boot.lmf: Bootstrap resampling for class "lmf" In lmf: Functions for estimation and inference of selection in age-structured populations

Description

Generates bootstrap replicates of the estimated parameters in a "lmf" model. Ordinary bootstrap is performed for the projection matrix, while both parametric and ordinary (non-parametric) resampling is available for the remaining parameters in the model. In addition, bootstrapping under any choosen null hypothesis is available for hypothesis testing.

Usage

 1 2 3 4 5 boot.lmf(object, nboot = 1000, what = c("projection", "alpha", "H0", "all"), asim = c("ordinary", "parametric"), sig.dj = TRUE, H0exp = list(alpha = NULL, M = NULL), H0con = c("fs", "nfs", "ds", "nds"), method = c("BFGS"), control = list(maxit = 500, reltol = sqrt(.Machine\$double.eps)), ...) 

Arguments

 object a fitted object of of class "lmf". nboot the number og bootstraps desired. what which set of parameters to bootstrap. Options are "projection" to only resample projection matrix, growth rate (λ), stable age distribution (u) and reproductive values (v). "alpha" to resample demographic and environmental variances as well as all the estimates selection parameters. "H0" to resample temporal coefficients of selection under a given null hypothesis (This options requires specification of the additional arguments H0exp and H0con). "all" (default) to resample all the above mentioned parameters (also here H0exp and H0con must be specified for hypothesis testing or only "projection" and "alpha" will be resampled). asim the type of bootstrap for the parameters other than the projection matrix (which is always ordinary bootstrapped). Options are "parametric" (default) and "ordinary". sig.dj logical, TRUE(default) to include uncertainty in the estimation of the demographic variance when bootstrapping alpha estimates. H0exp a list with the first element a vector containing the expected temproal mean coefficients of selection (alpha) and the second element a matrix containing the elements of the expected temporal variance-covariance matrix (M) under the null hypothesis. This argument needs to be specified to perform hypothesis testing. H0con the conditions under which the null hypothesis should be tested. Options are "fs" to assume fluctuating selection, "nfs" to assume no fluctuating selection, "ds" to assume directional selection and "nds" to assume no directional selection. "nds" is not implemented due to increased risk of Type I error if the assumption is not correct, but is included here for completeness. method defines what optimalization algorithm to be used in the maximization of the loglikelihood. Alternatives are: "Nelder-Mead", "BFGS" (default), "CG", "L-BFGS-B" and "SANN". Not all are applicable here. See ?optim for details. control a list of control parameters for the maximization of the likelihood. maxit sets the maximum number of iterations to use before convergence and reltol sets the relative threshold for improvement in the likelihood which desides whether to continue maximation or end. See ?optim for details. ... additional arguments to be passed to optim for the maximization of the loglikelihood. See ?optim for options.

Details

The resampling procedures preserve the observed ratios of the different age classes during resampling of the projection matrix.

Ordinary bootstrap will often be subject to bias due to few years of data in most available data sets within biology (generally << 40), thus the parametric bootstrap is recomended for most purposes.

The bootstrap procedure is closely associated with the method deployed in lmf and further details can be found in Engen et al. 2012.

Different from Engen et al. 2012, the sigma2.dj is defined as independent gamma distributed variables with shape = \frac{(EX)^2}{Var(X)} and rate = \frac{EX}{Var(X)}. Where X = \hatσ^2_{dj} and using the mean and variance from in the paper.

Value

boot.lmf returns a object of class "boot.lmf".

The function summary is used to obtain and print a summary of the bootstrap replicates and to print results from tests of hypotheses. For construction of confidene intervals for the parameters the function ci.boot.lmf is used.

An object of class "boot.lmf" is a list containing at most the following components:

 running.time the total time used for computation. optim.time the time used for maximation of the loglikelihood. call the matched call. asim the value specified of asim. nboot the number of bootstrap replicates generated. uage the unique age classes in the data set. nage the number of unique age classes in the data set. npar the number of parameters in the model. uyear the unique years in the data set. nyear the number of unique years in the data set. l the estimated projection matrix. lboot the bootstrap replicates of the projection matrix. lambda the deterministic multiplicative growth rate of the population. u the stable age distribution. v the vector of reproductive values for each age class. luvboot the bootstrap replicates of λ, u and v. sigma2.dj a list containing the demographic variance for each age class. Sorted by age class. djboot the bootstrap replicates of sigma2.dj. sigma2.d the total demographic variance of the population. dboot the bootstrap replicates of sigma2.d. Atboot the bootstrap replicates of the yearly variance-covariance matrices. The unscaled variance-covariance matrices are kept constant, but each set of yearly estimates are scaled by the bootstrapped sigma2.dj. atboot the bootstrap replicates of the yearly coefficients of selection. This can be performed "parametric"(default) or "ordinary". M the estimated temporal covariance matrix (fluctuating selection). aM the estimated temporal mean coefficients of selection. Mboot the bootstrap replicates of M. aMboot the bootstrap replicates of aM. atCboot the bootstrap replicates of the best linear predictor for the estimated yearly coefficients of selection (i.e. corrected for sampling errors). Anf the estimated temporal covariance matrix assuming no fluctuating selection. anf the estimated temporal mean selection coefficients assuming no fluctuating selection. Anfboot the bootstrap replicates of Anf. anfboot the bootstrap replicates of anf. sigma2.e the environmental variance of the population. eboot the bootstrap replicates of sigma2.e. eCboot the bootstrap replicates of sigma2.eC. H0aMboot the bootstrap replicates of aM under the specified null hypothesis H0exp and the assumption of fluctuating selection (Hexp = "fs"). H0anfboot the bootstrap replicates of anf under the specified null hypothesis H0exp and the assumption of no fluctuating selection (Hexp = "nfs"). H0atnfboot the bootstrap replicates of at under the specified null hypothesis H0exp and the assumption of directional selection (Hexp = "ds"). These bootstrap replicates are used to generate H0Mnfboot. H0Mnfboot the bootstrap replicates of M under the specified null hypothesis H0exp and the assumption of directional selection (Hexp = "ds").

Thomas Kvalnes

References

Engen, S., Saether, B.-E., Kvalnes, T. and Jensen, H. 2012. Estimating fluctuating selection in age-structured populations. Journal of Evolutionary Biology, 25, 1487-1499.

lmf, ci.boot.lmf
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 #Data set from Engen et al. 2012 data(sparrowdata) #Fit model lmf.1 <- lmf(formula = cbind(recruits, survival) ~ weight + tars, age = age, year = year, data = sparrowdata) #Bootstrap parameters b.1 <- boot.lmf(object = lmf.1, nboot = 10, sig.dj = TRUE, what = "all", asim = "parametric") #Print b.1 #Summary summary(b.1) #View density plots plot(b.1) #Test of hypoteses b.2 <- boot.lmf(object = lmf.1, nboot = 10, sig.dj = TRUE, what = "H0", H0exp = list(rep(0, 3), matrix(0, ncol = 3, nrow = 3)), asim = "parametric") #Summary summary(b.2)