WO2018222217A1 - Lifespan in regression-based estimation of phenomenon-specific probabilities of mortality or survivorship - Google Patents

Abstract

Methods, systems, and computer program products are hereby claimed for including and distinguishing independent variables that denote lifespan in regression-based estimation of phenomenon-specific probabilities of mortality or survivorship. The invention involves inclusion and distinction of at least one independent variable denoting lifespan in at least one multivariable binary regression analysis of one of mortality and survivorship. Results of said at least one regression analysis are utilized in the estimation of phenomenon-specific probabilities of one of mortality and survivorship, and said probabilities are specified or plotted.

Description

LIFESPAN IN REGRESSION-BASED ESTIMATION OF PHENOMENON- SPECIFIC PROBABILITIES OF MORTALITY OR SURVIVORSHIP

FIELD OF THE INVENTION

[0001] The invention relates, in general, to the estimation of probabilities of mortality or survivorship. The invention relates, more specifically, to including and distinguishing lifespan in regression-based estimation of phenomenon-specific probabilities of mortality or survivorship. BACKGROUND OF THE INVENTION

[0002] Ample evidence– from science, medicine, public affairs, industry, business, and other activities– shows that regression analyses of mortality or survivorship are usefully employed in the estimation of probabilities of mortality or survivorship. The“estimation of probabilities” is conceptualized here as being synonymous with the calculation, forecasting, prediction, extrapolation, and other methods or systems of formulating or making or constructing or measuring probabilities.“Phenomenon-specific probabilities” refer to probabilities that are specific to a specific phenomenon (e.g., age, size, sex). For example, age-specific probabilities of mortality refer to probabilities of mortality at diverse levels of age, size-specific probabilities of mortality refer to probabilities of mortality at diverse levels of size, sex-specific probabilities of mortality refer to probabilities of mortality for diverse categories of sex. Phenomenon-specific probabilities could be combined; for example, age- and-sex-specific probabilities of mortality refer to probabilities of mortality at diverse levels of age and diverse categories of sex.

[0003] Mortality refers to an individual’s cessation of existence, and survivorship refers to an individual’s continuation of existence. Therefore, M_ij∈ {0,1}, wherein M_ij denotes mortality of individual i at situation j, wherein∈ indicates that M_ij is a binary variable that adopts one of values 0 and 1, wherein M_ij = 1 denotes that individual i ceases to exist at situation j, and wherein M_ij = 0 denotes that individual i does not cease to exist at situation j. Similarly, Sij∈ {0,1}, wherein Sij denotes survivorship of individual i at situation j, wherein ∈ indicates that Sij is a binary variable that adopts one of values 0 and 1, wherein Sij = 1 denotes that individual i continues to exist at situation j, and wherein Sij = 0 denotes that individual i does not continue to exist at situation j. Therefore, the invention in this

Application focuses on multivariable binary regression analysis of one of mortality and survivorship.

[0004] The multivariable binary regression analysis of one of mortality and survivorship analyzes the relationship between a dependent variable Y_ij and independent variables X_vij for one of Y_ij = M_ij and Y_ij = S_ij, utilizing data on N individuals i at respective at least one situation j, ηij = β0 + βvXvij +… + βWXWij, binary link function B(ηij), and π(Yij) = F(ηij). Wherein: i = 1:N, indicating that i are sequential positive integers 1 through N; i denotes an individual; N denotes the total number of individuals i in said data; j = 1:J, indicating that j are sequential positive integers 1 through J; j denotes a situation of individual i in reference to said variables Xvij and Yij; J denotes the total number of situations of an individual i in said data, allowing distinct J for distinct individuals i; M_ij denotes the mortality status of individual i at situation j; S_ij denotes the survivorship status of individual i at situation j; Y_ij∈ {0,1}, indicating that Y_ij is a binary variable that adopts one of values 0 and 1 for one of Y_ij = Mij and Yij = Sij of individual i at situation j; Mij = 1 indicates cessation of existence of individual i at situation j; Mij = 0 indicates non-cessation of existence of individual i at situation j; S_ij = 1 indicates continuation of existence of individual i at situation j; S_ij = 0 indicates non-continuation of existence of individual i at situation j; v = 1:W, indicating that v are sequential positive integers 1 through W; W≥ 2; v denotes an index of each of the following: variables X_v, variables X_vij of individual i at situation j, and coefficients β_v; X_vij denotes a variable X_v of individual i at situation j; W denotes the total number variables X_v, variables Xvij of individual i at situation j, and coefficients βv; r = 1:R, indicating that r are sequential positive integers 1 through R; R≥ 2; W≥ R; r denotes an index of each of the following: variables K_r and variables K_rij; K_r denotes a variable that directly denotes a distinct phenomenon, and R respective variables Kr respectively directly denote R distinct phenomena; Krij denotes a variable Kr of individual i at situation j; T denotes a

transformation function, allowing identity transformation; q = 1:Q, indicating that q are sequential positive integers 1 through Q; Xv = Tq(Kr); Xvij = Tq(Krij); Q denotes the total number of transformations Tq(Kr) for a specific Kr, and Q also denotes the total number of transformations T_q(K_rij) for a specific K_rij of individual i at situation j (allowing distinct Q for distinct variables K_r, and allowing distinct Q for distinct variables K_rij of individual i at situation j); β denotes a regression coefficient; β0 denotes the regression coefficient for the intercept (allowing β0 to be any of the following: estimated, suppressed, and user-provided); F(η_ij) denotes a cumulative distribution function of (∙), and cumulative distribution function F(ηij) corresponds to binary link function B(ηij).

[0005] As noted, multivariable binary regression analysis of mortality and survivorship analyzes the relationship between independent variables X_vij and a dependent variable Y_ij for a respective individual i at a respective situation j. An“independent” variable is also known as a“predictor,”“explanatory,”“covariate,” or“right-hand side” variable. A“dependent” variable is also known as a“response,”“outcome,”“explained,” or“left-hand side” variable.

[0006] The following is noted in said consideration of the multivariable binary regression analysis of mortality and survivorship: T denotes a transformation function, allowing identity transformation; q = 1:Q, indicating that q are sequential positive integers 1 through Q; r = 1:R, indicating that r are sequential positive integers 1 through R; Q denotes the total number of transformations Tq(Kr) for a specific Kr and the total number of transformations Tq(Krij) for a specific Krij, allowing distinct Q for distinct variables Kr and distinct variables K_rij; and R denotes the total number of denoted phenomena and denoting variables K_rij. For example, where K_1ij = A_ij and where A_ij denotes the denoted variable age, relationships X_1ij = T1(K1ij) = (Aij)^0.5, X2ij = T2(K1ij) = (Aij)^-0.7, and X3ij = T3(K1ij) = ln(Aij) illustrate nonlinear transformations Xvij = Tq(Krij) in which respective variables X1ij, X2ij, and X3ij respectively equal the respective transformations T_q of a denoting variable A_ij into three distinct nonlinearly transformed forms of the denoted variable age. Relationship X4ij = T4(K1ij) = Aij illustrates an identity transformation of a denoting variable Aij; the identity transformation preserves the identity of the denoting variable (an identity transformation is sometimes conceptualized as no transformation or zero transformation). In these examples, A_ij and K_1ij directly denote the denoted variable age, but respective variables X1ij, X2ij, X3ij, and X4ij respectively indirectly denote age, wherein r = 1 and K_1ij = A_ij. However, X_5ij = T₅(K_2ij) = G_ij wherein G_ij∈ {0,1} illustrate a binary transformation in which an independent variable X_5ij equals a denoting binary variable G_ij that denotes– and gives a mathematical form to– a respective categorical variable gender, where, for example, Gij = 0 if individual i is not female, but Gij = 1 if individual i is female, wherein T5(K2ij) = Gij indicates that r = 2, q = 5, K_2ij directly denotes the denoted variable female, and X_5ij indirectly denotes the denoted variable female. These examples illustrate the sense in which independent variables X_vij indirectly denote the variables age and female. In these examples, age and female are the denoted variables, respective variables Krij, Aij, and Gij are the respective directly denoting variables K_rij that directly denote the respective denoted variables, whereas respective variables Xvij are the indirectly denoting variables that indirectly denote the respective denoted variables. These examples also illustrate the following kinds of transformation: nonlinear (i.e., T₁, T₂, and T₃ in these examples), identity and linear (i.e., T₄ in these examples), and binary (i.e., T₅ in these examples). Analysts employ diverse kinds of mathematical transformations of denoted variables; analysts utilize, for example, polynomial, fractional polynomial, logarithmic, diverse kinds of smoothers (e.g., lowess, splines), and other kinds of mathematical transformations, as considered in some detail in Royston and Sauerbrei (2008). In multivariable binary regression analyses, the analyst selects the following: the denoting variables, the denoted variables, and the transformations that are employed in the respective analyses.

[0007] Multivariable regression analyses require selection of respective binary link functions. B(ηij) denotes here the binary link function in the present consideration of the multivariable binary regression analysis of mortality and survivorship. The binary link function is the function that links the linear model (i.e., the model that is denoted here with ηij = β0 + βvXvij +… + βWXWij) to the conditional mean of the dependent variable (i.e., the conditional mean of the dependent variable that is denoted here with Yij for one of Yij = Mij and Y_ij = S_ij). Moreover, as will be further elucidated here, the binary link function (denoted here with B(ηij)) corresponds to a cumulative distribution function (denoted here with F(ηij)) that enables estimation of probabilities of mortality or survivorship. Diverse binary link functions and corresponding cumulative distribution functions are currently available for use in multivariable binary regression analysis (Gündüz and Fokoué 2015). For example, the logistic binary link function is commonly known as“logit” and is denotable with B(ηij) = ln{η_ij/(1 - logit is the most popular conventional binary link function. Also quite popular is the binary link function that is commonly known as“probit” and is denotable with B(ηij) = Φ^-1(ηij). Less popular– but quite prevalent and useful– is the complementary log- log binary link function that is commonly known as“compit” and is denotable with B(ηij) = ln{-ln(1 - less popular is the Cauchy binary link function that is commonly known as “cauchit” and is denotable with B(ηij) = tan{π(ηij)– π/2}; Gündüz and Fokoué (2015) provide detailed consideration of binary link functions and corresponding cumulative distribution functions. Most of the commonly adopted binary link functions have fixed skewness and they lack the flexibility to allow the data to determine the degree of skewness; to address these limitations, researchers have proposed, for example, parametric, semi- parametric, or non-parametric generalized extreme value binary link functions with unconstrained shape parameters (Li et al.2016).

