Fitting routines for the Vitality family of mortality models.

Description

This package provides support for fitting the vitality family of mortality models that characterize mortality in terms of the loss vitality, an abstract measure of survival capacity. Mortality occurs by two processes. Intrinsic mortality occurs when vitality is depleted by stochastic losses. Extrinsic mortality occurs when a random external challenge exceeds the available vitality. The package contains four model versions:

• vitality.k is a 3-parameter model. Intrinsic mortality is characterized by the mean (r) and variability (s) in vitality loss rate. Extrinsic mortality is characterized by the frequency (k) of lethal random challenges. Model is appropriate to animal mortality data (e.g. Anderson 2000).

• vitality.ku is a 4-parameter model. Intrinsic mortality is characterized by the mean (r) and variability (s) in the vitality loss rate and the standard deviation of initial vitality (u). Extrinsic mortality is characterized by the frequency (k) of lethal random challenges. Model is appropriate to animal mortality data (e.g. Li and Anderson 2009).

• vitality.4p is a 4-parameter model. Intrinsic mortality is characterized by the mean (r) and variability (s) in the vitality loss rate. Extrinsic mortality is characterized by random challenges of frequency (lambda) and random magnitude (beta) exceeding the remaining average vitality. Model is appropriate to adult human mortality data (e.g. Li and Anderson 2013).

• vitality.6p is a 6-parameter model. Intrinsic mortality is characterized by the mean (r) and variability (s) in the vitality loss rate. Adult extrinsic mortality is characterized by random challenges of frequency (lambda) and random magnitude (beta) exceeding the remaining average vitality. Child extrinsic mortality is characterized by childhood challenges of frequency (gamma) exceeding childhood vitality development rate (alpha). Model is appropriate to full lifespan of human mortality data (e.g. Anderson and Li 2015).

Model parameters are estimated on survival or mortality rate data using maximum log likelihood methods based on Salinger et al. (2003).

Version 1.1 adds the versions vitality.k, vitality.ku and vitality.6p to the original code for the vitality.4p model previously designated vitality.2ps.

Version 1.2 makes previously invisible functions to produce the survival and mortality rate functions from a given set of parameters (e.g. SurvFn.4p, SurvFn.6p, mu.vd.4p, mu.vd.6p) usable. The child mortality rate formula in mu.vd.6p, mu.vd2.6p, and mu.vd4.6p has been updated to improve fit.

Details

Support for package development was provided by the National Institute of Ageing Grant 1R21AG046760-01, the Bonneville Power Administration, and the University of Washington Center for Statistics and the Social Sciences and Center for Studies in Demography and Ecology.

References

• Anderson, J.J. (2000). "A vitality-based model relating stressors and environmental properties to organism survival." Ecological Monographs 70(3):445-470.

• Anderson, J.J. and Li, T. (2015). "A two-process mortality model with extensions to juvenile mortality, populations and evolution." Population Association of America Annual Meeting 2015 http://paa2015.princeton.edu/abstracts/153144

• Li, T. and J.J. Anderson. (2009). "The vitality model: A way to understand population survival and demographic heterogeneity." Theoretical Population Biology 76: 118-131.

• Li, T. and J.J. Anderson (2013). "Shaping human mortality patterns through intrinsic and extrinsiv vitality processes." Demographic Research 28(12): 341-372.

