SEARCH

SEARCH BY CITATION

Keywords:

  • Batrachochytrium dendrobatidis;
  • chytridiomycosis;
  • detection;
  • disease;
  • occupancy;
  • prevalence;
  • sensitivity;
  • specificity

Summary

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

1. Epidemiological studies are crucial for understanding the distribution and dynamics of emerging infectious diseases. To accurately assess infection states in wild populations, researchers need to account for observational uncertainty. We focus on two sources of uncertainty when estimating epidemiological parameters: nondetection of infection in sampled individuals and sampling error when quantifying infection intensity for infected individuals.

2. We developed new analytical methods to simultaneously estimate prevalence and the distribution of infection intensities based on repeated sampling of individuals in the wild. The methods are an extension of those used for occupancy estimation and address both sources of observation error. At the same time, we account for heterogeneity in detection probability that results from individual variation in infection intensity.

3. We use two estimation approaches to account for detection. The first is to use the complete likelihood in a hierarchical Bayesian model and fit using Markov chain Monte Carlo sampling. The second is to estimate the detection relationship using a mark–recapture abundance estimator and use those results to calculate weighted estimates for prevalence and mean infection intensities.

4. We use data from a field survey of Batrachochytrium dendrobatidis in Illinois amphibians to test these methods. We show that detection probability using quantitative PCR is strongly related to infection intensity, measured in zoospore equivalents. Sites in the study varied greatly in estimated prevalence and to a lesser extent in mean infection intensities of infected individuals. We did not find evidence of a relationship of snout-vent-length to infection intensity or prevalence. Naïve estimates of prevalence that do not account for detection were less than estimates for either of our methods, which yielded similar prevalence values for most sites.

5. Uncertainty when assessing disease state is a characteristic of most diagnostic tests. The estimators presented here account for this uncertainty and thus can improve accuracy when assessing the relationship of ecological factors to prevalence and infection intensity.


Introduction

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

Increased interest in the dynamics of emerging wildlife diseases and zoonotics has spurred a need for field investigations to study the dynamics of diseases in wild animals (Daszak, Cunningham & Hyatt 2000). Scientists and wildlife managers need accurate assessments of current disease states and infection to understand disease dynamics in wild populations. However, methods that do not incorporate the effects of imperfect detection ignore a serious source of potential bias when estimating disease parameters (Jennelle et al. 2007; Conn & Cooch 2009; Murray et al. 2009; McClintock et al. 2010; Beyer et al. 2011). Recently developed inferential methods that account for incomplete detection in epidemiological studies have the potential to improve the quality of inferences researchers make from field data (reviewed in Cooch et al. 2011).

We focus on two common measures of infection state used in epidemiological studies: (i) whether a pathogen is present or absent in individuals and (ii) conditional on being present, the intensity of infection. Measures of infection intensity could include pathogen load (e.g. abundance of the pathogen) and measures of pathogen effects (e.g. symptom intensity or immune response to the pathogen). We focus on measures of pathogen load in this paper, but the methods are applicable to other continuous measures of infection intensity. At the individual level, the two measures correspond to the probability an individual is infected (pathogen present) and the expected infection intensity for an infected individual. At the population level, pathogen presence corresponds to the infection prevalence (i.e. proportion of individuals infected). We describe infection intensity based on both the mean infection intensity of infected individuals in a sample population and variability in the intensity among infected individuals. Note that we have defined measures of infection intensity to be conditional on the infection being present.

For most sampling methods, both infection presence and intensity are measured with uncertainty. In the case of pathogen presence, uncertainty is related to the sensitivity (probability an infected individual will be classified as infected) and the specificity (probability an uninfected individual will be classified as uninfected) of the diagnostic method. We develop estimators under the assumption that specificity is 1, although we detail how they may be modified to deal with false positives in the discussion. We focus instead on accounting for false negatives, which occur when sensitivity is less than 1. When measuring infection intensity, there are two ways that observation error can occur. First, sampling error can occur when the disease is detected. For example, duplicate runs from the same individual using quantitative PCR will vary in the estimate of pathogen load. Second, false negatives, which occur when the pathogen is not detected at all, can cause observation error. For example, if a pathogen is less likely to be detected on individuals with low infection intensity, then missed detections will affect estimates of the mean and variance of infection intensity.

Even if nondetection is addressed in estimates, unaccounted for variation among individuals in the probability a pathogen is detected can still bias estimators of prevalence (Royle & Nichols 2003; MacKenzie et al. 2006). In most cases, infection intensity is likely to be the largest source of among-individual variation in the probability of detecting the pathogen (Lachish et al. 2012). To reduce potential bias, it is important to account for this relationship. One approach, demonstrated by Lachish et al. (2012), is to model the expected distribution of detection probabilities based on a latent distribution of pathogen abundances (Royle & Nichols 2003). We propose an alternative approach where detection is estimated as a direct function of infection intensity. The method is similar to estimators of abundance used in mark–recapture studies (Huggins 1991; Royle 2009). This allows us to directly incorporate information about measured infection intensity when making estimates and provides a flexible framework for examining factors related to both infection presence and intensity.

As a motivating example, we focus on the emerging disease chytridiomycosis in amphibians, caused by infection of the fungus Batrachochytrium dendrobatidis (Bd; Longcore, Pessier & Nichols 1999). Chytridiomycosis has had especially devastating consequences for amphibian populations worldwide (Fisher, Garner & Walker 2009), causing rapid declines and local extirpations of numerous species (e.g. Lips et al. 2006; Cheng et al. 2011). Significant effort has been expended to determine the geographical extent and history of spread of the disease organism. However, the utility of much of the data is limited because of opportunistic sampling and an inability to make comparisons across species and regions (Muths, Pedersen & Pedersen 2009). Studies of Bd dynamics have focused on both prevalence and infection intensity (e.g. Briggs, Knapp & Vredenburg 2010; Vredenburg et al. 2010). The standard assay for determining Bd infection is based on the detection of Bd zoospores on the skin of animals, measured by quantitative PCR (qPCR; Hyatt et al. 2007). The skin of an individual is swabbed and zoospore equivalents on the swab are measured, where zoospore equivalents are a proxy for infection intensity. Like most tests for disease presence, the sensitivity of the assay is <1, leading to results that include detection errors (Hyatt et al. 2007; Cheng et al. 2011).