[0008] Researchers typically search for optimal results in multivariable binary regression analyses of mortality or survivorship; researchs typically search, for example, for optimal data, models, denoting variables, denoted variables, transformations, and binary link functions. The search for optimal results typically involves iterations of analyses with diverse input multivariable regression models (i.e., iterations of analyses with diverse variables Xvij and diverse kinds of transformations Xvij = Tq (Zrij)) and iterations of analyses with diverse kinds of binary link functions B(ηij). The selection of optimal results is typically guided by attempts to optimize prediction, goodness of fit, replicability, or explanation. Diverse tests– employing prediction, goodness of fit, replicability, or explanation criteria– are currently available for the purpose of evaluating results of multivariable binary regression analyses; examples of such tests include pseudo-R² tests, deviance statistics, likelihood ratio tests, Hosmer-Lemeshow tests, mean predicted fit tests, information criteria tests (such as Akaike Information Criterion (AIC) tests or Bayesian Information Criterion (BIC) tests), simulation studies, and investigations of model stability by bootstrap resampling. These issues are considered in great detail in Hilbe (2009) and Royston and Sauerbrei (2008).

[0009] Multivariable binary regression analysis enables estimation of individualized probabilities of one of mortality and survivorship. An individualized probability of one of mortality and survivorship is denoted here by π(Y_ij) for one of π(Y_ij) = π(M_ij) and π(Y_ij) = π(Sij), wherein: π(Mij) denotes the probability of mortality of individual i at situation j, and π(S_ij) denotes the probability of survivorship of individual i at situation j. Estimation of π(Y_ij) for one of π(Y_ij) = π(M_ij) and π(Y_ij) = π(S_ij) utilizes the following: said data on variable Yij and variables Xvij, said model ηij = β0 + βvXvij +… + βWXWij, said binary link function B(ηij), cumulative distribution function F(ηij), and π(Yij) = F(ηij) wherein F(ηij) denotes the cumulative distribution function of η_ij, and wherein function F(η_ij) corresponds to function B(ηij). Respective cumulative distribution functions F(ηij) that correspond to respective popular binary link functions B(ηij) are denotable as follows: the logit binary link function B(ηij) = ln{ηij/(1 - ηij)} corresponds to a cumulative distribution function F(ηij)= 1/{1+ exp(- ηij)} = {exp(ηij)}/{1+ exp(ηij)}, the probit binary link function B(ηij) = Φ^-1(ηij) corresponds to a cumulative distribution function F(η_ij)= Φ(η_ij), the complementary log-log binary link function B(ηij) = ln{-ln(1 - ηij)} corresponds to a cumulative distribution function F(ηij)= 1 - exp{-exp(ηij)}, and the cauchit binary link function B(ηij) = tan{π(ηij)– π/2} corresponds to a cumulative distribution function F(η_ij)= (1π){tan^-1(η_ij) + π/2}. Therefore, for example, if a multivariable binary regression analysis is analyzed with a logit binary link function B(η_ij), then the probability of mortality of individual i at situation j that is denoted with π(Mij) is calculated with π(Mij)= F(ηij)= 1/{1+ exp(-ηij)} = {exp(ηij)}/{1+ exp(ηij)}, however, if a multivariable binary regression analysis is analyzed with a complementary log-log binary link function B(ηij), then the probability of mortality of individual i at situation j that is denoted with π(Mij) is calculated with π(Mij)= F(ηij)= 1 - exp{-exp(ηij)}. Less popular cumulative distribution functions that respectively correspond to less popular binary link functions are also available. As noted, Gündüz and Fokoué (2015) provide detailed consideration of binary link functions and cumulative distribution functions.

[0001] On 1/15/2014, the Inventor of the invention in this Application published a scientific article entitled "Lifespan and aggregate size variables in specifications of mortality or survivorship." This scientific article was published in PLoS ONE 9(1):e84156 and is cited here as“Epelbaum (2014).” Epelbaum (2014) is hereby incorporated herein by reference in its entirety.

[0002] Epelbaum (2014) introduces and makes public the invention of multivariable binary regression analyses of mortality or survivorship that include and distinguish independent variables that denote lifespan. Most conventionally, an individual’s lifespan refers to the total time span of an individual’s existence. Lifespan and age are distinct phenomena; an individual’s age is a contemporary measurement of an individual’s time of existence; age is conventionally calculated by A_iq = t_iq– t_i0, whereas lifespan is conventionally calculated by L_iq = L_i = t_iz– t_i0, wherein A_iq denotes the age of a natural or artificial individual i at time t_q, Liq denotes lifespan of a natural or artificial individual i at time tq, z≥ q, ti0 denotes the time of this individual’s initiation of existence, tiq denotes the current time of this individual’s existence, and t_iz denotes the time of this individual’s cessation of existence, such that A_iq varies in times ti0:tiz whereas Li denotes this individual’s constant length of time of existence at times ti0:tiz. The time of birth typically indicates time ti0, and the time of death typically indicates time tiz, but these typical notions of time of birth and time of death as respective limits of age or lifespan do not apply to all kinds of individuals.

[0003] The consideration of lifespan in terms of an individual’s constant length of time of existence at times ti0:tiz is related to– but differs from– the consideration of the lifespan in terms of the lifespan aggregate. The lifespan aggregate includes all the individuals that are identically characterized with respect to lifespan and every other condition in a data set. The individuals that are included in a lifespan aggregate begin their existence in coexistence at the beginning of a specific lifespan, they coexist through said lifespan, and they cease to exist and cease to coexist at the conclusion of this lifespan. Therefore, a lifespan aggregate’s size, composition, and other characteristics (e.g., beginning or ending time, density) are constant from the time of the initiation of existence of this aggregate (i.e., the initiation of existence of all the individuals that are included in this aggregate) to the time of this aggregate’s cessation of existence (i.e., the cessation of existence of all the individuals that are included in this aggregate). In some cases, the lifespan aggregate consists only of a respective single natural or artificial individual, but in many cases the lifespan aggregate consists of more than one individual. An individual’s lifespan aggregate is included in every contemporary aggregate of this individual. The lifespan aggregate differs from the contemporary aggregate. The contemporary aggregate includes all the individuals that are identically characterized with respect to every condition in a data set at a specific point (e.g., a point of cessation or continuation of existence), except that these individuals share or do not share an identical lifespan. These considerations indicate that the contemporary aggregate’s size, composition, or other characteristics (e.g., beginning or ending time, density) are time-specific and changeable through time. Thus, for example, age, lifespan (as time of existence since birth), lifespan aggregate size, and contemporary aggregate size are distinct characteristics that characterize every natural or artificial individual at every point of survivorship (i.e., continuation of existence) or mortality (i.e., cessation of existence).

Further details on age, lifespan, lifespan aggregate, and contemporary aggregate– and consideration of previous research on these and related phenomena– are available in Epelbaum (2014).

[0004] As noted, multivariable binary regression analysis utilizes η_ij = β₀ + β_vX_vij +… + βWXWij wherein: i = 1:N (indicating that i are sequential positive integers 1 through N); i denotes an individual; N denotes the total number of individuals i in said data; j = 1:J (indicating that j are sequential positive integers 1 through J); j denotes a situation of individual i in reference to said variables Xvij and Yij; J denotes the total number of situations of an individual i in said data (allowing distinct J for distinct individuals i); v = 1:W

(indicating that v are sequential positive integers 1 through W); W≥ 2; v denotes an index of variables Xvij and coefficients βv; W denotes the total number of variables Xvij of individual i at situation j and total number of coefficients β_v; and X_vij denotes a variable X_v of individual i at situation j.

[0005] Inclusion and distinction of lifespan as an independent variable in multivariable binary regression analyses of mortality or survivorship means that Xvij = Tq(Krij), K1ij = Aij, and K_2ij = L_ij in η_ij = β₀ + β_vX_vij +… + β_WX_Wij for the multivariable binary regression analysis of mortality or survivorship (indicating that one of at least two variables Krij directly denotes the age of individual i at situation j, also indicating that another of said at least two variables K_rij directly denotes the lifespan of individual i at situation j, further indicating that Xvij is a transformation Tq of Krij that directly denotes a specific phenomenon, and further indicating that Xvij indirectly denotes said specific phenomenon); wherein: r = 1:R

(indicating that r are sequential positive integers 1 through R); R≥ 2; W≥ R; r denotes an index of variables K_rij; K_r denotes a variable that directly denotes a distinct phenomenon, and R respective variables Kr respectively directly denote R distinct phenomena; Krij denotes a variable Kr of individual i at situation j; T denotes a transformation function (allowing identity transformation); q = 1:Q (indicating that q are sequential positive integers 1 through Q; Q denotes the total number of transformations Tq(Krij) for a specific Krij of individual i at situation j (allowing distinct Q for distinct variables Krij of individual i at situation j); R denotes the total number of variables K_rij of individual i at situation j; R also denotes the total number of phenomena denoted by variables K_rij of individual i at situation j; A directly denotes age; and L directly denotes lifespan.

[0006] Most previous research on lifespan focuses on lifespan as an explanandum, explicandum, left-hand side, outcome, dependent, or response variable that is caused, determined, explained, explicated, or predicted; such focus on lifespan is illustrated, for example, in Patent US 7,794,957 B2 (Kenyon et al.2010). The focus on lifespan as an explanandum, explicandum, left-hand side, outcome, dependent, or response variable that is caused, determined, explained, explicated, or predicted may have provided one possible motivation for the absence, exclusion, or omission of an explicit right-hand side independent variable that denotes lifespan and is distinct from another independent variable that denotes age in individualized multivariable binary regression analyses of mortality or survivorship.

[0007] Inclusions of explicit distinct independent variables that respectively denote distinct age, lifespan, lifespan aggregate, and contemporary aggregate variables offer considerable advantages in multivariable binary regression analyses of mortality or survivorship. The main advantages are that these inclusions and distinctions could increase explanatory and predictive efficacies and powers, deepen insights and understanding, and widen the scope of respective analyses - and corresponding conceptions, measurements, estimations, and descriptions - of mortality or survivorship.

[0008] Absence, exclusion, or omission of independent variables that denote lifespan in multivariable binary regression analyses of mortality or survivorship could lead to omitted variables bias, unobserved heterogeneity bias, and errors of confusion (e.g., age is often erroneously confounded with lifespan; and population size or density are often erroneously confounded with corresponding distinct lifespan aggregates and contemporary aggregates). Absence, exclusion, or omission of independent variables that denote lifespan aggregates in multivariable binary regression analyses of mortality or survivorship could also be disadvantageous and lead to diverse biases. These considerations reveal that absence, exclusion, or omission of independent variables that distinguish and denote lifespan could be disadvantageous and could lead to diverse biases or errors. Inclusion of explicitly distinct independent variables that respectively denote explicitly distinct age, lifespan, lifespan aggregate, and contemporary aggregate variables could reduce or eliminate omitted variables bias, unobserved heterogeneity bias, or confusion errors in multivariable binary regression analyses of mortality or survivorship.

[0009] The foregoing considerations show that shortcomings of the prior art could be overcome - and diverse advantages could be gained - by including explicitly distinct independent variables that respectively denote explicitly distinct age, lifespan, lifespan aggregate, and contemporary aggregate variables in multivariable binary regression models of mortality or survivorship .

