DOI: 10.5061/dryad.7h44j0ztw
title: Source code for R tutorials and dataset for empirical case study on Malurus elegans (red-winged fairy wren)

description: Biological processes exhibit complex temporal dependencies due to the
 sequential nature of allocation decisions in organisms’ life-cycles,
 feedback loops, and two-way causality. Consequently, longitudinal data
 often contain cross-lags: the predictor variable depends on the response
 variable of the previous time-step. Although statisticians have warned
 that regression models that ignore such covariate endogeneity in time
 series are likely to be inappropriate, this has received relatively little
 attention in biology. Furthermore, the resulting degree of estimation bias
 remains largely unexplored. We use a graphical model and numerical
 simulations to understand why and how regression models that ignore
 cross-lags can be biased, and how this bias depends on the length and
 number of time series. Ecological and evolutionary examples are provided
 to illustrate that cross-lags may be more common than is typically
 appreciated and that they occur in functionally different ways. We show
 that routinely used regression models that ignore cross-lags are
 asymptotically unbiased. However, this offers little relief, as for most
 realistically feasible lengths of time series conventional methods are
 biased. Furthermore, collecting time series on multiple subjects–such as
 populations, groups or individuals—does not help to overcome this bias
 when the analysis focusses on within-subject patterns (often the pattern
 of interest). Simulations (R tutorial 1 & 2), a literature search
 and a real-world empirical example on fairy wrens (data archived here with
 analyses presented in R-tutorial 3) together suggest that approaches that
 ignore cross-lags are likely biased in the direction opposite to the sign
 of the cross-lag (e.g. towards detecting density-dependence of vital rates
 and against detecting life history trade-offs and benefits of group
 living). Next, we show that multivariate (e.g. structural equation) models
 can dynamically account for cross-lags, and simultaneously address
 additional bias induced by measurement error, but only if the analysis
 considers multiple time series. We provide guidance on how to identify a
 cross-lag and subsequently specify it in a multivariate model, which can
 be far from trivial. Our tutorials with data and R code of the worked
 examples provide step‐by‐step instructions on how to perform such
 analyses. Our study offers insights into situations in which cross-lags
 can bias analysis of ecological and evolutionary time series and suggests
 that adopting dynamical models can be important, as this directly affects
 our understanding of population regulation, the evolution of life
 histories and cooperation, and possibly many other topics. Determining how
 strong estimation bias due to ignoring covariate endogeneity has been in
 the ecological literature requires further study, also because it may
 interact with other sources of bias.
description: The data was part of a long-term study on red-winged fariy wrens
 (Malurus elegans) in South-west Australia (Pemberton) from 2008-2016. In
 each year data was collected on group size, offspring production and
 survival of all group members. See description in Box 4 in the associated
 paper, and references therein.
description: Tutorials (Rmarkdown files), R function (R-file) and empirical
 data (asci-text file) associated with the paper (5 files in
 total). Tutorial1.rmd shows estimation bias in
 simulated dataset (see Box 1 & 2 in paper for details).
 Tutorial2.rmd illustrates bias due to measurement error and how
 to account for it (see Box 3 in paper for details).
 Tutorial3.rmd explains how to analyze the real-world case study
 of group living benefits in red-winged fairy wrens (see Box 4 in paper for
 details). Melegans.txt contains the emprical data for
 red-winged fairy wrens (Malurus elegans) for each of the 108 groups
 (SubjectID) across 9 years (time). Presented are 698 values for adult
 group size (GroupSize), the number of surviving adults till the next year
 (Survivors) and the group productivity in terms of number of offspring
 produced in a year that survives till the next year (Offspring), and their
 1-timestep-lagged variables (OffspringLagged & GroupSizeLagged,
 with LaggedUnavailable=1 meaning missing lagged value). There are no
 further missing values, see description in Box 4 in paper and references
 therein for details. simulation_functions.R contains
 the R functions used in Tutorials 1-3.