Methods originally developed to estimate site occupancy probabilities for species (MacKenzie et al. 2006) can be used to account for sampling uncertainty when estimating pathogen presence (Thompson 2007; Abad-Franch et al. 2010; Adams et al. 2010; McClintock et al. 2010; Walker et al. 2010; Lachish et al. 2012). We extend these methods using an approach that simultaneously estimates the probability individuals are infected and the infection intensity of infected individuals, while accounting for observation error in each. This allows us to quantify the relationship in infected individuals between infection intensity and the probability of detecting the pathogen, an important source of detection heterogeneity (Royle & Nichols 2003; Lachish et al. 2012). We show how to estimate infection prevalence and the mean and variance of infection intensity for sampled populations. In addition, we demonstrate how estimators can be used to determine factors correlated with individual and among-population variation in prevalence, infection intensity and detection using fixed- and random-effect modelling approaches. We construct the statistical model using Bayesian methods and using a two-step approach where initial detection relationships are estimated using maximum likelihood estimators developed for mark–recapture experiments. Methods are demonstrated using a sample data set for Bd infection in Illinois amphibians.

Materials and methods

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

Hierarchical Bayesian Estimator

First, we demonstrate how to estimate parameters using a hierarchical occupancy model and MCMC (Markov chain Monte Carlo) to fit the model. The method fully accounts for our two types of measurement error, uses a single step for analysis and readily allows for additional model structure to be incorporated. Thus, it will be the preferred approach in most cases.

Data come from a population where n individuals are sampled, at least a subset of individuals are sampled ≥2 times, and the ith individual is sampled Si times. We assume that whether the pathogen is present and the actual infection intensity remains constant for all samples from the same individual.

For the ith sampled individual, zi = 1 if the individual is infected and zi = 0 if it is not. Each infected individual in turn has an infection intensity xi. Both z and x are latent variables that are not directly observed. Instead, we must make inferences about each based on the observed data. For each sample, we can observe if infection was detected. The observation for the sth sample from the ith individual is denoted by yis, where yis = 1 if detected and yis = 0 if it is not. If the pathogen is detected, we observe a value for infection intensity for the sample wis, which is measured with sampling error.

Figure 1 describes the components of the hierarchical model. We denote the probability an individual is infected as ψ. To demonstrate the approach, we assume infection intensities come from a normal distribution, although other distributions may be more appropriate in other cases. We denote mean infection intensity of infected individuals as δ and the standard deviation of infection intensity for infected individuals as σ(δ). We assume a normally distributed sampling error in measuring infection intensities where the standard deviation is σ(e). The probability of detecting the pathogen in an infected individual is denoted as pi, where pi = Pr(= 1|zi = 1, xi).

image

Figure 1.  Relationship among true state variables, observed state variables and population parameters for our model used to estimate prevalence and infection intensity of Bd in an amphibian population. The true state of the ith sampled individual includes whether it was infected or not (zi; dark individuals are infected) and infection intensity for infected individuals (xi) measured in log-zoospore equivalents (LZE). This differs from the observed state variables, which are whether disease is detected in the sth sample (yis; dark samples indicate infection detected) and the measured intensity for positive samples (wis). In fitting a model, we estimate parameters related to the population being sampled, including the prevalence (ψ), the mean intensity of infected individuals (δ) and variance in infection intensity (σ[δ]). These parameters are dependent on the detection process including the probability of detecting disease as a function of infection intensity (pi[α,β]) and sampling error in measuring infection intensity (σ[e]).

Download figure to PowerPoint

First, we consider components related to the presence of disease in sampled individuals. The probability of the ith individual being infected is a Bernoulli random variable where

  • image

The probability of observing detection infection in the sth sample from the ith individual is

  • image

so that the probability is Bernoulli distributed with probability pi if the individual is infected (i.e. zi = 1) and 0 if it is not.

Next, we consider infection intensity. If the ith individual is infected, the true infection intensity comes from a normal distribution so that

  • image

Observations of intensity for the sth sample from the ith individual also include the measurement error so that

  • image

If infection is detected, then w is observed but w will be unobserved for infected individuals where infection is not detected in a sample. As part of the MCMC simulation, we calculate x and w for all samples. However, the values only contribute to the likelihood when = 1.

Finally, we specify the relationship between our occurrence model and our intensity model. The two are related through the detection parameter pi, which is a function of infection intensity xi, and conditional on presence of Bd on an individual. We use a generalized linear model to estimate this relationship with a logit-link function. Two additional parameters can be used to describe the relationship (intercept α and slope β) where

  • image

We implement the model using weak priors, selecting flat normal priors for δ, and β (N[0,0·001]) and uniform priors for ψ and logit(α) (U[0,1]) and for σ(δ) and σ(e) (U[0,10]).

The model described above constitutes the basic approach for dealing with a single population and serves as the backbone for other analyses. We provide code in Appendix A to simulate data and fit the basic model with winbugs (v. 1.4.3; Spiegelhalter, Thomas & Best 1999) using the r2winbugs package (Sturtz, Ligges & Gelman 2005) in r (v. 2.12.1; R Development Core Team 2011). We truncate extreme values of normal priors to improve efficiency of MCMC simulation (Royle & Dorazio 2008). Note the model is very similar to methods used for estimating abundance from mark–recapture data when detection is a function of an individual covariate (Royle & Dorazio 2008 Section 6.5; Royle 2009). This approach differs in that we estimate prevalence (i.e. occupancy) rather than abundance and also account for sampling error in measuring infection intensity.