[0010] Epelbaum (2014) presents the first published inclusion and distinction of at least one independent variable that denotes and distinguishes lifespan in multivariable binary regression analyses of mortality or survivorship; Epelbaum (2014) distinguishes between independent variables that respectively denote distinct age and lifespan in multivariable binary regression analyses of mortality or survivorship. Other previous multivariable binary regression analyses of mortality or survivorship did not consider independent variables that distinctly denote lifespan; said other previous investigations also failed to distinguish between independent variables that respectively denote distinct age and lifespan in multivariable binary regression analyses of mortality or survivorship. U.S. Patent US 8,417,541 B1 (cited here as Kramer 2013)) is one of many examples of the absence, exclusion, or omission of independent variables that denote lifespan in multivariable binary regression analyses of mortality or survivorship. Kramer (2103) is also one of many examples of the absence, exclusion, or omission of independent variables that distinguish between independent variables that denote respective distinct age and lifespan in

multivariable binary regression analyses of mortality or survivorship.

[0011] Epelbaum (2014) also presents the first published inclusion and distinction of an independent variable that denotes a lifespan aggregate independent variable in multivariable binary regression analyses of mortality or survivorship; Epelbaum (2014) distinguishes between independent variables that respectively denote distinct lifespan aggregates and contemporary aggregates in multivariable binary regression analyses of mortality or survivorship. Other previous multivariable binary regression analyses of mortality or survivorship did not include– and did not distinguish– independent variables that denote lifespan aggregate; said other previous analyses also failed to distinguish between independent variables that respectively denote respective distinct lifespan aggregates and contemporary aggregates in multivariable binary regression analyses of mortality or survivorship. Kramer (2013) is one of many examples of the absence, exclusion, or omission of independent variables that denote the lifespan aggregate in multivariable binary regression analyses of mortality or survivorship. Kramer (2103) is also one of many examples of the absence, exclusion, or omission of independent variables that distinguish between independent variables that denote respective distinct lifespan aggregate and contemporary aggregate in multivariable binary regression analyses of mortality or survivorship.

[0012] As noted, an individualized probability of one of mortality and survivorship is denoted here by π(Y_ij) for one of π(Y_ij) = π(M_ij) and π(Y_ij) = π(S_ij), wherein: π(M_ij) denotes the probability of mortality of individual i at situation j, and π(Sij) denotes the probability of survivorship of individual i at situation j. As also noted, estimation of π(Yij) for one of π(Yij) = π(Mij) and π(Yij) = π(Sij) utilizes the following: said data on variable Yij and variables Xvij, said model ηij = β0 + βvXvij +… + βWXWij, said binary link function B(ηij), cumulative distribution function F(η_ij), and π(Y_ij) = F(η_ij) wherein F(η_ij) denotes the cumulative distribution function of ηij, and wherein function F(ηij) corresponds to function B(ηij).

Epelbaum (2014) publishes the first presentation of individual probabilities of one of mortality and survivorship– i.e., probabilities that are denoted by π(Y_ij) for one of π(Y_ij) = π(M_ij) and π(Y_ij) = π(S_ij)– that are based upon multivariable binary regression analyses that include and distinguish independent variables that denote lifespan length and lifespan aggregate size. However, Epelbaum (2014) does not consider phenomenon-specific probabilities of mortality or survivorship. DISCLOSURE OF THE INVENTION

[0013] Methods, systems, and computer program products are hereby claimed for including and distinguishing lifespan in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship.

[0014] As noted,“phenomenon-specific probabilities” refer to probabilities that are specific to a specific phenomenon (e.g., age, size, sex). For example, age-specific probabilities of mortality refer to probabilities of mortality at diverse levels of age, size-specific probabilities of mortality refer to probabilities of mortality at diverse levels of size, sex-specific probabilities of mortality refer to probabilities of mortality for diverse categories of sex.

[0015] An individualized phenomenon-specific– e.g., an age-specific, a size-specific, a sex- specific– probability of one of mortality and survivorship is denoted here by π(YijZ*) for one of π(Y_ijZ*) = π(M_ijZ*) and π(Y_ijZ*) = π(S_ijZ*), wherein Z denotes the specific phenomenon (e.g., age, size, sex). An averaged phenomenon-specific probability of one of mortality and survivorship is denoted here by π(YZ*) for one of π(YZ*) = π(MZ*) and π(YZ*) = π(SZ*).

[0016] Estimation of at least one of π(YijZ*) and π(YZ*) utilizes ηijZ* = β0 + ΣβvZ* +

ΣβvXvij~Z, estimation of π(YijZ*) also utilizes π(YijZ*) = F(ηijZ*), and estimation of π(YZ*) also utilizes at least one of π(YZ*) = F{average(ηijZ*)} and π(YZ*) =

wherein: π(Y_ijZ*) denotes one of π(Y_ijZ*) = π(M_ijZ*) and π(Y_ijZ*) = π(S_ijZ*); π(Y_Z*) denotes one of π(Y_Z*) = π(M_Z*) and π(Y_Z*) = π(S_Z*); Z denotes a specifically selected variable K_r; said variable Z and said specifically selected variable Kr directly denote a specifically selected phenomenon; Z* denotes a specifically selected value of variable Z; π(M_ijZ*) denotes an individualized phenomenon-specific probability of mortality of individual i at situation j and at a specifically selected value Z* of variable Z; π(SijZ*) denotes an individualized phenomenon- specific probability of survivorship of individual i at situation j and at a specifically selected value Z* of variable Z; π(MZ*) denotes an averaged specified probability of mortality at a specifically selected value Z* of variable Z; π(SZ*) denotes an averaged specified probability of survivorship at a specifically selected value Z* of variable Z; coefficients β in η_ijZ* = β₀ + ΣβvZ* + ΣβvXvij~Z are coefficients β from ηij = β0 + βvXvij +… + βWXWij, βvZ* denotes a replacement of a βvXvij in said model ηij = β0 + βvXvij +… + βWXWij, wherein variable Kr in X_vij = T_q(K_rij) is said specific variable K_r that is denoted by Z, and wherein X_vij = T_q(Z_ij) for said replaced βvXvij; ΣβvZ* denotes the sum of all respective βvZ* replacements of respective βvXvij in said model ηij = β0 + βvXvij +… + βWXWij; Xvij~Z denotes an Xvij wherein variable Kr in Xvij = Tq(Krij) is not said specific variable Kr that is denoted by Z; ΣβvXvij~Z denotes the sum of all βvXvij~Z in said model ηij = β0 + βvXvij +… + βWXWij; F(∙) denotes a cumulative distribution function of (∙); said cumulative distribution function F(∙) corresponds to said binary link function B(η_ij); average(η_ijZ*) denotes an average of at least two η_ijZ* at a specifically selected value Z* of variable Z; and average{F(η_ijZ*)} denotes an average of at least two F(ηijZ*) at a specifically selected value Z* of variable Z.

[0017] The invention herein focuses upon the following phenomenon-specific probabilities of mortality and survivorship: π(Y_ijZ), π(Y_Z), π(MS_ijZ), π(MS_Z), π(MS_ijZ+), and π(MS_Z+).

π(YijZ) denotes one of π(YijZ) = π(MijZ) and π(YijZ) = π(SijZ); π(YZ) denotes one of π(YZ) = π(MZ) and π(YZ) = π(SZ); π(MijZ) denotes at least two π(MijZ*), wherein said at least two π(M_ijZ*) denote respective individualized phenomenon-specific probabilities of mortality of individual i at situation j and at at least two specifically selected Z* values of variable Z; π(SijZ) denotes at least two π(SijZ*), wherein said at least two π(SijZ*) denote respective individualized phenomenon-specific probabilities of survivorship of individual i at situation j and at at least two specifically selected Z* values of variable Z; π(M_Z) denotes at least two π(MZ*), wherein said at least two π(MZ*) denote respective averaged phenomenon-specific probabilities of mortality at at least two specifically selected Z* values of variable Z; π(SZ) denotes at least two π(S_Z*), wherein said at least two π(S_Z*) denote respective averaged phenomenon-specific probabilities of survivorship at at least two specifically selected Z* values of variable Z; π(MSijZ) denotes the combination of corresponding π(MijZ) and π(SijZ) at at least two specifically selected Z* values of variable Z; π(MSZ) denotes the combination of corresponding π(MZ) and π(SZ) at at least two specifically selected Z* values of variable Z; π(MS_ijZ+) denotes the combination of at least two corresponding π(MS_ijZ) at respective at least two specifically selected Z* values of respective at least two variables Z; and π(MSZ+) denotes the combination of at least two corresponding π(MSZ) at respective at least two specifically selected Z* values of respective at least two variables Z.

[0018] The invention herein claims that estimates of π(Y_ijZ), π(Y_Z), π(MS_ijZ), π(MS_Z), π(MSijZ+), and π(MSZ+) utilize ηij = β0 + βvXvij +… + βWXWij wherein Xvij = Tq(Krij), K1ij = Aij, K2ij = Lij, A directly denotes age, and L directly denotes lifespan (indicating that one of at least two variables K_rij directly denotes the age of individual i at situation j, also indicating that another of said at least two variables Krij directly denotes the lifespan of individual i at situation j, further indicating that Xvij is a transformation Tq of Krij that directly denotes a specific phenomenon, and further indicating that X_vij indirectly denotes said specific phenomenon).

[0019] As noted, the invention herein claims methods, systems, and computer program products for including and distinguishing lifespan in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship. Therefore, inclusions of explicit distinct independent variables that respectively denote distinct age, lifespan, lifespan aggregate, and contemporary aggregate variables offer considerable advantages in the estimation of phenomenon-specific probabilities of one of mortality and survivorship. The main advantages are that these inclusions and distinctions could increase explanatory and predictive efficacies and powers, deepen insights and understanding, and widen the scope of respective analyses– and corresponding conceptions, measurements, and descriptions– of estimations of phenomenon-specific probabilities of mortality or survivorship.

[0020] Absence, exclusion, or omission of independent variables that denote lifespan in the estimation of phenomenon-specific probabilities of one of mortality and survivorship could lead to omitted variables bias, unobserved heterogeneity bias, and errors of confusion (e.g., age is often erroneously confounded with lifespan; and population size or density are often erroneously confounded with corresponding distinct lifespan aggregates and contemporary aggregates). These considerations reveal that absence, exclusion, or omission of

independent variables that distinguish and denote lifespan could be disadvantageous and could lead to diverse biases or errors. Moreover, inclusion of explicitly distinct independent variables that respectively denote explicitly distinct age, lifespan, lifespan aggregate, and contemporary aggregate variables could reduce or eliminate omitted variables bias, unobserved heterogeneity bias, or confusion errors in the estimation of phenomenon-specific probabilities of one of mortality and survivorship.

[0021] The foregoing considerations show that shortcomings of the prior art could be overcome - and diverse advantages could be gained - by including explicitly distinct independent variables that respectively denote explicitly distinct age, lifespan, lifespan aggregate, and contemporary aggregate variables in the estimation of phenomenon-specific probabilities of one of mortality and survivorship.

[0022] Shortcomings of the prior art could be overcome - and diverse advantages could be gained - through the provision of methods, systems, and computer program products for including explicitly distinct independent variables that respectively denote explicitly distinct lifespan variables in the estimation of phenomenon-specific probabilities of one of mortality and survivorship. Such methods, systems, and computer program products could be usefully utilized in diverse computing environments.

