija * )} and π ( m a * )= f { average ( n ija * )} that are calculated utilizing respective arithmetic means , fig2 presents corresponding respective π ( m a * )= average { f ( n ija * )} and π ( m a * )= f { average ( n ija * )} that are calculated utilizing respective medians . respective plots of π ( m a * )= average { f ( n ija * )} and π ( m a * )= average { f ( n ija * )} can then be produced for the data that are presented in fig2 . fig1 to fig2 reveal specifications and plots that are based upon the best - fitting multivariable binary regression analyses of 188 , 087 data records of 79 , 164 , 608 events of death or survival of all individuals that were born in sweden in decennial years 1760 - 1930 and died between 1760 and 2008 . however , the specifications and plots in fig1 to fig2 are restricted to the 19 , 394 events of death or survival in sweden 1760 - 2008 that are shown in the tables of fig3 , fig6 and fig7 . methods and procedures for the specifications and plots of all events of death or survival in sweden 1760 - 2008 are those that have been shown here in reference to fig1 to fig2 . moreover , these methods and procedures are applicable to all other events of death or survival of all kinds of individuals , as will be apparent to those skilled in the art . in contradistinction with previous research on mortality or survivorship , the data and analyses that are presented here include lifespan and lifespan aggregate variables and distinguish here among age , lifespan , lifespan aggregate size , and contemporary aggregate size . excepting applicant &# 39 ; s u . s . provisional patent application no . 61 / 962 , 502 and epelbaum ( 2014 ), previous multivariable binary analyses of mortality or survivorship did not include distinct lifespan and lifespan aggregate variables . moreover , said previous analyses did not distinguish age , lifespan , lifespan aggregate , and contemporary aggregate . the following paragraphs present additional embodiments of the invention ; these embodiments include and distinguish age , lifespan , lifespan aggregate size , and contemporary aggregate size in multivariable binary regression analysis of medflies &# 39 ; mortality or survivorship . the data on mortality or survivorship of mediterranean fruit flies — ceratitis capitata , commonly known as medflies — were collected in 1991 at the moscamed medflies mass - rearing facility in metapa , a small village located about 20 kilometers from the city of tapachula in the state of mexico . the original moscamed data file contains information on numbers of age - cage - and - sex - specific deaths of 1 , 203 , 646 male and female medflies , where medflies are distributed in 167 cages , and the numbers of age - cage - sex - specific dead individuals are counted daily . the original moscamed data are stored in the memory of a computer in california , and — in this embodiment — these data are transmitted through the internet to the memory of a computer in tennessee . a processor of a computer in tennessee processes these moscamed data and compiles them into pluralized data on age - cage - and - sex - specific deaths of 1 , 203 , 646 male and female medflies that are distributed in 167 cages . computer intensive analyses impose restrictions on the size of the data file that is analyzed here . therefore , the analytic data file is restricted here to cases of physical size # 5 and birth aggregate batch # 2 . in these selected cases , individuals lived and died in one of thirteen cages , where the cages averaged 3 , 646 . 3 sex - specific medflies per cage at age 0 to 1 days . the analyst further instructed the processor to process these pluralized data and convert them to individualized data of daily events of each individual &# 39 ; s death or survival ; parts of these individualized data are illustrated here in fig2 , in accordance with an aspect of the present invention . as depicted in fig2 , each row depicts one data record containing data on the following : one individual in a specific situation ( i denotes the individual and j denotes the situation , the situation refers here to events occurring during a specific day in the life of the respective individual medfly ), this individual &# 39 ; s death or survival ( depicted in columns m ij and s ij , respectively denoting mortality or survivorship of individual i at situation j in fig2 ), this individual &# 39 ; s sex ( depicted in column g ij , wherein g ij = 1 denotes being female , and g ij = 0 denotes being male in fig2 ), this individual &# 39 ; s age during