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Keywords:

  • Bayesian statistics;
  • developmental stability;
  • fitness;
  • fluctuating asymmetry;
  • latent variable;
  • quality

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

Since the influential paper by Palmer and Strobeck in 1986, the statistical analysis of fluctuating asymmetry and developmental stability has received much attention. Most studies deal with one of the following four difficulties: (i) correcting for bias in asymmetry estimates due to measurement error; (ii) quantifying sampling error in the estimation of individual developmental stability using individual asymmetry; (iii) the detection of directional asymmetry and antisymmetry; and (iv) combining data from several traits. Yet, few studies have focused on statistical properties of estimating a relationship between individual developmental stability and other factors (e.g. fitness). In this paper I introduce a fully Bayesian model in which the unobservable individual developmental stability is treated as a latent variable. The latter is then related to individual fitness. I show by means of the analysis of simulated data that this approach has several advantages over traditional techniques. First, the method provides unbiased (but slightly less accurate) estimates of slopes between developmental stability and fitness taking all sources of error into account. Secondly, it allows proper investigation of non-linear associations. Finally, the model allows unbiased estimation of unobserved fitness of individuals that have been measured on left and right side.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

Developmental stability (DS) can be defined as an individual’s ability to buffer its development against random perturbations, and is often estimated by fluctuating asymmetry (FA, small random deviations from perfect symmetry) (Ludwig, 1932; Palmer & Strobeck, 1992). Measuring asymmetry is easy relative to the determination of (components of) individual fitness and/or quality. Therefore, if DS were closely related to fitness and quality, asymmetry could be a practical measure of the latter two. Yet, this association appears to be weak on average and highly heterogeneous, which hampers its general use (Leung & Forbes, 1996; Møller, 1997; Clarke, 1998; Bjorksten et al., 2000a). Therefore, there is a need for guidelines that predict if and when asymmetry accurately reflects fitness. This paper attempts to unify the statistical aspects of the determination of the DS–fitness association in a single modelling framework using concepts of Bayesian statistics.

FA estimates individual DS with seeming ease, yet statistical properties indicate otherwise. First, measurement error (ME) may make up a large part of the observed asymmetry (e.g. Palmer & Strobeck, 1986) and repeated measurements and mixed-model analysis are required to estimate this degree of ME and to obtain unbiased FA estimates at both the population and the individual level (e.g. Van Dongen, 2000). Second, individual single trait FA is often only weakly correlated with the presumed underlying DS even when ME is small (Whitlock, 1996; Houle, 1997). Because asymmetry estimates a variance (i.e. DS) with only two datapoints (i.e. left and right side), the amount of sampling variation is large. Consequently, associations between individual FA and other variables are biased downward. Whitlock (1996, 1998), Van Dongen (1998), Van Dongen et al. (1999b) and Gangestad & Thornhill (1999) developed closely related methods to estimate the proportion of variation in FA that can be attributed to between-individual variation in DS and/or to correct for this downward bias. Finally, the occurrence of other forms of asymmetry (i.e. directional asymmetry and/or antisymmetry) obstructs interpretation since their association with DS is unclear (Palmer & Strobeck, 1992; Graham et al., 1993).

It is generally assumed that the development of a trait is disturbed by random factors (i.e. developmental noise, further denoted as DN) and that some usually unknown processes correct for these developmental errors (i.e. the process of DS). As a result, if these corrections are made accurately, i.e. when DS is high, the observed phenotype will deviate only little from its expected value (under the given environmental conditions and the individuals’ genotype). Otherwise, within-individual phenotypic variation is expected to be higher. The phenotype can be modelled as a sample from a normal distribution with mean equal to its expected value and a variance reflecting the joint action of DN and DS (further called VDS). For bilaterally symmetrical traits the means of both sides are equal and asymmetry (left minus right) will follow a normal distribution with zero mean and variance equal to the double of VDS (Whitlock, 1996). Thus, FA primarily estimates the joint and opposite action of developmental noise (DN) and DS, which is referred to as developmental instability (DI) (e.g. Whitlock, 1996).