[0023] A computing environment provides an efficient and fast environment for processing regression analyses of diverse data, including very large data. In a computing environment, a plurality of local or remote devices are connected through at least one network. In this environment, regression analyses are conducted employing one or more processors of the computing environment to process the following: instructions for the regression analysis, specifications of models and procedures, and respective data. Regression analyses and other statistical applications can be conducted in diverse kinds of computing environments.

Diverse kinds of computing environments - including the cloud environment - for statistical applications are considered, for example, in Patent US 8,645,966 B2 (Andrade et al. 2014), Patent US 9,152,921 B2 (Chu et al. 2015), and Patent US 9,443, 194 B2 (Chu et al. 2016).

[0024] Features, embodiments, and aspects of the invention are described in detail herein and are considered to be a part of the claimed invention.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0025] One or more aspects of the present invention are particularly pointed out and distinctly claimed in the appended claims. The foregoing and other objects, features, and advantages of the invention are apparent from the detailed description taken in conjunction with the accompanying drawings in which:

[0026] FIG.1 depicts features of relationships among major aspects of the invention, in accordance with diverse aspects of the present invention.

[0027] FIG.2 depicts a plot of individualized lifespan-specific probabilities of mortality by lifespan and a plot of averaged lifespan-specific probabilities of mortality by lifespan, in accordance with an aspect of the present invention.

[0028] FIG.3 depicts a plot of individualized age-specific probabilities of mortality by age and a plot of averaged age-specific probabilities of mortality by age, in accordance with an aspect of the present invention.

[0029] FIG.4 depicts a plot of individualized lifespan-specific probabilities of mortality and survivorship by lifespan, in accordance with an aspect of the present invention.

[0030] FIG.5 depicts a plot of individualized age-specific probabilities of mortality and survivorship by age, in accordance with an aspect of the present invention.

[0031] FIG.6 depicts a plot of one kind of averaged lifespan-specific probabilities of mortality and survivorship by lifespan, in accordance with an aspect of the present invention.

[0032] FIG.7 depicts a plot of one kind of averaged age-specific probabilities of mortality and survivorship by age, in accordance with an aspect of the present invention.

[0033] FIG.8 depicts a plot of another kind of averaged lifespan-specific probabilities of mortality and survivorship by lifespan, in accordance with an aspect of the present invention.

[0034] FIG.9 depicts a plot of another kind of averaged age-specific probabilities of mortality and survivorship by age, in accordance with an aspect of the present invention.

[0035] FIG.10 depicts a plot of individualized lifespan-specific and age-specific probabilities of mortality and survivorship by time of age and lifespan, in accordance with an aspect of the present invention.

[0036] FIG.11 depicts a plot of one kind of averaged lifespan-specific and age-specific probabilities of mortality and survivorship by time of age and lifespan, in accordance with an aspect of the present invention. [0037] FIG. 12 depicts a plot of another kind of averaged lifespan-specific and age-specific probabilities of mortality and survivorship by time of age and lifespan, in accordance with an aspect of the present invention.

DETAILED DESCRIPTION OF EMBODFMENTS OF THE INVENTION

[0038] Multivariable regression analyses of mortality or survivorship employ a computing environment as schematically described in FIG. 1, in accordance with diverse aspects 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, Texas; 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 Table I, in accordance with an aspect of the present invention. TABLE I illustrates the setting of part of a computing environment.

[0039]

TABLE I

[0040] As shown in FIG. 1, the computing environment of the embodiments herein also includes respective instructions for a regression model. In the embodiments herein, 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 FIG. 1, in accordance with major aspects of the present invention. It is, however, important to note - as already noted herein - that there are diverse kinds of computing environments for regression analyses as considered, for example, in Patent US 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, it is also important to note - as already noted herein - that there are diverse aspects of multivariable binary regression analysis; as considered, for example, in Patent US 8,417,541 B l (Kramer 2013), Epelbaum (2014), Hilbe (2009), Gunduz and Fokoue (2015), and Royston and Sauerbrei (2008). These diverse kinds of computing environments and diverse aspects of multivariable binary regression analysis can incorporate and use one or more aspects of the present invention.

[0041] The following paragraphs present embodiments of the invention; these embodiments include and distinguish age, lifespan, gender, lifespan aggregate, and contemporary aggregate in multivariable binary regression analyses of humans' mortality and survivorship.

[0042] In embodiments of the invention that are presented here, the data on mortality or survivorship of samples of humans are compiled from the deaths lxl and exposures lxl tables (last modified on 14 July, 2010) from the Human Mortality Database, known as HMD. The HMD 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 HMD 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'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.

[0043] TABLE II depicts variables, denotations, and selected data employed in

multivariable binary regression analyses of humans’ mortality and survivorship. The first row of TABLE II presents a variable’s denotation, as this variable’s denotation is denoted in the regression model. The first row of TABLE II presents the variable name as it appears in the regression model. The second row of TABLE II presents a variable’s denotation, as this variable’s denotation is denoted in the computer instructions. Each subsequent row in TABLE II depicts one data record. TABLE II illustrates 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’s death or survival (depicted in columns Mij and Sij, respectively denoting mortality or survivorship of individual i at situation j in TABLE II), this individual’s sex (depicted in column G_ij, wherein G_ij = 1 denotes being female, and G_ij = 0 denotes being male in TABLE II), this individual’s age during this situation (during the mid-year, depicted in column Aij in TABLE II), this individual’s lifespan (depicted in column Lij in TABLE II), this individual’s historic context (depicted in column Hij, denoting a specific year, starting at year 1760 which is coded as year 0.5 in TABLE II), the individual’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 TABLE II), the individuals’ 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 TABLE II). 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’s lifespan aggregate size (i.e., as depicted in column Λij in TABLE II) 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.

[0044] TABLE II

[0045] 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 Aij, Lij, Cij, Λij, Hij, and Gij 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 TABLE II. Transformations of these denoted variables are included as independent variables Xvij in multivariable binary regression model η_ij = β₀ + β_vX_vij +… + β_WX_Wij for one of Y_ij = M_ij and Y_ij = S_ij of humans in Sweden 1760– 2008.

[0046] The analysis in the present embodiment employs forward selection methods in iterative multivariable binary regression analyses of humans’ mortality or survivorship to select transformations of each of denoted variables Aij, Lij, Cij, Λij, Hij, and Gij 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, the analyst conducted iterative multivariable binary regression analyses of humans’ mortality or survivorship. The initial iterating multivariable binary regression analyses utilized diverse input models ηij = β0 + β1X1ij + β2X2ij with respective logit, probit, and complementary log-log binary link functions– as well as X₁ = Gij and diverse values of power coefficient p in X2 = (Aij)^p wherein Gij denotes sex and Aij denotes age– for one of Yij = Mij and Yij = Sij; 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 X2ij = (Aij)^p, the analyst proceeded to conduct further iterative multivariable binary regression analyses with input model η_ij = β₀ + β₁X_1ij + β₂X_2ij + β₃X_3ij and logit, probit, and complementary log-log binary link functions for one of Yij = Mij and Yij = Sij to find optimal AIC and BIC values for diverse values of power coefficient p in X3ij = (Lij)^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

= (Cij)^p, X5ij = (Λij)^p, and X6ij = (Hij)^p in respective input models until reaching an optimal best-fitting first-degree polynomial input model ηij = β0 + β1X1ij + β2X2ij + β3X3ij + β4X4ij + β5X5ij + β6X6ij. 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 X7ij = {(Aij)^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 η_ij = β₀ + β₁X_1ij + β2X2ij + β3X3ij + β4X4ij + β5X5ij + β6X6ij + β7X7ij + β8X8ij + β9X9ij + β10X10ij + β11X11ij.

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 X12ij = {(Aij)^p}³, X12ij = {(Lij)^p}³, X12ij = {(Cij)^p}³, and X12ij = {(Λij)^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 ηij = ^^0 + β1X1ij + β2X2ij + β3X3ij + β4X4ij + β5X5ij + β6X6ij + β7X7ij + β8X8ij + β9X9ij + β10X10ij + β11X11ij + β12X12ij and a logit binary link function that incorporate an identity transformation of denoted binary variable Gij 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 Hij. Computer instructions for said input model of humans’ mortality are depicted in TABLE III.

[0047] TABLE III illustrates the computer instruction for calling the data that are used in the analysis; the computer instruction also name and specify the variables, and provide the instructions for the multivariable regression analysis. In iterative analyses that are presented herein, 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 X1 = ln(Aij) instead of X1 = (Aij)⁰), and allowing identity transformation (i.e., when p = 1, e.g., using X1 = (Aij)¹ = Aij), 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).

[0048] In the best-fitting multivariable binary regression analysis of human’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 Yij = Mij, X1ij = Gij, X2ij = (Aij)^0.16, X3ij = (Lij)^0.88, X4ij = (Cij)^0.75, X5ij = (Λij)^0.30, X6ij = (Hij)^1.41, X7ij = {(Aij)^0.16}², X8ij = {(Lij)^0.88}², X9ij = {(Cij)^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 TABLE II, i denotes an individual, j is a consecutive number of the year of life of this individual, Mij = 1 when the individual is dead, and Mij = 0 when the individual is not dead, Aij denotes the individual’s age (in years) at situation j, Lij denotes the individual’s lifespan (in years) at situation j, C_ij denotes the individual’s contemporary aggregate size at situation j, Λij denotes the individual’s lifespan aggregate size at situation j, Hij denotes the individual’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’ mortality the analyst employed an input model ηij = β0 + β1X1ij + β2X2ij + β3X3ij + β4X4ij +β5X5ij + β6X6ij + β₇X_7ij + β₈X_8ij + β₉X_9ij + β₁₀X_10ij + β₁₁X_11ij +β₁₂X_12ij with a logit binary link function B(η_ij) = ln{η_ij/(1 - η_ij)}. Computer instructions for said input model and said logit binary link function are shown in TABLE III.

TABLE III

[0049] A computer output of the execution of said estimated model is depicted in TABLE IV.

TABLE IV

[0050] TABLE IV shows that a random-effects logistic regression analysis of humans’ mortality has been conducted on 188,087 observations in 3,653 groups, where the variable “smalln” that is depicted in TABLE III is the grouping variable, and where there were at least 1 observation per group, a maximum of 110 observations per group, and an average of 51.5 observations per group. TABLE IV shows that random effects in this analysis are assumed to be Gaussian. Most importantly, TABLE IV shows the estimated regression coefficients for this analysis, with respective standard errors and significance coefficients, as well as the log likelihood and the likelihood test for this analysis.

[0051] Utilizing the estimated regression coefficients that are depicted in TABLE IV, and utilizing respective denoting variables that correspond to variables X_vij, the best-fitting estimated model is specified with ηij = 511.78– 1074.55(Aij^0.16) + 546.12 (Aij^0.16)²– 17.12(Lij^0.88) + 0.101(Lij^0.88)² + 0.006 (Cij^0.75)– (4.39e-7) (Cij^0.75)² + 6.19 (Λij^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(ηij) = ln{ηij/(1 - ηij)}. As noted, an individualized probability of mortality refers to the probability of an individual’s mortality; π(Mij) denotes here the probability of mortality of individual i at situation j, π(M_ij) is calculated here with π(M_ij)= F(η_ij).