this situation ( during the mid - day , depicted in column a ij in fig2 ), this individual &# 39 ; s lifespan ( depicted in column l ij and including the mid - day of the last day of life of this individual , in fig2 ), this individual &# 39 ; s environmental context ( depicted as a specific cage in column e ij in fig2 ), the individual &# 39 ; s lifespan aggregate size ( i . e ., number of corresponding age - lifespan - day - cage - sex - specific identical individuals , this is the number of age - lifespan - day - cage - sex - specific individuals with identical birth day and identical death day to the criterion individual , as depicted in column λ ij in fig2 ), the individual &# 39 ; s contemporary aggregate size ( i . e ., the number of age - day - cage - sex - specific - identical individuals that are exposed to the risk of death and prospect of survival during this situation , i . e ., during the specific day , as depicted in column c ij in fig2 ). the resultant data file contains 50 , 716 data records , wherein each record is weighted for depicting 2 , 211 , 782 situations j involving respective individuals i , wherein the individual &# 39 ; s lifespan aggregate size ( i . e ., as depicted in column a ij in fig2 ) is the weighting variable in the analyses , and wherein j ( i . e ., the total number of situations j of each specific individual ) varies among individuals . based upon previous research , theoretical knowledge , and the available data , the analyst selected the following denoted variables : age , lifespan , lifespan aggregate size , contemporary aggregate size , environmental context ( i . e ., cage ), and sex . these denoted variables are respectively denoted here with a ij , l ij , c ij , a ij , e ij , and g ij for an individual i at situation j ; in these denotations a denotes age ( in years ), l denotes lifespan ( in years ), c denotes contemporary aggregate size , λ ( the greek capital letter lambda ) denotes lifespan aggregate size , e denotes environmental context , and g denotes sex . numerical values for these denoted variables for individuals i at situations j are illustrated here in fig2 . analyst employed forward selection methods in iterative multivariable binary regression analyses of medflies &# 39 ; mortality or survivorship to select transformations of each of denoted variables a ij , l ij , c ij , λ ij , e ij , and g ij for an individual i at situation /. in these iterative analyses analyst tested power transformations of each of these denoted variables , selecting transformation that improved aic and bic . in these analyses , analyst conducted iterative multivariable binary regression analyses of medflies &# 39 ; mortality or survivorship . the initial iterating multivariable binary regression analyses utilized diverse input models n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij with respective logit , probit , and complementary log - log binary link functions — as well as x 1 = g ij , x 2 = e ij and diverse values of power coefficient p in x 3 =( a ij ) p wherein g denotes sex , e denotes environmental context , and a denotes age — for one of y ij = m ij and y ij = s ij ; these iterative analyses sought to optimize aic and bic best - fitting values , selecting the power coefficient p beyond which aic and bic cease to improve , ensuring that all regression coefficients in the selected model are significant beyond the 0 . 05 level . utilizing said selected power coefficient p in x 3ij =( a ij ) p , analyst proceeded to conduct further iterative multivariable binary regression analyses with input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij and logit , probit , and complementary log - log binary link functions for one of y ij = m ij and y ij = s ij to find optimal aic and bic values for diverse values of power coefficient p in x 4ij =( l ij ) p . utilizing these forward selection methods per additional respective denoted variable , the analyst continued to select optimal power coefficients p in such iterative transformations of respective denoted variables in x 5ij =( c ij ) p and x 6ij =( λ ij ) p in respective input models until reaching an optimal best - fitting first - degree polynomial input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij . utilizing said respective selected optimal first - degree polynomial power coefficients p , the analyst continued to test the optimality of second degree polynomial transformations for these respective selected power coefficients p , optimizing optimal aic and bic best - fitting criteria in such