• Salinger, D.H., J.J. Anderson, and O.S. Hamel. (2003). "A parameter estimation routine for the vitality-based survival model." Ecological Modelling 166 (3): 287-29

Examples

# vitality.k
data(daphnia)
time <- daphnia$days
survival_fraction <- daphnia$lx

results.modk <- vitality.k(time = time,
                           sdata = survival_fraction,
                           rc.data=TRUE, 
                           se=FALSE,
                           gfit=FALSE, 
                           datatype="CUM", 
                           ttol=.000001, 
                           init.params=FALSE, 
                           lower=c(0,-1,0), upper=c(100,50,50), 
                           pplot=TRUE, 
                           tlab="days", 
                           lplot=TRUE, 
                           cplot=TRUE, 
                           Iplot=TRUE, 
                           silent=TRUE)

# vitality.ku
data(rainbow_trout_for_k)
time <- rainbow_trout_for_k$days
survival_fraction <- rainbow_trout_for_k$survival

results.modku <- vitality.ku(time = time,
                             sdata = survival_fraction,
                             rc.data=TRUE,
                             se=FALSE,
                             gfit=FALSE,
                             datatype="CUM",
                             ttol=.000001,
                             init.params=FALSE,
                             lower=c(0,-1,0,0),upper=c(100,100,50,50),
                             pplot=TRUE,
                             tlab="days",
                             lplot=TRUE,
                             cplot=TRUE,
                             Iplot=TRUE,
                             silent=TRUE,
                             L=0)
# vitality.4p
data(swedish_females)
swe <- swedish_females
initial_age <- 20 # Could be adjusted
time <- initial_age:max(swedish_females$age)
survival_fraction <- swe$lx / swe$lx[1]
survival_fraction <- survival_fraction[time] # when first element <1 data is adjusted
sample_size <- swe$Lx[initial_age] #sample size

results.4par <- vitality.4p(time = time,
                            sdata = survival_fraction,
                            #init.params=FALSE,
                            init.params=c(0.012, 0.01, 0.1, 0.1),
                            lower = c(0, 0, 0, 0), upper = c(100,50,1,50),
                            rc.data = TRUE, 
                            se = sample_size, 
                            datatype = "CUM", 
                            ttol = 1e-06,
                            pplot = TRUE,
                            Iplot = TRUE,
                            Mplot = TRUE,
                            tlab = "years",
                            silent = FALSE)

# vitality.6p
data(swedish_females)
swe <- swedish_females
initial_age <- 0 
time <- swedish_females$age
survival_fraction <- swe$lx / swe$lx[1]
sample_size <- swe$Lx[1] #sample size

results.6par <- vitality.6p(time = time,
                            sdata = survival_fraction,
                            #init.params=FALSE,
                            init.params=c(0.012, 0.01, 0.1, 0.1, 0.1, 1),
                            lower = c(0, 0, 0, 0, 0, 0), upper = c(100,50,1,50,50,50),
                            rc.data = TRUE, 
                            se=FALSE,
                            #se = sample_size, 
                            datatype = "CUM", 
                            ttol = 1e-06,
                            pplot = TRUE,
                            Iplot = TRUE,
                            Mplot = TRUE,
                            tlab = "years",
                            silent = FALSE)

The cumulative survival distribution function for 2-process 4-parameter vitality model

Description

Gives the cumulative survival proportions at xx from all processes for a given set of parameter values.

Usage

  SurvFn.4p(xx, r, s, lambda, beta)

Arguments

Details

Used within vitality.4p for estimating model parameters based on the observed cumulative survival function.

Value

vector of cumulative survival proportions at xx from all processes

See Also

vitality.4p, survProbInc.4p, SurvFn.in.4p, SurvFn.ex.4p

The cumulative survival distribution function for 2-process 6-parameter vitality model

Description

Gives the cumulative survival proportions at xx from all processes for a given set of parameter values.

Usage

  SurvFn.6p(xx, r, s, lambda, beta, gamma, alpha)

Arguments

Details

Used within vitality.6p for estimating model parameters based on the observed cumulative survival function.

Value

vector of cumulative survival proportions at xx from all processes

See Also

vitality.6p, survProbInc.6p, SurvFn.in.6p, SurvFn.ex.6p

The extrinsic cumulative survival distribution function for 2-process 4-parameter vitality model

Description

Gives the cumulative survival proportions at xx from extrinsic process for a given set of parameter values.