Studies at the individual level make use of traditional regression or correlation (either parametric or non-parametric) analysis to investigate the relationship between DI and other factors, like components of fitness. Individual DI is thereby often estimated by single trait asymmetry (e.g. Van Dongen et al., 1999c). In spite of the fact that the unsigned asymmetry is half-normally distributed, parametric tests are often applied. Gangestad & Thornhill (1998) showed that Type I error rates of parametric correlations are robust against this type of deviation and encourage using parametric tests over their non-parametric counterparts because of increased power, thereby assuming that the relationship between FA and DI is linear. However, this relationship is non-linear if DI is expressed as a variance and asymmetry as the traditional unsigned FA (i.e. absolute value of left minus right trait value) (Van Dongen & Lens, 2000). Therefore, if the DI–fitness association is non-linear, it is not possible to predict the shape of the observable FA–fitness relationship. In addition, there is no a priori reason to accept that the DI–fitness association is a linear one, and the shape of the relationship is difficult to determine via FA. Furthermore, if individual FA is to be used as a predictor of fitness, the latter is the dependent and asymmetry the so-called independent or explanatory variable. Yet, traditional regression analysis assumes that the independent variable can be observed without error (e.g. Neter et al., 1990). This is clearly not the case for asymmetry, which is subject to different sources of error as identified above. If individual FA is to be used as a (crude) predictor of fitness there is a need for statistical models that allow investigation of the shape of the relationship between fitness and DI directly, taking all sources of variability and measurement error into account. In this way, unbiased parameter estimates can be obtained and meaningful predictions will be possible.

In this paper, a latent variable model to investigate the association between DI and fitness is proposed in a Bayesian framework. Although traditional likelihood approaches can be used to obtain parameter estimates of latent variable models (Arminger & Muthén, 1998) I have opted for a fully Bayesian approach because of four advantages. First, in a Bayesian analysis all parameters are considered to be stochastic rather than fixed. This stochasticity is captured by the so-called posterior distributions. In this way, the uncertainty associated with each parameter value is taken into account in the analysis and in the estimations of all parameters. This will turn out to be especially important for the analysis of individual DI considering its weak association with asymmetry. Second, posterior distributions do not depend on large sample properties, as is often the case for likelihood approaches (Arminger & Muthén, 1998; Dunson, 2000). Third, using informative prior distributions, information from other studies or meta-analyses can be assigned to parameters resulting in improved estimates. However, the use of informative priors is debatable since it involves a subjective decision. To avoid this, so-called uninformative or weak priors can be used to allow the data to drive the analyses (Gelman et al., 1995). Finally, estimation of unobserved data is a natural application of the Bayesian approach as it is a by-product of the technique to obtain the posterior distributions of the parameters of interest. In summary, the Bayesian approach allows the proper propagation of uncertainty at the various stages of the model and offers a flexible engine to model a variety of (complex) problems (Carlin & Louis, 2000).

The following sections briefly introduce Bayesian analyses, and describe a fully Bayesian latent variable model to study the association between DI and fitness. Then, a number of simulated datasets are analysed to explore convergence and model-fitness properties of the proposed models and to illustrate the advantage of this approach over traditional techniques. Estimation accuracy of slopes between traditional regression and the Bayesian model are compared, and the estimation of fitness of individuals with known asymmetry is studied. In a final section, assumptions, limitations and possible extensions of the model are discussed. Analyses are performed in WINBUGS (freely available at http://www.mrc-bsu.cam.ac.uk/bugs). Therefore, the specific features and notations of this package are used throughout this paper. A worked out example from this paper is available at http://www.blackwell-science.com/products/journals/suppmat/JEB/JEB315/JEB315sm.htm.

