Mathieu Buoro
email: mathieu.buoro@cefe.cnrs.fr

Olivier Gimenez
email: olivier.gimenez@cefe.cnrs.fr

Etienne Prévost
email: eprevost@st-pee.inra.fr

Assessing adaptive phenotypic plasticity by means of conditional strategies from empirical data: the latent environmental threshold model

Keywords: Bayesian modeling, Conditional strategies, Environmental Threshold Model, Measurement error, Phenotypic plasticity

Conditional strategies are the most common form of discrete phenotypic plasticity.

In a conditional strategy, an expressed phenotype is determined by the difference between an environmental cue and a threshold, both of which may vary among individuals. The Environmental Threshold model (ETM) has been proposed for understanding the evolution of conditional strategies. Surprisingly, the ETM has little been applied to empirical studies.

Here, we extending the ETM that allows assessment of conditional strategies from observational data. Our model accommodates likely situation where an observable aspect of the environment ('observable cue') is distinct from the unobserved proximate cue. First, we show that ignoring the observable vs. proximate cue distinction can lead to overestimate the variation in the threshold. Second, despite of identifiability issue, we show that the LETM allow to estimate the proximate cue and the threshold at individual level. Third, we propose the use of genetic data to cope with identifiability issues. Finally, we illustrate our approach with empirical data on the size-dependent smolting process for stream-dwelling Atlantic salmon juveniles in the Scorff River (Brittany, France). We argue that coupling our model with quantitative genetic methods could allow disentangling the genetic and environmental components of the threshold phenotypic variance.

INTRODUCTION

Phenotypic plasticity -the ability of a genotype to produce different phenotypes according to environmental conditionscan be an adaptive response of organisms to selection in stochastic environments [START_REF] Ghalambor | Adaptive versus non-adaptive phenotypic plasticity and the potential for contemporary adaptation in new environments[END_REF]. When plasticity refers to a discrete trait (e.g., maturation at given age, polymorphism in defensive structures, and alternative mating tactics), the concept of the conditional strategy [START_REF] Gross | Alternative reproductive strategies and tactics: diversity within sexes[END_REF]) is a popular framework to study the ability of a genotype to express alternative phenotypes (or tactics). [START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF] define the conditional strategy as a genetically determined decision rule containing a conditional clause. For binary traits, i.e. traits two possible categorical phenotypic states, the phenotype expressed by an individual may be dependent on an environmental cue and the choice between phenotypes may result from a physiological 'comparison' between the cue and a threshold (or switchpoint). [START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF] critically assess the Status Dependent Selection model (SDSM) initially formulated by [START_REF] Gross | Alternative reproductive strategies and tactics: diversity within sexes[END_REF] for representing a conditional strategy. They propose to replace the SDSM with the Environmental Threshold Model (ETM; [START_REF] Hazel | A polygenic model for the evolution and maintenance of conditional strategies[END_REF][START_REF] Hazel | The ecological genetics of conditional strategies[END_REF]). The ETM is an extension of the SDSM. First, the main difference between the two models lies in that the ETM assumes the threshold can vary among individuals according to their genotype. Second, the ETM facilitates the use of quantitative genetic theory for understanding the evolution of conditional strategies [START_REF] Hazel | The ecological genetics of conditional strategies[END_REF]).

In the ETM the threshold is considered as a trait that varies among individuals, and in this quantitative genetic framework the threshold is considered to be controlled by a trait that is modeled as continuously and normally distributed on some unobserved latent scale. In the ETM model, the threshold can thus be treated as a heritable trait where variation may come from multiple sources, including environmentally-induced variation and heritable differences among individuals [START_REF] Roff | Evolutionary Quantitative Genetics[END_REF]. At the individual level, the phenotype expressed results from the comparison between the cue and the threshold both in the SDSM and the ETM. At the population level, the proportion of the alternative phenotypes depends only on the cue distribution in the SDSM -the threshold is fixed -whereas it depends both on the cue and the threshold distributions in the ETM. The ETM formally distinguishes phenotypic plasticity from evolution of the phenotype proportions [START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF]. A change in the cue distribution immediately translates into a change in the phenotype proportions, therefore reflecting phenotypic plasticity. Evolutionary changes modify the threshold distribution and thus shift the phenotype proportions as well, and potentially independently of changes in the distribution of the cue.

In the context of environmental change, understanding how organisms can respond to varying environmental conditions is of particular importance. Understanding this potential for phenotypic change requires simultaneous consideration of the consequences of phenotypic plasticity and of evolutionary change, and in the current context, of the evolution of plasticity. The ability of the ETM to accommodate both phenotypic plasticity and its evolution is most appealing for assessing conditional strategies in the wild. Indeed, it allows addressing jointly in a single model the two mechanisms by which adaptation can occur, namely environmental cues that trigger phenotype expression and genetic variation that control phenotype expression. However, confronting the ETM with observation data collected in the wild requires embedding the ETM into a statistical model to deal with uncertainty. Although the thresholds and the parameters of their distribution are clearly not observable, they are conceptual and unknown quantities for which statistical inference can be derived from observed data. In contrast, the phenotypes expressed by individuals are observable. The status of the environmental cue is more ambiguous. Indeed, the proximate mechanism by which an individual assesses its environment to adjust its phenotype remains in most instances unknown [START_REF] Metcalfe | The interaction between behavior and physiology in determining life history patterns in Atlantic salmon (Salmo salar)[END_REF][START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF].

The environmental cue of the ETM should then be split in two distinct but related quantities: the proximate cue, for which the comparison with a threshold would trigger the phenotype expressed by an individual, is hidden; however it may be correlated with an observable cue which can be readily measured.

The proposed distinction between observable and (unobservable) proximate cue is supported by [START_REF] Ostrowski | Quantitative genetics of sexual plasticity: the environmental threshold model and genotype-by-environment interaction for phallus development in the snail Bulinus truncatus[END_REF] study on the snail Bultinus truncatus. They tested the ETM in a set of experiments where both the genotype and the environment were controlled. In contrast with what was expected under the ETM, they observed significant random variation in the phenotype expression for fixed environment X genotype combinations. They hypothesized a micro-environmental and uncontrolled variation in the threshold to explain this residual randomness. Avoiding the rather inconsistent statement that environment varies significantly even when it is experimentally fixed, we contend the proximate vs. observable cue dichotomy is a more sensible alternative hypothesis. In the case [START_REF] Ostrowski | Quantitative genetics of sexual plasticity: the environmental threshold model and genotype-by-environment interaction for phallus development in the snail Bulinus truncatus[END_REF]'s study, it would mean that the organisms assess temperaturethe observable environmental cue experimentally controlled -through an unknown proximate mechanism with some random "measurement error". The proximate cue would then only be correlated with temperature, which could be used as the observed cue. Note that any observable quantity traits can also be used as environmental cues. In fish, growth rate or size at a given age are known to be strongly influenced by the environment and are thus considered as integrating various environmental factors [START_REF] Hutchings | Norms of Reaction and Phenotypic Plasticity in Salmonid Life Histories[END_REF][START_REF] Dieckmann | Probabilistic maturation reaction norms: their history, strengths, and limitations. Disentangling the causes of maturation trends in exploited fish populations[END_REF]. Body size may be particularly useful as an observed cue, because it may integrate many important aspects of the environment an individual has experienced in a single tractable and obtainable measurement. However body size or any other cue should nonetheless still be distinguished from the proximate cue triggering the phenotype expression because there must be an (unknown) inner mechanism by which an organism gets informed about its size or growth rate, because a perfect correlation with the proximate cue is unlikely.

The purpose of this paper is to develop a statistical model extending the ETM that incorporates the proximate vs. observable cue distinction. It allows the estimation of latent variables of primary interest, i.e. the proximate cue and the threshold trait. We refer to this new model as the Latent Environmental Threshold Model (LETM). We first describe the model and show that ignoring the proximate vs. observable cue distinction leads to overestimation of the variance of the distribution of the threshold. This result is crucial since the potential for evolution of the conditional strategy directly depends on this the existence of this variability in the threshold and its heritability [START_REF] Hazel | The ecological genetics of conditional strategies[END_REF]). Second, we investigate the properties of our new model using simulated data. In particular, we explore the conditions in which the parameters of interest are, or can be rendered, identifiable. Particularly, we show that the use ofgenetic data render parameters of interest identifiable. Our LETM approach is applicable to any situation where the conditional strategy framework is relevant and when individual data are available for at least the alternative phenotypes involved and a related observable cue. Third, because Atlantic salmon is a conditional strategist (sensu [START_REF] Gross | Alternative reproductive strategies and tactics: diversity within sexes[END_REF] with respect to status-dependent alternative life-history tactics (migrating to sea or not, delaying reproduction or not), we illustrate our approach with a case study on the size-dependent smolting process for stream-dwelling juvenile Atlantic salmon in the Scorff River (Southern Brittany, France).

THE LATENT ENVIRONMENTAL THRESHOLD MODEL (LETM)

We use the notation A| B ~Dist f B to denote a set of random variables A distributed conditionally on the set of variables B according to a probability distribution Dist which parameters are a function f of B. Observable quantities are denoted with capital roman letters and unknowns with greek letters.

