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Keywords:

  • conditional model;
  • generalized estimating equations;
  • generalized linear-mixed models;
  • marginal model;
  • mixed effects;
  • random effects;
  • sandwich estimators

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

1. Statistical methods that assume independence among observations result in optimistic estimates of uncertainty when applied to correlated data, which are ubiquitous in applied ecological research. Mixed effects models offer a potential solution and rely on the assumption that latent or unobserved characteristics of individuals (i.e. random effects) induce correlation among repeated measurements. However, careful consideration must be given to the interpretation of parameters when using a nonlinear link function (e.g. logit). Mixed model regression parameters reflect the change in the expected response within an individual associated with a change in that individual’s covariates [i.e. a subject-specific (SS) interpretation], which may not address a relevant scientific question. In particular, a SS interpretation is not natural for covariates that do not vary within individuals (e.g. gender).

2. An alternative approach combines the solution to an unbiased estimating equation with robust measures of uncertainty to make inferences regarding predictor–outcome relationships. Regression parameters describe changes in the average response among groups of individuals differing in their covariates [i.e. a population-averaged (PA) interpretation].

3. We compare these two approaches [mixed models and generalized estimating equations (GEE)] with illustrative examples from a 3-year study of mallard (Anas platyrhynchos) nest structures. We observe that PA and SS responses differ when modelling binary data, with PA parameters behaving like attenuated versions of SS parameters. Differences between SS and PA parameters increase with the size of among-subject heterogeneity captured by the random effects variance component. Lastly, we illustrate how PA inferences can be derived (post hoc) from fitted generalized and nonlinear-mixed models.

4.Synthesis and applications. Mixed effects models and GEE offer two viable approaches to modelling correlated data. The preferred method should depend primarily on the research question (i.e. desired parameter interpretation), although operating characteristics of the associated estimation procedures should also be considered. Many applied questions in ecology, wildlife management and conservation biology (including the current illustrative examples) focus on population performance measures (e.g. mean survival or nest success rates) as a function of general landscape features, for which the PA model interpretation, not the more commonly used SS model interpretation may be more natural.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Ecological studies naturally result in correlated data. Repeated measurements within individuals tend to be more similar than those taken among individuals, and temporally or spatially proximate measurements are more similar than temporally or spatially disparate ones. Ignoring these correlations typically results in underestimation of the true uncertainty. Methods for modelling correlated response data can generally be categorized by the type of estimation procedure (i.e. likelihood-based or semi-parametric method) and by whether one models a conditional mean (where conditioning is performed with respect to one or more random effects) or a marginal mean (which averages over individuals or dependencies in the data). Conditional models operate under the assumption that latent or unobserved characteristics of individuals (i.e. random effects) induce correlation among repeated measurements and are almost always fit using likelihood-based methods. By contrast, marginal models for non-Gaussian data have typically been fit by combining the solution to an unbiased estimating equation with robust measures of uncertainty (e.g. sandwich or cluster-level bootstrap variances). These two approaches differ fundamentally in how they model correlation arising from individual heterogeneity, their assumptions and their robustness properties. In models with a nonlinear link function (e.g. logit), regression coefficients from the two methods even have different interpretations (Zeger, Liang & Albert 1988; Pendergast et al. 1996; Carriere & Bouyer 2002). Ecologists have recognized the importance of accounting for spatial and temporal correlation, as well as correlation arising from phylogenetic dependencies when fitting regression models (e.g. Ives & Zhu 2006; Dormann et al. 2007 and references therein). However, little attention has been given in the ecological literature to the differences between conditional and marginal modelling approaches, or the interpretation of parameters when modelling non-Gaussian data.