Background of Bayesian inference

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

Introduction

Statistical inference is concerned with drawing conclusions from observed variables (data) to unobserved variables (parameters). Bayesian statistical conclusions about a parameter θ or unobserved data are made in terms of probability statements, conditional on the observed data, captured by the posterior distribution [p|y)]. Let us start with a model providing the joint distribution of θ and the data y (i.e. p,y), further called the full probability model). This model can be written as the product of the prior distribution p(θ) (reflecting what is known about the distribution of parameters prior to the analysis) and the sampling distribution (or traditional likelihood p(y|θ), i.e. the likelihood of observing the dataset conditional on the unknown true values of the unobserved parameters):

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Conditioning on the data and making use of Bayes’ rule yields the posterior distribution:

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The normalizing factor p(y) does not depend on θ and can therefore be omitted leading to the unnormalized posterior distribution:

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The posterior distribution of the parameters θ of the model is a combination of prior knowledge [as captured in the prior distribution p(θ)] and what the data ‘tell’ us [as obtained from the likelihood of the observed dataset p(y|θ)]. Thus, Bayesian statistical techniques update current knowledge of the parameter-set θ with the new information obtained from the data and capture this in a posterior distribution (Gelman et al., 1995; Carlin & Louis, 2000). This information can be summarized by the mean and standard deviation and/or by 95% credibility intervals (i.e. the analogue of traditional confidence intervals) (Carlin & Louis, 2000).

The three steps in a Bayesian analysis

The first step in a fully Bayesian analysis is to build a full probability model [p,y)] with parameters of interest and a structure that reflects the underlying system as closely as possible. The entire model can be decomposed in a number of submodels and must be completed with prior distributions of the different parameters.

In a second step, the full joint distribution of the parameters and the data can be obtained by simulation techniques (like Gibbs sampling and the Metropolis–Hastings algorithm, Gelman et al., 1995). These simulations require that the submodels are constructed in such a way that the product of their distributions is equal to full joint distribution. This is provided by the so-called directed Markov assumption (Lauritzen et al., 1990) which holds when all the edges are directed and it is impossible to return to a node after leaving it just by following the directed links (a so-called directed acyclic graph,Spiegelhalter, 1998). Simulation of the posterior distribution starts with a set of initial values for all parameters and missing data in the model. In each iteration parameter values are updated and represent a sample from the posterior distribution. When repeated many times, the distribution of this so-called Monte Carlo Markov Chain (MCMC) approximates the exact posterior distribution (Gelman et al., 1995; Carlin & Louis, 2000).

Finally, checks of convergence of the MCMC and model-fitness must be carried out. Convergence relates to the decision when it is safe to stop the MCMC iteration procedure, or in other words when it is reasonable to assume that the observed posterior distribution approximates the true underlying distribution. There are many different tools to assess convergence, yet no single method can detect all types of convergence failure in all cases (Cowles & Carlin, 1996). Following Cowles & Carlin (1996), I use the so-called shrink factor (Gelman & Rubin, 1992) in combination with visual inspection of the MCMCs to assess convergence. Unless mentioned otherwise, in all analyses five independent MCMCs of 2000 iterations are run after a burn-in period of 4000 iterations (i.e. to avoid dependence on the initial values), which lead to good convergence behaviour. Compared to the convergence diagnostics, model-checks have received relatively little attention in the literature. The most commonly used methods are the posterior predictive checks (Gelman et al., 1996). This simulation-based technique can be performed during the run of the MCMCs. In each iteration a replicate dataset is generated, conditional on the current parameter values. Using some discrepancy measure – and the appropriate choice depends on the particular features of the model – for both the data and the replicate dataset, model-fitness can be tested. If the model under consideration is ‘adequate’, the discrepancy of the replicated dataset is expected to be on average equal to the one for the observed data (i.e. the model generates data with a discrepancy comparable to that of the observed data). Otherwise, the discrepancy of the replicate dataset is expected to be higher. Alternatively, the deviance or likelihood can be used as an informal way to investigate model-fitness of different alternatives. Below, specific model checks are proposed and evaluated.

