exactLTRE_fixed {exactLTRE}

Exact LTRE analysis: fixed design

Description

Life Table Response Experiments (LTREs) are a method of comparative demographic analysis. The purpose is to quantify how the difference or variance in vital rates (stage-specific survival, growth, and fertility) among populations contributes to difference or variance in the population growth rate, "lambda." The exact method of LTRE is based on the principles of functional ANOVA. The equations and descriptions for the exact method will be published in a forthcoming paper, which we will link to.

Usage

exactLTRE_fixed(Aobj, maxint = "all", fixed.directional = FALSE)

Arguments

Aobj

An object containing all the population projection matrices to be included in the analysis. It should either be a list, or a matrix where each row is the column-wise vectorization of a matrix. For fixed design, exactly 2 matrices must be provided, ordered as ⁠[reference matrix, treatment matrix⁠].

maxint

The maximum interaction order to be evaluated. The default input is "all" but this input can take any integer value. If maxint=3, then the output will include contributions terms up to 3-way interactions.

fixed.directional

A true/false switch that allows the user to specify whether a directional LTRE should be used. The default behavior is to calculate a symmetric LTRE, where the mean matrix is used as the baseline. See details for more guidance.

Details

Lambda is the asymptotic population growth rate, defined as the largest eigenvalue of the population projection matrix. A fixed design LTRE decomposes the difference in lambda due to differences at each position of the matrices. For a fixed design LTRE, exactly 2 matrices must be provided, ordered as ⁠[reference matrix, treatment matrix⁠].

fixed.directional=FALSE is most appropriate for comparisons where it is not entirely obvious which population should be the reference and which should be the treatment (for example, when comparing a wet and a dry year). In this case, the difference in lambda is decomposed using the mean matrix as the baseline. The decomposition is symmetric, meaning that if the treatment and reference matrix are swapped, the contributions from the vital rates will be equal in magnitude, but positive contributions will become negative and vice versa.

fixed.directional=TRUE is most appropriate for comparisons between a control and treatment population in a controlled experiment. In this case, the reference matrix is treated as the baseline. This is a directional analysis, meaning that if the reference and treatment matrices were to be swapped, the contributions of the vital rates would change.

We set fixed.directional=FALSE as the default behavior because most population projection models are built with field-collected data rather than controlled experiment data.

Value

This returns a list object, with 3 items: (1) a vector of the matrix indices where the parameters vary between/among the matrices in Aobj; (2) a list of the indices varying for each of the contribution terms provided; (3) a vector of the contribution terms. For fixed design LTRE these are contributions to the difference in lambda.

indices.varying is a vector with the indices of parameters that vary. The numeric indices count down the columns of a given population projection matrix. For example, in a 3x3 matrix, the (2,2) position would be identified with a 5.

varying.indices.list is a list object, where each entry is a vector containing the indices (matching the indices.varying part of the output) that differed or varied for the corresponding entry in the epsilon vector.

epsilon is a vector of contributions to the difference in lambda due to the observed values of the various life history parameters. For example, the contribution to the difference in lambda of adult survival is determined by setting all parameters except adult survival to their mean values, and then calculating the difference in lambda in this manipulated set of matrices.

Examples

A1<- matrix(data=c(0,0.8,0, 0,0,0.7, 5,0,0.2), nrow=3, ncol=3)
A2<- matrix(data=c(0,0.9,0, 0,0,0.5, 4,0,0.3), nrow=3, ncol=3)
A3<- matrix(data=c(0,0.4,0, 0,0,0.6, 6,0,0.25), nrow=3, ncol=3)
cont_diff<- exactLTRE_fixed(list(A1,A2), maxint="all") # contributions to the difference in lambda
cont_diff<- exactLTRE_fixed(list(A1,A2), maxint=2) # only first- and second-order terms

# if A1 represents a control and A2 is a treatment:
cont_diff<- exactLTRE_fixed(list(A1,A2), maxint="all", fixed.directional=TRUE)