We compare marginal and conditional approaches to modelling correlated data, using data from a repeated measures study of mallard Anas platyrhynchos (Linnaeus) nest structure use (Zicus, Fieberg & Rave 2003; Zicus et al. 2006) in illustrative examples. Following a brief overview of the nest structure data, we review linear models for correlated response data, where parameters have both a subject-specific (SS) and population-averaged (PA) interpretation. The former describes how the response changes within an individual as one changes that subject’s covariate values, whereas the latter describes differences among groups of individuals that differ with respect to one or more covariates. We then observe that SS and PA parameters diverge with generalized linear and nonlinear-mixed effects models, and because many research questions do not lend themselves to a SS interpretation, we show how marginal (PA) inferences can be derived (post hoc) from conditional generalized linear and nonlinear models. We then discuss regression methods for non-Gaussian data that directly model marginal means using GEE. We end with a general discussion of the relative strengths and limitations of these different approaches. Data and R code (R Development Core Team 2007) for performing all analyses are included in Appendices S1–S3 (Supporting Information).

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Data

The data in our illustrative examples come from a study investigating the relationship between land use and mallard nest occupancy and clutch size in two types of nest structures (single- and double-cylinders) on a 658 km2 (254 mile2) study area in Western Minnesota, from 1997 to 1999 (Zicus et al. 2003, 2006). In the spring of 1996, 110 nesting structures were placed in 104 wetlands (the largest eight wetlands included two structures each). Wetlands were chosen to represent differing amounts of cropland surrounding the structure, and the type of nest structure (single- or double-cylinder) was chosen randomly, resulting in the deployment of 53 single- and 57 double-cylinders. Nest structures were inspected ≥4 times annually to record the timing of all nesting attempts and hatch dates. In addition, clutch size was recorded for 139 nests during the course of the study. Because repeated observations were recorded on the same set of nest structures, one might expect observations (structure occupancy or clutch size) from the same structure to be more similar than observations from different structures.

Linear response models

We begin by considering 106 clutch sizes from 52 successful nests. Extreme outliers reflecting parasitized nests were dropped, and only nests initiated before 30 May of each year were included to avoid the need to consider a nonlinear effect of Julian Date on clutch size (as in Zicus et al. 2003). To account for correlation among repeated observations from the same nest structure, we fit a simple linear-mixed effects model with a single random effect to account for individual (nest-structure) heterogeneity in clutch sizes:

  • image( eqn 1a)
  • image( eqn 1b)

where Yij and Dij represent the clutch size and Julian nest initiation date for the jth observation at the ith structure respectively, Xi is an indicator variable for nest structure type (equal to 0, 1 for single, double cylinders respectively), βs are (fixed effects) regression parameters, τi is the random nest structure effect and inline image measures within-structure variation (within-structure residuals, εij, are assumed to be independent of τi). The random effects (τi) are used to capture response dependence arising from latent or unobserved characteristics of the nesting structures. Their variance component, inline image, measures the extent of between-structure heterogeneity not captured by the fixed effects (Julian nest initiation date and nest structure type). The intracluster-correlation coefficient, ρ, describes the proportion of total response variation arising from between-cluster variation, and is given by inline image Thus, inline image can also be used to provide an intuitively appealing measure of the extent of within-structure response dependence (relative to total variance).

Marginal model parameters can be derived analytically in the case of linear-mixed effects models. Let Yi = (Yi1, Yi2, …, inline image) denote the response vector for structure i (at each of mi observation times) and Di and Xi be mi × 1 matrices containing the Julian nest initiation dates and cylinder types for structure i. The marginal model for (1) is given by:

  • image( eqn 2)

One could arrive at model (2) directly without first postulating the existence of random effects, i.e. by thinking in terms of the population of responses and specifying a correlation structure directly for Yi (in addition to the model for the mean response vector). Most software packages offer the capability of fitting this model using either a conditional or marginal specification (e.g. we obtained identical answers using lme and gls functions in the r programming language; Pinheiro & Bates 2000). However, the marginal specification permits negative within-individual correlation (e.g. as might arise from competition among litter mates) whereas the conditional formulation does not (Molenberghs & Verbeke 2005:39).