A fully Bayesian latent variable model relating developmental instability with fitness

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

Building the full probability model in three steps

The entire model can be decomposed in three parts. In a mixed regression, unbiased estimates of individual asymmetry are obtained (Van Dongen et al., 1999a, 2001). These signed asymmetry values will then be linked to the unobservable and presumed underlying individual DI. Finally DIs will be related to fitness.

(i) Mixed regression model.

On the left side of Fig. 1 individual asymmetry is modelled using a mixed regression model which in itself can be split up in three submodels. Each observation (i.e. measurement of a trait) is assumed to be a sample from a normal distribution with mean equal to the expected value (or true unknown underlying trait value) and variance equal to the degree of measurement error (σ2ME):

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Figure 1.  Graphical representation of the fully Bayesian latent variable model determining the association between developmental instability (DI) and fitness. Parameters and data are represented by ellipses which are either stochastic when they are given a distribution or deterministic when they are logical functions. Observed constants are represented by rectangles. Arrows interconnect parameters and data. Solid arrows indicate stochastic dependence and hollow edges indicate a logical function. Individual DI (Devinst[i], σ2FA[i] in text) is a latent variable acting as explanatory variable causing variation in fitness and asymmetry. Here, variation in DI is modelled by a gamma distribution with parameters shape and scale. Three alternatives are described in the text.

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where i=1 …N and j=1 …K with N=number of individuals and K=number of within-subject repeats.

The expected value of a single measurement (i.e. expected[i,j]) is modelled by the sum of a population specific part (i.e. the fixed-effects) and an individual specific part (i.e. the random-effects). As fixed-effects an intercept and a slope are included, modelling directional asymmetry. An individual-specific random intercept (interc_ind[i]) and slope (slope_ind[i]) model individual variation in size and asymmetry around this regression line, reflected in Ind_profile[i,j]. Random effects are assumed to follow a normal distribution with zero means and variances reflecting variation in size (σ2size) and asymmetry (σ2FA, but see below), respectively. Formally, these submodels can be denoted as:

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The model is then completed with appropriate prior distributions (Gelman et al., 1995):

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The parameter estimates obtained from this model have exactly the same interpretation as for the empirical Bayes approach applied by Van Dongen et al. (1999a). However, because each parameter is treated as a stochastic variable, standard deviations become slightly larger (Van Dongen et al. 2001).

(ii) Modelling developmental instability.

Asymmetry is assumed to be a result of the underlying DI. When all individuals have the same degree of DI each asymmetry value can be viewed as a sample from a normal distribution with zero mean and variance equal to σ2FA (see above). Thus σ2FA is an estimator of population level DI, yet all individual variation in asymmetry reflects sampling variation (Whitlock, 1996). When individuals differ in their degree of DI, the distribution of the asymmetry values (i.e. slope_ind[i] in eqn 6) becomes a mixture of normal distributions all with zero mean, but different variances (which reflect the different levels of DI). As a result it becomes leptokurtic (Van Dongen, 1998). This can be included in the mixed regression model by replacing submodel eqn 8 by:

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where the σ2FA[i]’s reflect the individual-specific levels of DI. In this way, the individual-specific variance of the individual slope needs to be estimated by two datapoints, i.e. the left and right side of a trait. As a consequence sampling variation will be very large (see Whitlock, 1996) causing convergence problems (data not shown). Therefore, additional variables are included which impose a distribution on the σ2FA[i] values (Gelman et al., 1995), assuming three alternatives of increasing complexity. In the simplest case, a discrete number of different levels of DI are assumed. With two levels the distribution of σ2FA[i] is modelled as:

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where prob[i] follows a Bernoulli distribution and equals 1 with probability q:

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The following priors complete the model:

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The proportion q reflects the fraction of the individuals that exhibit a level of DI equal to σ2FA2, while the others exhibit a level of DI of σ2FA1. This model (further called discrete instability model, DIM) can be easily extended to more discrete levels of DI.