The statistical model can be extended to incorporate more complex study designs and to draw additional inferences. Infection (i.e. ψ, δ and σ[δ]) and detection parameters (p and σ[e]) can be specified to be functions of both fixed effect and random effect, using generalized mixed models to specify relationships (Bolker et al. 2009). For example, infection intensity could be a function of fixed effects for age (= 0 for juveniles and = 1 for adults) and length (l) where

  • image

with priors specified for each of the coefficients (α, βa, βl). Likewise, if multiple sites are sampled, one might consider modelling variation in site-specific prevalence ψs as a random effect such as

  • image

where ψ is the mean prevalence among sites and σ(ψ) is the standard deviation of among-site variation. We provide code to simulate data and fit a model for multiple sites that vary in prevalence in Appendix B. The variation has both a random and a fixed component that are accounted for in the prevalence model.

Distributional Assumptions

As we noted, a distribution needs to be specified for the expected values of infection intensity for infected individuals. In cases where infection intensity is a measure of pathogen load, a log-normal distribution is a useful starting point. To implement this using our formulation, values of w should be the log-transformed measure of load. The mean log-load and standard deviation of log-load are given by δ and σ[δ]. The gamma distribution is another potential continuous distribution restricted to positive values. Both the Poisson and negative binomial distribution are common for count data and may fit well in some cases. Both distributions include 0 as a possible value, and estimates of prevalence must account for these additional zeros. Royle & Nichols (2003) and Lachish et al. (2012) demonstrate calculations for this case.

We examined how estimates of parameters were affected when data were generated under a different distribution than specified in the model. Our goal was to provide a basic sense of how misspecifying the distribution could affect estimates, using a focused set of examples. We analysed data assuming a log-normal distribution, and because our interest was in bias rather than precision, we minimized sampling error by simulating a very large data set (50 000 individuals with three each) for each sampling distribution. For all simulations, we generated data where ψ = 0·50, mean log-infection intensity of 4, σ(e) = 0·5, α = −4 and β = 1. We simulated data from log-normal (log(μ) = 4, σ(log[μ]) = 1·50), gamma (α = 0·0563, β = 0·0125), Poisson (λ = 4) and negative binomial distributions (r = 1, p = 0·98201). To provide insights when violations were severe, we also simulated a weakly bimodal and strongly bimodal distribution. These were simulated from a mixture of 2 log-normal distributions with half of individuals coming from each. For the weakly bimodal distribution, we used log(μ) = 2·8 and 5·2 and σ(log[μ]) = 0·9. For strongly bimodal distribution, we used log(μ) = 2·5 and 5·5 and σ(log[μ]) = 0·5. The simulated proportion of individuals with at least one positive sample ranged from 0·36 to 0·42 among distributions, and the mean observed infection intensity ranged from 4 to 4·6. Both measures were positively related to σ(δ). For all simulated distributions, prevalence and mean infection intensity were much smaller than if we had not accounted for detection (Table 1). The log-normal distribution fits the data well in many cases, with lack of fit mostly at low intensity values (Fig. 2). Bias was negligible for the log-normal and for the Poisson distribution. Because the mean and variance are equal for the Poisson distribution, detection heterogeneity was minimal. Bias was still small for each of the other unimodal distributions and the weakly bimodal distribution. The strongly bimodal distribution resulted in larger bias in prevalence, although a cursory examination of the data would show how poorly the distributional assumption would be likely to hold.

Table 1.   Results of simulations to determine the effect of misspecification of the true distribution of infection intensities. Data were simulated using six different distributions and fit using a log-normal model. For each parameter, the difference between the estimated and expected value is shown
Simulated value difference from expectedψδσ(δ)σ(e)αβ
0·504·0010·50−4·001·00
  1. 1The simulated standard deviation of δ differed for each distribution. Expected values for each distribution were log-normal – 1·5, Poisson – 0·14, negative binomial – 1·21, gamma – 1·19, weak bimodal – 1·50 and strong bimodal 1·58.

Log-normal−0·0070·050−0·0160·0000·085−0·021
Poisson−0·001−0·009−0·011−0·001−0·8710·220
Negative binomial−0·0250·169−0·2240·0030·159−0·036
Gamma−0·0220·152−0·2290·0070·030−0·002
Weak bimodal−0·0210·085−0·008−0·0020·212−0·043
Strong bimodal−0·0540·1950·0130·0030·500−0·096
image

Figure 2.  Comparison of the simulated cumulative distribution for six distributions for infection intensity (black line) to the estimated cumulative distribution assuming infection intensities was log-normally distributed (red dashed lines).

Download figure to PowerPoint

Postpredictive checks are a useful goodness-of-fit method to determine whether hierarchical models are properly specified (Gelman & Hill 2007). This involves simulating a new data set based on parameter values for each MCMC iteration and comparing them to the actual data to see whether they are consistent. We focus on the cumulative distribution of observed values of infection intensity w because we could not directly observe true infection intensities. However, we expect distributional violations to be reflected in the observed infection intensity values. For each MCMC iteration, we calculate new values of w and examine the cumulative distribution for samples where = 1.