The LETM as an extension of the ETM model

For an individual i, the threshold modeling framework stipulates that if the value of a cue η i is larger (resp. lower) than a critical threshold θ i , then it triggers the expression of a phenotype, say migrant (resp. resident). If Y i is the binary variable indicator of the phenotype (e.g., 1 for migrant and 0 for resident), then we have:

Y i = { 1 if η i >θ i 0 if η i ≤ θ i } (1)
In line with the ETM, the threshold varies among individuals and is a polygenic quantitative trait which is normally distributed with mean μ θ and standard deviation σ θ , as typically assumed in quantitative genetics [START_REF] Hazel | A polygenic model for the evolution and maintenance of conditional strategies[END_REF][START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF][START_REF] Lynch | Genetics and Analysis of Quantitative Traits[END_REF]:

θ i | μ θ ,σ θ ~N μ θ ,σ θ (2)
where η i and θ i .is the value of the proximate cue and the threshold of individual i, respectively. The proximate cue η i is to be compared with the threshold θ i . η i is assumed to vary among individuals as a function of the environment, and θ i is considered an intrinsic property of the individuals. Although η i is not observable, an observable cue X i can be measured which is correlated with η i . Indeed, the distribution of the unknown proximate cue η i can be expressed conditionally on the observable cue X i with some error.

Specifically, η i is assumed to be normally distributed around X i with standard deviation σ η :

η i | X i ,σ η ~N X i ,σ η (3)
Note that it corresponds to the Berkson measurement error model in the statistical literature [START_REF] Congdon | Bayesian Statistical Modelling[END_REF]. This formulation has the advantage of making no assumption regarding the distribution of η i , therefore making the statistical analysis convenient since irrespective of the origin of the X i distributions.

We refer to the model defined by Eqns 1-3 as the Latent Environmental Threshold Model (LETM) since (i) in the statistical modeling terminology, both η i and θ i are latent variables [START_REF] Congdon | Bayesian Statistical Modelling[END_REF]) and (ii) it is an extension of the ETM which corresponds to the limit case where σ η is null, i.e., the proximate cue and the observable cue are equal (X i = η i ). The conditional structure of the LETM can be summarized by a Directed Acyclic Graph (DAG; Fig. 1).

[Figure 1 about here]

The latent data formulation of the LETM

In the LETM, we assume that Y i the binary variable indicator of the phenotype (e.g., 1 for migrant and 0 for resident) depends on continuous latent variables, i.e. the proximate cue and the threshold. To estimate the relevant parameters, we need to derive the distribution of Y i . First, Y i is distributed according to a Bernoulli distribution with probability p i :

Y i ~Bernoulli p i (4)
Second, using that the random variable

θ i -μ θ σ θ is distributed as a standardized normal
distribution N(0,1) (recall that θ i is normally distributed; Eqns 2), we have:

p i =Pr Y i = 1 =Pr η i >θ i =Pr η i -μ θ σ θ θ i -μ θ σ θ =F η i -μ θ σ θ ( 5
)
where F is the cumulative function of a standardized normal distribution. The inverse of F is often referred to as the probit function and is often used for parameters that lie in the unit interval i.e., it connects the dichotomous variable Y i and the corresponding continuous latent variable η i through the proportion p i . Third, conditionally on the proximate cue η i and integrating over the latent variable θ i , we obtain:

Y i |η i ,μ θ ,σ θ ~Bernoulli F η i -μ θ σ θ (6)
This formulation corresponds to the standard ETM formulation (e.g., [START_REF] Ostrowski | Quantitative genetics of sexual plasticity: the environmental threshold model and genotype-by-environment interaction for phallus development in the snail Bulinus truncatus[END_REF].

To make the distinction between the proximate and observable cues , the LETM formulation can be obtained using the same reasoning as above by conditioning on the observable cue X i and by integrating over the latent variables η i and θ i :

Y i | X i ,μ θ ,σ θ ,ση ~Bernoulli F X i -μ θ σ θ 2 +σ η 2 (7)
The comparison of the two formulations of the ETM and LETM (respectively Eqns [6] and

[7]) shows that confounding the proximate and the observable cue makes the variance component of the proximate cue falsely attributed to the variation in the threshold and artificially inflates the threshold variability. Given that the variability in the threshold is of primary interest for studying conditional strategy, ignoring the proximate vs. observable cue distinction could result in flawed inference regarding the potential evolution of the conditional strategy [START_REF] Hazel | The ecological genetics of conditional strategies[END_REF][START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF].

Identifiability issues

The LETM is a statistical model which makes the connection between the conditional strategy conceptual framework and observable quantities at the individual level. The unknowns that need to be estimated are both the individual latent variables (the proximate cue η i and the threshold θ i ) and the parameters controlling the distribution of these latent variables (the mean μ θ and the standard deviation σ θ of the individual threshold distribution and the standard deviation σ η of the proximate cue distribution around the observable cue X i ). However, when only the observable cues X i 's and the alternative phenotypes Y i 's are available, the LETM is not fully identifiable, that is not all parameters are separately estimable. More specifically, only the total variance

σ T 2 =σ θ 2 +σ η 2 is identifiable while the ratio κ=σ θ 2 / σ T 2
is not. The ratio  which corresponds to the proportion of total variance explained by the variability in the threshold is of primary interest because if it is not identifiable, then the variance of the threshold is not identifiable either. Indeed, from Eqns [5], the likelihood of the observations (Y i , X i ) conditionally on (μ θ , σ θ , σ η ) (and consequently σ T ) remains unchanged for any combination of (σ θ , σ η ). To circumvent this issue, additional information is required. When studying wild populations or when performing controlled experiments, genetic data may be available providing additional information about the structure of the θ i 's variability. Assuming the threshold is heritable [START_REF] Hazel | A polygenic model for the evolution and maintenance of conditional strategies[END_REF][START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF], batches of related individuals have the same threshold value, thus enhancing our ability to estimate the value of the threshold for each individual of each batch and the variability of the threshold in the population.

Consequently, knowledge of the individuals kinship or pedigree inform about the relative degree of similarity of the θ i 's between individuals and about the variability of the threshold θ in the population. We explore the introduction of the genetic information in the LETM and its influence on the estimation of parameters below (see "Simulation study" section and "Results" section).

Bayesian statistical inference

To fit our LETM to the data, we adopted a Bayesian approach using Markov chain Monte Carlo (MCMC) algorithms because it provides a flexible framework for analyzing latent variables models and their conditional structure (Clark 2005). The Bayesian approach combines the likelihood (i.e., information derived from the observed data) and the prior distribution of the unknown quantities (i.e., knowledge available before the data were observed) to produce a joint probability distribution of all model unknowns conditionally on the observed data, the so called joint posterior distribution (see [START_REF] Gelman | Bayesian Data Analysis, Second Edition[END_REF][START_REF] Ellison | Bayesian inference in ecology[END_REF][START_REF] Mccarthy | Bayesian Methods for Ecology Cambridge[END_REF] for more details about the Bayesian statistical modeling). If the prior and the posterior distributions of a given parameter look similar, , then there is not enough information in the data to estimate this parameter. The joint posterior distribution of all the model unknowns, i.e., the parameters (μ θ , σ θ , σ η ) and the individual thresholds and proximate cues (θ i , η i ), was obtained by means of MCMC sampling as implemented in the OpenBUGS software [START_REF] Spiegelhalter | WinBUGS user manual. Version 1.4[END_REF]. The OpenBUGS code of our model is available at (http://www.cefe.cnrs.fr/biom/zips/LETM_Buoro et al.txt). We ran 3 parallel MCMC chains and retained 50,000 iterations after an initial burn-in of 10,000 iterations. Convergence of MCMC sampling was assessed by means of the Brooks-Gelman-Rubin diagnostic [START_REF] Brooks | General methods for monitoring convergence of iterative simulations[END_REF]. A Bayesian analysis requires specifying prior probability distributions for the model parameters, i.e. the unknown quantities which are not conditioned by any other quantity in the model (μ θ , σ θ , σ η ; Fig. 1). The prior on the mean of the threshold distribution μ θ was a normal distribution with mean 0 and a large variance (1000). To make the assessment of identifiability issues easier, priors on the standard deviations σ θ and σ η were not defined directly as is usual but rather on the total variance σ T 2 and the ratio .

Because there is a one-to-one transformation relating ( σ T 2 ,) to (σ θ , σ η ), assigning a prior to ( σ T 2 ,) induces a prior on (σ θ , σ η ) as well [START_REF] Gelman | Bayesian Data Analysis, Second Edition[END_REF]). We used a scaled inverse-χ 2 with one degree of freedom for σ T 2 [START_REF] Gelman | Bayesian Data Analysis, Second Edition[END_REF]) and a uniform distribution between 0 and 1 for . The empirical variance of the observed cue X i was used to scale the prior on σ T 2 to the level of the observed cue.

SIMULATION STUDY

To evaluate the performance of the LETM, we carried out a simulation study. The data were generated according to the LETM using known parameter values and including a simple genetic structure for the θ i 's. Statistical inference was derived from the simulated data to check whether the LETM was able to provide accurate estimates of the known values of the parameters (μ θ , σ θ , σ η ) and of the individual latent variables (i.e., the thresholds θ i and proximate cues η i ). To assess the added value of bringing genetic knowledge about the θ i 's in the fitting process, inferences were successively conducted considering the genetic (batch) structure of the θ i 's was unknown and known. In the first instance, the model assumes the θ i 's are potentially all different. In the second instance, it is explicitly included in the model that the θ i 's are equal within a batch while being potentially different between batches (Eqn [8]).