Since between-individual variation in DI can potentially be affected by a wide variety of factors (pollution, nutrition, inbreeding, genomic co-adaptation, …), it probably follows a continuous distribution and it seems unlikely that a few discrete levels of DI describe the distribution properly. Variation in DI may be better described on a continuous scale. The gamma distribution is a good candidate distribution for this purpose since it has zero density for negative values and can take many different shapes (Houle, 2000). A shape and scale parameter α and β, respectively, determine the gamma distribution which is incorporated in our model as:

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The model is completed with the following priors:

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This model will be further denoted as the continuous instability model (CIM).

In spite of the fact that the gamma distribution can take many different forms, its flexibility has limitations. One could imagine that the distribution of σ2FA[i] is even more complex. Basically there are no limitations to the complexity one can model, but it should be kept in mind that the distribution of a latent unobservable variable is modelled, and that very high sample sizes will be required to obtain reliable results. In this paper the complexity of the models is limited to a two-component mixture of two gamma distributions:

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where as for the DIM model prob2[i] follows a Bernoulli distribution and equals 1 with probability q2:

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This mixed continuous instability model (MCIM) is completed with the following priors:

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(iii) Modelling the association between fitness and developmental instability.

Traditional regression analysis is inappropriate when the independent variable is subject to error (e.g. Sokal & Rohlf, 1995). This results in a downward bias of estimated slopes. In the above models, individual DIs (i.e. σ2FA[i]) are represented as distributions. The distributions reflect the uncertainty and/or error associated with the estimation of individual DI and can be used to perform a regression analysis with other factors of interest (e.g. fitness). Taking all variability into account in the analysis, both the obtained estimate of the slope and the estimate of unobserved fitness of individuals of which left and right side were measured are unbiased.

In its most simple form, the expected individual fitness (mean_fitness[i]) is modelled by a linear regression with developmental instability as independent variable. The observed fitness is assumed to be normally distributed with mean equal to the expected value and a variance of σ2fitness:

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The model is completed with the following priors:

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In this simple regression model (in fact a kind of model II regression using traditional terminology since both the dependent and independent variables are subject to error) a linear association between individual DI and fitness is assumed. More complex alternatives are of course possible like a quadratic relationship replacing eqn 19 by:

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Obtaining posterior distributions and checking model fitness: an analysis of simulated data

(i) Variation in individual DI.

Modelling the unobservable underlying DI by a set of alternatives aims at describing its variation as adequately as possible. The statistical significance is therefore not of interest. Rather we want to select a model that is complex enough to capture (nearly) all variation in DI, but is not too complex such that the computational efficiency decreases drastically. An ‘adequate’ model should be able to generate asymmetry values that closely reflect the observed values. To investigate model adequacy a discrepancy measure is estimated. Therefore, in each iteration of the MCMC, a new dataset of asymmetry values is generated conditional on the current parameter values as:

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A discrepancy measure that summarizes how much these new values deviate from the observed asymmetries (i.e. the random slopes) can be calculated as:

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where n equals the number of individuals. In addition, the likelihood (further called λ) of the model will be monitored (Spiegelhalter, 1998):

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To evaluate these two measures of model adequacy four datasets of size 500 with different types of variation in DI were simulated (see Table 1 for details) and the four alternative models were fitted (no variation, DIM, CIM and MCIM). Table 1 summarizes the posterior distributions of the parameters of interest, the discrepancy and λ. For the datasets with no variation in DI and those that follow DIM and CIM, the posterior means approximate the underlying parameter values closely when the respective models are fitted. The discrepancy hardly decreases when a more complex model is fitted, but increases rapidly for models with a less complex structure of variation in DI (Table 1). A comparable but inverse pattern is observed for λ. This suggests that both measures of adequacy are informative, at least in the ideal situations of these simulations. For the MCIM model, however, the estimated posterior distributions of the shape and scale parameters differ a lot from their expected values. In addition, it took much longer to obtain convergence (i.e. 10 000 iterations). This could suggest that the sample size of 500 as used in these simulations is too small to construct the full-joint distribution of such a complex model. Indeed, the analysis of a dataset of 2000 individuals showed much better convergence properties, and posterior means more closely reflected the underlying parameter values (Table 2). Nevertheless, the more complex MCIM model showed only slightly lower discrepancy and a comparable likelihood as the CIM. Although more simulations are required, this suggests that the gamma distribution is flexible enough to approximate more complex distributions like mixtures of two gamma distributions.

