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The authors have declared that no competing interests exist.

Conceived and designed the experiments: MB, MM, BB. Performed the experiments: MB, MM, BB. Analyzed the data: MB, MM, BB. Contributed reagents/materials/analysis tools: MB, MM, BB. Wrote the paper: MB, MM, BB.

Current address: Institute for Evolutionary Biology and Environmental Studies, University of Zurich, Zurich, Switzerland.

Current address: Department of Biology, East Carolina University, Greenville, North Carolina, United States of America.

Current address: Department of Mathematics & Statistics and Department of Biology, McMaster University, Hamilton, Ontario, Canada.

In most ecological studies, within-group variation is a nuisance that obscures patterns of interest and reduces statistical power. However, patterns of within-group variability often contain information about ecological processes. In particular, such patterns can be used to detect ^{2} = 0.5 in a cohort of 100 individuals measured on 16 occasions). We present a case study of growth in the red-eyed tree frog. Better quantification of the processes driving size variation will help ecologists improve predictions of population dynamics. This work will help researchers to detect growth autocorrelation in cases where marking is logistically infeasible or causes unacceptable decreases in the fitness of marked individuals.

Ecologists and evolutionary biologists have long been interested in growth in body size. Studies of growth typically focus on differences among means of populations or treatment groups, striving for low variability around the mean to increase statistical power; variation within groups is often treated as noise obscuring the phenomena of interest. However, ecological studies are increasingly considering among-individual variation as either a treatment or a response variable

Many ecological and evolutionary processes depend on body size

Among-individual variation also modifies population and community dynamics

Three main growth processes lead to growth depensation (increasing size variation within a cohort through time): within-individual variation in growth rate, among-individual variation in growth rate (i.e., positive growth autocorrelation), and size-dependent growth (

A - Patterns of growth rate Growth rates of five individuals are represented in each graph by five separate lines. B - Patterns of size variation Average cohort variance in 2000 cohorts of 50 indviduals: mean (solid line) and 2.5% and 97.5% quantiles (grey ribbons).

Within-individual variation in growth rate occurs when environmental heterogeneity causes uncorrelated temporal variation in individuals' growth rates through time. Here, we take the pattern of growth depensation caused by within-individual variation to be the null expectation.

Among-individual variation in growth rate, or positive growth autocorrelation, is defined as positive temporal correlation in the growth rate of individuals. Many ecological processes can generate positive growth autocorrelation. The proactive and reactive behavior types discussed above generate permanent autocorrelation (autocorrelation that applies throughout the entire life stage), as individuals consistently express the same behavior pattern

Size-dependent growth, where larger individuals have higher expected growth rates, can result from size-dependent gape limitation or size-dependent range size in animals. In plants, size-dependent growth often results from size-dependent resource uptake and asymmetric competition

These three classes of mechanisms lead to different patterns of variation among individuals in a cohort through time (

Because methods to separate the contributions of all three mechanisms acting simultaneously would be both complicated and data-hungry, we focus on the relative contribution to growth depensation of within- and among-individual variation in growth. We thus assume that individual growth rates are independent of size, or equivalently that a cohort's mean body size grows linearly through time. Although this assumption may seem restrictive, many organisms grow approximately linearly in size over some window in their ontogeny

In the past, teasing apart the relative importance of within- and among-individual variation in growth for growth depensation has required scientists to mark individuals and follow each individual's growth pattern

Our first “data” set is simulated growth data for a range of experimental designs (number of evenly spaced sampling times, number of individuals sampled across times) and growth parameters (increase in variance, ^{2}; all other parameters can be set to 1.0 without loss of generality). The increase in variance (_{t}

Growth autocorrelation was simulated by assigning each individual a normally distributed mean growth rate with mean _{t}_{t}_{t}

We also analyzed data from an experiment designed to quantify density dependent growth for red-eyed treefrogs (

Permission to conduct this research in Panama was granted by Autoridad Nacional del Ambiente de Panamá (permiso no. SE/A-41-08) and the Smithsonian Tropical Research Institute (STRI). This research was conducted under Boston University Institutional Animal Care and Use Comittee (IACUC) protocol 08-011 and STRI's IACUC protocol 2008-04-06-24-08.

We first derive the equations for the changes in cohort variance over time as a function of average growth rate, total variance in growth rate, and level of growth autocorrelation (non-technical readers can safely skip this section). We then discuss our protocol for simulating cohort growth dynamics to test the statistical power of our approach and summarize the practical aspects of the estimation procedure for researchers interested in applying the method to their own data. We compare our method to standard repeated measures methods that are available only when individuals are marked. Finally, we add size-dependent mortality to the data simulations and describe its effects on parameter estimates.

