### Abstract

- Top of page
- Abstract
- Introduction
- Background of Bayesian inference
- A fully Bayesian latent variable model relating developmental instability with fitness
- Accuracy of estimation
- Estimation of unobserved fitness
- Discussion
- Acknowledgments
- References
- 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

- Top of page
- Abstract
- Introduction
- Background of Bayesian inference
- A fully Bayesian latent variable model relating developmental instability with fitness
- Accuracy of estimation
- Estimation of unobserved fitness
- Discussion
- Acknowledgments
- References
- 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 V_{DS}). 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 V_{DS} (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.

### Accuracy of estimation

- Top of page
- Abstract
- Introduction
- Background of Bayesian inference
- A fully Bayesian latent variable model relating developmental instability with fitness
- Accuracy of estimation
- Estimation of unobserved fitness
- Discussion
- Acknowledgments
- References
- 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√[σ^{2}/μ^{2}]) 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.

### Estimation of unobserved fitness

- Top of page
- Abstract
- Introduction
- Background of Bayesian inference
- A fully Bayesian latent variable model relating developmental instability with fitness
- Accuracy of estimation
- Estimation of unobserved fitness
- Discussion
- Acknowledgments
- References
- 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. ### Discussion

- Top of page
- Abstract
- Introduction
- Background of Bayesian inference
- A fully Bayesian latent variable model relating developmental instability with fitness
- Accuracy of estimation
- Estimation of unobserved fitness
- Discussion
- Acknowledgments
- References
- 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.