Table 1.   Mean (SD) values of posterior distributions of the parameters of the four fully Bayesian models proposed in this paper (see text for details). Analyses were performed on four datasets (N = 500) that were simulated in SAS under the conditions of the four models (no variation, Discrete Instability (DIM), Continuous Instability (CIM) and Mixed Continuous Instability Model (MCIM), see text for details). For each combination the discrepancy and −2LogLikelihood (λ) of the model are given. Underlying parameter values applied to generate the data are given in the first column. NC indicates that the Monte Carlo Markov Chain did not converge. Thumbnail image of
Table 1.   (Continued) Thumbnail image of
Table 2.   Mean (SD) values of posterior distributions of the parameters of the four fully Bayesian models proposed in this paper (i.e. no variation, Discrete Instability (DIM), Continuous Instability (CIM) and Mixed Continuous Instability Model (MCIM), see text for details). Analyses were performed on a dataset (N = 2000) that was simulated in SAS under the conditions of MCIM. For each combination the discrepancy and −2LogLikelihood (λ) of the model are given. Underlying parameter values applied to generate the data are given in the first column. Thumbnail image of
(ii) The association between DI and fitness.

I first consider a linear association between individual DI and fitness. A dataset of 1000 individuals was simulated and analysed by traditional regression and the fully Bayesian model. Traditional regression was performed with fitness as the dependent variable and either the unsigned asymmetry or the square of the unsigned asymmetry as independent variables. The first analysis is used most frequently, whereas the second preserves the scale as DI is expressed as a variance. During simulation of the data, individual fitness followed a normal distribution with mean 50 and variance 0.25 for individuals with perfect developmental stability (i.e. σ2FA[i]=0). With increasing instability, fitness decreased with 5 units per unit of DI (i.e. slope_fitness=−5). Variation in DI was assumed to follow the CIM (details in Table 3). As expected, the slope obtained by the traditional simple linear regression underestimated the true slope, whereas the fully Bayesian approach yielded an estimate very close to the expected value (Table 3). To investigate the importance of adequate modelling variation in DI, the fitness–DI association was also estimated assuming a DIM. This inadequate way of describing variation in DI (see above and Table 1) led to an underestimation of the fitness–DI association (Table 3) and thus stresses the importance of the appropriate incorporation of variation in DI in the model.

Table 3.   Parameter estimates and posterior distributions of traditional regression and fully Bayesian analysis, respectively, of a simulated fitness – DI association. Underlying parameter values applied to generate the data are given in the first column. Thumbnail image of

In a second simulation a quadratic association between fitness and DI is incorporated and modelled (details in Table 4). In summary, the traditional regression either fails to detect a non-linear relationship (i.e. with unsigned asymmetry as independent) or indicates a significant positive quadratic effect (i.e. with the square of the unsigned asymmetry as independent variable) while a negative effect was imposed during simulation. The Bayesian approach, in contrast, yielded results that were consistent with the underlying parameter values of the data generation process. This example shows that traditional regression is inappropriate and may be misleading when studying non-linear fitness–DI associations.