Usage

  SurvFn.ex.4p(xx, r, s, lambda, beta)

Arguments

Details

Used within vitality.4p for estimating model parameters based on the observed cumulative survival function.

Value

vector of cumulative survival proportions at xx from extrinsic process

See Also

vitality.4p, survProbInc.4p, SurvFn.4p, SurvFn.in.4p

The extrinsic cumulative survival distribution function for 2-process 6-parameter vitality model

Description

Gives the cumulative survival proportions at xx from the extrinsic processes for a given set of parameter values.

Usage

  SurvFn.ex.6p(xx, r, s, lambda, beta, gamma, alpha)

Arguments

Details

Used within vitality.6p for estimating model parameters based on the observed cumulative survival function.

Value

vector cumulative survival proportions at xx from extrinsic processes

See Also

vitality.6p, survProbInc.6p, SurvFn.6p

Intrinsic cumulative survival distribution for 4 parameter model

Description

Gives the intrinsic cumulative survival distribution at xx.

Usage

  SurvFn.h.4p(xx, r, s, u)

Arguments

Details

For use in vitality.4p.

Value

intrinsic cumulative survival distribution

See Also

vitality.4p

Intrinsic cumulative survival distribution for 6 parameter model

Description

Gives the intrinsic cumulative survival distribution at xx.

Usage

  SurvFn.h.6p(xx, r, s)

Arguments

Details

For use in vitality.6p.

Value

intrinsic cumulative survival distribution

See Also

vitality.6p

The intrinsic cumulative survival distribution function for 2-process 4-parameter vitality model

Description

Gives the cumulative survival proportions at xx from intrinsic process for a given set of parameter values.

Usage

  SurvFn.in.4p(xx, r, s)

Arguments

Details

Used within vitality.4p for estimating model parameters based on the observed cumulative survival function.

Value

vector of cumulative survival proportions at xx from intrinsic process

See Also

vitality.4p, survProbInc.4p, SurvFn.4p, , SurvFn.ex.4p

The intrinsic cumulative survival distribution function for 2-process 6-parameter vitality model

Description

Gives the cumulative survival proportions at xx from the intrinsic process for a given set of parameter values.

Usage

  SurvFn.in.6p(xx, r, s)

Arguments

Details

Used within vitality.6p for estimating model parameters based on the observed cumulative survival function.

Value

vector of cumulative survival proportions at xx from intrinsic process

See Also

vitality.6p, survProbInc.6p, SurvFn.6p

The cumulative survival distribution function for 3-parameter vitality model

Description

Gives the cumulative survival proportions at xx for a given set of parameter values.

Usage

  SurvFn.k(xx, r, s, k)

Arguments

Details

Used within vitality.k for estimating model parameters based on the observed cumulative survival function.

Value

cumulative survival proportions at xx

See Also

vitality.k, survProbInc.k

The cumulative survival distribution function for 4-parameter vitality model

Description

Gives the cumulative survival proportions at xx for a given set of parameter values.

Usage

  SurvFn.ku(xx, r, s, k, u)

Arguments

Details

Used within vitality.ku for estimating model parameters based on the observed cumulative survival function.

Value

cumulative survival proportions at xx

See Also

vitality.ku, survProbInc.ku

Sample Daphnia Data

Description

Sample survival data for daphnia. Columns include "days" and "lx" (cumulative survival proportion by day).

Format

data frame

Source

http://cbr.washington.edu/analysis/vitality

Anderson, J.J. (2000). "A vitality-based model relating stressors and environmental properties to organism survival." Ecological Monographs 70(3):445-470 (Figure 5)

Function for data preparation

Description

Function to deal with NAs, right truncated data, and datatype (i.e. cumulative survival or incremental mortality).

Usage

  dataPrep(time, sdata, datatype, rc.data,
    returnMatrix = FALSE)

Arguments

Details

This function is designed for use in the primary vitality model fitting functions in this package. See package documentation.