Ad Hoc Approach

An alternative approach is to treat the estimation of prevalence as a closed population abundance estimation problem (Otis et al. 1978; Huggins 1991), which has previously been suggested for occupancy estimation problems (Nichols & Karanth 2002). We use Huggins (1991) conditional likelihood approach, which allows inclusion of individual covariates such as infection intensity and is readily implemented in the free software package mark (White & Burnham 1999). Estimates of prevalence are then calculated by weighting observations by the estimated probability they would have been detected in the sample (Horvitz & Thompson 1952). Standard errors and confidence intervals can be estimated via bootstrap (Huggins 1991).

Estimates of ψ and δ and their standard errors are generated as follows. From the closed population estimator, we estimate parameters related to detection (α, βw). We suggest that all observations of intensity be averaged for an individual with at least one positive observation of Bd and denote the value as wi*. Then, for each of these K individuals, we calculate the detection probability for the individual as

  • image

The probability that Bd would be detected at least one time for the ith individual, pi*, if it is sampled S times is

  • image

The inverse probability inline image can be thought of as representing the average number of infected individuals of a given infection intensity that would need to be sampled to observe (i.e. detect) one infected individual. An estimator for the prevalence in a population of n sampled individuals is given by

  • image

and the mean infection intensity as

  • image

Estimates of standard errors and confidence intervals for inline image, inline image and inline image can be calculated via bootstrap simulations. For inline image sampling with replacement should occur from the population of individuals with ≥1 positive detection. The standard error of inline image can be calculated by resampling from the combined pool of both Bd positive and Bd negative individuals.

Example

We analysed a portion of data from a multi-year multi-species study of Bd presence on amphibians in Illinois. We focus on data for 366 Hyla chrysoscelis/versicolor (hereafter, gray treefrog samples) from 11 sites, sampled in 2008. The species are morphologically indistinguishable in the field and therefore are combined. We only had repeated samples for 58 individuals. To improve our sample size for estimating detection probabilities, we included 287 additional individuals of multiple species that were double-sampled (hereafter, multi-species samples). This consisted of 75 Acris crepitans, 12 Hyla avivoca, 11 Hyla cinerea, six Pseudacris crucifer, 34 Pseudacris streckeri, nine Pseudacris triseriata, 51 Lithobates catesbeianus, 39 Lithobates pipiens, 20 Lithobates sphenocephalus, 1 Bufo americanus, 20 Bufo fowleri and one Ambystoma texanum. We assume there is a common relationship for all species between measured infection intensity and detection probabilities. Estimates of infection intensity were based on qPCR, and infection intensity was measured in log-zoospore equivalents. We assume a log-normal distribution of intensities for infected individuals. For double-sampled individuals, a second independent run using the same DNA extract was conducted.

Teams of 2–6 researchers visually searched and captured animals by hand between 19·00 and 01·00 h, when anurans were most active. A new set of latex gloves was worn for each individual, and each animal was immediately placed into a separate plastic bag. Frogs were swabbed following the techniques described by Hyatt et al. (2007; five sweeps to each thigh, each foot and the belly). We used PCR primers developed in previous Bd studies for qPCR assays (Hyatt et al. 2007). In each 96-well reaction plate, we included a negative control and standards in triplicate (i.e. 100-zoospore, 10-zoospore, 1-zoospore and 0·1-zoospore). Sample plates were run on an ABI 7300 PCR machine for 50 amplification cycles. Based on the dilution factor, we calculated zoospore equivalents from the ‘Qty’ value, which is the same as genomic equivalents.

Our objectives were threefold: (i) to demonstrate the full flexibility of the MCMC analysis, including accounting for factors affecting observation error and disease parameters, using random and fixed effects, building functions for continuous and discrete predictors, and accounting for predictors of individual and group (e.g. site) effects; (ii) to determine the relationship of detection probability to infection intensity; and (iii) to determine how accounting for detection affects estimates of prevalence.

We first analysed the data using the MCMC estimator. Our focus was on the infection status of the gray treefrogs. The multi-species data were included to improve estimates of observation parameters. Therefore, we did not focus on understanding process variation for these data. We assumed all data had a common observation process, estimating a single value for α, β and σ(e) for both the gray treefrog and multi-species samples. For the multi-species data, we estimated a single value for ψ, δ and σ(δ). For gray treefrog samples, we focused on among-site differences in infection parameters and whether snout-vent-length (SVL) predicted among-individual variation in infection probability and intensity of infected individuals. We used a random-effect formulation to constrain differences among sites in δ and σ(δ) to come from a random probability distribution (normal and gamma distributions, respectively). To make estimates of ψ for each site most comparable to other estimators, we treated each site as a fixed effect. We specified the relationship of SVL to ψ and δ using a linear and logit-linear model, respectively. Before fitting the model, we standardized SVL to have a standard deviation of 1 for all sites and a mean of 0 for each site. We simulated three chains of 15 000 iterations, keeping values from every 10th simulation for the final 10 000 iterations to estimate the posterior distributions of parameters. We checked chains for convergence and plotted the cumulative probability function of w for the repeat-sample and gray treefrog data to simulated values as a postpredictive check of our distributional assumptions (Gelman & Hill 2007).

We also compared estimates of prevalence for each of the 11 sites from the MCMC model to estimates that do not account for detection (hereafter, naïve estimates) and to estimates using our ad hoc estimator. For naïve estimates, prevalence was estimated as the proportion of individuals for which disease was detected, and the mean and variance of detection intensity were calculated for the set of positive samples. For the ad hoc estimator, we estimated the detection function based on all repeat-sample individuals and used that function to calculate prevalence for gray treefrog samples from each site. Means and confidence intervals for naïve and ad hoc estimates were calculated from 10 000 bootstrap simulations.