[0052] Utilizing said best-fitting estimated model of humans’ mortality in Sweden 1760- 2008, the individualized probability of mortality π(Mij) is calculated with π(Mij)= F(ηij)= 1/{1+ exp(-η_ij)} = {exp(η_ij)}/{1+ exp(η_ij)} wherein cumulative distribution function F(η_ij) corresponds to said best-fitting logit binary link function B(ηij) = ln{ηij/(1 - ηij)}.

[0053] Utilizing said model and said cumulative distribution function F(ηij), TABLE V shows selected η_ij values and selected π(M_ij) values. The selected η_ij values and selected π(M_ij) values in TABLE V correspond to the data that are shown in TABLE II, these values also correspond to the computer program that is shown in TABLE III, and these values further correspond to the results that are shown in TABLE IV.

TABLE V

[0054] As noted, examples of a phenomenon-specific probabilities 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(MijZ*), wherein Z denotes a specific phenomenon, and Z* denotes a specific selected value of this specific phenomeon. As further noted, π(MijZ*) is estimated here by utilizing ηijZ* = β0 + ΣβvZ* + ΣβvXvij~Z and π(MijZ*) = F(HijZ*). Therefore, utilizing L = Z wherein L denotes lifespan, π(MijL*) 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’ mortality in Sweden 1760-2008, η_ijL* = 511.78–

1074.55(Aij^0.16) + 546.12 (Aij^0.16)²– 17.12(L*^0.88) + 0.101(L*^0.88)² + 0.006 (Cij^0.75)– (4.39e-7) (Cij^0.75)² + 6.19 (Λij^0.30)– 0.35(Λij^0.30)²– 0.008(Hij^1.41) + (1.92e-6)(Hij^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 ηijL*, 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 η_ij has been estimated utilizing a logit binary link function; therefore, π(M_ijL*) = F(ηijL*) = 1/{1+ exp(-ηijL*)} = {exp(ηijL*)}/{1+ exp(ηijL*)} for individual i at situation j and at a specific lifespan level L*.

[0055] Utilizing said best-fitting model for η_ijL* and said π(M_ijL*) = F(η_ijL*), and utilizing the data from TABLE V, TABLE VI presents values of individualized lifespan-specific probabilities of mortality π(MijL*) of selected individuals i at selected situations j 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.

TABLE VI

[0056] As noted, a plot of π(M_ijZ) is a plot of at least two individualized specific probabilities of mortality π(MijZ*), wherein π(MijZ) denotes at least two probabilities π(MijZ*) (wherein said at least two probabilities π(MijZ*) 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’ mortality in Sweden 1760-2008, TABLE VI specifies π(MijL*) values of individual i = 1 at situation j = 1 in TABLE V and TABLE VI; this individual was a less than 1 year old Swedish male who died in 1760; TABLE V and TABLE VI indicate that there were an estimated 6,519 Swedish males who died that year at less than 1 year of age, further indicating that these males are included in the estimated 27,866 Swedish males who were less than 1 year old at that year.

[0057] The π(M_ijL*) data about individual i = 1 at situation j = 1 and at lifespans 0.5, 40, 60, and 90 in TABLE VI is specified in greater detail in TABLE VII. TABLE VII

[0058] TABLE VII adds π(M_ijL*) data at lifespans 1, 3, and 6 of this individual. TABLE VII shows individualized lifespan-specific probabilities of mortality or survivorship π(YijL*) and averaged lifespan-specific probabilities of mortality or survivorship π(YL*) of selected individuals at selected situations and at selected levels of lifespan. Based upon the π(M_ijL*) data about individual i = 1 at situation j = 1 in TABLE VII, panel A of FIG.2 shows a scatterplot and a corresponding supersmoothed Friedman line of π(MijL) by lifespan L, depicting the trajectory of the individualized lifespan-specific probabilities of mortality π(MijL) of the less than 1 year old male who died in Sweden in 1760, wherein π(MijL) denotes individualized lifespan-specific probabilities of mortality.

[0059] 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 η_ijZ* = β₀ + _Σβ_vZ* + _Σβ_vX_vij~Z and at least one of π(Y_Z*) = F{average(η_ijZ*)} and π(Y_Z*) =

average{F(ηijZ*)}. TABLE VII illustrates π(YL*) of each of π(ML*) = π(YL*) and π(SL*) = π(Y_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 TABLE VI.

[0060] 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(MZ*) denote respective averaged specific probabilities of mortality at at least two specifically selected Z* values of variable Z). Utilizing the respective π(ML*) = average{F(η_ijL*)} data in TABLE VII, panel B of FIG.2 illustrates a scatterplot and a supersmoothed Friedman lineplot of averaged lifespan-specific probabilities of mortality that are denoted by π(ML), wherein π(ML) denotes more than one selected averaged lifespan- specific probabilities of mortality π(M_L*). Similarly, utilizing the respective π(M_L*) = F{average(ηijL*)} data in TABLE VII, panel C of FIG.2 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*).

[0061] The individualized age-specific probability of mortality of individual i at situation j and at a specific level of age is estimated here utilizing η_ijA* = β₀ + _Σβ_vA* + _Σβ_vX_vij~A, wherein A denotes age and wherein A* denotes the specific level of age, and further utilizing π(MijA*) = F(HijA*), wherein π(MijA*) 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’ mortality in Sweden 1760-2008, η_ijA* = 511.78 – 1074.55(A*^0.16) + 546.12 (A*^0.16)²– 17.12(Lij ^0.88) + 0.101(Lij ^0.88)² + 0.006 (Cij^0.75)– (4.39e-7) (Cij^0.75)² + 6.19 (Λij^0.30)– 0.35(Λij^0.30)²– 0.008(Hij^1.41) + (1.92e-6)(Hij^1.41)²– (7.97e- 10)(H_ij ^1.41)³– 1.13(G_ij) and π(M_ijA*) = F(η_ijA*) = 1/{1+ exp(-η_ijA*)} = {exp(η_ijA*)}/{1+ exp(η_ijA)} for individual i at situation j and at a specific age A*, TABLE VIII presents values of individualized age-specific probabilities of mortality π(MijA*) of selected individuals i at selected situations j and at the following specific levels of age A*: A* = 0.5 year (denoting less than 1 year), A* = 40 years, A* = 60 years, and A* = 90 years.

TABLE VIII

[0062] TABLE VIII illustrates that if age is set at 0.5 year (denoting less than 1 year) or 40, 60, or 90 years then the individualized age-specific probability of mortality π(MijA*) 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; TABLE VIII 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;

TABLE VIII illustrates that if age is set at less than 1 year then individualized age-specific probability of mortality π(MijA*) 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; TABLE VIII also illustrates that if age is set at 40, 60, or 90 years then individualized age- specific probability of mortality π(MijA*) 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.

[0063] TABLE IX shows individualized age-specific probabilities of mortality or survivorship π(Y_ijA*) and averaged age-specific probabilities of mortality or survivorship π(Y_A*) of selected individuals at selected situations and at selected levels of age A*. TABLE IX

[0064] TABLE IX illustrates π(YA*) of each of π(MA*) = π(YA*) and π(SA*) = π(YA*) at selected specific levels of age A*; these π(YA*) were calculated utilizing arithmetic means of the 19,394 events of death or survival in Sweden 1760-2008 that are shown in TABLE VIII. The π(MijA*) data about individual i = 1 at situation j = 1 and at ages 0.5, 40, 60, and 90 in TABLE VIII are specified in greater detail in TABLE IX. TABLE IX also adds π(MijA*) 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 TABLE IX, panel A in FIG.3 shows a scatterplot and a corresponding supersmoothed Friedman line of π(MijA) by age A, depicting the trajectory of the

individualized age-specific probabilities of mortality π(MijA) 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.

[0065] Utilizing the respective π(MA*) = average{F(ηijA*)} data in TABLE IX, panel B in FIG.3 illustrates a scatterplot and a supersmoothed Friedman lineplot of averaged age- specific probabilities of mortality that are denoted by π(MA), wherein π(MA) denotes more than one especially selected averaged age-specific probabilities of mortality π(MA*).

Similarly, utilizing the respective π(M_A*) = F{average(η_ijA*)} data in TABLE IX, panel C in FIG.3 illustrates a scatterplot and a supersmoothed Friedman lineplot of averaged age- specific probabilities of mortality that are denoted by π(MA), wherein π(MA) denotes more than one especially selected averaged age-specific probabilities of mortality π(MA*).

[0066] The invention is also applied here in multivariable binary regression analysis of humans’ survivorship. In the best-fitting multivariable binary regression analysis of human’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’ survivorship the analyst also employed an input model η_ij = β₀ + β₁X_1ij + β₂X_2ij + β₃X_3ij + β₄X_4ij +β₅X_5ij + β₆X_6ij + β₇X_7ij + β₈X_8ij + β₉X_9ij + β₁₀X_10ij + β₁₁X_11ij +β₁₂X_12ij with a logit binary link function B(ηij) = ln{ηij/(1 - ηij)}. The best-fitting multivariable binary regression analyses of humans’ survivorship also yielded a best-fitting estimated model ηij = ^^0 + β1X1ij + β2X2ij + β3X3ij + β4X4ij + β5X5ij + β6X6ij + β7X7ij + β8X8ij + β9X9ij + β10X10ij + β11X11ij + β12X12ij with an F(ηij)= 1/{1+ exp(-η_ij)} = {exp(η_ij)}/{1+ exp(η_ij)} cumulative distribution function corresponding to said logit binary link function B(η_ij) = ln{η_ij/(1 - η_ij)}. Utilizing corresponding best fitting estimated regression coefficients, and utilizing respective denoting variables that correspond to variables Xvij, the best-fitting estimated model for humans’ survivorship is specified with η_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(Cij^0.75) + (4.39e-7)(Cij^0.75)²– 6.19(Λij^0.30) + 0.35(Λij^0.30)² + 0.008(Hij^1.41)– (1.92e- 6)(Hij^1.41)² + (7.97e-10)(Hij^1.41) + 1.13(Gij) for which the probability of survivorship π(Sij) is calculated with π(S_ij)= F(η_ij)= 1/{1+ exp(-η_ij)} = {exp(η_ij)}/{1+ exp(η_ij)} corresponding to said logit binary link function B(η_ij) = ln{η_ij/(1 - η_ij)}.

[0067] As noted, π(MSijZ) denotes the combination of corresponding π(MijZ) and π(SijZ) 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 TABLE VII, FIG.4 shows π(MSijL) plots of the combination of corresponding π(MijL) and π(SijL) trajectories. Utilizing respective supersmoothed Friedman lineplots for respective π(MijA*) and π(SijA*) data from TABLE IX, FIG.5 shows π(MS_ijA) plots of the combination of corresponding π(M_ijA) and π(SijA) trajectories.