The equivalence between the marginal and conditional models in this case also implies that the regression parameters have both SS and PA interpretations. The conditional mean, E[Yi | Xi,Di, τi] = (βτi) + DijβXijβ2, describes the expected clutch size (as a function of Julian nest initiation date and nest structure type) at a particular location, whereas the marginal mean, E[Yi | Xi,Di] = βDiβXiβ2, describes the expected clutch size for a group of structures. The effects of Julian nest initiation date and structure type are linear in both cases. Thus, β1 measures the change in the expected clutch size resulting from postponing nest initiation at a particular structure by 1 day (SS interpretation) as well as the expected difference in clutch size among groups of structures initiated on any two consecutive days (PA interpretation). Similarly, β2 measures the expected change in clutch size resulting from changing the structure type from a single-cylinder structure to a double-cylinder structure (at a particular location) as well as the difference in the expected clutch size across the population of single- and double-cylinder nest structures.

Models for non-Gaussian responses

Mixed effects models

Mixed effects models for non-Gaussian data are almost always formulated using a conditional or hierarchical approach, in which the distribution of Yi is written in terms of the conditional distribution of Yi given the random effects (τ), inline image, combined with a distribution of the random effects, f(τ|γ), where γ represents additional parameters that determine the distribution of the random effects. In our example, let Yij be a binary indicator variable taking the value 1 if a nest was initiated in structure i during time point j (= 1,…,n; = 1,…,t), and 0 otherwise. If four chronological periods are defined for each study year, each nest structure could have up to 12 observations (four periods × three study years). In some cases, missing data resulted from prior occupancy of a structure within a year by a different bird species. In this case, we considered the structure unavailable for mallard nesting. For binary data, a popular model is the logistic-normal model (Stiratelli, Laird & Ware 1984):

  • image( eqn 3a)
  • image(eqn 3b)

As in Zicus et al. (2006), we included in Xij (1) the amount and attractiveness of nesting cover [as indexed by average visual obstruction measurements (VOM) in a 1·6-km-radius buffer around the nest structure], the effects of which were allowed to vary from period to period; (2) the type of nesting structure (i.e. single- vs. double-nest cylinder); and (3) linear and squared terms for the size of the open-water area in which the structure was deployed. The model also included fixed effects for period, year, and their interaction to allow the expected occupancy rates to vary from year to year, and from period to period within each year.

In conditionally specified nonlinear models, the marginal distribution cannot usually be determined analytically (Pinheiro & Bates 2000). Two notable exceptions are: (1) Yi | Xi,τi ∼ Poisson (λi) with log(λi) = Xiβ + τi and inline image and (2) Yi | Xi,τi∼ Bernoulli(pi) with pi = Φ(Xiβ +τi), inline image, and Φ is the cumulative distribution function of the standard normal distribution (Ritz & Spiegelman 2004; Young et al. 2007). In the former case, the PA response pattern is determined by adjusting the intercept (marginal βo = conditional βo + inline image, with all other (covariate) βs having both PA and SS interpretations. In the latter case, marginal parameters (βM) can be determined from the estimated conditional parameters (βC) and the random effects variance component: βM =(inline image+1)−1/2βC. By contrast, in our example, the probability of nest occupancy (averaged over individuals) in eqn 3 is given by:

  • image( eqn 4)

which has no closed form solution. Similarly, to estimate β one must approximate integrals involving the distribution of the random effects. Several approaches exist, including the use of Taylor’s series expansions, numerical integration, and Markov Chain Monte Carlo methods for Bayesian implementations (Bolker et al. 2009), and the relative merits of each are outside the scope of this paper; however, maximization of the likelihood can be computationally challenging (more so than standard nonlinear regression problems), particularly for large data sets with multiple random effects. We fit this model using the glmmML function in the gmmML R library (Broström 2008).

We use the fitted model to explore the importance of VOM and nest structure type (single- or double-cylinder) in determining occupancy rates during period 1 in the first year of the study for structures in wetlands with no open water around the structure (25% of the structures were in wetlands dominated by emergent vegetation and had zero for open water size). Let βVOM and βD refer to the (period × year)-specific regression coefficients associated with VOM and a nest structure type indicator (0 for single and 1 for double cylinders) respectively. Holding the random structure effect (τi) constant, we see from eqn 3a that βVOM presents the change in log odds of occupancy as we increase the value of VOM at a particular structure by 1 U. Similarly, βD presents the change in log odds of occupancy resulting from replacing a single-cylinder structure with a double-cylinder structure at a particular location, i.e. the regression parameters have a SS interpretation. However, unlike linear-mixed effects models, parameters in generalized and nonlinear-mixed effects models do not generally also have a PA or marginal interpretation. To determine the general (unconditional) relationship between VOM and the probability of occupancy (across the population of structures), or to estimate how occupancy rates differ on average for a group of single- versus double-cylinder structures, we need to evaluate the integral in eqn 4 (see Appendix S1 in Supporting Information for r code using adaptive quadrature methods). In other words, these research questions require the marginal distribution of Y, which has no closed form expression.