Suppose that individuals in a cohort grow linearly with mean growth rate

The cohort's size variance increases quadratically through time when ^{2}>0:

Without loss of generality, we scale the units of

Because our model assumes only process and not measurement error, we fit the parameters by step-ahead prediction, equivalent for a normal response to fitting the between-step changes in variance as independent and normally distributed values with mean

For our simulated data sets, we used the basic approach above and calculated statistical power (fraction of the time that the null hypothesis of

Our red-eyed treefrog data showed clearly nonlinear (and decelerating) patterns of increasing size over time, We fitted a saturating-exponential model (size =

A step-by-step protocol for quantifying growth autocorrelation with our method is as follows: (1) Confirm that the mean growth rates of the cohort are roughly independent of the mean body size (or equivalently that growth is approximately linear), transforming the data (e.g. by taking logarithms, or fitting a growth model and applying the inverse growth-curve function as described above) if necessary. (2) Calculate the cohort's variance at each time step and take the differences to find the change in variance at each time step. (3) Estimate

When it is possible to mark individuals, more traditional repeated measures analyses can be used to estimate growth autocorrelation. To compare our method to repeated measures methods we fit a linear mixed model (LMM) to individual growth rates with a random effect of individual, using the same simulated data described above. We fit the model using ^{2} from the variance of the random effect of individual and the residual variance of the fitted model:

Our model assumes that all individuals survive throughout the experiment. However, this assumption may be violated in experimental and especially in observational studies. The worst-case scenario is when individual mortality rates depend on size; we tested our method's performance in this scenario, specifically assuming that smaller individuals have a higher mortality rate (_{0}_{0}

Sampling more individuals improved point estimates and narrowed confidence intervals (^{2} were 0.07 to 0.13 units narrower for 6 time points compared to 32 time points). In simulations with 6 or 8 time points, confidence intervals contained the true value of

Estimates of ^{2} (solid lines), true values of ^{2} (dashed horizontal lines) and 95% confidence intervals (gray ribbons), averaged over 1000 replicates for each parameter combination. Number of time points _{t}

Our simulation results can be used to guide experimental designs for detecting growth autocorrelation in cohorts of unmarked individuals. Preliminary growth data from pilot lab or field studies, or data from the literature, can be used to guess an approximate ^{2}. Given this information, researchers can use

The line represents the minimum number of individuals in a cohort needed to statistically detect that ^{2} is greater than 0 at least 80 percent of the time, based on the 95% confidence intervals of 1000 simulations with _{t}^{2}<0.36, more than 512 individuals are needed, beyond the simulated range.

At a minimum, researchers will want to confirm whether observed growth depensation is the result of growth autocorrelation (i.e., to test the null hypothesis that ^{2} equal to 0.64, only 30 individuals are needed for 80% power (although with fewer than 50 individuals, estimates of ^{2} may be biased: ^{2} = 0.36 and ^{2} greater than 0.36 were ever distinguishable from zero with 80% power, regardless of sample size.

When individuals of the study species can be marked, traditional repeated-measures analyses can be used to estimate growth autocorrelation. As more individuals are sampled, both variance-pattern and repeated measures methods approach 100% power, although repeated measures power is always higher and increases more rapidly (^{2}.

Our method for detecting growth autocorrelation (dotted line) is less powerful than exact restricted likelihood ratio tests on linear mixed models fit to data on marked individuals (solid line).

In simulations incorporating size-dependent mortality, estimates of _{0}

While we were successful in linearizing the growth curves for the red-eyed treefrog data (^{2}>0.7 at best. When we fitted the model at the level of individual tank replicates (i.e. for each of 4 block-tank combinations in each resource-density combination), we found the confidence regions generally spanned the entire range of autocorrelation. In three cases (one replicate each at densities of 5, 25, and 50 per tank in the good-resource treatment) the 95% lower bound was greater than 0 (^{2}>0.35, 0.77, and 0.63 respectively), and in each of the cases the MLE was at ^{2} = 1.0. When we pooled the data either to the level of the treatment (density∶resource combination), or to a single overall data set, we were unable to make any definitive statements about growth autocorrelation despite the larger effective sample size, probably due to the variation within and among treatments. In principle it would be possible to try to fit random effects models to try to squeeze slightly more information out of the data set without pooling, but with only 4 replicates per treatment we suspect the data set would still be insufficient to make any definitive conclusions.

Points represent tank means; red lines are estimated Michaelis-Menten growth curves; blue dotted lines are estimated asymptotic sizes.

Color scale represents the fraction of individual measurements dropped because they exceeded the estimated asymptotic size for the treatment combination (and hence could not be used in our linearizing transformation).

Previous methods for detecting autocorrelation in individual growth trajectories require the marking of individuals, which is logistically or ethically infeasible in many ecological systems. We have shown that, with a large enough sample, one can detect growth autocorrelation observationally by analyzing the patterns of increasing variance in body size over time. This new technique allows researchers to choose how to best allocate their effort: they can sample more individuals without marking them (a cheaper and faster design) or mark fewer individuals (a design that gains more information per individual).

Our method does have some limitations – it is reliable only where mortality is relatively low (<5% based on simulations in

In circumstances where large numbers of organisms can easily be sampled and measured at repeated intervals through time, but the same individuals cannot be recovered or identified, our method should provide a reasonably powerful method for quantifying growth autocorrelation. Better quantification of the patterns and genesis of size variation will help improve management through better predictions of population dynamics as well as furthering ecologists' basic understanding of ecological systems.

^{2} = 0.25, 0.49, 0.81).^{2} for each simulation are plotted as grey dots. Red lines represent the true value of ^{2}. Blue lines summarize the simulations grouped by the total increase in size variation that would have been realized without mortality (^{2}). Smooth functions were fit with B-splines with five degrees of freedom.

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The authors thank the University of Florida High-Performance Computing Center (