Value

Returns a data.frame or matrix with columns time, sfract, x1, x2, Ni (incremental survival fraction), rc.data.

Density function for 3-parameter (r, s, u)

Description

This function is used in the calculation of the fitted intrinsic (mu.vd1.4p) and total (mu.vd.4p) mortality rate in the 4-parameter model.

Usage

  ft.4p(xx, r, s, u)

Arguments

Value

density

See Also

vft.4p, ft.6p

Density function for 2-parameters (r, s)

Description

This function is used in the calculation of the fitted intrinsic (mu.vd1.6p) and total (mu.vd.6p) mortality rate in the 6-parameter model.

Usage

  ft.6p(xx, r, s)

Arguments

Value

density

See Also

vft.6p

Finds the first value of a vector that is less than a value.

Description

For use in the primary vitality model fitting functions in this package. See package documentation.

Usage

  indexFinder(x, val)

Arguments

Value

Gives the index of the first value of x that is <= val. returns -1 if no value satisfies the condition

Log likelihood of 2-process 4-parameter model

Description

Gives the log likelihood of 2-process 6 parameter vitality model.

Usage

  logLikelihood.4p(par, xx1, xx2, NNi)

Arguments

Details

For use in vitality.4p.

Value

log likelihood

See Also

logLikelihood.6p

Log likelihood of 2-process 6-parameter vitality model

Description

Gives the log likelihood of 6-parameter vitality model.

Usage

  logLikelihood.6p(par, xx1, xx2, NNi)

Arguments

Details

For use in vitality.6p.

Value

log likelihood

See Also

vitality.6p

Log likelihood of 3-parameter (r,s,k) model

Description

Gives the log likelihood of 3-parameter vitality model.

Usage

  logLikelihood.k(par, xx1, xx2, NNi)

Arguments

Details

For use in vitality.k.

Value

log likelihood

See Also

vitality.k

Log likelihood of 4-parameter (r,s,k,u) model

Description

Gives the log likelihood of 4-parameter vitality model.

Usage

  logLikelihood.ku(par, xx1, xx2, NNi)

Arguments

Details

For use in vitality.ku.

Value

log likelihood

See Also

vitality.ku

Total mortality rate for the 2-process 4-parameter vitality model

Description

Gives the total age-specific mortality rates for a given set of the four parameters. See mu.vd1.4p for calculation of intrinsic age-specific mortality rates. See mu.vd2.4p for calculation of extrinsic age-specific mortality rates.

Usage

  mu.vd.4p(t, r, s, lambda, beta)

Arguments

Value

Total age-specific mortality rates

See Also

mu.vd1.4p, mu.vd2.4p

Total mortality rate for the 2-process 6-parameter vitality model

Description

Gives the total age-specific mortality rates for a given set of the six parameters. See mu.vd1.6p for calculation of intrinsic age-specific mortality rates. See mu.vd2.6p for calculation of extrinsic age-specific mortality rates.

Usage

  mu.vd.6p(t, r, s, lambda, beta, gamma, alpha)

Arguments

Value

Total age-specific mortality rates

See Also

mu.vd1.6p, mu.vd2.6p

Intrinsic mortality rate for the 2-process 4-parameter vitality model

Description

Gives the intrinsic age-specific mortality rates for a given set of r and s, the intrinsic parameters.

Usage

  mu.vd1.4p(x, r, s)

Arguments

Value

Intrinsic age-specific mortality rates

See Also

mu.vd.4p, mu.vd2.4p

Intrinsic mortality rate for the 2-process 6-parameter vitality model

Description

Gives the intrinsic age-specific mortality rates for a given set of r and s, the intrinsic parameters.

Usage

  mu.vd1.6p(x, r, s)

Arguments

Value

Vector of intrinsic age-specific mortality rates at age x

See Also

mu.vd.6p, mu.vd2.6p

Extrinsic mortality rate for the 2-process 4-parameter vitality model

Description

Gives the extrinsic age-specific mortality rates for a given set of r and the extrinsic parameters.