We can illustrate these points graphically by plotting the conditional mean [E(Yi | VOM, wetland size = 0, period 1, year = 1999, τi = τ) for a range of τ’s] along with the unconditional (marginal) mean, [E(Yi | VOM, wetland size = 0, period 1, year = 1999], as a function of VOM (Fig. 1a). The marginal mean, which is calculated by taking the average value of the conditional means at each value of VOM, has a different shape from the SS lines. In fact, it is no longer linear on the logit scale (Fig. 1b). The marginal relationship between occupancy and VOM also differs from that of a ‘typical’ or ‘average’ structure (defined by setting τi = 0 in eqn 3a, blue dotted line). This difference can be attributed to Jenson’s inequality (i.e. E[f(x; τ, β)] ≠ f(x; E[τ], β)] for convex or concave functions, f(·), which include log(·) and logit(·) transformations). For logistic-normal models, the marginal relationship can be approximated fairly well by a logistic regression model with marginal parameters = (0·346inline image+1)−1/2β (Zeger et al. 1988), as provided by the black dashed line in Fig. 1a,b. Thus, marginal regression parameters behave like attenuated versions of the SS parameters (i.e. they will be closer to 0), with the degree of attenuation determined by the variance component associated with the random effect. Similarly, traditional odds ratio summaries estimated from conditional models (formed by exponentiating regression parameters) have SS interpretations because they control for individual random effects (in addition to the fixed effects covariates). Thus, exp(βD) = 1·11 presents the odds ratio of occupancy for a double (versus single) cylinder at a particular location, and is larger than the approximate marginal odds ratio, exp[(0·346inline image + 1) 1/2βD] = 1·08. These two approaches estimate different quantities (i.e. SS and PA effects), and therefore it is not surprising to see that they differ, sometimes substantially (Molenberghs & Verbeke 2005, Chapter 16).

image

Figure 1.  Comparison of estimated subject-specific (E[Yij | τi, Xij]) and population-averaged (E{E[Yij | τi, Xij])}) relationships between visual obstruction measurement (VOM), a measure of the amount and attractiveness of nesting cover surrounding the structure, and the probability of nest-structure occupancy during period 1 in year 1999 (in wetland size = 0). Estimates from the logistic-normal random effects model are depicted on the original (a) and logit (b) scales. Dotted lines show estimates of single-cylinder subject-specific regression lines for τi = 0 (blue line) and (±στ, ±2στ) (black lines). The red line depicts the corresponding population-average (marginal) relationship determined by integrating over the random effects distribution, and the bold dashed line shows the approximation in Zeger et al. (1988). Panel (c) compares predictions from the logistic-normal model (GLMM-PA and GLMM-SS correspond to marginal and subject-specific means respectively; the latter corresponding to a ‘typical’ individual with random effect = 0) and from generalized estimating equations (GEE) with three different working correlation structures: GEE-Ind, GEE-AR1 and GEE-EXC (independence, exchangeable and ar1).

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Importantly, some research questions do not lend themselves to a SS interpretation (Zeger et al. 1988; Neuhaus, Kalbfleish & Hauck 1991; Heagerty 1999). For example, a SS effect of gender would only be appropriate for those rare species in which individuals can change their gender. Even when a SS regression parameter is interpretable, its estimation arguably involves extrapolating outside the range of observed data when the covariate does not vary within a subject (Zeger et al. 1988). For instance, the change in probability of occupancy resulting from replacing a single-cylinder structure with a double-cylinder structure (at a particular location) is interpretable. However, structure types were held constant for the duration of this study, and therefore a within subject nest structure effect must be estimated solely from contrasts involving single- and double-cylinder structures placed in different locations (i.e. from different subjects). Wetland size and year- and period-specific VOM effects suffer from the same problem and are also difficult to interpret.