iterative transformations of respective denoted variables in x 7ij ={( a ij ) p } 2 , x 8ij ={( l ij ) p } 2 , x 9ij ={( c ij ) p } 2 , and x 10ij ={( λ ij ) p } 2 in respective input models until reaching an optimal second - degree powered polynomial input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij . utilizing said respective selected optimal first - degree and second - degree polynomials power coefficients p , the analyst continued to test the optimality of third - degree polynomial transformations for these respective selected power coefficients p , finding that respective x 11ij ={( a ij ) p } 3 , x 11ij ={( l ij ) p } 3 , x 11ij ={( c ij ) p } 3 , and x 11ij ={( λ ij ) p } 3 failed to improve aic and bic best - fitting criteria in such iterative transformations . in all these iterative analyses , the analyst used non - negative values of power coefficient p ( in the interest of investigating power laws ), utilizing the natural logarithmic transformation when p = 0 ( e . g ., using x 1 = ln ( a ij ) instead of x 1 =( a ij ) 0 ), and allowing identity transformation ( e . g ., when p = 1 , e . g ., using x 1 =( a ij ) 1 = a ij ), ensuring that all regression coefficients in respective selected best - fitting models are significant beyond the 0 . 05 level of significance . moreover , all analyses employed random effects input models . further information on these analyses is available in epelbaum ( 2014 ). in the best - fitting multivariable binary regression analysis of medflies &# 39 ; mortality , the analyst created a data set consisting of variable y ij and 10 variables x vij for 50 , 716 data records , wherein each respective record is variably weighted for depicting a total of 2 , 211 , 782 situations j involving all individuals of physical size # 5 and birth aggregate batch # 2 in thirteen specific cages in the moscamed study . these best - fitting analyses employed a best - fitting input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij and a logit link function b ( n ij )= ln {− ln ( 1 − n ij )}, employing an identity transformation of denoted binary variable g ij , an identity transformation of denoted categorical variable e ij , as well as first - degree and second - degree polynomial transformations of denoted powered variables a ij , l ij , c ij , and λ ij for a respective individual i in a respective situation j , wherein y ij = m ij , x 1ij = g ij , x 2ij = e ij , x 3ij =( a ij ) 0 . 13 , x 4ij =( l ij ) 0 . 98 , x 5ij =( c ij ) 1 . 02 , x 6ij =( λ ij ) 0 . 95 , x 7ij ={( a ij ) 0 . 13 } 2 , x 8ij ={( l ij ) 0 . 98 } 2 , x 9ij ={( c ij ) 1 . 02 } 2 , and x 10ij ={( λ ij ) 0 . 95 } 2 , wherein , as noted and as illustrated in fig2 , i denotes an individual , j is a consecutive number of the day of life of this individual , m ij = 1 when the individual is dead , and m ij = 0 when the individual is not dead , a ij denotes the individual &# 39 ; s age ( in days ) at situation j , l ij denotes the individual &# 39 ; s lifespan ( in days ) at situation j , c ij denotes the individual &# 39 ; s contemporary aggregate size at situation j , λ ij denotes the individual &# 39 ; s lifespan aggregate size at situation j , e ij denotes the individual &# 39 ; s environmental context ( i . e ., indicated by cage , where e denotes environmental context transformed to a sequential number ) at situation j , g ij = 1 when the individual is female , and g ij = 0 when the individual is male . the best - fitting multivariable binary regression analyses of medflies &# 39 ; mortality yielded a best - fitting estimated model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij with an f ( n ij )= 1 /{ 1 + exp (− n ij )}={ exp ( n ij )}/{ 1 + exp ( n ij )} cumulative distribution function corresponding to said logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. utilizing corresponding best - fitting estimated regression coefficients , and utilizing respective denoting variables , the best - fitting estimated model is specified with n ij = 1391 . 92 − 2 , 648 . 52 ( a ij 0 . 13 )+ 1295 . 76 ( a ij 0 . 13 ) 2 − 16 . 67 ( l ij 0 . 98 )+ 0 . 095 ( l ij 0 . 98 ) 2 − 0 . 006 ( c ij 1 . 02 )+( 6 . 85e − 07 )( c ij 1 . 02 ) 2 − 0 . 09 ( λ ij 0 . 95 )+ 0 . 00026 ( λ ij 0 . 95 ) 2 − 1 . 83 ( g ij )+{− 1 . 42 ( e i1 )+ 0 . 63 ( e i2 )+ 0 . 76 ( e i3 )+ 1 . 59 ( e i4 )+ 2 . 80 ( e i5 )+ 0 . 997 ( e i6 )− 4 . 42 ( e i7 )+ 2 . 043 ( e i8 )+ 3 . 96 ( e i9 )+ 0 . 863 ( e i10 )+ 2 . 069 ( e i11 )+ 1 . 65 ( e i12 )+ 1 . 