Usage

  mu.vd2.4p(x, r, lambda, beta)

Arguments

Value

Extrinsic age-specific mortality rates

See Also

mu.vd.4p, mu.vd1.4p

Extrinsic mortality rate for the 2-process 6-parameter vitality model

Description

Gives the extrinsic age-specific mortality rates for a given set of r and the extrinsic parameters.

Usage

  mu.vd2.6p(x, r, lambda, beta, gamma, alpha)

Arguments

Value

Vector of extrinsic age-specific mortality rates at ages x

See Also

mu.vd.6p, mu.vd1.6p, mu.vd3.6p, mu.vd4.6p

Adult extrinsic mortality rate for the 2-process 6-parameter vitality model

Description

Gives the extrinsic age-specific mortality rates for a given set of r and the adult extrinsic parameters.

Usage

  mu.vd3.6p(x, r, lambda, beta)

Arguments

Value

Vector of adult extrinsic age-specific mortality rates at ages x

See Also

mu.vd.6p, mu.vd1.6p, mu.vd2.6p, mu.vd4.6p

Childhood extrinsic mortality rate for the 2-process 6-parameter vitality model

Description

Gives the childhood extrinsic age-specific mortality rates for a given set of the childhood extrinsic parameters.

Usage

  mu.vd4.6p(x, gamma, alpha)

Arguments

Value

Vector of childhood extrinsic age-specific mortality rates at ages x

See Also

mu.vd.6p, mu.vd1.6p, mu.vd2.6p, mu.vd3.6p

Plotting function for 2-process 4-parameter vitality model

Description

This function plots the estimated results from the 4 parameter vitality model. It is used within the function vitality.4p.

Usage

  plotting.4p(r.final, s.final, lambda.final, beta.final,
    mlv, time, sfract, x1, x2, Ni, pplot, Iplot, Mplot,
    tlab, rc.data)

Arguments

Details

See vitality.4p for further description of function arguments.

Only one of Iplot or Mplot should be set to TRUE at once.

See Also

vitality.4p, mu.vd.4p, mu.vd1.4p, mu.vd2.4p

Plotting function for 2-process 6-parameter vitality model

Description

This function plots the estimated results from the 6 parameter vitality model. It is used within the function vitality.6p.

Usage

  plotting.6p(r.final, s.final, lambda.final, beta.final, 
    gamma.final, alpha.final, mlv, time, sfract, x1, x2, Ni, pplot, Iplot, Mplot,
    tlab, rc.data)

Arguments

Details

See vitality.6p for further description of function arguments.

Only one of Iplot or Mplot should be set to TRUE at once.

See Also

vitality.6p, mu.vd.6p, mu.vd1.6p, mu.vd2.6p, mu.vd3.6p, mu.vd4.6p

Plotting function for 3-parameter vitality model

Description

This function plots the estimated results from the 3 parameter vitality model. It is used within the function vitality.k.

Usage

  plotting.k(r.final,s.final,k.final,mlv,time,sfract,x1,x2,Ni,
  pplot,tlab,lplot,cplot,Iplot,gfit,rc.data)

Arguments

Details

See vitality.k for further description of function arguments.

See Also

vitality.k

Plotting function for 4-parameter vitality model

Description

This function plots the estimated results from the 4 parameter vitality model. It is used within the function vitality.ku.

Usage

  plotting.ku(r.final,s.final,k.final,u.final,mlv,time,sfract,x1,x2,Ni,
    pplot,tlab,lplot,cplot,Iplot,gfit)

Arguments

Details

See vitality.ku for further description of function arguments.

See Also

vitality.ku

Sample Rainbow Trout Data

Description

Sample survival data for rainbow trout. Columns include "days" and "survival" (cumulative survival proportion by day).