We considered a set of 200 individuals assuming they were made of 20 batches of 10 individuals, each batch corresponding to the same genotype and a single threshold.First, 20 threshold values were generated for the 20 batches from a normal distribution with mean μ θ =0 and standard deviation σ θ =0.25 (Eqns [2]).

We then allocated these latent threshold values to the 200 individuals i so that for two distinct individuals i and i' :

B i =B i , ⇔θ i =θ i , ( 8
)
where B i and B i' denotes the batch number of the individuals i and i'. If individuals i and i' are from the same batch B i , then they have the same threshold value. It means that the variability of the threshold is exclusively genetic and the heritability of this trait is 1. Our aim here is not to identify the genetic component of the threshold trait but the structure of the variability in the threshold through the relatedness between individuals.

Second, we generated an observed cue X i value for each individual i by drawing from a normal distribution with mean 0 and standard deviation 1. For each individual i, given the value of the observed cue, X i , we generated its proximate cue η i from a normal distribution with mean X i and standard deviation σ η =0.5 (Eqn [3]). Finally, given the values of the proximate cue η i and of the threshold θ i , we assigned the phenotype indicator values Y i (Eqn [1]). Note that simulated data are much more variable in the proximate cue than in their threshold, as should usually be the case [START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF]. A graphical representation of the resulting data is provided in Figure 2.

[Figure 2 about here]

APPLICATION TO ALTERNATIVE LIFE HISTORY TACTICS IN ATLANTIC SALMON

Atlantic salmon is an anadromous species that shares its life cycle between freshwater and the ocean [START_REF] Guéguen | Costs, benefits and the evolution of inducible defences: a case study with Daphnia pulex[END_REF]. The juvenile phase takes place in freshwater and lasts one or two years in Brittany (Baglinière et al., 1993). Thereafter, fish migrate to the ocean and return after one or two years to their native stream to breed.

Atlantic salmon are conditional strategists with state-dependent choice among alternative life history tactics [START_REF] Thorpe | Is smolting a positive or a negative developmental decision?[END_REF]. During their first year of life in their natal river, young of the year (YOY; i.e., individuals less than one year old, counting from fertilization)

have to 'decide' whether to migrate to the ocean the next spring or to reside in freshwater an additional year [START_REF] Thorpe | Is smolting a positive or a negative developmental decision?[END_REF]. The choice between the migrant vs. the resident alternative tactics (i.e., phenotypes) is related to the size of the individuals at their first autumn [START_REF] Nicieza | Development of length-bimodality and smolting in wild stocks of Atlantic salmon, Salmo salar L., under different growth conditions[END_REF][START_REF] Thorpe | Is smolting a positive or a negative developmental decision?[END_REF]. Although size is an observable cue, it is probably best considered as a proxy for energetic status [START_REF] Thorpe | Is smolting a positive or a negative developmental decision?[END_REF], i.e. likely a more biologically proximate cue, which is to be compared to a threshold for triggering seaward migration the next spring [START_REF] Mangel | Combining proximate and ultimate approaches to understand lufe history variation in salmonids with application to fisheries, conservation and aquaculture[END_REF][START_REF] Thorpe | Is smolting a positive or a negative developmental decision?[END_REF][START_REF] Satterthwaite | Steelhead Life History on California's Central Coast: Insights from a State-Dependent Model[END_REF]). The individual energetic status influences this life-history choice [START_REF] Jonsson | Lipid energy reserves influence life-history decision of Atlantic salmon (Salmo salar) and brown trout (S. trutta) in fresh water[END_REF]) because migration to the ocean is preceded by the smolting process preparing individuals for sea water life which is energetically costly [START_REF] Mccormick | Movement , migration , and smolting of Atlantic salmon (Salmo salar)[END_REF][START_REF] Thorpe | Is smolting a positive or a negative developmental decision?[END_REF]. The energetic status reflects the way energy is acquired, stored and used and is strongly influenced by the environmental conditions experienced by each individual (e.g., food availability, temperature regime or density of fellow beings; [START_REF] Elliott | A functional model for maximum growth of Atlantic salmon parr, Salmo salar, from two populations in northwest England[END_REF][START_REF] Forseth | Functional models for growth and food consumption of Atlantic salmon parr, Salmo salar, from a Norwegian river[END_REF][START_REF] Imre | Density-dependent growth of young-of-theyear Atlantic salmon in Catamaran Brook, New Brunswick[END_REF][START_REF] Jones | Seasonal patterns of growth, expenditure and assimilation in juvenile Atlantic salmon[END_REF][START_REF] Murphy | Evaluation of winter severity on growth of young-of-the-year atlantic salmon[END_REF]. Under the LETM, we consider migrant vs. resident (at one year of age) as alternative phenotypes, YOY size in autumn as an observable cue indicative of the individual energetic status, the proximate cue triggering phenotype expression.

Data collection

In autumn 2005, YOY juveniles were sampled by electrofishing at 39 stations along the main course of the Scorff. Every fish captured was measured (fork length, to the nearest 1mm) and individually marked with a Passive Integrative Transponder (PIT) tag (11 mm long, 2.2 mm in diameter) inserted into the peritoneal cavity according to the protocol described in [START_REF] Acolas | Laboratory experiment on survival, growth and tag retention following PIT injection into the body cavity of juvenile brown trout (Salmo trutta)[END_REF].

One year old seaward migrating juveniles (smolts) previously PIT tagged were identified during their downstream migration in the spring of 2006. They were captured at two successive traps located at the lower end of the river system below all sites where YOY were marked. At both facilities, their individual PIT tags were identified. Eventually, PIT tagged anadromous salmon were recaptured in 2007 and 2008 when returning into the Scorff river after one or two years at sea. They were sampled at the Princes Mill facility in a trap designed to catch upstream migrating adults. PIT tagged resident juveniles, i.e., future two years old smolts, were identified in autumn 2006 using sampling by electrofishing according to same protocol used for the YOY the previous year. Two year old smolts we also recaptured the following spring ( 2007) and were also identified by their PIT tags.

Here, we considered the set of YOY juveniles marked in autumn 2005 and recaptured later on (N=124). For each of them, we observed its alternative phenotype (migrant vs. resident) and its observable cue (fork length at first autumn).

Modeling

For each individual i, given the observed cue (fork length at first autumn) Fl i , the proximate cue η i (energetic status) was assumed to be normally distributed with mean Fl i and standard deviation σ η (Eqn [3]). For each individual i, the alternative phenotype indicator Y i (Eqn.

[1]) takes the value 1 if the individual migrates to sea at one year of age, and 0 if it stays an additional year in fresh water.

We considered two scenarios as in the simulation study. First, we assumed we have no information available about the structure of the threshold variability. Second, we assumed that YOY captured in the same station in autumn were closely related genetically and had the same threshold value. The latter hypothesis was unrealistic but was used as an (extreme) illustration of the likely greater genetic similarity of YOY salmon within a site than between sites. Indeed, YOY juveniles tend to stay close to their natal spawning nest [START_REF] Beall | Dispersal patterns and survival of Atlantic salmon (Salmo salar L.) juveniles in a nursery stream[END_REF][START_REF] Einum | Nest distribution shaping within-stream variation in Atlantic salmon juvenile abundance and competition over small spatial scales[END_REF][START_REF] Foldvik | Relating juvenile spatial distribution to breeding patterns in anadromous salmonid populations[END_REF].

RESULTS

Simulation study

The comparison of posterior to prior distributions suggested that the information contained in the data led to considerable updating of the prior distributions. Whether genetic information was included or not, the LETM properly estimated the threshold mean μ θ and the total variance σ T

2

; the true value of these parameters were close to their posterior median (fig. 3). As expected,  was not identifiable when the genetic data was lacking; its posterior distribution was the same as its prior distribution. When genetic information was included,  became identifiable as its posterior distribution was much narrower than the prior and the true value was very close to the posterior median (Fig. 3).

The standard deviations of the threshold σ θ and of the proximate cue σ η were poorly estimated without the genetic information. Although the true values fell within their corresponding 95% PPI (Posterior Probability Interval (PPI) is defined as the posterior probability that the true value of the parameter lies in an interval of two given values with probability 0.95; [START_REF] Congdon | Bayesian Statistical Modelling[END_REF], they were at the edge of the posterior inter-quartile range. The posterior distributions of σ θ and σ η were the same, reflecting the essentially uniform posterior distribution of . With the inclusion of genetic information, the posterior distributions of σ θ and σ η were substantially smaller, and the posterior medians coincided with the true values. Moreover, the posterior distributions of σ θ and σ η were well contrasted indicating that the parameters were identifiable, given the inclusion of genetic information.

[Figure 3 about here]

Whether genetic information was included or not, the proximate cue η i and the threshold θ i estimates were estimated with little bias, the latter falling within or being close to the 95% PPI (Fig. 4). The uncertainty in the threshold estimates was much reduced when integrating genetic information while the proximate cue estimates remained almost unchanged.

[Figure 4 about here]

Case study

Using the LETM framework, we were able to obtain precise estimates of the mean latent threshold μ θ and the total variance σ T 2 (Fig. 5). The posterior distributions of these parameters were little affected by the addition or exclusion (pseudo) genetic information.