Results

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

Of the 345 repeat-sample individuals, 90 individuals tested positive in both samples, 29 individuals were positive in only a single sample, and the rest were negative for both. Chains converged quickly for the MCMC analysis, and our postpredictive check indicated good fit of a log-normal distribution for intensities (Fig. 3). The cumulative distribution of our data was within the range for data predicted from our fitted model.

image

Figure 3.  Postpredictive check for the cumulative distribution of observed infection intensities (w) for data from multi-species samples and for gray treefrog samples. The solid black line is the cumulative distribution for the true data. Grey lines are the cumulative distributions for each of the simulations from the Markov chain and show the upper and lower extent of expected values. Red lines are for 10 randomly selected simulations and give an idea of the expected smoothness of a single data set. Overall the fit of the model seems adequate, although there is some indication of poor fit for gray treefrog at the lowest infection intensities.

Download figure to PowerPoint

The probability of detecting Bd increased with increasing infection intensity (Fig. 4). Means and credible intervals for the posterior distributions of model parameters are summarized in Table 2 and Fig. 5. There was significant variation among sites in Bd prevalence and mean infection intensity of infected individual, although we did not observe obvious correlation among sites between the two measures (Fig. 5). The results do not support a relationship of infection probability or intensity to SVL, although credible intervals for the effect of each were wide (Table 2).

image

Figure 4.  The estimated probability of detecting Bd in a sample given the infection intensity of the individual measured in log-zoospore equivalents (ZE). The solid line is parameter values at the mean of the posterior distribution, and dashed lines represent 95% credible interval.

Download figure to PowerPoint

Table 2.   Posterior distribution of estimated parameters using the MCMC estimator
 Posterior distribution
Mean2·5%97·5%
  1. SVL, snout-vent-length.

Multi-species
 δ3·152·583·70
 σ(δ)2·742·343·19
 ψ0·340·280·40
Gray treefrog
 σ of among-site variation in δ1·290·272·56
 Effect of SVL on δ−0·10−0·640·44
 Effect of SVL on ψ−0·06−0·420·30
 σ(δ) site 12·471·763·30
 σ(δ) site 22·611·793·58
 σ(δ) site 33·172·364·05
 σ(δ) site 41·410·182·82
 σ(δ) site 52·671·364·28
 σ(δ) site 63·401·955·40
 σ(δ) site 71·130·072·73
 σ(δ) site 82·300·206·48
 σ(δ) site 92·200·116·05
 σ(δ) site 102·180·055·93
 σ(δ) site 112·631·633·86
Shared parameters
 σ(error)1·711·471·97
 α0·770·251·32
 β0·320·140·51
image

Figure 5.  Estimated prevalence and mean infection intensity of infected individuals for gray treefrogs from 11 sites sampled in 2008. Plotted points are the mean of the posterior distribution, and error bars are 95% credible intervals. Sample size was small (3–4 individuals), and no positives were detected for the three sites plotted in red. The mode of the posterior distribution for prevalence in each of these sites is near 0. However, large uncertainty about the estimate means that the upper credible interval for prevalence is as high as 0·58 to 0·72.

Download figure to PowerPoint

Estimated prevalences were greater using our MCMC estimator and the ad hoc method than from naïve estimates (Fig. 6). Consistent with the proper accounting of uncertainty, confidence and credible intervals were wider for the methods where detection was estimated relative to naïve estimates. If detection is imperfect, the naïve estimator will both be biased low and overestimate precision of estimates.

image

Figure 6.  Comparison of estimated prevalences of Bd on gray treefrogs individuals sampled from 11 sites sampled in 2008. Estimated values include those from a naïve estimator (detection assumed to be perfect) and our MCMC and ad hoc estimators of prevalence that account for imperfect detection. Sample sizes for site 8–10 were small (n = 3–4), and because no positive detections occurred, bootstrap confidence intervals could not be generated for the naïve and ad hoc estimate.

Download figure to PowerPoint

Discussion

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

Application to Epidemiological Studies

As we demonstrate, nondetection leads to underestimation of prevalence and can cause overestimation of mean infected intensity of infected individuals. The magnitude of this bias is not necessarily consistent, as we show for among-site differences in estimates of Bd prevalence on gray treefrogs. This type of bias is likely to cause misleading conclusions about ecological hypotheses and misallocation of management or conservation efforts. Less obvious, but equally important, precision in estimates of infection presence and intensity is reduced when heterogeneity in detection occurs. This is not accounted for in estimates that do not explicitly consider detection. As a result of the combination of bias and underestimated sampling error, the probability of false inference will be greater when estimators do not account for observational uncertainty.

The starting point for many epidemiological investigations, such as with the example data set we use, is to predict the static distribution of infection. We found large variation among sites in Bd prevalence and to a lesser extent among-site variation in infection intensities of infected individuals. Within sites, individual size was a poor predictor of infection probability and intensity, although precision was low for estimating this relationship. Other studies of Bd have focused on additional predictors such as species, latitude, thermal environment, habitat type (e.g. Piotrowski, Annis & Longcore 2004; Kriger & Hero 2007; Hossack et al. 2010). Accounting for observation error is important for this research because even very small frequencies of detection errors can inflate Type I error rates if unaccounted for (Tyre et al. 2003; Archaux, Henry & Gimenez 2011). This problem will be further exacerbated when comparisons are made across populations monitored by different researchers, in different seasons, and when samples are analysed in different laboratories.

Dynamic models are useful in making predictions about the distribution and effects of Bd (e.g. Briggs, Knapp & Vredenburg 2010; Johnson & Briggs 2011). The ability to directly parameterize and fit models using field data is a powerful tool (e.g. Beyer et al. 2011; Johnson & Briggs 2011). However, ignoring detection errors can lead to substantial bias in both estimates of the proportion of individuals in different disease states (e.g. susceptible, infected, recovering) and the transition probabilities among infection states (Conn & Cooch 2009; McClintock et al. 2010; Cooch et al. 2011). Consider the example of determining the infection dynamics for Bd that precedes die offs. Bd results in death when the pathogen load increases beyond a threshold where the host cannot maintain normal skin function (Voyles et al. 2009). As we show, bias associated with nondetection is likely to be greatest for populations where infected individuals are in a low-intensity infection state. This has the potential to affect conclusions about dynamics preceding die offs because infected individuals could have transitioned from a low- to high-load state or susceptible individuals could have become newly infected (Rachowicz et al. 2005).