Format

matrix

Source

http://cbr.washington.edu/analysis/vitality

Standard errors for 4-parameters: r, s, lambda, beta

Description

Gives the standard errors for the 4 parameter model. Primarily used within vitality.4p.

Usage

  stdErr.4p(r, s, k, u, x1, x2, Ni, pop)

Arguments

Value

standard error for r, s, lambda, beta

Note

if k <= 0, cannot find standard error for k

See Also

vitality.4p

Standard errors for 6-parameters: r, s, lambda, beta, gamma, alpha

Description

Gives the standard errors for the 6 parameter model. Primarily used within vitality.6p.

Usage

  stdErr.6p(r, s, k, u, g, a, x1, x2, Ni, pop)

Arguments

Value

standard error for r, s, lambda, beta, gamma, and alpha.

Note

if k <= 0, cannot find standard error for k

See Also

vitality.6p

Standard errors for 3-parameters: r, s, k

Description

Gives the standard errors for the 3 parameter model. Primarily used within vitality.k.

Usage

  stdErr.k(r, s, k, x1, x2, Ni, pop)

Arguments

Value

standard error for r, s, k.

Note

k is restricted to be >0.

See Also

vitality.k

Standard errors for 4-parameters: r, s, k, u

Description

Gives the standard errors for the 4 parameter model. Primarily used within vitality.ku.

Usage

  stdErr.ku(r, s, k, u, x1, x2, Ni, pop)

Arguments

Value

standard error for r, s, k, u.

Note

k is restricted to be >0.

See Also

vitality.ku

Incremental survival probability for 2-process 4-parameter model

Description

Calculates the incremental survival probabilities (between xx1 and xx2) for 2-process 4-parameter model.

Usage

  survProbInc.4p(r, s, lambda, beta, xx1, xx2)

Arguments

Details

For use in vitality.4p.

Value

incremental survival probabilities

See Also

vitality.4p, logLikelihood.4p

Incremental survival probability for 2-process 6-parameter model

Description

Calculates the incremental survival probabilities (between xx1 and xx2) for 2-process 6-parameter model.

Usage

  survProbInc.6p(r, s, lambda, beta, gamma, alpha, xx1, xx2)

Arguments

Details

For use in vitality.6p.

Value

incremental survival probabilities

See Also

vitality.6p, logLikelihood.6p

Incremental survival probability for 3-parameter model

Description

Calculates the incremental survival probabilities (between xx1 and xx2) for 3-parameter model.

Usage

  survProbInc.k(r, s, k, xx1, xx2)

Arguments

Details

For use in vitality.k.

Value

incremental survival probabilities

See Also

vitality.k, logLikelihood.k

Incremental survival probability for 4-parameter model

Description

Calculates the incremental survival probabilities (between xx1 and xx2) for 4-parameter model.

Usage

  survProbInc.ku(r, s, k, u, xx1, xx2)

Arguments

Details

For use in vitality.ku.

Value

incremental survival probabilities

See Also

vitality.ku, logLikelihood.ku

Swedish Female Mortality Data

Description

Period life table for Swedish females in the year 2000. Columns follow standard life-table naming conventions.

Format

A data.frame object

Source

Human Mortality Database

Vectorized density function

Description

This function is used in the calculation of the fitted intrinsic (mu.vd1.4p) and total (mu.vd.4p) mortality rate in the 4-parameter model.

Usage

  vft.4p(xx, r, s, u)

Arguments

Value

vector of densities

See Also

ft.4p

Vectorized density function

Description

This function is used in the calculation of the fitted intrinsic (mu.vd1.6p) and total (mu.vd.6p) mortality rate in the 6-parameter model.

Usage

  vft.6p(xx, r, s)

Arguments

Value

vector of densities

See Also

ft.6p

Fitting routine for the 2-process, 4-parameter vitality model (no childhood hook).