The ratio  was identifiable only when the genetic information was included, although it remained imprecisely estimated. In this case, the variance of the threshold corresponds approximately to 30% of the total variance σ T

2

. Recall that σ T 2 would have been equated

with the threshold variance if the observed vs. proximate cue distinction was ignored. The posterior distributions of standard deviations of the proximate cue σ η and of the threshold σ θ were narrowed down when incorporating the (pseudo) genetic information. The posterior distributions of σ θ and σ η were well contrasted indicating that the identifiability issue was alleviated.

[Figure 5 about here]

Estimates of the proximate cue and of the threshold at the individual level were also obtained (Fig. 6). YOY salmon appeared much more variable in the proximate cue than in their threshold. As for the simulated data, the uncertainty of the threshold estimates was reduced when integrating (pseudo) genetic information while the proximate cue estimates remained almost unchanged.

The proximate cue is a conceptual quantity and as such its scale is arbitrary. Here, given the measurement error structure of the LETM, its scale is the same as that of the observed cue. Then a proximate cue of say 90 can be interpreted as the mean energetic status of a YOY of 90 mm fork length in autumn. For the same reason, the mean threshold μ θ can be either interpreted as the energetic status (proximate cue; eq. [5]) or the fork length in autumn (observed cue; Eq. [6]) of a YOY salmon having equal odds to become migrant or resident.

[Figure 6 about here]

DISCUSSION

Conditional strategies are the most common form of discrete phenotypic plasticity (phenotype or tactics) within species [START_REF] Gross | Alternative reproductive strategies and tactics: diversity within sexes[END_REF]. Understanding how these strategies evolve and are maintained by natural selection is a challenge. [START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF] critically reviewed the theoretical models that have been proposed to understand the evolution of phenotypic plasticity in the conditional strategy framework. They argued that the ETM is "the best model available currently for understanding the evolution and maintenance of conditional strategies" because it accounts for both genetic variation and environmental cues that affect phenotype expression. Surprisingly, the ETM has little been used to study adaptive phenotypic plasticity either in the wild (with theoretical approach;

Edeline 2007) or under controlled experimental conditions [START_REF] Ostrowski | Quantitative genetics of sexual plasticity: the environmental threshold model and genotype-by-environment interaction for phallus development in the snail Bulinus truncatus[END_REF]. In this paper, we propose a statistical model extension of the ETM, the Latent ETM, which should make the assessment of conditional strategies from observational data easier. The originality of our proposal lies in the proximate vs. observable cue distinction which accounts for the underlying proximate mechanism triggering the phenotype expression (by means of a Berkson type measurement error structure). It allows the estimation of not only the parameters of the threshold distribution but also the proximate cue and the threshold at the individual level.

When data are available only for the alternative phenotypes and the observable cue, the LETM is not fully identifiable. However, this issue does not prevent the LETM from providing reliable estimates of the mean threshold and of the individual thresholds as

shown by our simulation study (Fig. 3 and4). To circumvent the identifiability issue, we recommend integrating supplementary information about the structure of the individual threshold variability. The threshold is considered as a heritable phenotypic trait being under polygenic control and can vary among individuals according to their genotype [START_REF] Tomkins | The status of the conditional evolutionarily stable strategy[END_REF]. The genetic component in the threshold variability has been evidenced by [START_REF] Ostrowski | Quantitative genetics of sexual plasticity: the environmental threshold model and genotype-by-environment interaction for phallus development in the snail Bulinus truncatus[END_REF] and [START_REF] Piché | Genetic variation in threshold reaction norms for alternative reproductive tactics in male Atlantic salmon, Salmo salar[END_REF] show that probabilistic maturation reaction norms vary genetically. Consequently, considering the genetic basis about the threshold trait, knowledge about the relative degree of similarity between individuals' genotype (by means of kinship or pedigree) could inform about the structure of the threshold variability. We simulated genetic data in this way, by having batches of individuals with the same threshold. We showed that using this additional information allows separation of the threshold variance from the proximate cue variance (Fig. 3 and5)

and can improve the precision of the individual threshold estimates (Fig. 4 and6).

Alternatively, the Bayesian approach with informative priors could be used to separate the threshold variance from the proximate cue variance [START_REF] Congdon | Bayesian Statistical Modelling[END_REF].

Wrongly ignoring the distinction between proximate vs. observable cue would lead to overestimation of the variability of the latent threshold since the variance component of the proximate cue is falsely attributed to the variation in the threshold and thus artificially inflates the threshold variability. Even when the threshold variance cannot be separated from the proximate cue variance, the LETM protects against overestimating the threshold variability while acknowledging a greater uncertainty in this key parameter. This variability conditions the ability of the conditional strategy to evolve under selective pressure [START_REF] Hazel | The ecological genetics of conditional strategies[END_REF]). Hence, the LETM can be seen as a cautious approach preventing erroneous inference regarding the potential for an adaptive evolutionary response of the threshold components of reaction norms to environmental change.

Within the conditional strategy framework, the LETM opens up interesting prospects in the study of phenotypic plasticity from observational data. Our new approach is a generic tool and could be applied to a wide range of taxa and to different forms of conditional strategies, including the induction of defences against predators [START_REF] Guéguen | Costs, benefits and the evolution of inducible defences: a case study with Daphnia pulex[END_REF], polyphenic traits in insects [START_REF] Moczek | Phenotypic plasticity and diversity in insects[END_REF][START_REF] Tomkins | Patterns of threshold evolution in polyphenic insects under different developmental models[END_REF], sex-ratio investment, filial cannibalism [START_REF] Takeyama | Filial cannibalism as a conditional strategy in males of a paternal mouthbrooding fish[END_REF]) and alternative reproductive tactics [START_REF] Piché | Genetic variation in threshold reaction norms for alternative reproductive tactics in male Atlantic salmon, Salmo salar[END_REF][START_REF] Pitnick | Size-dependent alternative male mating tactics in the yellow dung fly, Scathophaga stercoraria[END_REF][START_REF] Gross | Alternative reproductive strategies and tactics: diversity within sexes[END_REF]. For the sake of simplicity, only binary traits were considered in this paper. However, the extension to more than two phenotypes can be envisaged by considering multiple thresholds [START_REF] Gianola | Theory and analysis of threshold characters[END_REF][START_REF] Sorensen | Bayesian inference in threshold models using Gibbs sampling[END_REF].

Further development of the means for including additional information, including genetic data, will be highly profitable. The LETM could be adapted to take advantage of the knowledge about the relatedness of individuals in a population in a more realistic way than we did in the simulation study. Pedigree data could be incorporated to account for the genetic structure of the threshold and consequently, to ensure model identifiability and to improve estimation precision. Coupling the LETM with quantitative genetic methods (such as the animal model; Kruuk 2004[START_REF] Wilson | An ecologist's guide to the animal model[END_REF]) would allow the estimation of the genetic and environmental components of phenotypic variance of the threshold (i.e., its 5 heritability) and a better assessment of the evolutionary potential of the conditional strategy. In a context of rapid and global environmental change, both evolution and plasticity can be critical for species adaptation [START_REF] Gienapp | Climate change and evolution: disentangling environmental and genetic responses[END_REF]. The joint appraisal of both phenomena from observational data is required, for which the use and further developments of the LTEM should help. Observable data are presented in squares and unknown quantities to be estimated are in circles. For an individual i, the threshold θ i is normally distributed with mean μ θ and standard deviation σ θ . Its proximate cue η i is assumed to be normally distributed around the observable cue X i with standard deviation σ η . Finally, Y i is a binary indicator variable of the observed phenotype and is modeled as a joint function of the threshold and the proximate cue at the individual level, such that Y i =1 when θ i <η i ]. Solid and broken arrows stand for stochastic and logical dependence, respectively. The model is fit over observations at phenotyped individuals, i.e., the boxes denote a loop over i=1,2, ...… N.

Figure captions

FIGURES FIGURE 1

Figure 1 :

 1 Figure 1: Directed Acyclic Graph of the Latent Environmental Threshold Model (LETM).

Figure 2 :

 2 Figure 2: Representation of the simulated data for 200 individuals (circles). Upper panel:

Figure 3 :

 3 Figure 3: Posterior distributions of Latent Environmental Threshold Model (LETM)

Figure 4 :

 4 Figure 4: Posterior distributions of proximate cue η i and threshold θ i for one individual

Figure 5 :

 5 Figure 5: Posterior distributions of latent Environmental Threshold Model (LETM)

Figure 6 :

 6 Figure 6: Posterior distributions of proximate cue η i and threshold θ i for one individual

Acknowledgments

We thank N. Jeannot (INRA, U3E, Pont-Scorff), J. Rives, F. Lange, and F. Guéraud (INRA, Ecobiop, St Pée sur Nivelle), Y. Guilloux (Federation de pêche du Morbihan, Pont-Scorff) and other technical staff members for their help in collecting field data.

Mathieu Buoro and Olivier Gimenez were supported by a grant from the French Research 5

National Agency (ANR), reference ANR-08-JCJC-0028-01.


Jean-Dominique Lebreton

Olivier Gimenez

DETECTING AND ESTIMATING DENSITY DEPENDENCE IN WILDLIFE POPULATIONS

Keywords: Animal demography, Bayesian methods, Density-dependence, Kalman filter, Population dynamics, State-space models, Wildlife

We review methods for detecting and assessing the strength of density dependence (DD) based on two types of approaches: surveys of population size and studies of life history traits, in particular demographic parameters.