The best-studied cases of anuran-Bd disease dynamics are those that have already undergone or are currently undergoing declines [e.g. Panama (Lips et al. 2006), the California Sierra Nevada (Vredenburg et al. 2010), Australia (Retallick, McCallum & Speare 2004) and Colorado (Muths et al. 2003)]. During a Bd epizootic, declining populations typically have both high prevalence and high intensities. However, less understood are the long-term dynamics of Bd once established, or the patterns of prevalence and intensity prior to an epizootic. Understanding whether certain combinations of low or high prevalence and high or low infection intensity are consistent with a low-impact enzootic disease state vs. indicative of recurring epizootics will be useful in prioritizing management in areas where Bd is established (Briggs, Knapp & Vredenburg 2010). Properly accounting for observational uncertainty is especially important in this case, given the relationship between detection and infection intensities. In our example, we see that mean Bd loads of infected gray treefrogs are relatively consistent across sites, while prevalence appears to vary greatly.

Sampling Design Considerations

We recognize some areas where our data collection and laboratory analysis could be improved. First, our sampling method only accounts for laboratory uncertainty, and future work should also address uncertainty in field sampling. Our repeated sampling was carried out from DNA extracts taken from a single swab. Using samples from multiple swabs collected from the same individual would account for additional missed detection and sampling error owing to variation in the field sampling techniques. It is also important that genetic samples for the same individual are dispersed among plates during analyses as we have done and preferably that assignment is completely random. If large numbers of samples are being analysed, additional laboratory variation could be accounted for by including random effects for plate ID in models for detection and intensity sampling error.

Optimal sampling design should balance the number of samples per individuals vs. the number of individuals sampled. MacKenzie & Royle (2005) and Bailey et al. (2007) provide general guidance for occupancy studies. In cases where infected individuals, even those with low infection intensity, have high probabilities of being classified as infected from a single sample (>0·6), double-sampling may be sufficient. In cases where detection is lower, greater numbers of samples per individual may yield more precise estimates of prevalence even if fewer total individuals are sampled. When considering sampling designs, it is also important to consider heterogeneity. Greater variation in infection intensity and a stronger relationship between infection intensity and detection will both lower precision of estimates.

Another source of observation error that we did not explicitly address is the potential for unequal sampling of infected and noninfected individuals (Murray et al. 2009). Thus, our estimates are conditional on the sampled population, which may or may not be representative of the actual population of interest. To minimize differences between our sample and target population, we employed a standardized protocol to sample individuals. However, methods that account for differential sampling should be considered in combination with methods such as those presented here whenever possible (Cooch et al. 2011).

Future Work

Although we focus on Bd and amphibians, the methods we propose will be relevant for measuring infection status in other host–pathogen systems. Data collected using quantitative PCR are especially suited to the framework we developed, but the same methods can be used for other diagnostics that provide measures of presence–absence and infection intensity. Researchers will need to select distributions for infection intensity and determine how to specify the relationship between infection intensity and detection based on the properties of the pathogen and diagnostic method. In addition, incorporating multiple diagnostic methods may be helpful for further reducing detection heterogeneity and improving estimates if there is greater independence between diagnostic tests.

We have assumed that false positive detections are negligible. It may be useful to relax this assumption and to use approaches that estimate presence when both false negatives and false positives occur (e.g. Royle & Link 2006; McClintock et al. 2010; Hanks, Hooten & Baker 2011; Miller et al. 2011). When a subset of samples can be tested using a method for which false positives can be assumed to be negligible, estimators that incorporate multiple sampling methods may be especially relevant (Hanks, Hooten & Baker 2011; Miller et al. 2011). Alternatively, if independent estimates of false positive probabilities are available, such as from controls, it should be possible to use these data to inform model-based estimates and improve estimation.

The model presented here can serve as the basis for modelling the detection component for other hierarchical structures. Our approach could be used to estimate disease detection at the individual level using the approach developed by McClintock et al. (2010) to estimate how prevalence changes across time at multiple hierarchical levels. Likewise, uncertainty in disease state is a typical component of multi-state mark–recapture models used to assess changes in disease status and its relationship to survival across time (Jennelle et al. 2007; Cooch et al. 2011). Adapting the approach we take here to model the relationship between intensity and detection could improve inferences in some cases. Similarly, our methods could be used to account for heterogeneity when estimating pathogen detection for state-space models (e.g. Royle & Dorazio 2008; McClintock et al. 2010; Beyer et al. 2011).