Table 4.   Parameter estimates and posterior distributions of traditional regression and fully Bayesian analysis, respectively, of a simulated quadratic fitness–DI association. Underlying parameter values applied to generate the data are given in the first column. Thumbnail image of

Accuracy of estimation

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

The proposed Bayesian latent variable model provides unbiased estimates of the association between fitness and DI. This section investigates estimation accuracy by comparing bias and the coefficient of variation (CV, defined as 100√[σ22]) of the estimated slopes of seven simulated datasets between the fully Bayesian model and traditional regression. Data were generated assuming the CIM model detailed in Table 3, yet with slope between fitness and DI ranging between –5 and –0.025 (Fig. 2). In each case, Bayesian estimates are nearly unbiased, whereas the traditional regression between fitness and AFA underestimate the association. Both estimates of the slope are linearly related, indicating that the degree of bias is a constant proportion across the range of possible strengths of associations (Fig. 2). Estimation accuracy of both approaches, as measured by the CV, appeared to be of comparable magnitude (Fig. 3). Yet, in all but one case, the traditional regression yields more accurate (i.e. lower CV) slopes. This increased variability is not a drawback but an inherent property of Bayesian statistics and can be viewed as an advantage over maximum likelihood methods. Because Bayesian techniques consider parameters as stochastic, it incorporates the uncertainty associated with the estimation of each (i.e. by obtaining the full-joint distribution). Therefore, the slightly more accurate maximum likelihood estimates create a false sense of security because not all sources of variability are incorporated.

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Figure 2.  Bias and accuracy of the estimation of the slope of the regression between fitness on the one hand and both developmental instability (i.e. DI, applying the fully Bayesian model) and the unsigned asymmetry (i.e. AFA, applying traditional regression) on the other hand. Error bars indicate 95% credibility and confidence intervals, respectively. The solid line represents the linear association between the two estimates. The dashed line is the expected regression if both estimates would yield unbiased estimates. Numbers indicate the underlying values of the slope between fitness and DI used to simulate the data.

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Figure 3.  Coefficients of variation (CV) of estimates of the slope of the regression between fitness and both developmental instability (DI) and the unsigned asymmetry (AFA). Estimates and their accuracy were obtained from a fully Bayesian and traditional regression, respectively (estimates in Fig. 2).

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Estimation of unobserved fitness

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

Estimation of unknown observations is a natural application of Bayesian statistics. If one wants to estimate the fitness of an individual from which left and right side have been measured, then the measurements should be added to the dataset and the unknown fitness values treated as missing values. During the MCMC, the distribution of these unobservable fitness values is estimated conditional on the model parameters and the data (i.e. the asymmetry and thus the DI). To illustrate this application, observations for three individuals with relative low, intermediate, and high asymmetry were simulated from the CIM in Table 3 and their individual fitness-values were entered in the dataset as missing values. Next, estimates of individual fitness were obtained by traditional regression (unsigned FA and square of unsigned FA as independent variables) and the Bayesian approach (results in Table 5). The difference in mean fitness is small between the three approaches, where the regression approach of the square of the unsigned asymmetry appears to overestimate fitness relative to the other two (Table 5). However, investigating the distribution of fitness more carefully, there appears a more subtle difference (Fig. 4). Whereas the regression approach assumes normality (e.g. Neter et al., 1990) the Bayesian model takes the distributional characteristics of the individual DI estimates into account. The latter is not a normal but a skewed distribution (Fig. 5). Thus, although on average very similar, the distributional characteristics differ substantially, and the median differs relatively more between the two methods (Table 5). For example, the data were simulated such that individual fitness maximally equals 50, yet the regression-based predictive distribution would indicate otherwise. It is important to note that even for the individual with a small asymmetry value, the distribution of the individual DI still exhibits a tail to the right (Fig. 5). This is because highly symmetrical traits can develop in individuals with low stability (Whitlock, 1996; Houle, 1997). Estimating individual fitness on the basis of left and right trait values should take the resulting skewness of the distribution into account.