For the first type of studies, methods neglecting uncertainty in population size should definitely be abandoned. Bayesian approaches to simple state-space models accounting for uncertainty in population size are recommended, with some caution because of numerical difficulties and risks of model misspecification. Realistic state-space models incorporating features such as environmental covariates, age-structure, etc., may lack power because of the shortness of the time series and the simultaneous presence of process and sampling variability. In all cases, complementing the population survey data with some external information, in priority on the intrinsic growth rate, is highly recommended.

Methods for detecting DD in life history traits are generally conservative, i.e. tend to underestimate the strength of DD. Among approaches to correct for this effect, the state-space formulation of capture-recapture models is again the most promising. Foreseeable developments will exploit integrated monitoring combining population size surveys and individual longitudinal data in refined state-space models, for which a Bayesian approach is the most straightforward statistical treatment. One may thus expect an integration of various types of models that will make it possible to look at DD as a complex biological process interacting with other processes rather than in terms of a simple equation: modern statistical and modeling tools make such a synthesis within reach.

INTRODUCTION

The i-state philosophy (Diekmann 1985) describes demography at the individual level based on life-cycle stages. This philosophy naturally leads to describe death, fecundity, and more generally any transition between stages such as recruitment or dispersal, through per capita parameters. If such individual parameters are constant or at least stationary (in the sense of stochastic processes: affected by random effects that can be shifted over time, and, as such, in particular, not affected by a trend), the population growth is exponential, generally after damping of the effect of initial population structure, as well known for instance for matrix models [START_REF] Caswell | Matrix population models[END_REF] and their stochastic generalizations [START_REF] Tuljapurkar | Population dynamics in variable environments[END_REF]. The key paradigm of exponential growth in population dynamics (Turchin 1981) is thus intimately linked to the i-state philosophy.

It has been recognized for more than two centuries [START_REF] Malthus | An essay on the principle of population[END_REF]) that exponential growth is impossible in the long term (Turchin 1981), as exponential growth leads to large population sizes that unavoidably induce depletion of resources and in turn of individual performance. This is a broad definition of Density-Dependence (DD hereafter), as a phenomenon that explicitly has to do with resource and individual performance. In passing, as most authors in this context, we loosely use the word density for population size or local population size, e.g. in the term DD itself.

Density-Dependence became a key subject in population dynamics with the well known logistic growth curve [START_REF] Verhulst | Notice sur la loi que suit la population dans son accroissement (English translation)[END_REF]). One of the discrete time counterparts is the discrete time Gompertz model to mechanistic models such as, e.g., matrix models that translate individual parameters into population level consequences. There has been a continuous tension since the XIX th century between phenomenological and mechanistic approaches to density-dependence (see [START_REF] Krebs | Two paradigms of population regulation[END_REF][START_REF] Murdoch | Population regulation in theory and practice[END_REF]) and continuing discussions (e.g. [START_REF] Berryman | Limiting factors and population regulation[END_REF]) on the role of limitation by resource availability -an individual level phenomenonas supposedly opposed to regulation by DD, a population level result.

A key question is whether DD is present in a population, and if so, how to measure its intensity. As usual, even if one relies on some statistical significance for detection, one has indeed in mind some biological significance, i.e. the idea that DD is or is not negligible for the future fate of the population or relative to other sources of variation, and thus bears a relationship to its management [START_REF] Guthery | Density-Dependence: applications in widlife management[END_REF]. Indeed the measurement phase is often linked with the idea that population projections accounting for DD go one step beyond those based on constant parameters: there is still a strong assumption of stationarity in the projection but rather than "as if parameters were the same as up to now", it is "as if parameters were varying with density as they have been shown to do up to now". Estimating the strength of DD is also central in discussions of the relative role and magnitude of density-dependent and density-independent variation in population size, the subject of a famous controversy in the 50s [START_REF] Andrewartha | The distribution and abundance of animals[END_REF]Birch 1954, Lack 1954).

Not surprisingly, the tension between phenomenological / population and mechanistic / individual approaches has been pervasive in the issue of detection / estimation of DD, would it be only because the data themselves can be at the population level only (e.g. population surveys) or at the level of individual traits only (e.g. body weight data). Finally, statistical difficulties arise as the arrow of time induces dependencies that cannot be handled by naive statistical approaches. While a comprehensive review of the literature would be beyond reach, it is clear that the issue of detection / measurement of DD has alternatively raised optimistic and pessimistic points of view, and has been a subject of "frustration" [START_REF] Dennis | Density-dependence in time-series observations of natural populations -Estimation and testing[END_REF]. Can we take series of data and estimate in a simple fashion how many show densitydependence, as done by [START_REF] Brook | Strength of evidence for density dependence in abundance time series of 1198 species[END_REF] for 1198 species, or should we conclude with [START_REF] Krebs | Two paradigms of population regulation[END_REF] that DD is an unattainable holy grail?

The purpose of this paper is to attempt to review the subject of detection and estimation of DD with specific reference to these difficulties and the resulting confusing state of the art.

Although we will use many equations, as "if you are faced by a difficulty or a controversy in science, an ounce of algebra is worth a ton of verbal argument" (a conviction of J.B.S.Haldane, according to [START_REF] Smith | [END_REF], this review will attempt to avoid technical developments (for a more technical review, see [START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF]).

We will first review the methods for detection and estimation of DD based on population size estimates, in three steps: a) an attempt to formalize the role of resource and density, b) methods in the absence of uncertainty in population size, c) methods accounting for uncertainty in population size, a key feature for wild animal populations. Then we will review methods based on analyses of individual traits, demographic or not. We will then discuss new modeling opportunities and perspectives.

Calculations were done using our own Matlab ® code, carefully validated by a series of crosschecks, and WinBUGS [START_REF] Lunn | WinBUGS --a Bayesian modelling framework: concepts, structure, and extensibility[END_REF]. The Matlab ® code is made available as supplementary material.

DETECTING AND ESTIMATING DENSITY DEPENDENCE BASED ON

POPULATION SIZE ESTIMATES

The Gompertz model

t b t t N N N     1 becomes linear on a log scale. Denoting ) N ( Ln x t t  and ) ( Ln r   : t t x ) b ( r x     1 1 [1]
As recalled by [START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF], is the growth rate for , while the growth rate for is infinite. This bears no consequences on our treatment of this model and related ones, and we will speak of r as the intrinsic growth rate. Under 0  b , i.e. density independence, the model reduces to

t t x r x   1 [2]
As the world is not deterministic, a straightforward stochastic generalization considers some extra random variation to represent demographic and environmental stochasticity, leading under density-independence and density-dependence, respectively, to:

t t t x ) b ( r x       1 1 [3] t t t x r x     1 [4]
The random terms, represented by Greek letters, have expectations equal to 0 and, unless otherwise stated, are assumed to be normally distributed. The latter model is a random walk as, besides the shift r , t x varies through independent additive increments. Because of their simplicity, these models have been used by a number of authors over the years (see [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF][START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF]

t t t t x ) bx r ( x       1 [5]
As DD occurs through the depletion of resource of some kind, in the per capita growth rate, the log population size t x is a proxy for something else. Let's assume a single latent variable t l , such as the amount of a key resource left, is the actual determinant of the growth rate. Assuming t l can be expressed on the same scale as t x , the actual growth rate is then

t t t t t x ) ) x ( b r ( x         1 [6]
While model [5] (or equivalently [3]) is analogous to a linear regression of 1  t x on t x (to an extent that will be discussed later), model [6] is affected by an "error-in-variable" (see e.g. [START_REF] Fuller | Measurement error models[END_REF]). In practice, the latent variable t l is unknown and so is ) var( t  : model [6] cannot be readily used instead of [3]. What are the consequences of the error-in-variable problem on detection and estimation of DD when using [3]? Under H 0 0  b , model [6] reduces to model [4]. The distribution of any estimator of b under H 0 remains thus unaffected by the error-in-variable problem. So, although density-dependence is strictly a model concept, testing for it is equivalent to a test of dependence on resource depletion, at least in the simplistic setting considered here. As far as we know, the consequences on the estimated slope under H 1 0  b , i.e. the estimated strength of DD when present, remain to be explored.

t t t x ) r ( x      1 , i.e.

Methods for detecting DD in the absence of uncertainty on population size

Model [3] t t t x ) b ( r x       1 1
can easily be treated by maximum likelihood, preferably conditional on 1

x which plays no role in the estimation of b (Hamilton 1994 p.123;Dennis and Taper 1994 p. 209;[START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF]. Because of the linear structure of the model, the Maximum Likelihood Estimator (MLE) of b  1 is obtained by the formula for estimating the slope of the ordinary linear regression of 1  t x wrt t x . This apparent simplicity is quite unfortunate, as the estimator does not benefit from the properties of the regression estimator, because the underlying statistical model is not at all the usual linear regression model: the same variable appears both as independent and dependent variable, with a shift in time, i.e. we are in a so-called autoregressive model. In figure 1 . Samples of this distribution thus have on the average a major axis with slope 1. The regression estimate is then necessarily on the average below 1. As a consequence, the estimator of b is positively biased, i.e. the naïve approach tends to detect too often DD. Despite repeated warnings as early as the 70s (e.g. [START_REF] Maelzer | The regression of log(N t+1 ) on Log(N t ) as a test of density-dependence: an exercise with computer constructed density independent populations[END_REF], few people realize this bias is present indeed in the absence of uncertainty on population size (e.g. [START_REF] Freckleton | Census error and the detection of density dependence[END_REF], item 2 of summary) as can be seen for abscissa 0 in figure 2. Asymptotically, the estimate is not biased, just because the regression line in Figure 1 comes closer and closer to the line x r y   as the scatter of points become more and more elongated since

  ) x var( t when   t (Figure 1) while ) var( ) / var( 1 t t t x x    remains constant.
As, in studies of wild animal populations, the time series are usually fairly short, the asymptotic absence of bias offers little consolation.