Acknowledgements

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

Field and laboratory work were supported by the Illinois Department of Resources State Wildlife Grant (#T-56 R-1) to K.R.L. K.R.L. was supported by NSF (DEB 0917653). Thanks to Jody Shimp and Scott Ballard at IDNR for support throughout this process and numerous IDNR and USFWS land managers for site access. We thank Vance Vredenburg (SFSU) and Ed Heist (SIUC) for access to genetics laboratories and equipment, and Jake Kerby (USD) for additional insight into reliable genetics methodology for Bd testing. This is contribution 404 of the US Geological Survey’s Amphibian Research and Monitoring Initiative. Any use of trade names does not imply endorsement by the U.S. government.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information
  • Abad-Franch, F., Ferrz, G., Campos, C., Palomeque, F.S., Grijalva, M.J., Aguilar, H.M. & Miles, M.A. (2010) Modeling disease vector occurrence when detection is imperfect: infestation of Amazonian Palm Trees by Triatomine bugs at three spatial scales. PLOS Neglected Tropical Diseases, 43, e620.
  • Adams, M.J., Chelgren, N.D., Reinitz, D., Cole, R.A., Rachowicz, L.J., Galvan, S., Mccreary, B., Pearl, C.A., Bailey, L.L., Bettaso, J., Bull, E.L. & Leu, M. (2010) Using occupancy models to understand the distribution and prevalence of an amphibian pathogen, Batrachochytrium dendrobatidis. Ecological Applications, 20, 289302.
  • Archaux, F., Henry, P.Y. & Gimenez, O. (2012) When can we ignore the problems of imperfect detection in comparative studies? Methods in Ecology and Evolution, 3, 188194.
  • Bailey, L.L., Hines, J.E., Nichols, J.D. & MacKenzie, D.I. (2007) Sampling design trade-offs in occupancy studies with imperfect detection: examples and software. Ecological Applications, 17, 281290.
  • Beyer, H.L., Hampson, K., Lembo, T., Cleaveland, S., Kaare, M. & Haydon, D.T. (2011) Metapopulation dynamics of rabies and the efficacy of vaccination. Proceedings of the Royal Society B, 278, 21822190.
  • Bolker, B., Brooks, M.E., Clark, C.J., Geange, S.W., Poulsen, J.R., Stevens, M.H.H. & White, J.S. (2009) Generalized linear mixed models: a practical guide for ecology and evolution. Trends in Ecology and Evolution, 24, 127135.
  • Briggs, C.J., Knapp, R.A. & Vredenburg, V.T. (2010) Enzootic and epizootic dynamics of the chytrid fungal pathogen of amphibians. Proceedings of the National Academy of Sciences of the United States of America, 107, 96959700.
  • Cheng, T.L., Rovito, S.M., Wake, D.B. & Vredenburg, V.T. (2011) Coincident mass extirpation of neotropical amphibians with the emergence of the infectious fungal pathogen Batrachochytrium dendrobatidis. Proceedings of the National Academy of Sciences of the United States of America, 108, 95029507.
  • Conn, P.B. & Cooch, E.G. (2009) Multi-state capture–recapture analysis under imperfect state observation: an application to disease models. Journal of Applied Ecology, 46, 486492.
  • Cooch, E.G., Conn, P.B., Ellner, S.P., Dobson, A.P. & Pollock, K.H. (2011) Disease dynamics in wild populations: modeling and estimation: a review. Journal of Ornithology, 152, S485S509.
  • Daszak, P., Cunningham, A.C. & Hyatt, A.D. (2000) Emerging infectious diseases of wildlife – threats to biodiversity and human health. Science, 287, 443449.
  • Fisher, M.C., Garner, T.W.J. & Walker, S.F. (2009) Global emergence of Batrachochytrium dendrobatidis and amphibian chytridiomycosis in space, time, and host. Annual Review of Microbiology, 63, 291310.
  • Gelman, A. & Hill, J. (2007) Data analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, New York, NY.
  • Hanks, E.M., Hooten, M.B. & Baker, F.A. (2011) Reconciling multiple data sources to improve accuracy of large-scale prediction of forest disease incidence. Ecological Applications, 21, 11731188.
  • Horvitz, D.G. & Thompson, D.J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663685.
  • Hossack, B.R., Adams, M.J., Grant, E.H.C., Pearl, C.A., Bettaso, J.B., Barichivich, W.J., Lowe, W.H., True, K., Ware, J.L. & Corn, P.S. (2010) Low prevalence of chytrid fungus (Batrachochytrium dendrobatidis) in amphibians of U.S. headwater streams. Journal of Herpetology, 44, 253260.
  • Huggins, R.M. (1991) Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics, 47, 725732.
  • Hyatt, A.D., Boyle, D.G., Olsen, V., Boyle, D.B., Berger, L., Obendorf, D., Dalton, A., Kriger, K., Hero, M., Hines, H., Phillot, R., Cambell, R., Marantelli, G., Gleason, F. & Colling, A. (2007) Diagnostic assays and sampling protocols for the detection of Batrachochytrium dendrobatidis . Diseases of Aquatic Organisms, 73, 175192.
  • Jennelle, C.S., Cooch, E.G., Conroy, M.J. & Senar, J.C. (2007) State-specific detection probabilities and disease prevalence. Ecological Applications, 17, 154167.
  • Johnson, L.R. & Briggs, C.J. (2011) Parameter inference for an individual based model of chytridiomycosis in frogs. Journal of Theoretical Biology, 277, 9098.
  • Kriger, K.M. & Hero, J. (2007) The chytrid fungus Batrachochytrium dendrobatidis is non–randomly distributed across amphibian breeding habitats. Diversity and Distribution, 13, 18.
  • Lachish, S., Gopalaswamy, A.M., Knowles, S.C.L. & Sheldon, B.C. (2012) Site-occupancy modeling as a novel framework for assessing test sensitivity and estimating wildlife disease prevalence from imperfect diagnostic tests. Methods in Ecology and Evolution, 3, 339348.