Description

This function provides the fitting routine for the 4-parameter 2-process vitality model. Intrinsic mortality is characterized by the mean (r) and variability (s) in the vitality loss rate. Extrinsic mortality is characterized by random challenges of frequency (lambda) and random magnitude (beta) exceeding the remaining average vitality. Model is appropriate to adult human mortality data (e.g. Li and Anderson 2013).

Usage

  vitality.4p(time = 0:(length(sdata)-1), sdata, init.params = FALSE, 
    lower = c(0, 0, 0, 0), upper = c(100,50,100,50),rc.data = FALSE, 
    se = FALSE, datatype = c("CUM", "INC"), ttol = 1e-06, pplot = TRUE,
    Iplot = FALSE, Mplot = FALSE, tlab = "years", silent = FALSE)

Arguments

Value

vector of final MLE r, s, lambda, beta parameter estimates. standard errors of MLE parameter estimates (if se = <population> is specified).

References

D.H. Salinger, J.J. Anderson and O. Hamel (2003). "A parameter fitting routine for the vitality based survival model." Ecological Modeling 166(3): 287–294.

Li, T. and J.J. Anderson (2013). "Shaping human mortality patterns through intrinsic and extrinsic vitality processes." Demographic Research 28(12): 341-372.

Examples

data(swedish_females)
swe <- swedish_females
initial_age <- 20 # Could be adjusted
time <- initial_age:max(swedish_females$age)
survival_fraction <- swe$lx / swe$lx[1]
survival_fraction <- survival_fraction[time] # when first element <1 data is adjusted
sample_size <- swe$Lx[initial_age] #sample size

results.4par <- vitality.4p(time = time,
                            sdata = survival_fraction,
                            #init.params=FALSE,
                            init.params=c(0.012, 0.01, 0.1, 0.1),
                            lower = c(0, 0, 0, 0), upper = c(100,50,1,50),
                            rc.data = TRUE, 
                            se = sample_size, 
                            datatype = "CUM", 
                            ttol = 1e-06,
                            pplot = TRUE,
                            Iplot = TRUE,
                            Mplot = TRUE,
                            tlab = "years",
                            silent = FALSE)

Fitting routine for the 2-process, 6-parameter vitality model (with childhood hook).

Description

This function provides the fitting routine for the 6-parameter 2-process vitality model. Intrinsic mortality is characterized by the mean (r) and variability (s) in the vitality loss rate. Adult extrinsic mortality is characterized by random challenges of frequency (lambda) and random magnitude (beta) exceeding the remaining average vitality. Child extrinsic mortality is characterized by childhood challenges of frequency (gamma) exceeding childhood vitality development rate (alpha). Model is appropriate to full lifespan of human mortality data (e.g. Anderson and Li 2015).

Usage

  vitality.6p(time = 0:(length(sdata)-1), sdata, init.params = FALSE, 
    lower = c(0, 0, 0, 0, 0, 0),upper = c(100,50,100,50,50,10), rc.data = FALSE, 
    se = FALSE, datatype = c("CUM", "INC"), ttol = 1e-06, pplot = TRUE,
    Iplot = FALSE, Mplot = FALSE, tlab = "years", silent = FALSE)

Arguments

Value

vector of final MLE r, s, lambda, beta, gamma and alpha parameter estimates. standard errors of MLE parameter estimates (if se = <population> is specified).

References

D.H. Salinger, J.J. Anderson and O. Hamel (2003). "A parameter fitting routine for the vitality based survival model." Ecological Modeling 166(3): 287–294.