Two approaches have been proposed to account for this bias: a)

Using simulation (i.e. parametric bootstrap) [START_REF] Dennis | Density-dependence in time-series observations of natural populations -Estimation and testing[END_REF]. Basically, one generates pseudo-samples using estimates under H 0 0  b and use the appropriate percentiles of the empirical distribution of the resulting estimates of b to accept or reject H 0 . In our model, this approach would amount to a bias correction, but [START_REF] Dennis | Density-dependence in time-series observations of natural populations -Estimation and testing[END_REF] use it with a slightly different, nonlinear, model in which the approach also overcomes the absence of distributional results.

b)

Obtaining an expression for the bias. [START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF], reformulating results by Saint-Amant (1970), develops a bias corrected t-test of H 0 0  b , and checks the resulting test-level is sufficiently close to the nominal  . A bias-corrected estimate can indeed be also directly derived from the general study of the bias of the autocorrelation coefficient by Kendall (1954, in Sawa 2002) for a time series of length T , as

T b b b ~3 4    .
Naive approaches are still commonly used (e.g. [START_REF] Saether | Generation time and temporal scaling of bird population dynamics[END_REF], Sibly et al. 2005, among others). For a particular study they can lead to grossly misleading results. In meta-analyses, the bias on the DD coefficient induces an overestimation of the prevalence of densitydependence. Based on the impossibility of long term exponential growth, one gets then a faked statistical confirmation of a plausible result! However the corrections to bias mentioned above do not appear to be a general solution to be recommended. Indeed, they largely amount to a useless statistical exercise, as the assumption of no uncertainty in t x is very stringent and far from met for most animal population surveys. We thus strongly recommend approaches not accounting for uncertainty in population size are abandoned. Handling this uncertainty is the subject of the next section. [START_REF] Bulmer | The statistical analysis of density-dependence[END_REF] was the first author to formulate the linearized Gompertz model with uncertainty in population size. In this formulation, the state of the population is described by model [3] above, and only a time series of estimated log-population size t y is observed: [START_REF] Bulmer | The statistical analysis of density-dependence[END_REF], in a direct treatment of the model, rewrote it as

Methods accounting for uncertainty in population size

t t t x b r x       ) 1 ( 1 [7] 1 1 1      t t t x y  [8]
t t t t t ) b ( y ) b ( r y             1 1 1 1 [10]
which clearly shows a further dependency over time induced by the presence of t  both in the equation for 1  t y and that for t y .

Neglecting it, i.e. treating t y by the previous regression-like method induces a severe bias [START_REF] Bulmer | The statistical analysis of density-dependence[END_REF]) that has rarely if ever been illustrated. , as presented in a clear and detailed way by [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF]. The MLE adequately exploits the statistical information available and is definitely preferable to any ad hoc estimate such as the regression estimate from the previous model or the explicit formulas proposed by [START_REF] Bulmer | The statistical analysis of density-dependence[END_REF]. The MLE has to be obtained numerically, with, according to [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF], special care because of potential multiple minima in the likelihood (see also [START_REF] Polansky | Likelihood ridges and multimodality in population growth rate models[END_REF]). This approach removes the substantial bias that would have been present by ignoring the uncertainty in population size (Figure 2). However, the MLE is again only asymptotically unbiased, i.e. suffers from some bias for realistic sample sizes. As in the absence of sampling error, some bias corrections could be obtained from the results on the autocorrelation coefficient bias, extended to autocorrelated random errors by Sawa (2002), based on the model presentation in equation ( 10).

Moreover, the distribution of the ML estimate under H 0 0  b cannot be easily obtained [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF]. It cannot thus be used without care for a test of DD. [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF] also discuss alternative estimation methods, based in particular on the first-order differences of the estimated log-population sizes t y . Knape and de Valpine (2011) go one step further with this approach by implementing a test of DD based on the likelihood ratio statistic between model [7] and its density-independent version, obtaining its distribution under H 0 0  b by parametric bootstrap, exactly as [START_REF] Dennis | Density-dependence in time-series observations of natural populations -Estimation and testing[END_REF] did for the model without uncertainty on population size. To avoid distributional problems with the test statistics, one can use a Bayesian approach, easily implemented in WinBUGS [START_REF] Lunn | WinBUGS --a Bayesian modelling framework: concepts, structure, and extensibility[END_REF] or in R via JAGS (Plumer 2003), and, e.g., use a 95 % credible interval as an admissible set of parameter values. Bayesian approaches also allow for flexibility in the functional form of the state equation -i.e. the "shape" of DD -and in the distribution of the random terms. These approaches currently seem the most reliable and the handiest ones, even if the underlying properties of the models, e.g., the degree to which the various parameters are separately identifiable do not depend on the estimation method and still remain often problematic.

Let us briefly examine the performance of these approaches, first by simulation. Deviance contours from simulated data (figure 3) show no evidence of local minima at the scale considered. However the estimates of r and b are strongly correlated for realistic lengths of the time series. They behave as the estimated intercept and slope of a regression when the dependent variable values are on a same side of the origin, to an even greater extent because of the two levels of uncertainty (process + sampling) and the dependency over time (Eq. Let us use as an illustrative example the spring population surveys of Greater Snow Geese Chen caerulescens atlantica in Quebec from 1973 to 2002 [START_REF] Gauthier | Population growth in Snow Geese: A modeling approach integrating demographic and survey information[END_REF]). The deviance did not show local minima (Figure 4): over 1000 iterative searches starting from random initial values, only 21 did not fully converge in 1000 iterations; 15 had indeed practically converged, and 6 were still not at the global optimum. They concerned searches starting from large sampling standard error and small process standard error, a combination mimicking a short, instable, time series.

The MLEs of r and b are highly correlated (0.9990), and the MLE of r , 0.3191, seems unrealistic (Table 2, second row) as it corresponds to a 37.6 % increase per year. These two features combined make the estimate of the coefficient of DD b suspicious.

The correlations between estimates (Table 1) emphasize the difficulty to separately estimate the sampling and process variance, and even more, r and b , that are indeed nearly not separately estimable. The latter point makes practical sense: similar population trajectories can arise as the result of a high intrinsic growth rate and a strong DD, or a weaker intrinsic growth rate and a low DD (see figure 5, below). The set of four parameters nearly appears as a set of two macro-parameters, one for the deterministic trajectory, the other one for the uncertainty. Based on our simulations (Figure 3) and a variety of unpublished examples, we think this situation is general. It immediately leads to consider the use of external information to improve the identifiability. Bayesian approaches using appropriate prior distributions on the parameters appear then as a necessity, rather than just an algorithmic convenience.

An estimate of the sampling standard error will often be available, e.g., when the population size estimates come from capture-recapture (e.g. Dennis and Oten 2000, who do not account for this uncertainty in their analysis), from stratified sampling (as it is the case with the Greater snow Goose), or more directly, through replicated population sampling (Dennis et al. 2010). However, because of the high correlation with the estimated coefficient of DD, and because of the unrealistic value often obtained, as in the snow Goose example, the main target for external information is certainly the intrinsic growth rate r . We strongly suspect that a Assuming in the snow Goose example a Gaussian prior distribution for r with mean 0.20 and standard error 0.06 (i.e. with 95 % of the density between 0.12 and 0.28) to constrain r to a realistic value for such a species (Niel and Lebreton 2005 Table 2) did improve the results

(Table 1, second and third row) and provided evidence of very weak DD, with nearly no biological significance.

The population size survey appeared as fairly precise, as the MLE of the standard error of the log population size, i.e. of its coefficient of variation is 0.0607, comparable in magnitude to the field estimate (10 % or less, [START_REF] Gauthier | Population growth in Snow Geese: A modeling approach integrating demographic and survey information[END_REF]Gauthier et al. p. 1422)). This fairly good precision limited the bias of the naïve regression approach (Table 1 first (2006) a "minimal model". The improvement brought by using a prior for the intrinsic population growth rate encourages refining the model and combining it with further pieces of external information, to improve its moderate performance. It gives a central role to Bayesian approaches, which also bring a great deal of flexibility. We do not think reasonable to consider as a fair evidence for DD a value of b ˆ significantly differing from 0 without checking if the associated r ˆ value makes sense. We currently recommend as a minimal step a careful look at the estimated intrinsic growth rate, and the use of a prior for r based on comparative demography approaches (e.g; Niel and Lebreton 2005).