[0068] As noted, π(MSZ) denotes the combination of corresponding π(MZ) and π(SZ) at at least two specifically selected Z* values of variable Z. Utilizing respective supersmoothed Friedman lineplots for respective π(M_L*)= F{average(η_ijL*)} and π(S_L*)= F{average(η_ijL*)} data from TABLE VII, FIG.6 shows π(MSL) plots of the combination of corresponding π(ML) and π(SL) trajectories. Utilizing respective supersmoothed Friedman lineplots for respective π(M_A*)= F{average(η_ijA*)} mortality data and π(S_A*)= F{average(η_ijA*)} survivorship data from TABLE IX, FIG.7 shows π(MSA) plots of the combination of corresponding π(MA) and π(SA) trajectories.

[0069] Utilizing respective supersmoothed Friedman lineplots for respective π(M_L*)= F{average(η_ijL*)} and π(S_L*)= F{average(η_ijL*)} data from TABLE VII, FIG.8 shows π(MS_L) plots of the combination of corresponding π(ML) and π(SL) trajectories. Utilizing respective supersmoothed Friedman lineplots for respective π(M_A*)= average{F(η_ijA*)} mortality data and π(S_A*)= average{F(η_ijA*)} survivorship data from TABLE IX, FIG.9 shows π(MS_A) plots of the combination of corresponding π(MA) and π(SA) trajectories.

[0070] As noted, π(MSijZ+) denotes the combination of corresponding π(MSijZ) at at least two specifically selected Z* values of at least two variables Z). Utilizing respective

supersmoothed Friedman lineplots for respective π(MijL*) and π(SijL*) data from TABLE VII and FIG.4, and utilizing respective supersmoothed Friedman lineplots for respective π(MijA*) and π(SijA*) data from TABLE IX and FIG.5, FIG.10 shows π(MSijZ+) plots of the combination of corresponding π(MijL), π(SijL), π(MijA), and π(SijA) trajectories.

[0071] 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(ML*)= F{average(ηijL*)} and π(SL*)= F{average(ηijL*)} data from TABLE VII and FIG.6, and utilizing respective supersmoothed Friedman lineplots for respective π(M_A*)= F{average(η_ijA*)} mortality data and π(S_A*)= F{average(η_ijA*)} survivorship data from TABLE IX and FIG.7, FIG.11 shows π(MSZ+) plots of the combination of corresponding π(ML), π(SL), π(MA), and π(SA) trajectories.

[0072] Utilizing respective supersmoothed Friedman lineplots for respective π(M_L*)= F{average(ηijL*)} and π(SL*)= average{F(ηijL*)} data from TABLE VII and FIG.8, and utilizing respective supersmoothed Friedman lineplots for respective π(MA*)=

average{F(η_ijA*)} mortality data and π(S_A*)= average{F(η_ijA*)} survivorship data from TABLE IX and FIG.9, FIG.12 shows π(MSZ+) plots of the combination of corresponding π(ML), π(SL), π(MA), and π(SA) trajectories.

[0073] 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 TABLE IX presents respective π(MA*) = average{F(ηijA*)} and π(MA*) = F{average(ηijA*)} that are calculated utilizing respective arithmetic means, TABLE X presents corresponding respective π(M_A*) = average{F(η_ijA*)} and π(M_A*) =

F{average(ηijA*)} that are calculated utilizing respective medians. Respective plots of π(MA*) = average{F(ηijA*)} and π(MA*) = F{average(ηijA*)} can then be easily produced for the data that are presented in TABLE X.

TABLE X

APPLICABILITY, DISTINCTIVENESS, AND GENERALIZ ABILITY

[0074] The embodiments that are presented here focus 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. These best-fitting analyses yield the best-fitting models that are presented here and provide the foundation for the specifications and plots in the tables and plots herein. The specifications and plots in the tables and plots herein are restricted to the 19,394 events of death or survival in Sweden 1760-2008 that are shown in TABLE II, TABLE V and TABLE VI. Methods and procedures for the specifications and plots of all events of death or survival in Sweden 1760 - 2008 are the same as those that have been shown here in reference to the tables and plots herein.

[0075] Epelbaum (2014) shows best-fitting multivariable analyses and models for

Mediterranean fruit flies as well as humans. These analyses and models indicate that the methods, computer programs, and systems that are shown here are applicable to diverse kinds of individuals. [0076] As noted, mortality refers to cessation of existence, and survivorship refers to continuation of existence. That which exists is an entity, that which ceases to exist is an entity, and that which continues to exist is an entity. An entity can be simple or complex, natural or artificial, living or non-living. A particle, bubble, droplet, cell, virus, insect, human, dinosaur, plant, rock, lake, mountain, planet, celestial system, universe, city, sculpture, bicycle, airplane, basketball team, nation-state, language, book, or poem are some of many examples of an entity. The methods, systems, and computer program products that have been presented here are applicable to the cessation or continuation of existence of any entity, as will be apparent to those skilled in the art.

[0077] The data and analyses that are presented here include lifespan and lifespan aggregate size variables and distinguish here among age, lifespan, lifespan aggregate size, and contemporary aggregate size in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship. Previous investigations of mortality and survivorship did not include - and did not distinguish - independent variables that denote lifespan in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship.

[0078] The analyses that are presented here show that inclusions of lifespan and lifespan aggregate size variables - and distinctions among age, lifespan, lifespan aggregate size, and contemporary aggregate size variables - provide valuable explanatory insights and improve goodness-of-fit in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship.

[0079] The invention has been described herein in some detail by way of illustration and example for purposes of clarity of understanding. The examples describe and detail embodiments and aspects of the invention discussed above; the examples are intended to be purely exemplary of the invention; the examples should, therefore, not be considered to limit the invention in any way; the foregoing examples and descriptions are offered by way of illustration and not by way of limitation.

[0080] All publications, patent applications, and patents cited in this specification are herein incorporated by reference as if each of said publications, patent applications, or patents were specifically and individually indicated to be incorporated by reference. In particular, all publications cited herein are expressly incorporated herein by reference for the purpose of describing and disclosing methods, systems, and computer program products which might be used in connection with the invention. However, it will be readily apparent to those of ordinary skill in the arts in light of the teachings of this invention that certain changes and modifications may be made thereto without departing from the spirit or scope of the invention.

[0081] The appended claims present the spirit or scope of the invention. Every appended claim herein includes unique, unobvious, innovative, and new subject matter that has not yet been published.

[0082] The appended claims present methods, systems, and computer program products for including and distinguishing independent variables that denote lifespan in regression-based estimation of phenomenon-specific probabilities of mortality or survivorship. FIG.1 shows that the methods, systems, and computer program products involve inclusion and distinction of at least one independent variable denoting lifespan in at least one multivariable binary regression analysis of one of mortality and survivorship. FIG.1 also shows that results of said at least one regression analysis are utilized in the estimation of phenomenon-specific probabilities of one of mortality and survivorship. Furthermore, the appended claims clarify that said probabilities can also be specified or plotted.

REFERENCES

[0083] Andrade, H., B. Gedik, V. Kumar, and K.L. Wu.2014. Managing resource allocation and configuration of model building components of data analysis applications, Patent US 8,645,966 B2.

[0084] Chu, Yea J, Sler Han, Jing-Yun Shyr, and Jing Xu. 2016. Missing value imputation for predictive models, Patent US 9,443,194 B2.

[0085] Chu, Yea J, Dong Liang, and Jing-Yun Shyr.2015. Computing regression models, Patent US 9,152,921 B2.

[0086] Epelbaum, M.2014. Lifespan and aggregate size variables in specifications of mortality or survivorship. PLoS ONE 9(1):e84156.

[0087] Gündüz, Necla and Ernest Fokoué.2015.“On the predictive properties of binary link functions.” arXiv:1502.04742v1.

[0088] Hilbe, Joseph M.2009. Logistic Regression Models. CRC Press: Boca Raton. [0089] Kenyon, C, J. Apfeld, A. Dillin, D. Garigan, A.L.A. Hsu, J. Lehrer-Graiwer, and C. Murphy. 2010. Eukaryotic genes involved in adult lifespan regulation, Patent US 7, 794,957 B2.

[0090] Kramer, A. A. 2013. Multi-stage model for predicting probabilities of mortality in adult critically ill patients, Patent US 8,417,541 Bl.

[0091] Li, Dan, Xia Wang, Lizhen Lin, and Dipak K. Dey. 2016. Flexible link functions in nonparametric binary regression with Gaussian process priors. Biometrics 72(3): 707-719.

[0092] Royston, Patrick and Willi Sauerbrei. 2008. Multivariable model-building. Wiley: West Sussex, England.

Claims

CLAIMS What is claimed is:

1. A method for including and distinguishing independent variables that denote lifespan in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship, said method comprising:

at least one of obtaining and creating data, said data being about N individuals i at respective J situations j, said data including variable Y_ij and at least two variables Xvij;

conducting a multivariable binary regression analysis of said data, analyzing the relationship between a dependent variable Y_ij and independent variables X_vij for one of Yij = Mij and Yij = Sij, said analyzing utilizing ηij = β0 + βvXvij +… + βWXWij, binary link function B(ηij), and π(Yij) = F(ηij);

estimating at least one of π(Y_ijZ*) and π(Y_Z*), said estimating of at least one of

π(Y_ijZ*) and π(Y_Z*) utilizing η_ijZ* = β₀ + _Σβ_vZ* + _Σβ_vX_vij~Z, said estimating of π(YijZ*) also utilizing π(YijZ*) = F(ηijZ*), and said estimating of π(YZ*) also utilizing at least one of π(Y_Z*) = F{average(η_ijZ*)} and π(Y_Z*) =

average{F(ηijZ*)};

wherein at least one of the steps is carried out by a computer; and

wherein:

i = 1:N, indicating that i are sequential positive integers 1 through N;

i denotes an individual;

N denotes the total number of individuals i in said data;

j = 1:J, indicating that j are sequential positive integers 1 through J;

j denotes a situation of individual i in reference to said variables Xvij and Yij; J denotes the total number of situations of an individual i in said data, allowing distinct J for distinct individuals i;

M_ij denotes the mortality status of individual i at situation j;

Sij denotes the survivorship status of individual i at situation j;

Yij∈ {0,1}, indicating that Yij is a binary variable that adopts one of values 0 and 1 for one of Yij = Mij and Yij = Sij of individual i at situation j;

v = 1:W, indicating that v are sequential positive integers 1 through W;

W≥ 2; v denotes an index of each of the following: variables Xv, variables Xvij of individual i at situation j, and coefficients βv;

W denotes the following: total number of variables X_v, total number of variables Xvij of individual i at situation j, and total number of coefficients βv;

Xvij denotes a variable Xv of individual i at situation j;

r = 1:R, indicating that r are sequential positive integers 1 through R;

R≥ 2;

W≥ R;

r denotes an index of each of the following: variables Kr and variables Krij; K_r denotes a variable that directly denotes a distinct phenomenon, and R

respective variables Kr respectively directly denote R distinct phenomena; Krij denotes a variable Kr of individual i at situation j;

T denotes a transformation function, allowing identity transformation;

q = 1:Q, indicating that q are sequential positive integers 1 through Q;

