title: Effects of sample size on estimates of population growth rates calculated with matrix models.
abstract: BACKGROUND
  Matrix models are widely used to study the dynamics and demography of populations. An important but overlooked issue is how the number of individuals sampled influences estimates of the population growth rate (lambda) calculated with matrix models. Even unbiased estimates of vital rates do not ensure unbiased estimates of lambda-Jensen's Inequality implies that even when the estimates of the vital rates are accurate, small sample sizes lead to biased estimates of lambda due to increased sampling variance. We investigated if sampling variability and the distribution of sampling effort among size classes lead to biases in estimates of lambda.
  
  
  RESULTS
  Using data from a long-term field study of plant demography, we simulated the effects of sampling variance by drawing vital rates and calculating lambda for increasingly larger populations drawn from a total population of 3842 plants. We then compared these estimates of lambda with those based on the entire population and calculated the resulting bias. Finally, we conducted a review of the literature to determine the sample sizes typically used when parameterizing matrix models used to study plant demography.
  
  
  CONCLUSIONS
  We found significant bias at small sample sizes when survival was low (survival = 0.5), and that sampling with a more-realistic inverse J-shaped population structure exacerbated this bias. However our simulations also demonstrate that these biases rapidly become negligible with increasing sample sizes or as survival increases. For many of the sample sizes used in demographic studies, matrix models are probably robust to the biases resulting from sampling variance of vital rates. However, this conclusion may depend on the structure of populations or the distribution of sampling effort in ways that are unexplored. We suggest more intensive sampling of populations when individual survival is low and greater sampling of stages with high elasticities.
authors.affiliation: Department of Wildlife Ecology & Conservation, University of Florida, Gainesville, Florida, USA. ianfiske@gmail.com
authors.forename: Ian J
authors.initials: IJ
authors.lastname: Fiske
authors.forename: Emilio M
authors.initials: EM
authors.lastname: Bruna
authors.forename: Benjamin M
authors.initials: BM
authors.lastname: Bolker