An alternative interpretation, sometimes put forth, for the regression parameters in generalized linear and nonlinear-mixed effects models is that they represent the population median response (e.g. Rabe-Hesketh & Skrondal 2008). Indeed, setting τi = 0 in eqn 3 yields exponentiated parameters that can be interpreted as comparing the median odds in a population with + 1 to the median odds in a population with x. Similarly, setting τi = −1·96 for a N(0,1) random effect yields comparisons between the 2·5th percentiles of the two populations. We still view these interpretations as SS, as they compare one individual in a population with X+ 1 to another individual in a population with x, where the two individuals have exactly the same random effect. With the assumed continuous Gaussian random effects distribution, all subjects have unique random effects, and thus, the parameters necessarily have a within subject or SS interpretation. However, we recognize that a comparison between ‘average’ (i.e. median) individuals from two populations, motivated by considering the SS response with τi = 0, may have some appeal as an alternative form of population-level inference. The more important point is that parameters in GLMM estimate fundamentally different quantities than marginal (mean) model parameters, e.g. estimated by GEE fit to non-Gaussian data (covered in the following section). In particular, GLMM parameters will be larger in absolute value than their GEE counterparts when using a logit link (Diggle, Liang & Zeger 1994).

Marginal model estimation for non-Gaussian data

Because the multivariate normal distribution is defined completely by parameters describing its first two moments (i.e. means, variances and covariances), it is easy to directly postulate an appropriate model for the multivariate response vector in the linear case (e.g. eqn 2). By contrast, most other distributions can be difficult to generalize to dependent data without reverting to a conditional approach (e.g. using random effects). In particular, for the exponential family of distributions, which includes the Binomial, Gamma, Poisson and Multinomial (as well as the Gaussian distribution with constant variance) as special cases, variances and covariances are themselves constrained by mean parameters in complicated ways, and as a result, multivariate analogues often lead to complicated likelihoods that are difficult to parameterize or result in parameters that are difficult to interpret (Diggle et al. 1994; Molenberghs & Verbeke 2005). Therefore, marginal regression models have most often been fit using generalized estimating equations (GEE; Liang & Zeger 1986).

Rather than identifying a complete multivariate distribution p(Yi | Xi), GEE only require a model for the first two moments of the distribution (i.e. mean and covariance), and estimate regression parameters by solving:

  • image( eqn 5)

where Yi = (Yi1, Yi2, …, inline image) denotes the response vector for individual i, μi = f(Xi,β) represents the mean response as a function of covariates (Xi is an mi × p matrix holding the individual i’s covariate values at each of mi response times, with p denoting the length of the vector β), and ∂μi/∂βi is an mi × p matrix of first derivatives of μi with respect to β. The variance-covariance matrix for subject i, Vi(α) = Ai1/2Ri(α)Ai1/2, is typically modelled by adopting a common form for the variance (based on an appropriate member of the exponential family, e.g. Ai = diag{μi(1 − μi)} for binary data), and using a ‘working correlation model’, Ri(α), to describe within-subject dependencies. These dependencies are characterized by one or more additional parameters, α, which can themselves be ‘formulated and estimated in a number of ways’ (Diggle et al. 1994:151). In any number of settings, proper specification of Vi(α) can be challenging. However, even when the model for Vi(α) is incorrect, inline image will be (asymptotically) unbiased and its distribution will approach that of a normal distribution as the number of clusters (i) goes to infinity, provided that the model for the marginal mean is correct. Precision can be gained with proper specification of Vi(α). Separate analyses using multiple working correlation matrices could be used as a sort of sensitivity analysis, or one can use various tools (e.g. lorelograms; Heagerty & Zeger 1998) to determine an appropriate correlation structure. Confidence intervals and hypothesis tests should employ robust (‘sandwich’) standard errors, as is standard practice for most applications of GEE (Liang & Zeger 1986).