Anderson, J.J. and T. Li. (2015). "A two-process mortality model with extensions to juvenile mortality, populations and evolution." Population Association of America Annual Meeting 2015 http://paa2015.princeton.edu/abstracts/153144

Examples

data(swedish_females)
swe <- swedish_females
initial_age <- 0 
time <- swedish_females$age
survival_fraction <- swe$lx / swe$lx[1]
sample_size <- swe$Lx[1] #sample size

results.6par <- vitality.6p(time = time,
                            sdata = survival_fraction,
                            #init.params=FALSE,
                            init.params=c(0.012, 0.01, 0.1, 0.1, 0.1, 1),
                            lower = c(0, 0, 0, 0, 0, 0), upper = c(100,50,1,50,50,50),
                            rc.data = TRUE, 
                            se=FALSE,
                            #se = sample_size, 
                            datatype = "CUM", 
                            ttol = 1e-06,
                            pplot = TRUE,
                            Iplot = TRUE,
                            Mplot = TRUE,
                            tlab = "years",
                            silent = FALSE)

Fitting routine for the 3-parameter vitality model.

Description

This function provides the fitting routine for the 3-parameter vitality model. Intrinsic mortality is characterized by the mean (r) and variability (s) in vitality loss rate. Extrinsic mortality is characterized by the frequency (k) of lethal random challenges. Model is appropriate to animal mortality data (e.g. Anderson 2000).

Usage

vitality.k(time, sdata, rc.data=F, se=F, gfit=F, datatype="CUM", ttol=.000001, 
    init.params=F, lower=c(0,-1,0), upper=c(100,50,50), pplot=T, tlab="days", 
    lplot=F, cplot=F, Iplot=F, silent=F)

Arguments

Value

vector of final MLE r, s, k parameter estimates. standard errors of MLE parameter estimates (if se = <population> is specified).

References

• Anderson, J.J. (2000). "A vitality-based model relating stressors and environmental properties to organism survival." Ecological Monographs 70(3):445-470.

Examples

data(daphnia)
time <- daphnia$days
survival_fraction <- daphnia$lx

results.modk <- vitality.k(time = time,
                           sdata = survival_fraction,
                           rc.data=TRUE, 
                           se=FALSE,
                           gfit=FALSE, 
                           datatype="CUM", 
                           ttol=.000001, 
                           init.params=FALSE,
                           #init.params=c(0.075, 0.15, 0.001),
                           lower=c(0,-1,0), upper=c(100,50,50), 
                           pplot=TRUE, 
                           tlab="days", 
                           lplot=TRUE, 
                           cplot=TRUE, 
                           Iplot=TRUE, 
                           silent=FALSE)

Fitting routine for the 4-parameter vitality model.

Description

This function provides the fitting routine for the 4-parameter vitality model. Intrinsic mortality is characterized by the mean (r) and variability (s) in the vitality loss rate and the standard deviation of initial vitality (u). Extrinsic mortality is characterized by the frequency (k) of lethal random challenges. Model is appropriate to animal mortality data (e.g. Li and Anderson 2009)

Usage

vitality.ku(time, sdata, rc.data=F, se=F, gfit=F, datatype="CUM", ttol=.000001, 
  init.params=F, lower=c(0,-1,0,0), upper=c(100,100,50,50), pplot=T, tlab="days", 
  lplot=F, cplot=F, Iplot=F, silent=F, L=0)

Arguments

Value

vector of final MLE r, s, k, u parameter estimates. standard errors of MLE parameter estimates (if se = <population> is specified).

References

• Li, T. and J.J. Anderson. (2009). "The vitality model: A way to understand population survival and demographic heterogeneity." Theoretical Population Biology 76: 118-131.

Examples

data(rainbow_trout_for_k)
time <- rainbow_trout_for_k$days
survival_fraction <- rainbow_trout_for_k$survival

results.modku <- vitality.ku(time = time,
                             sdata = survival_fraction,
                             rc.data=TRUE,
                             se=FALSE,
                             gfit=FALSE,
                             datatype="CUM",
                             ttol=.000001,
                             init.params=FALSE,
                             lower=c(0,-1,0,0),upper=c(100,100,50,50),
                             pplot=TRUE,
                             tlab="days",
                             lplot=TRUE,
                             cplot=TRUE,
                             Iplot=TRUE,
                             silent=FALSE,
                             L=0)