Xv = Tq(Kr), indicating that Xv is a transformation Tq of variable Kr that directly denotes a specific phenomenon, further indicating that X_v indirectly denotes said specific phenomenon;

Xvij = Tq(Krij), indicating that Xvij is a transformation Tq of Krij that directly

denotes a specific phenomenon, further indicating that Xvij indirectly denotes said specific phenomenon;

Q denotes the total number of transformations Tq(Kr) for a specific Kr, and Q also denotes the total number of transformations Tq(Krij) for a specific Krij of individual i at situation j, allowing distinct Q for distinct variables K_r, and allowing distinct Q for distinct variables K_rij of individual i at situation j;

R denotes the following: the total number of variables K_r, the total number of variables K_rij of individual i at situation j, the total number of phenomena denoted by variables Kr, and the total number of phenomena denoted by variables Krij of individual i at situation j;

A directly denotes age;

L directly denotes lifespan; K1 = A and K2 = L, indicating that one of at least two variables Kr directly denotes age, and indicating that another of said at least two variables Kr directly denotes lifespan;

K1ij = Aij and K2ij = Lij, indicating that one of at least two variables Krij directly denotes the age of individual i at situation j, and indicating that another of said at least two variables K_rij directly denotes the lifespan of individual i at situation j;

β denotes a regression coefficient;

β0 denotes the regression coefficient for the intercept, allowing β0 to be any of the following: estimated, suppressed, and user-provided;

π(Yij) denotes one of π(Yij) = π(Mij) and π(Yij) = π(Sij);

π(Mij) denotes the probability of mortality of individual i at situation j;

π(S_ij) denotes the probability of survivorship of individual i at situation j;

ηij denotes ηij resulting from said multivariable binary regression analysis; Z denotes a specifically selected variable Kr;

said variable Z and said specifically selected variable K_r directly denote a

specifically selected phenomenon;

Z* denotes a specifically selected value of variable Z;

π(YijZ*) denotes one of π(YijZ*) = π(MijZ*) and π(YijZ*) = π(SijZ*);

π(Y_Z*) denotes one of π(Y_Z*) = π(M_Z*) and π(Y_Z*) = π(S_Z*);

π(MijZ*) denotes an individualized specific probability of mortality of individual i at situation j and at a specifically selected value Z* of variable Z;

π(S_ijZ*) denotes an individualized specific probability of survivorship of

individual i at situation j and at a specifically selected value Z* of variable Z;

π(M_Z*) denotes an averaged specified probability of mortality at a specifically selected value Z* of variable Z;

π(SZ*) denotes an averaged specified probability of survivorship at a

specifically selected value Z* of variable Z;

coefficients β in ηijZ* = β0 + ΣβvZ* + ΣβvXvij~Z are respective coefficients β resulting from said regression analysis; βvZ* denotes a replacement of a βvXvij in said model ηij = β0 + βvXvij +… + βWXWij, wherein variable Kr in Xvij = Tq(Krij) is said specific variable Kr that is denoted by Z, and wherein X_vij = T_q(Z_ij) for said replaced β_vX_vij;

Σβ_vZ* denotes the sum of all respective β_vZ* replacements of respective β_vX_vij in said model ηij = β0 + βvXvij +… + βWXWij;

X_vij~Z denotes an X_vij wherein variable K_r in X_vij = T_q(K_rij) is not said specific variable Kr that is denoted by Z;

Σβ_vX_vij~Z denotes the sum of all β_vX_vij~Z in said model η_ij = β₀ + β_vX_vij +… + β_WX_Wij;

F(∙) denotes a cumulative distribution function of (∙);

cumulative distribution function F(∙) corresponds to binary link function B(ηij); average(η_ijZ*) denotes an average of the respective η_ijZ* of at least two

individuals i at situation j and at a specifically selected value Z* of variable Z; and

average{F(ηijZ*)} denotes an average of the respective F(ηijZ*) of at least two individuals i at a situation j and at a specifically selected value Z* of variable Z.

2. The method of claim 1, further comprising at least one of specifying and plotting, said at least one of specifying and plotting being of any of the following: π(Y_ijZ), π(Y_Z), π(MS_ijZ), π(MSZ), π(MSijZ+), and π(MSZ+); wherein: π(YijZ) denotes one of π(YijZ) = π(MijZ) and π(YijZ) = π(SijZ);

π(Y_Z) denotes one of π(Y_Z) = π(M_Z) and π(Y_Z) = π(S_Z);

π(M_ijZ) denotes the π(M_ijZ*) of an individual i at a situation j and at at least two specifically selected Z* values of variable Z;

π(SijZ) denotes the π(SijZ*) of an individual i at a situation j and at at least two specifically selected Z* values of variable Z;

π(MZ) denotes the π(MZ*) of individuals i at a situation j and at at least two specifically selected Z* values of variable Z;

π(S_Z) denotes at least two π(S_Z*) of individuals i at a situation j and at at least two specifically selected Z* values of variable Z; π(MSijZ) denotes corresponding π(MijZ) and π(SijZ) at at least two specifically selected Z* values of variable Z;

π(MS_Z) denotes corresponding π(M_Z) and π(S_Z) at at least two specifically

selected Z* values of variable Z;

π(MSijZ+) denotes corresponding π(MSijZ) at at least two specifically selected Z* values of a variable Z and at respective at least two specifically selected Z* values of each of at least one other variable Z; and

π(MSZ+) denotes corresponding π(MSZ) at at least two specifically selected Z* values of a variable Z and at respective at least two specifically selected Z* values of each of at least one other variable Z.

3. A system for including and distinguishing independent variables that denote lifespan in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship, said system in a computing environment, said system comprising: a processor;

a non-transitory computer readable storage medium connected to the processor, wherein the storage medium has stored thereon at least one of data and instructions, and wherein the processor is configured to execute said instructions;

said processor configured to at least one of obtain and create data, said data to be about N individuals i at respective J situations j, said data to include variable Y_ij and at least two variables Xvij;

said processor further configured to conduct a multivariable binary regression

analysis of said data, said analysis to analyze the relationship between a dependent variable Y_ij and independent variables X_vij for one of Y_ij = M_ij and Y_ij = Sij, said analysis to utilize ηij = β0 + βvXvij +… + βWXWij, binary link function B(η_ij), and π(Y_ij) = F(η_ij);

said processor further configured to estimate at least one of π(Y_ijZ*) and π(Y_Z*), said estimating of at least one of π(Y_ijZ*) and π(Y_Z*) to utilize η_ijZ* = β₀ + _Σβ_vZ* + ΣβvXvij~Z, said estimating of π(YijZ*) also to utilize π(YijZ*) = F(ηijZ*), and said estimating of π(Y_Z*) also to utilize at least one of π(Y_Z*) = F{average(η_ijZ*)} and π(YZ*) = average{F(ηijZ*)}; and

wherein: i = 1:N, indicating that i are sequential positive integers 1 through N;

i denotes an individual;

N denotes the total number of individuals i in said data;

j = 1:J, indicating that j are sequential positive integers 1 through J;

j denotes a situation of individual i in reference to said variables Xvij and Yij; J denotes the total number of situations of an individual i in said data, allowing distinct J for distinct individuals i;

Mij denotes the mortality status of individual i at situation j;

Sij denotes the survivorship status of individual i at situation j;

Y_ij∈ {0,1}, indicating that Y_ij is a binary variable that adopts one of values 0 and 1 for one of Y_ij = M_ij and Y_ij = S_ij of individual i at situation j;

v = 1:W, indicating that v are sequential positive integers 1 through W;

W≥ 2;

v denotes an index of each of the following: variables X_v, variables X_vij of

individual i at situation j, and coefficients β_v;

W denotes the following: total number of variables Xv, total number of variables Xvij of individual i at situation j, and total number of coefficients βv;

X_vij denotes a variable X_v of individual i at situation j;

r = 1:R, indicating that r are sequential positive integers 1 through R;

R≥ 2;

W≥ R;

r denotes an index of each of the following: variables K_r and variables K_rij; Kr denotes a variable that directly denotes a distinct phenomenon, and R

respective variables Kr respectively directly denote R distinct phenomena; K_rij denotes a variable K_r of individual i at situation j;

T denotes a transformation function, allowing identity transformation;

q = 1:Q, indicating that q are sequential positive integers 1 through Q;

X_v = T_q(K_r), indicating that X_v is a transformation T_q of variable K_r that directly denotes a specific phenomenon, further indicating that Xv indirectly denotes said specific phenomenon; Xvij = Tq(Krij), indicating that Xvij is a transformation Tq of Krij that directly denotes a specific phenomenon, further indicating that Xvij indirectly denotes said specific phenomenon;

Q denotes the total number of transformations Tq(Kr) for a specific Kr, and Q also denotes the total number of transformations Tq(Krij) for a specific Krij of individual i at situation j, allowing distinct Q for distinct variables K_r, and allowing distinct Q for distinct variables K_rij of individual i at situation j;

R denotes the following: the total number of variables Kr, the total number of variables K_rij of individual i at situation j, the total number of phenomena denoted by variables Kr, and the total number of phenomena denoted by variables Krij of individual i at situation j;

A directly denotes age;

L directly denotes lifespan;

K1 = A and K2 = L, indicating that one of at least two variables Kr directly

denotes age, and indicating that another of said at least two variables K_r directly denotes lifespan;

K1ij = Aij and K2ij = Lij, indicating that one of at least two variables Krij directly denotes the age of individual i at situation j, and indicating that another of said at least two variables K_rij directly denotes the lifespan of individual i at situation j;

β denotes a regression coefficient;

β₀ denotes the regression coefficient for the intercept, allowing β₀ to be any of the following: estimated, suppressed, and user-provided;

π(Yij) denotes one of π(Yij) = π(Mij) and π(Yij) = π(Sij);

π(M_ij) denotes the probability of mortality of individual i at situation j;

π(S_ij) denotes the probability of survivorship of individual i at situation j;

resulting from said multivariable binary regression analysis; Z denotes a specifically selected variable Kr;

said variable Z and said specifically selected variable K_r directly denote a

specifically selected phenomenon;

Z* denotes a specifically selected value of variable Z; π(YijZ*) denotes one of π(YijZ*) = π(MijZ*) and π(YijZ*) = π(SijZ*);

π(YZ*) denotes one of π(YZ*) = π(MZ*) and π(YZ*) = π(SZ*);

π(M_ijZ*) denotes an individualized specific probability of mortality of individual i at situation j and at a specifically selected value Z* of variable Z;

π(SijZ*) denotes an individualized specific probability of survivorship of

individual i at situation j and at a specifically selected value Z* of variable Z;

π(MZ*) denotes an averaged specified probability of mortality at a specifically selected value Z* of variable Z;

π(S_Z*) denotes an averaged specified probability of survivorship at a

specifically selected value Z* of variable Z;

coefficients β in η_ijZ* = β₀ + _Σβ_vZ* + _Σβ_vX_vij~Z are respective coefficients β resulting from said regression analysis;

βvZ* denotes a replacement of a βvXvij in said model ηij = β0 + βvXvij +… + βWXWij, wherein variable Kr in Xvij = Tq(Krij) is said specific variable Kr that is denoted by Z, and wherein X_vij = T_q(Z_ij) for said replaced β_vX_vij;

ΣβvZ* denotes the sum of all respective βvZ* replacements of respective βvXvij in said model ηij = β0 + βvXvij +… + βWXWij;

X_vij~Z denotes an X_vij wherein variable K_r in X_vij = T_q(K_rij) is not said specific variable Kr that is denoted by Z;

Σβ_vX_vij~Z denotes the sum of all β_vX_vij~Z in said model η_ij = β₀ + β_vX_vij +… + β_WX_Wij;

F(∙) denotes a cumulative distribution function of (∙);

cumulative distribution function F(∙) corresponds to binary link function B(ηij); average(η_ijZ*) denotes an average of the respective η_ijZ* of at least two

individuals i at situation j and at a specifically selected value Z* of variable Z; and

average{F(ηijZ*)} denotes an average of the respective F(ηijZ*) of at least two individuals i at a situation j and at a specifically selected value Z* of variable Z.