We fit a marginal response model using the same set of predictors as in eqn 3 using the GEE package in r (Højsgaard, Halekoh, & Yan 2005):

  • image( eqn 6)

Note that (6) differs from the logistic normal model (eqn 3) in that the (link transformed) marginal means, logit(E[Yi X]), rather than the conditional means, logit(E[Yi | X,τi,]), are assumed to be a linear function of covariates (Xijβ). We used three different working correlation assumptions: independence (equivalent to logistic regression), exchangeable (equal correlation among all observations from the same structure, similar to the logistic-normal model) and an AR1 structure (i.e. serial dependence). The three GEE models yielded similar conclusions (e.g. regarding the effect of VOM) and also agreed well with the marginalized relationship estimated from the logistic normal model (eqn 3; Fig. 1c). Marginal parameters from the fitted GEE models were relatively insensitive to the assumed correlation structure and were closer to 0 than conditional parameters estimated from the logistic-normal model (Table 1).

Table 1.   Estimates of regression parameters and (SE) for models fit to the nest structure occupancy data. Generalized estimating equation (GEE) regression models were fit using independence, exchangeable and ar1 working correlation structures (GEE-Ind, GEE-EXC and GEE AR1 respectively) and the GLMM model was fit using a random intercept for nest structure
ParameterGEE-IndGEE-EXCGEE-AR1GLMM
Intercept−3·78 (0·54)−3·63 (0·63)−3·79 (0·64)−4·82 (0·75)
Year 19981·03 (0·39)1·02 (0·36)1 (0·36)1·23 (0·44)
Year 19990·96 (0·41)0·95 (0·31)0·94 (0·31)1·17 (0·46)
Period 21·2 (0·73)1·29 (0·71)1·26 (0·75)1·6 (0·83)
Period 32·2 (1·98)2·44 (1·77)2·45 (1·89)3·25 (2·25)
Period 44·87 (2·35)4·84 (2·48)4·83 (2·71)6·17 (2·72)
Structure type (double)0·12 (0·19)−0·02 (0·29)0·12 (0·3)0·1 (0·37)
Visual obstruction measurement (VOM)4·21 (1·41)3·9 (1·67)4·29 (1·73)5·01 (1·93)
Wetland size (linear effect)0·22 (0·04)0·23 (0·06)0·22 (0·05)0·31 (0·08)
Wetland size (quadratic effect)−0·01 (0)−0·01 (0)−0·01 (0)−0·01 (0)
Year 1998: period 2−1·99 (0·63)−1·85 (0·57)−1·92 (0·6)−2·25 (0·7)
Year 1999: period 2−1·32 (0·59)−1·21 (0·45)−1·28 (0·47)−1·46 (0·66)
Year 1998: period 3−1·59 (0·96)−1·48 (0·83)−1·5 (0·87)−1·75 (1·06)
Year 1999: period 3−1·24 (0·95)−1·12 (0·8)−1·17 (0·85)−1·3 (1·05)
Year 1998: period 4−1·58 (0·72)−1·62 (0·68)−1·59 (0·7)−1·9 (0·78)
Year 1999: period 4−1·27 (0·68)−1·26 (0·63)−1·27 (0·66)−1·51 (0·73)
Period 2: VOM−2·29 (2·27)−2·41 (2·23)−2·5 (2·35)−2·96 (2·66)
Period 3: VOM−5·28 (2·63)−5·11 (2·5)−5·57 (2·67)−6·68 (3·17)
Period 4: VOM−6·33 (1·86)−6·02 (2·03)−6·39 (2·21)−7·7 (2·4)

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Most statistical software packages provide the capability of fitting mixed effects and GEE regression models, and both approaches have been applied in ecological research. Although differences between the two approaches, including the interpretation of model parameters, have been well-established in the statistical and biomedical literature (e.g. Zeger et al. 1988; Neuhaus et al. 1991; Pendergast et al. 1996), ecologists do not appear to be cautious in choosing between SS and PA inferences. In many cases, the differences between SS and PA parameters can be substantial (Molenberghs & Verbeke 2005, Chapter 16). Thus, choice of an analysis method may influence management decisions and also complicate comparative studies that include both types of analyses. For example, estimates of SS effects can be expected to be larger in absolute value than their PA counterparts when using a logit link to analyse binary response data (Table 1). Therefore, managers may have inflated expectations of population-level responses (e.g. mean structure occupancy rates) when relying on SS inferences.