Table 5.   Estimation of fitness from asymmetry values using traditional regression and the fully Bayesian latent variable model. In the traditional analysis the independent variable is the unsigned asymmetry (AFA) or the square of AFA. Fitness can be obtained from the estimated regression equation (see Table 3 for parameter estimates). In the fully Bayesian approach individual asymmetry (FA), developmental instability (DI) and fitness are estimated in the MCMC and thus represented as posterior distributions. Thumbnail image of
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Figure 4.  Distributions of individual fitness as obtained by traditional regression and the Bayesian latent variable model for three individuals differing in their degree of asymmetry (see Table 4 for details).

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Figure 5.  Posterior distribution of developmental instability for three individuals differing in their degree of asymmetry (see Table 5 for details) as obtained from the Bayesian latent variable model.

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Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

Since the influential review paper by Palmer & Strobeck (1986), the statistical aspects of the analysis of DI as estimated by FA has received a remarkable amount of attention dealing with bias, sampling error, other forms of asymmetry and multivariate analyses. In this paper I introduce a fully Bayesian approach that simultaneously corrects both for bias due to measurement error and for sampling variation. The proposed model treats DI as a latent unobservable variable and between-individual variation is quantified after correction for measurement error. Individual DI can then be modelled further in relation to other factors, such as fitness. The analysis of simulated data showed that this approach: (i) provides unbiased (but slightly less accurate) estimates of regression slopes; (ii) allowed for non-linear analyses; and (iii) provided better estimation of expected fitness.

The introduced set of models can be extended in several ways. First, heterogeneity in measurement error can be incorporated (see Van Dongen et al., 2001, for details). Secondly, different traits can be analysed simultaneously (making use of multivariate distributions) and thirdly more complex associations between DI and fitness can be incorporated. Still, convergence properties and required sample sizes need to be addressed further in these contexts.

The introduction of Bayesian statistics in this area of research is indispensable considering the different levels of stochasticity that need to be taken into account. Nevertheless, as recently pointed out, the use of sophisticated statistical tools will not remove the observed heterogeneity in fitness–DI associations that hampers the general applicability of FA as a bio-monitoring tool (Bjorksten et al., 2000b). More specifically, the developmental aspects of DS are only poorly understood, and consequently the link between asymmetry and DI is a black box. More research in this area should allow us to evaluate to what extent the normal and independent development model [as assumed by Whitlock (1996) and adopted here] forms a suitable link between DI and the observed asymmetry. In addition, it is at present not clear to what extent other forms of asymmetry (directional asymmetry and antisymmetry) can result from DI. Nevertheless, when more insights in this area are gained, and when it might be needed to formulate the asymmetry–DI link in a more complex way relative to eqn 10, Bayesian analysis has the flexibility to do so. Furthermore, when competing models are available, it is possible to statistically compare different alternatives.

As a final point, it is worth noting that relatively high sample sizes (N > 250) are needed in order to obtain good convergence behaviour of the proposed models. For smaller samples I often observed that the shrink factor did not approach one and that individual MCMCs ended outside the range of biologically meaningful parameter values. It might be possible to improve on this convergence problem by using more informative priors. Yet, since this involves a subjective decision more research with respect to sensitivity of the obtained results on the choice of a prior is required. On the other hand, the incorporation of data from several traits could partly solve this problem, but further research is required to develop those models. Samples of more than 250 individuals are relatively rare in empirical studies and one could argue that this makes this approach useless. However, the high requirements simply indicate that if one wants to incorporate all sources of variability in a statistical analysis, like the Bayesian approach does, a large sample is indispensable.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

S.V.D. holds a postdoctoral fellowship from the Science Fund Flanders (FWO-Vlaanderen). I thank André A. Dhondt, Luc Lens, Erik Matthysen and two anonymous reviewers for their suggestions that helped to improve earlier versions of this manuscript.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Background of Bayesian inference
  5. A fully Bayesian latent variable model relating developmental instability with fitness
  6. Accuracy of estimation
  7. Estimation of unobserved fitness
  8. Discussion
  9. Acknowledgments
  10. References
  11. Supporting Information
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