4. In the system of claim 3, said processor further configured to at least one of specify and plot, said at least one of specifying and plotting to be of any of the following: π(YijZ), π(Y_Z), π(MS_ijZ), π(MS_Z), π(MS_ijZ+), and π(MS_Z+); wherein:

π(YijZ) denotes one of π(YijZ) = π(MijZ) and π(YijZ) = π(SijZ);

π(YZ) denotes one of π(YZ) = π(MZ) and π(YZ) = π(SZ);

π(M_ijZ) denotes the π(M_ijZ*) of an individual i at a situation j and at at least two specifically selected Z* values of variable Z;

π(SijZ) denotes the π(SijZ*) of an individual i at a situation j and at at least two specifically selected Z* values of variable Z;

π(M_Z) denotes the π(M_Z*) of individuals i at a situation j and at at least two specifically selected Z* values of variable Z;

π(SZ) denotes at least two π(SZ*) of individuals i at a situation j and at at least two specifically selected Z* values of variable Z;

π(MSijZ) denotes corresponding π(MijZ) and π(SijZ) at at least two specifically selected Z* values of variable Z;

π(MS_Z) denotes corresponding π(M_Z) and π(S_Z) at at least two specifically

selected Z* values of variable Z;

π(MSijZ+) denotes corresponding π(MSijZ) at at least two specifically selected Z* values of a variable Z and at respective at least two specifically selected Z* values of each of at least one other variable Z; and

π(MSZ+) denotes corresponding π(MSZ) at at least two specifically selected Z* values of a variable Z and at respective at least two specifically selected Z* values of each of at least one other variable Z.

5. The system of claim 3 further comprising a plurality of local or remote devices connected through one or more networks.

6. The system of claim 4 further comprising a plurality of local or remote devices connected through one or more networks.

7. A computer program product for including and distinguishing independent variables that denote lifespan in regression-based estimation of phenomenon-specific probabilities of one of mortality and survivorship, said computer program product stored in a non- transitory computer storage medium, said storage medium storing at least one of data and instructions for processing by a processor, said instructions comprising: instructions to at least one of obtain and create data, said data to be about N individuals i at respective J situations j, said data to include variable Yij and at least two variables X_vij;

instructions to conduct a multivariable binary regression analysis of said data, said analysis to analyze the relationship between a dependent variable Yij and independent variables X_vij for one of Y_ij = M_ij and Y_ij = S_ij, said analysis to utilize η_ij = β₀ + β_vX_vij +… + β_WX_Wij, binary link function B(η_ij), and π(Y_ij) = F(ηij); and

instructions to estimate at least one of π(YijZ*) and π(YZ*), said estimating of at least one of π(YijZ*) and π(YZ*) to utilize ηijZ* = β0 + ΣβvZ* + ΣβvXvij~Z, said estimating of π(YijZ*) also to utilize π(YijZ*) = F(ηijZ*), and said estimating of π(YZ*) also to utilize at least one of π(YZ*) = F{average(ηijZ*)} and π(YZ*) = average{F(η_ijZ*)}; and

wherein:

i = 1:N, indicating that i are sequential positive integers 1 through N;

i denotes an individual;

N denotes the total number of individuals i in said data;

j = 1:J, indicating that j are sequential positive integers 1 through J;

j denotes a situation of individual i in reference to said variables X_vij and Y_ij; J denotes the total number of situations of an individual i in said data, allowing distinct J for distinct individuals i;

Mij denotes the mortality status of individual i at situation j;

S_ij denotes the survivorship status of individual i at situation j;

Y_ij∈ {0,1}, indicating that Y_ij is a binary variable that adopts one of values 0 and 1 for one of Y_ij = M_ij and Y_ij = S_ij of individual i at situation j;

v = 1:W, indicating that v are sequential positive integers 1 through W;

W≥ 2;

v denotes an index of each of the following: variables X_v, variables X_vij of

individual i at situation j, and coefficients βv;

W denotes the following: total number of variables Xv, total number of variables X_vij of individual i at situation j, and total number of coefficients β_v;

Xvij denotes a variable Xv of individual i at situation j; r = 1:R, indicating that r are sequential positive integers 1 through R;

R≥ 2;

W≥ R;

r denotes an index of each of the following: variables Kr and variables Krij; Kr denotes a variable that directly denotes a distinct phenomenon, and R

respective variables K_r respectively directly denote R distinct phenomena; K_rij denotes a variable K_r of individual i at situation j;

T denotes a transformation function, allowing identity transformation;

q = 1:Q, indicating that q are sequential positive integers 1 through Q;

X_v = T_q(K_r), indicating that X_v is a transformation T_q of variable K_r that directly denotes a specific phenomenon, further indicating that Xv indirectly denotes said specific phenomenon;

X_vij = T_q(K_rij), indicating that X_vij is a transformation T_q of K_rij that directly denotes a specific phenomenon, further indicating that Xvij indirectly denotes said specific phenomenon;

Q denotes the total number of transformations T_q(K_r) for a specific K_r, and Q also denotes the total number of transformations T_q(K_rij) for a specific K_rij of individual i at situation j, allowing distinct Q for distinct variables Kr, and allowing distinct Q for distinct variables Krij of individual i at situation j;

R denotes the following: the total number of variables Kr, the total number of variables Krij of individual i at situation j, the total number of phenomena denoted by variables K_r, and the total number of phenomena denoted by variables K_rij of individual i at situation j;

A directly denotes age;

L directly denotes lifespan;

K₁ = A and K₂ = L, indicating that one of at least two variables K_r directly

denotes age, and indicating that another of said at least two variables Kr directly denotes lifespan;

K_1ij = A_ij and K_2ij = L_ij, indicating that one of at least two variables K_rij directly denotes the age of individual i at situation j, and indicating that another of said at least two variables Krij directly denotes the lifespan of individual i at situation j;

β denotes a regression coefficient;

β0 denotes the regression coefficient for the intercept, allowing β0 to be any of the following: estimated, suppressed, and user-provided;

π(Y_ij) denotes one of π(Y_ij) = π(M_ij) and π(Y_ij) = π(S_ij);

π(M_ij) denotes the probability of mortality of individual i at situation j;

π(Sij) denotes the probability of survivorship of individual i at situation j;

ηij denotes ηij resulting from said multivariable binary regression analysis; Z denotes a specifically selected variable K_r;

said variable Z and said specifically selected variable Kr directly denote a

specifically selected phenomenon;

Z* denotes a specifically selected value of variable Z;

π(YijZ*) denotes one of π(YijZ*) = π(MijZ*) and π(YijZ*) = π(SijZ*);

π(YZ*) denotes one of π(YZ*) = π(MZ*) and π(YZ*) = π(SZ*);

π(M_ijZ*) denotes an individualized specific probability of mortality of individual i at situation j and at a specifically selected value Z* of variable Z;

π(SijZ*) denotes an individualized specific probability of survivorship of

individual i at situation j and at a specifically selected value Z* of variable Z;

π(MZ*) denotes an averaged specified probability of mortality at a specifically selected value Z* of variable Z;

π(S_Z*) denotes an averaged specified probability of survivorship at a

specifically selected value Z* of variable Z;

coefficients β in η_ijZ* = β₀ + _Σβ_vZ* + _Σβ_vX_vij~Z are respective coefficients β resulting from said regression analysis;

β_vZ* denotes a replacement of a β_vX_vij in said model η_ij = β₀ + β_vX_vij +… + βWXWij, wherein variable Kr in Xvij = Tq(Krij) is said specific variable Kr that is denoted by Z, and wherein X_vij = T_q(Z_ij) for said replaced β_vX_vij;

ΣβvZ* denotes the sum of all respective βvZ* replacements of respective βvXvij in said model ηij = β0 + βvXvij +… + βWXWij; Xvij~Z denotes an Xvij wherein variable Kr in Xvij = Tq(Krij) is not said specific variable Kr that is denoted by Z;

ΣβvXvij~Z denotes the sum of all βvXvij~Z in said model ηij = β0 + βvXvij +… + βWXWij;

F(∙) denotes a cumulative distribution function of (∙);

cumulative distribution function F(∙) corresponds to binary link function B(η_ij); average(ηijZ*) denotes an average of the respective ηijZ* of at least two

individuals i at situation j and at a specifically selected value Z* of variable Z; and

average{F(η_ijZ*)} denotes an average of the respective F(η_ijZ*) of at least two individuals i at a situation j and at a specifically selected value Z* of variable Z.

8. In the computer program product of claim 7, said instructions further comprising

instructions to at least one of specify and plot, said at least one of specifying and plotting to be of any of the following: π(YijZ), π(YZ), π(MSijZ), π(MSZ), π(MSijZ+), and π(MSZ+); wherein:

π(YijZ) denotes one of π(YijZ) = π(MijZ) and π(YijZ) = π(SijZ);

π(YZ) denotes one of π(YZ) = π(MZ) and π(YZ) = π(SZ);

π(M_ijZ) denotes the π(M_ijZ*) of an individual i at a situation j and at at least two specifically selected Z* values of variable Z;

π(SijZ) denotes the π(SijZ*) of an individual i at a situation j and at at least two specifically selected Z* values of variable Z;

π(M_Z) denotes the π(M_Z*) of individuals i at a situation j and at at least two specifically selected Z* values of variable Z;

π(SZ) denotes at least two π(SZ*) of individuals i at a situation j and at at least two specifically selected Z* values of variable Z;

π(MS_ijZ) denotes corresponding π(M_ijZ) and π(S_ijZ) at at least two specifically selected Z* values of variable Z;

π(MS_Z) denotes corresponding π(M_Z) and π(S_Z) at at least two specifically

selected Z* values of variable Z; π(MSijZ+) denotes corresponding π(MSijZ) at at least two specifically selected Z* values of a variable Z and at respective at least two specifically selected Z* values of each of at least one other variable Z; and

π(MSZ+) denotes corresponding π(MSZ) at at least two specifically selected Z* values of a variable Z and at respective at least two specifically selected Z* values of each of at least one other variable Z.