Although we only considered models with random intercepts, the same interpretation issues apply to random coefficient models (i.e. average slope parameters describe SS response curves for an ‘average’ subject with random effects set = 0). Recent papers arguing for the use of GLMM to study resource selection have not clearly distinguished between conditional and marginal models (Gillies et al. 2006;Hebblewhite & Merrill 2008). Both of these papers suggest marginal inference can be accomplished by setting all random effects = 0, when in fact these ‘marginal’ models are actually SS response models for a ‘typical’ or ‘average’ individual. Additionally, both include comparisons of logistic regression and logistic-normal (mixed) models without explicitly recognizing that these two approaches are estimating different quantities. Not surprisingly, the marginal model predictions (in these papers) were attenuated relative to SS predictions for an individual with all random effects = 0.

Mixed effects models are likely to appeal to ecologists as they are formulated using a conditional approach that can often match the way in which ecological data are structured (e.g. individuals sampled within different population segments). However, one should also consider the estimation target (i.e. whether a SS or PA interpretation is desired) when choosing between marginal and conditional (random effects) approaches to modelling non-Gaussian data. Many applied questions in ecology, wildlife management and conservation biology (including the current illustrative examples) focus on the importance of general landscape features as they relate to population-level performance measures (Aarts et al. 2008). GEE may be preferable for addressing these questions since they provide direct estimates of PA parameters. On the other hand, conditional models may be more appropriate for research questions in evolutionary ecology as natural selection occurs at the level of individuals (Cooch, Cam & Link 2002; Nussey et al. 2008). Further, some problems will probably benefit from both SS and PA interpretations – e.g. in habitat selection studies a SS approach may be useful for understanding factors that influence individual behavioural choices whereas a PA approach may provide insight regarding space use patterns at the population level. A distinct advantage of conditional models is that they offer a means for developing individual-level inferences and predictions, while marginal relationships can still be recovered by integrating (numerically or by simulation) over the random effects distribution(s) (Fig. 1a; Lee & Nelder 2004; Molenberghs & Verbeke 2005:301; Aarts et al. 2008). Although this is rarely performed (probably because it is not a standard option in most software packages), it involves only a few extra programming steps. Estimates of uncertainty require more work, but could be explored using a parametric bootstrap. Alternatively, the approximation established by Zeger et al. (1988) could be used to determine marginal relationships from logistic-normal models.

For many problems, the degree of dependence in the data may itself be of interest. This is particularly true in among-species comparative studies, where the degree of response dependence can be informative with regards to evolutionary processes (Freckleton, Harvey & Pagel 2002; Butler & King 2004). Response dependence in conditional models is quantified by the variance components associated with the random effects, whereas GEE largely treat the dependence structure as a nuisance that must be accounted for when estimating regression parameters and their uncertainty. While variance components in linear-mixed effects models are easy to interpret, the same is not the case when the random effects enter on a transformed (e.g. logit) scale (Preisser, Arcury & Quandt 2003; Larsen & Merlo 2005). Larsen & Merlo (2005) suggest an odds ratio approach to measuring the strength of dependence in logistic-normal models. Specifically, they quantify the median of the distribution of odds ratios (MOR) for two randomly chosen individuals having the same covariate values. The relative propensity of each individual to have an event is determined by the individual random effects (τi, and τj), with the odds ratio between the individual with the higher propensity and the individual with the lower propensity given by exp(|τiτj|). The median of this distribution, inline imageinline image (where Φ−1 is the quantile function of a standard normal distribution) is always >1, with larger values suggesting a higher degree of dependence (for the nest structure data, MOR = 3·82). Alternatively, Carey, Zeger & Diggle (1993) proposed a variation of GEE in which the within-cluster response dependence is parameterized using pairwise log odds ratios rather than correlations. This approach, sometimes referred to as an alternating logistic regression (ALR) model, provides a more natural description of response dependence for binary data and is also amenable to graphical displays (Heagerty & Zeger 1998). Further comparisons between traditional GEE, ALR and GLMM for binary data, as well as measures of response dependence from these models can be found in Preisser et al. (2003), Chaix et al. (2004) and Preisser (2004).