From a minimal to realistic models

Besides the difficulties just mentioned, the minimal model above may be inappropriate in practice for a variety of reasons [START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF]. The most prominent issues are:

 Environmental covariates have to be taken into account to reduce the residual standard error and enhance power, and, if negatively autocorrelated, to distinguish their effect from that of DD (e.g. [START_REF] Lebreton | Modelling density-dependence, environmental variability, and demographic stochasticity from population counts: an example about Wytham Great tits[END_REF]).

 Age structure, e.g. with delayed recruitment, may require considering dependency over several time steps in components of population size.

 Further structure such as spatial cells may have to be considered, e.g. to model local density-dependence [START_REF] Murdoch | Population regulation in theory and practice[END_REF] or dependence on some components of population size.

 The response to density may nonlinear and more complex functional forms may have to be considered (e.g. [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF]. Comparative explorations of the functional form based on naïve regressions neglecting uncertainty in population size [START_REF] Sibly | On the regulation of populations of mammals, birds, fish, and insects[END_REF]) appear as highly questionable.

Neglecting any of these issues may result in a model misspecification, and thus in biases of the estimate of the intensity of DD and the corresponding tests or diagnostics of DD.

While in the original presentation by [START_REF] Bulmer | The statistical analysis of density-dependence[END_REF] the minimal model was presented as quite a specific statistical object, the state-space model point of view opens a number of perspectives [START_REF] De Valpine | Review of methods for fitting time-series models with process and observation error and likelihood calculations for nonlinear, non-Gaussian state-space models[END_REF]. For instance, it is relatively easy to generalize the state equation to account for an environmental covariate t z , as

t t t t cz x b r x        ) 1 ( 1 .
Good examples are provided by [START_REF] Jacobson | Climate forcing and density-dependence in a mountain ungulate population[END_REF] and [START_REF] Pasinelli | Impact of density and environmental factors on population fluctuations in a migratory passerine[END_REF]. It is also possible to adapt the model to particular life cycles. A good example of a model with a seasonal life cycle is provided by [START_REF] Stenseth | Seasonality, density dependence, and population cycles in Hokkaido voles[END_REF].

In all such models, the main potential bias of the slope for DD is removed by incorporating the population size uncertainty in the observation equation. However, the slope remains biased for finite sample sizes to an unknown degree. The likelihood for such models can be multimodal [START_REF] Polansky | Likelihood ridges and multimodality in population growth rate models[END_REF]) and the MLE as a consequence often difficult to obtain.

Parametric bootstrap for such realistic models (e.g., with age structure, with covariates) may be tedious and remain thus confidential until a specialized user-friendly piece of software is produced. REML approaches based on first order differences found promising by [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF] could be useful, but have still to be investigated in such a more complex setting. A Bayesian approach seems the most straightforward treatment. As an alternative to MCMC algorithms, the posterior distributions can be obtained through numerical integration (de [START_REF] De Valpine | Fitting population models incorporating process noise and observation error[END_REF], and explicitly in simple cases under Gaussian distributions.

An overview of methods based on time series of estimated population sizes

The first clear recommendation is to abandon all methods not accounting for uncertainty in population size, as they are unavoidably biased when used with real world, "noisy", estimates of population size. This conclusion, although developed here based on a simple model, applies to any such approach such as the regression of "observed growth rate" t t y y  1 vs observed population size t y as well illustrated by Freckleton et al. (2006 Figure 1c). The resulting bias is particularly vicious in meta-analyses, in which the estimated proportion of populations subject to DD will unavoidably be overestimated. This was suspected in the analysis by [START_REF] Brook | Strength of evidence for density dependence in abundance time series of 1198 species[END_REF] by [START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF] and demonstrated for these same data by Knape and De Valpine (2011). Brook and Bradshaw conclude to DD in more than 75% of the cases studied, while, accounting for uncertainty in population size estimates and using a Bayesian approach, Knape and de Valpine conclude to significant DD in only 16 % of the case studied. Similarly, based on reliable tools, [START_REF] Jamieson | Density dependence in North American ducks[END_REF] conclude to weak to moderate DD among American ducks. We still suspect part of these conclusions would not resist a close examination of the resulting intrinsic growth rate estimates.

The state-space model formulation provides a sensible approach to account for uncertainty in population size [START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF]) and to develop specific models accounting for environmental covariates, age structure etc.., and testing thus for DD in presence of such potentially confounding effects.

Two intermingled issues severely limit the potential of such models if used by themselves:

1) Numerical issues (multiple maxima to the likelihood or sensitivity to priors, depending on the type of algorithms used) and problems of bias associated with finite sample size which currently make such models difficult to use without specific help from a specialist;

2) Low power and identifiability problems as a result of the shortness of the time series and the two-level uncertainty, in particular if further complexities are brought in the model, at an unavoidable cost in terms of number of parameters to be estimated.

We would rarely expect evidence for DD with fewer than 30 points and we think external information on the intrinsic growth rate has to be seriously considered in any such analysis unless the grail to remain inaccessible.

The situation is somewhat similar to that of capture-recapture models for closed populations, whose status rapidly moved in the last few years from an innocuous fairly standard approach to that of delicate, specialized models with severe issues of bias and robustness [START_REF] Link | Nonidentifiability of population size from capture-recapture data with heterogeneous detection probabilities[END_REF]).

We will come back in greater detail in the general discussion to the potential of state-space models.

DETECTING AND ESTIMATING DENSITY DEPENDENCE BASED ON TRAITS

By contrast, one clear type of evidence for DD concerns the response of life history traits to changes in population size. Surprisingly, most such studies are observational correlative studies relating particular traits to observed changes in population size over time. Some qualify as quasi-experiments, as based on a clear population crash or explosion (e.g. Aslhey et al. 1998). Despite the current development of experimental ecology, few really manipulative experiments of densities have ever taken place. The few such studies with nest boxes [START_REF] Torok | Density dependence in reproduciton of the collared flycatcher (Ficedula alboicollis) at high population levels[END_REF]Toth, 1988, Alatalo and[START_REF] Alatalo | Density-Dependence in breeding success of the pied flycatcher (Ficedula hypoleuca)[END_REF], although quite convincing, do not take full advantage of an experimental setting, notably in terms of block design and replication (see also [START_REF] Newton | Experiments on the limitation of bird breeding densities -a review[END_REF]. [START_REF] Bartmann | Compensatory mortality in a Colorado Mule Deer population[END_REF] provide a good example of an experimental study of density dependent compensatory mortality in the Mule Deer with several replicates.

The traits investigated in a search for DD can be morphological, such as body weight (e.g. [START_REF] Gaillard | Body mass of roe deer fawns during winter in 2 contrasting populations[END_REF], and are then relatively simple to study. Demographic traits such as fecundity can be studied in a similar fashion (e.g. [START_REF] Arcese | Effects of population density and supplemental food on reproduction in song sparrows[END_REF]. Demographic traits less directly measurable, such as survival, require more sophisticated approaches such as capture-recapture models (Lebreton et al. 1992, Roe Deer C.capreolus;Catchpole et al. 2000, Soay Sheep). A full review of studies relating estimated population size to life history traits, whether demographic or not, is beyond the purpose of this paper. [START_REF] Bonenfant | Empirical Evidence of Density-Dependence in Populations of Large Herbivores[END_REF] provide a broad review of density-dependence in mammals, discussing trait response to density, as well as different responses by different segments of the population, such as males and females or age classes. [START_REF] Newton | Population limitation in birds[END_REF] reviews a number of trait -density or traitresource relationships in birds.

In all analyses of traits, one has to deal with some kind of linear model between the density or population size (possibly transformed, to log, to discrete categories…), t

x , and the trait at time t in individual j , tj z :

tj t t tj bx a z       [11]
The model may have to incorporate further effects, as the two-level sampling (individuals and years) raises some specific issues (whether for instance the same individuals are sampled over several years or not). Environmental covariates can easily be incorporated as additive effects [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies[END_REF] Clearly, any test on the slope will be conservative, i.e., contrary to methods based on population size, the presence and intensity of DD is not overestimated, which is good news. As we usually have not much control on the uncertainty in population size, several possibilities are available to account for the error-in-variable problem, of which most remain to be explored:

 Use an error-in-variable model [START_REF] Fuller | Measurement error models[END_REF]; however, all such models are weakly identifiable, through stringent distributional assumptions, as they exploit differences in distribution in the  and the  terms to estimate their relative effect on the variation in the response variable;

 Correct for bias. [START_REF] Barker | Measuring density dependence in survival from mark-recapture data[END_REF], develop such an approach in the context of capture-recapture survival models. They develop two examples with seabird data that provide no evidence for DD.

 Use information on the precision of population estimates, i.e. in the notation above an estimate (or a prior distribution) for ) var( t  to improve identifiability in the error-in- variable model.

 Recast the capture-recapture model as a state-space model [START_REF] Gimenez | State-space modelling of data on marked individuals[END_REF]) and consider besides the observation equations needed for representing the recapture process an observation equation

t t t x y   
. In such an approach, survival for instance is represented as state equations made of Bernoulli (0/1) random variables, and the detection/recapture process as observation equations also made of 0/1 random variables.

 Use instrumental variables, a tool commonly used in econometrics (e.g. Stock 2001).

An instrumental variable will be in our context a variable uncorrelated with the random term for estimated population size, t  and correlated with true population size. Alternative, independent estimates of population size could be good candidates (for an example using multiple surveys, see [START_REF] Fromentin | Effects of density-dependent and stochastic processes on the regulation of cod populations[END_REF].