00 ( e i13 )}, where only one of the e ic terms within applies in each corresponding specification of a specific cage ; the probability of mortality π ( m ij ) for this estimated model is calculated with π ( m ij )= f ( n ij )= 1 /{ 1 + exp (− n ij )}={ exp ( n ij )}/{ 1 + exp ( n ij )} wherein said cumulative distribution function f ( n ij ) corresponding to said logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. in contradistinction with previous research on mortality , the data and models distinguish age , lifespan , lifespan aggregate , and contemporary aggregate ; the data and variables include specific independent variables x vij denoting respective age , lifespan , lifespan aggregate size , and contemporary aggregate size variables . excepting applicant &# 39 ; s u . s . provisional patent application no . 61 / 962 , 502 and applicant &# 39 ; s epelbaum ( 2014 ), previous multivariable binary regression analyses of mortality did not include the lifespan and lifespan aggregate variables , and did not distinguish age , lifespan , lifespan aggregate size , and contemporary aggregate size . in the best - fitting multivariable binary regression analysis of medflies &# 39 ; survivorship , the analyst employed the corresponding data set consisting of variable y ij = s ij and 10 variables x vij for 50 , 716 data records , wherein each record is relatively weighted for depicting a total of 2 , 211 , 782 situations j involving all individuals of physical size # 5 and birth aggregate batch # 2 in thirteen specific cages in the moscamed study . these best - fitting analyses of medflies &# 39 ; survivorship employed a best - fitting input model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij and a complementary log - log binary link function b ( n ij )= ln {− ln /( 1 − n ij )}, wherein y ij = s ij , x 1ij = g ij , x 2ij = e ij , x 3ij =( a ij ) 0 . 16 , x 4ij =( l ij ) 0 . 94 , x 5ij =( c ij ) 1 . 02 , x 6ij =( λ ij ) 0 . 88 , x 7ij ={( a ij ) 0 . 16 } 2 , x 8ij ={( l ij ) 0 . 94 } 2 , x 9ij ={( c ij ) 1 . 02 } 2 , and x 10ij ={( λ ij ) 0 . 88 } 2 ; fig2 presents computer instructions for said input model and said complementary log - log link function . as illustrated in fig2 , i denotes an individual , j is a consecutive number of the day of life of this individual , s ij = 1 when the individual is alive , and s ij = 0 when the individual is not alive , a ij denotes the individual &# 39 ; s age ( in days ) at situation j , l ij denotes the individual &# 39 ; s lifespan ( in days ) at situation j , c ij denotes the individual &# 39 ; s contemporary aggregate size at situation j , λ ij denotes the individual &# 39 ; s lifespan aggregate size at situation j , e ij denotes the individual &# 39 ; s environmental context ( i . e ., indicated by cage ) at situation j ( where e denotes environmental context transformed to a sequential number ), and g ij = 1 when the individual is female , and g ij = 0 when the individual is male . the best - fitting multivariable binary regression analyses of medflies &# 39 ; survivorship yielded a best - fitting estimated model n ij = β 0 + β 1 x 1ij + β 2 x 2ij + β 3 x 3ij + β 4 x 4ij + β 5 x 5ij + β 6 x 6ij + β 7 x 7ij + β 8 x 8ij + β 9 x 9ij + β 10 x 10ij with an f ( n ij )= 1 − exp {− exp ( n ij )} cumulative distribution function corresponding to said complementary log - log binary link function b ( n ij )= ln {− ln /( 1 − n ij )}. computer instructions for said input model and said complementary log - log binary link function are shown in fig2 . a computer output for this estimated model is depicted in fig2 . utilizing corresponding best - fitting estimated regression coefficients that are depicted in fig2 , and utilizing respective denoting variables that correspond to variables x vij , the best - fitting estimated model is specified with n ij =− 732 . 74 + 1402 . 49 ( a ij 0 . 16 )− 706 . 62 ( a ij 0 . 16 ) 2 + 19 . 24 ( l ij 0 . 94 )− 0 . 12 ( l ij 0 . 94 ) 2 + 0 . 0041 ( c ij 1 . 02 )−( 4 . 03e − 07 )( c ij 1 . 02 ) 2 + 0 . 11 ( λ ij 0 . 88 )− 0 . 000491 ( λ ij 0 . 88 ) 2 + 1 . 28 ( g ij )+{+ 0 . 96 ( e i1 )− 0 . 669 ( e i2 )− 1 . 03 ( e i3 )− 0 . 75 ( e i4 )− 1 . 53 ( e i5 )− 0 . 0 . 52 ( e i6 )+ 3 . 25 ( e i7 )− 1 . 15 ( e i8 )− 2 . 63 ( e i9 )+ 0 . 83 ( e i10 )− 1 . 12 ( e i11 )− 0 . 53 ( e i12 )+ 1 . 