Lastly, other operating characteristics can be important when choosing between marginal and conditional modelling approaches. In particular, likelihood-based approaches often have several advantages associated with them, including better small sample properties, more widely available tools for models selection, and less stringent requirements associated with missing data (and thus, better protection against heavily unbalanced data). Whereas standard applications of GEE require data to be missing completely at random, likelihood approaches are still appropriate if missingness depends only on the individual’s observed data (i.e. their covariates and their other response values) and not on data that were not observed (e.g. the missing response) (Carriere & Bouyer 2002; Rubin & Little 2002); the latter type of missingness is referred to as missing at random (MAR) in the statistical literature (Rubin & Little 2002). GEE can be modified using inverse probability weights (i.e. reflecting the probability missingness) (Robins, Rotnitzky & Zhao 1995) or doubly robust estimators (Scharfstein, Rotnitzky & Robins 1999; Bang & Robins 2005) to apply to the MAR case. Alternatively, multiple imputation techniques may be used in conjunction with GEE or likelihood-based models when data are MAR (Rubin & Little 2002; Molenberghs & Verbeke 2005). In our example, missing data occasionally resulted from nest structures being occupied by a different bird species. This prior occupancy could conceivably reflect an overall attractiveness of the nest structure. If missingness only depended on measured covariates and prior occupancy status, logistic-normal parameters would remain unbiased. Parameters estimated from traditional GEE (without imputation), by contrast, would not be unbiased since they will effectively present more weight to those (non-random) individuals with no missing data. In our example, however, the difference between SS and PA estimated effects was considerably larger than the difference in marginal estimates from the two approaches (logistic-normal model, GEE; Fig. 1c). This result supports the general conclusion of Heagerty & Kurland (2001) and Lee & Nelder (2004) that marginal relationships are typically more robust than conditional ones to assumptions regarding the dependence structure, which makes intuitive sense since the interpretation of conditional parameters itself depends on these assumptions. On the other hand, conditional models often have more power to detect within-subject effects when the dependence structure can be correctly specified.

Our focus has been on conditional (SS) and marginal (PA) models as well as the most commonly used estimation procedures (maximum likelihood for conditional models and GEE for marginal models). While we discussed indirect estimation of marginal model parameters once conditional model parameters have been estimated, this is rarely performed in practice. Thus, historically, for binary response data, the model (SS or PA) has been intrinsically linked to the estimation procedure (maximum likelihood or GEE). An emerging class of models, known as marginalized models (Heagerty 1999; Schildcrout & Heagerty 2007), allows us to decouple the model from the estimation procedure by permitting likelihood-based estimation of marginal model parameters. Thus, if interested in marginal model estimation, we may choose between likelihood-based and GEE estimators, and if we prefer likelihood-based estimation, then we may choose between marginal and conditional models. Explicit discussion of marginal model estimation with likelihood based methods is beyond the scope of this research; however, it will be addressed in future research along with extensive characterizations of operating characteristics among the three classes of model by estimation procedure combinations (e.g. likelihood-based estimation or marginal and conditional model parameters and semi-parametric GEE estimation of marginal model parameters).

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

The authors would like to thank J. Giudice and D. Staples for comments on earlier drafts as well as critical reviews by two anonymous referees.

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  3. Introduction
  4. Materials and methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Appendix S1.  R code for running all analyses in the paper (Microsoft Word).

Appendix S2.  Clutch size data (.csv file).

Appendix S3.  Nest structure occupancy data (.csv file).

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JPE_1692_sm_appendixS1.doc42KSupporting info item
JPE_1692_sm_appendixS2.csv2KSupporting info item
JPE_1692_sm_appendixS3.csv38KSupporting info item

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.