  • Lips, K.R., Brem, F., Brenes, R., Reeve, J.D., Alford, R.A., Voyles, J., Carey, C., Livo, L., Pessier, A.P. & Collins, J.P. (2006) Emerging infectious disease and the loss of biodiversity in a neotropical amphibian community. Proceedings of the National Academy of Sciences of the United States of America, 103, 31653170.
  • Longcore, J.E., Pessier, A.P. & Nichols, D.K. (1999) Batrachochytrium dendrobatidis gen. et sp. nov., a chytrid pathogenic to amphibians. Mycologia, 91, 219227.
  • MacKenzie, D.I. & Royle, J.A. (2005) Designing occupancy studies: general advice and allocating survey effort. Journal of Applied Ecology, 42, 11051114.
  • MacKenzie, D.I., Nichols, J.D., Royle, J.A., Pollock, K.H., Bailey, L.L. & Hines, J.E. (2006) Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence. Academic Press, New York, NY.
  • McClintock, B.T., Nichols, J.D., Bailey, L.L., MacKenzie, D.I., Kendal, W.L. & Franklin, A.B. (2010) Seeking a second opinion: uncertainty in disease ecology. Ecology Letters, 13, 659674.
  • Miller, D.A., Nichols, J.D., McClintock, B.T., Grant, E.H.C., Bailey, L.L. & Wier, L.A. (2011) Improving estimates of occupancy when two forms of observation error occur: non-detection and species misidentification. Ecology, 92, 14221428.
  • Murray, K.A., Skerratt, L.F., Speare, R. & McCallum, H. (2009) Impact and dynamics of disease in species threatened by the amphibian chytrid fungus, Batrachochytrium dendrobatidis. Conservation Biology, 23, 12421252.
  • Muths, E., Pedersen, B.S. & Pedersen, F.S. (2009) How relevant is opportunistic Bd sampling: are we ready for the big picture? Herpetological Review, 40, 183184.
  • Muths, E., Corn, P.S., Pessier, A.P. & Green, D.E. (2003) Evidence for disease related amphibian decline in Colorado. Biological Conservation, 110, 357365.
  • Nichols, J.D. & Karanth, K.U. (2002) Statistical concepts: assessing spatial distributions. Monitoring Tigers and Their Prey (eds K.U. Karanth & J.D. Nichols), pp. 2938. Centre for Wildlife Studies, Bangalore, India.
  • Otis, D.L., Burnham, K.P., White, G.C. & Anderson, D.R. (1978) Statistical inference from capture data on closed animal populations. Wildlife Monographs, 62, 1135.
  • Piotrowski, J.S., Annis, S.L. & Longcore, J.E. (2004) Physiology of Batrachochytrium dendrobatidis, a chytrid pathogen of amphibians. Mycologia, 96, 915.
  • R Development Core Team. (2011) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org .
  • Rachowicz, L.J., Hero, J., Alford, R.A., Taylor, J.W., Morgan, J.A.T., Vredenburg, V.T., Collins, J.P. & Briggs, C.J. (2005) The novel and endemic pathogen hypotheses: competing explanations for the origin of emerging infectious diseases of wildlife. Conservation Biology, 19, 14411448.
  • Retallick, R., McCallum, H. & Speare, R. (2004) Endemic infection of the amphibian chytrid fungus in a frog community post-decline. PLOS Biology, 2, 17.
  • Royle, J.A. (2009) Analysis of capture–recapture models with individual covariates using data augmentation. Biometrics, 65, 267274.
  • Royle, J.A. & Dorazio, R.M. (2008) Hierarchical Modeling and Inference in Ecology: The Analysis of Data From Populations, Metapopulations, and Communities. Academic Press, San Diego, CA.
  • Royle, J.A. & Link, W.A. (2006) Generalized site occupancy models allowing for false positive and false negative errors. Ecology, 87, 835841.
  • Royle, J.A. & Nichols, J.D. (2003) Estimating abundance from repeated presence-absence data or point counts. Ecology, 84, 777790.
  • Spiegelhalter, D.J., Thomas, A. & Best, N.G. (1999) WinBUGS Version 1.2 User Manual. MRC Biostatistics Unit, Cambridge.
  • Sturtz, S., Ligges, U. & Gelman, A. (2005) R2WinBUGS: a package for running WinBUGS from R. Journal of Statistical Software, 12, 116.
  • Thompson, K.G. (2007) Use of site occupancy models to estimate prevalence of Myxobolus cerebralis infection in trout. Journal of Aquatic Animal Health, 19, 813.
  • Tyre, A.J., Tenhumberg, B., Field, S.A., Niejalke, D., Parris, K. & Possingham, H.P. (2003) Improving precision and reducing bias in biological surveys: estimating false-negative error rates. Ecological Applications, 13, 17901801.
  • Voyles, J., Young, S., Berger, L., Campbell, C., Voyles, W.F., Dinudom, A., Cook, D., Webb, R., Alford, R.A., Skerratt, L.F. & Speare, R. (2009) Pathogenesis of chytridiomycosis, a cause of catastrophic amphibian declines. Science, 326, 582585.
  • Vredenburg, V.T., Knapp, R.A., Tunstall, T.S. & Briggs, C.J. (2010) Dynamics of an emerging disease drive large-scale amphibian population extinctions. Proceedings of the National Academy of Sciences of the United States of America, 107, 96899694.
  • Walker, S.F., Bosch, J., Gomez, V., Garner, T.W.J., Cunningham, A.A., Schmeller, D.S., Ninyerola, M., Henk, D.A., Ginestet, C., Arthur, C.-P. & Fisher, M.C. (2010) Factors driving pathogenicity vs. prevalence of amphibian panzootic chytridiomycosis in Iberia. Ecology Letters, 13, 372382.
  • White, G.C. & Burnham, K.P. (1999) Program MARK: survival estimation from populations of marked animals. Bird Study, 46(Suppl), 120138.

Supporting Information

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

Appendix S1. R code to simulate data and fit basic model.

Appendix S2. R code to simulate data and fit model with fixed and random effects.

As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials may be re-organized for online delivery, but are not copy-edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.

FilenameFormatSizeDescription
MEE3_216_sm_AppendixA-B.doc34KSupporting info item
MEE3_216_sm_AppendixA-B.pdf217KSupporting 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.