00 ( e i13 )}, where only one of the e ic terms within applies in each corresponding specification for a specific cage . the probability of survivorship π ( s ij ) corresponding to said best - fitting estimated model is calculated with π ( s ij )= f ( n ij )= 1 − exp {− exp ( n ijj )} wherein said cumulative distribution function corresponds to said complementary log - log binary link function b ( n ij )= ln {− ln /( 1 − n ij )}. utilizing said model and said cumulative distribution function f ( n ij ), fig2 shows selected n ij values and selected π ( s ij ) values that correspond to the data that are shown in fig2 . these values also correspond to the computer program that is shown in fig2 , these values further correspond to the results that are shown in fig2 . utilizing said model and said cumulative distribution function f ( n ij ), fig2 shows individualized lifespan - specific probabilities of survivorship π ( s ijl * ) and averaged lifespan - specific probabilities of survivorship π ( s l * ) of selected individual medflies at selected situations and at specific lifespan levels l *. these values also correspond to the computer program that is shown in fig2 , these values further correspond to the results that are shown in fig2 . as noted , in correspondence to the respective best - fitting multivariable binary regression analyses , the individualized probability of medflies &# 39 ; survivorship π ( s ij ) is calculated with π ( s ij )= f ( n ij )= 1 − exp {− exp ( n ijj )} wherein said cumulative distribution function f ( n ij ) corresponds to the best - fitting complementary log - log binary link function b ( n ij )= ln {− ln /( 1 − n ij )}. in contrast , in correspondence to the respective best - fitting multivariable binary regression analyses , the respective individualized probability of medflies &# 39 ; mortality π ( m ij ) and the individualized probability of humans &# 39 ; mortality π ( m ij ) and survivorship π ( s ij ) were calculated with f ( n ij )= 1 /{ 1 + exp (− n ij )}={ exp ( n ij )}/{ 1 + exp ( n ij )} wherein said cumulative distribution function f ( n ij ) corresponds to the best - fitting logit binary link function b ( n ij )= ln { n ij /( 1 − n ij )}. these considerations illustrate that the methods and procedures that are shown here are applicable to diverse kinds of binary link functions . the foregoing best - fitting multivariable binary regression analyses of medflies &# 39 ; mortality and survivorship provide the foundation for specifications and plots of medflies &# 39 ; individualized z - specific probabilities of mortality or survivorship π ( y ijz ), averaged z - specific probabilities of mortality or survivorship π ( y z ), individualized z - specific probabilities of mortality and survivorship π ( ms ijz ), averaged z - specific probabilities of mortality and survivorship π ( ms z ), individualized z +- specific probabilities of mortality and survivorship π ( ms ijz + ), and averaged z +- specific probabilities of mortality and survivorship π ( ms z + ). the methods and procedures for these specifications and plots of medflies &# 39 ; mortality and survivorship are same as the methods and procedures that have been applied here in the presentation of these specifications and plots of humans &# 39 ; mortality and survivorship , as shown here in reference to fig1 to fig2 . moreover , these methods and procedures are applicable to all other events of death or survival of all kinds of individuals , as would be apparent to those skilled in the art . excepting applicant &# 39 ; s u . s . provisional patent application no . 61 / 962 , 502 and epelbaum ( 2014 ), previous investigations of mortality and survivorship did not include — and did not distinguish — independent variables that denote lifespan in multivariable binary regression analyses of mortality and survivorship . the analyses that are presented show that these inclusion and distinction of such independent variables provide valuable explanatory insights about mortality and survivorship in diverse species . the analyses that are presented here also provide hitherto unavailable explanatory insights about lifespan aggregates in mortality and survivorship in respective distinctions from age and contemporary aggregates . moreover , the analyses that are presented here improve goodness - of - fit and prediction . finally , while the invention has been described by way of examples of multivariable binary regression analyses of humans &# 39 ; and medflies &# 39 ; mortality or survivorship and in terms of the above , it is to be understood that the invention is not limited to the disclosed embodiments . on the contrary , it is intended to cover diverse kinds of multivariable binary regression analyses and computing environments and various modifications that would be apparent to those skilled in the 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