The impact of the error-in-variable problem is clearly relatively limited, and, although some statistical care should be exercised, notably for what concerns the use of capture-recapture methodology, the assessment of DD based on traits is relatively straightforward.

DISCUSSION

The first part of our review, on methods for detecting DD based on population size surveys, leads first to several straightforward recommendations, to avoid the pervasive risk of overestimating the strength of DD. The first one is that methods neglecting uncertainty in population size should definitely be abandoned. The second is that the Bayesian approach to simple state-space models, such as the linearized Gompertz model, accounting for uncertainty in population size should be used, with some caution because of practical difficulties. We recommend also a reasonable prior based on external comparative information is used for the population growth rate. Even when these difficulties are correctly handled, which may require the assistance of an applied statistician with good knowledge of these models and their tricks, the simplest models may remain strongly misspecified because they neglect a number of features such as environmental covariates, age structure, particular functional form of DD, etc... Moving away from such minimal models is alike moving from simple regression to more sophisticated linear or nonlinear models: one cannot avoid a trade-off between the improvement brought by a better adaptation of a more complex model to the data (such as a decrease in residual variance), and the loss of precision implied by a greater number of parameters: modeling remains the art of oversimplification. Again, in this context, a Bayesian approach is probably the preferable approach, with caution exercised. In spite of their statistical and practical difficulties, methods based on population size surveys remain very attractive as they immediately translate into population projections. They are often the only practicable analyses of DD when population size surveys only are available, a common situation for managed wildlife populations. There is however little power for time series with fewer than, say, 30 time steps, i.e., in most cases, years. One has to recall that when the power drops down to values close to the test level , the analysis boils down to deciding for DD or no DD on the basis of a random number.

When moving from methods based on population size surveys to methods for detecting DD on traits, one is, in some sense, moving from pattern to process [START_REF] Swihart | Gray matters in ecology: dynamics of pattern, process and scientific progress[END_REF]. The most straigthforward methods for traits, such as using density as a covariate in capturerecapture models, are conservative, i.e. tend to underestimate the strength of DD, which means that the evidence for DD is reliable when present. Several perspectives to correct for the conservative effect of uncertainty in population size will probably be explored in the near future. The most promising is the use of the state-space formulation of capture-recapture models [START_REF] Gimenez | State-space modelling of data on marked individuals[END_REF]), completed by an observation equation for population size to properly model DD. While they can thus be used more confidently than approaches based on population size surveys, methods for DD in traits do not easily lead to population projections.

Even when the traits under study are demographic, they can only be translated into projections through a projection model such as a matrix model. [START_REF] Nevoux | Population regulation of territorial species: both site dependence and interference mechanisms matter[END_REF] examine the consequences of DD in different traits of the Mauritius Kestrel Falco punctatus by integrating these different traits and their DD relationships into a nonlinear discrete time model, studied independently from the trait analyses. In the absence of correction for attenuation, the results of such models should be looked at with a critical eye. Moreover, deterministic models do not account for the interplay of DD and various forms of stochasticity.

As the state-space formulation can easily encompass projection models (e.g. [START_REF] Gauthier | Population growth in Snow Geese: A modeling approach integrating demographic and survey information[END_REF]) and as it brings decisive advantages for assessing DD whether from population surveys or in studies of traits, it is clear that the future of DD modeling lies with state-space modeling.  Equations to iterate a population vector submitted to demographic stochasticity.

The corresponding observation equations are:

 Discrete variables such as Bernoulli for the capture-recapture process;

 Equations for the uncertainty in the observation for population size (as a vector or total number).

A model on these lines is proposed by Abadi et al. (submitted). [START_REF] Rotella | An evaluation of density-dependent and density-independent influences on population growth rates in Weddell seals[END_REF][START_REF] Lebreton | Assessing density-dependence: where are we left? Modeling demographic processes in marked populations[END_REF] go also one step in this direction by examining Weddell Seal population size estimates derived together with demographic parameters from a capture-recapture analysis within a stochastic Gompertz model. The demographic parameters are however not examined for DD and the estimated intrinsic growth rate is not given.

The first consequence of this evolution is that detecting and estimating DD has already started moving from a push-button procedure to a full size modeling exercise. A second consequence is that the gap between projection models and statistical models is progressively filled. In such a promising state of the art one may also expect a progressive integration of models based on mechanism at the individual levels, in the spirit of individual-based models, with state-space models derive from the classical phenomenological approach to population dynamics reviewed here (e.g., [START_REF] Stephens | Model complexity and population predictions. The alpine marmot as a case study[END_REF].

This evolution is fortunate as DD as a complex biological phenomenon has no reason to be uniformly reduced to a simple omnibus model, apart from a theoretical point of view that makes the logistic growth curve (Verhuslt 1838) so useful. We have to think of DD as a complex biological process interacting with other processes rather than in terms of a simple equation: modern statistical and modeling tools make such a synthesis within reach.

Legends of figures

  models are phenomenological, as they describe only the population level and do not explicitly consider the individual level, contrary

  , we illustrate what happens under H

  Figure 2, based on simulations, clearly shows how the faked evidence for DD (i.e. the bias in b ˆ) increases with the uncertainty in population size, over a range of realistic values of the coefficient of variation for the estimated population size.The ad hoc tests proposed by Bulmer were criticized by Den Boer and[START_REF] Reddingius | On the stabilization of animal numbers -problems of testing. 1. Power estimates and estimation errors[END_REF], in particular because they are not optimal, statistically speaking. However one clearly distinguishes in the model above a state equation[7] and an observation equation [8], i.e. a linear Gaussian state-space model, which can be treated by specific methods such as the Kalman filter, as noted by[START_REF] Lebreton | Statistical methodology for the study of animal populations[END_REF]. The Kalman filter makes it possible to obtain the likelihood of the model, based on a series of observations

  [10]) they induce. The estimate of the process and sampling standards errors, correlated, although to a lesser degree than the estimates of r and b , while the correlation among the two parameter subsets remain moderate. How does this translate in a real world example?

  fair part of published examples correspond to unrealistic estimates of r , without being able to check, as the estimates of r are rarely given in published examples because of the focus on the coefficient of DD b .

  row), with a 95 % confidence interval for the coefficient of DD encompassing 0, despite its bias. Accounting for uncertainty in population size not only removes a bias, but also increases the precision on the coefficient of DD by reducing the estimate for the process standard error.Contrary to[START_REF] Dennis | Estimating density dependence, process noise, and observation error[END_REF] andKnape and de Valpine (2011), we found no pervasive presence of local minima in the likelihood, although we agree that care should be exercised on this point[START_REF] Knape | Estimability of density-dependence in models of time-series data[END_REF]. The sampling correlation between the estimates of r and b and the resulting near non-identifiability is for us the main problem. Results from simulated data (Figure 3, and unpublished results on shorter time series) and a few other unpublished examples indicate similar general properties of what remains in the words of Dennis et al.



  One can easily foresee what could be a state-space model incorporating DD in a wildlife population, based on integrated monitoring covering both population size surveys and individual longitudinal data ( i.e. capture-mark-recapture data in the broad sense). Such a model has to combine different types of state equations, with possibly at some stage the need to account for the lack of independence between the marked individuals and the overall population: Multinomial distributions of individual trajectories on a Markov chain (reducing to Bernoulli equations in the case of survival), with parameters possibly dependent on population size, i.e. density-dependent;

Figure

  Figure 1: Current log population size 1  t x vs previous one

Figure 2 :

 2 Figure 2: Estimated strength of density dependence (ordinate) in absence of density

Figure 3 :

 3 Figure 3: Deviance (-2 x log-likelihood) contours of simulated trajectories of increasing

Figure 4 :

 4 Figure 4: Profile Deviance for parameter r of the density dependent model

Figure 5 :.

 5 Figure 5: reconstructed log spring population size for the Greater Snow Goose, based on the

  Figure 1

  . One can equivalently test for DD either by a test of model [3] vs model [4], such as a likelihood ratio test, or by a test of H 0

	test. The latter approach is usually preferred as it can easily be easily implemented as a one-
	tailed test of H 0	b		0	vs H 1	0 b , the alternative of biological interest. 
	Density-dependence and resource-dependence
	By rewriting [3] one can make a per capita growth rate	r		bx			appear:
								t	t
								r 	Ln	(	)	N		1
	N		0
								b		0	in model [3], e.g. as a Wald

  , Roe Deer example;Catchpole et al. 2000 for survival;[START_REF] Crespin | Recruitment to a seabird population depends on environmental factors and on population size[END_REF] for recruitment;[START_REF] Gaillard | Body mass of roe deer fawns during winter in 2 contrasting populations[END_REF] for morphometric traits). The prominent point common to all approaches is again that the density is always a proxy for something else and is always known with some uncertainty, i.e. we are again faced with an error-in-variable problem.Model [11] should be accompanied by an observation equation

	using the regression model	tj z		a		t by		tj    t	t , attenuation of the slope estimate towards 0 t t x y    . When
	(McArdle 2003) is unavoidable.

ACKNOWLEDGMENTS

We thank Gilles Gauthier for having authorized me to use the Greater Snow Goose Spring survey data. We thank Bret Collier and Bob Steidl for helpful comments on an earlier version.