Christopher S. Jennelle, Evan G. Cooch, Michael J. Conroy, and Juan Carlos Senar. 2007. State-specific detection probabilities and disease prevalence. Ecological Applications 17:154–167.

Appendix C. Bias evaluation of an approach to prevalence estimation in a special case.

As the approach presented in Senar and Conroy (2004) and in this paper are dependent upon the temporal resolution allowable from the data (i.e., resolution in the sense that there is sufficient data to estimate demographic parameters and detection probabilities with time variation if present, which in turn become the constituent inputs that produce the prevalence estimator), we were particularly interested in situations where a true underlying temporal trend in infection probability existed, while only time invariant infection probability was estimable given the data. To this end, we evaluated bias in estimated prevalence with respect to this special condition under two scenarios (which represent truth) with state-specific and time invariant apparent survival, detection, and recovery probabilities over 15 time steps (= 0.9, = 0.6; = 0.5, = 0.7; = 0.3). In scenario (A) true prevalence is a function of a decreasing trend in (infection probability) of the functional form, i = 1 to 15, which corresponds to a 10% decrease in time-specific infection probability per time step (Fig. C1A). This scenario is representative of a newly introduced pathogen into a naïve population of susceptible individuals, where less resistant genotypes are removed from the susceptible pool earlier in the disease cycle. In scenario (B) true prevalence is a function of an increasing trend in (infection probability) of the functional form, i = 1 to 15, which corresponds to a 5% increase in time-specific infection probability per time step (Fig. C1B). This scenario could arise when there is an increase in the virulence or transmission of some pathogen in a population. The decreasing trend in ‘true’ prevalence under scenario (A) would also be observed in distinct scenarios with a trend in increasing recovery probability or a decreasing trend in survival probability of diseased animals. Likewise, the increasing trend in ‘true’ prevalence under scenario (B) would also be observed in distinct scenarios with a decreasing trend in recovery probability or decreasing trend in survival probability of healthy animals. As such, the two scenarios we consider here can be considered qualitatively representative of the trends in prevalence expected under a variety of demographic circumstances (assuming all other parameters are set constant).

Using SAS software (v.9 SAS Institute), we simulated encounter histories under the aforementioned demographic scenarios starting each simulated population with 15% diseased individuals and for convenience assigned 1000 newly released (newly marked) individuals at each time step. We were not interested in determining a lower bound for the number of marked individuals necessary for adequate support of models with time-dependent variation. Rather, we used 1000 newly released ‘simulated’ individuals in each cohort in our bootstrap samples only to facilitate the convergence of time-invariant parameter estimates. We simulated 200 encounter histories under each scenario and estimated state-specific survival and transition probabilities using program MARK v.4.2 (White and Burnham 1999) under the model {(disease state) p(disease state)(.) (.)} (note the period surrounded by parenthesis indicates time-invariance). We used program MARKWAIT designed by James E. Hines of Patuxent Wildlife Research Center in Laurel, Maryland, to automate the creation of sample encounter histories in SAS, and subsequently estimate parameters under the aforementioned model in MARK. We suspect that in a fair number of wildlife disease studies, a sufficient number of recaptures and observed state transitions will not be available for convergence of mark–recapture models with time-dependent survival and/or state transition parameters. As previously mentioned, multistate models are considerably ‘data hungry’, so we have conducted these simulations under the special case that estimates of survival and state transitions are time invariant. With time invariant parameter estimates obtained from the bootstrap samples, we estimated the value of disease prevalence at each time step using the Senar and Conroy (2004) approach and compared this quantity with ‘true’ prevalence calculated using the two exponential models of infection probability (with all other parameters equal) at each respective time step under both scenarios. For each scenario we plotted and calculated the two maximal values of percent relative bias in estimated prevalence.

Projecting disease prevalence over 15 time steps under scenario (A) produced a declining trend in ‘true’ prevalence (Fig. C1A), while projection under scenario (B) produced an increasing trend in ‘true’ prevalence (Fig. C1B). The expected value of infection probability under scenario (A) was 0.1740, and that under scenario (B) was 0.3854. Using the given values of survival and recovery probabilities, along with the ‘true’ value of infection probability obtained from the exponential equations under each scenario we calculated ‘true’ prevalence, and estimated prevalence using the Senar and Conroy (2004) multistate model approach using the same given set of parameter values along with the expected value of infection probability generated using a bootstrap. We compared ‘true’ prevalence with estimated prevalence under the Senar and Conroy (2004) approach, and found significant bias in the pattern and magnitude of estimated prevalence when time dependence in infection probability was not taken into account (Fig. C1). Our simulations show that when time invariant infection probability is used in the calculation of disease prevalence with the Senar and Conroy (2004) approach in situations when there is a decreasing or increasing trend in ‘true’ infection probability, respectively, that important temporal trends in disease prevalence will not be detected. If a trend in true disease prevalence is mediated by a concomitant trend in infection probability, recovery probability, or state-specific survival, and if CMR data are not sufficient (plentiful) to support models with time variation in the trending parameter, then this type of inferential error will occur. This phenomenon could also occur when using the prevalence estimator presented in the body of the paper if a true trend in state-specific detection probability is not discernable given the data. As such, we urge researchers to be very cautious when making inferences about patterns in estimated disease prevalence when data is sparse.

FIG. C1. Evaluation of a special case of the Senar and Conroy (2004) multistate model approach to prevalence estimation reveals that under certain conditions researchers may falsely interpret patterns of estimated prevalence to be time invariant, when in fact a temporal trend exists. The figures show the potential bias in estimated prevalence when a temporal trend in 'true' prevalence is mediated by a trend in decreasing (C1A) or increasing infection probability (C1B). Points a and b in each panel indicate where maximal levels of percent relative bias (%RB) in estimated prevalence occur; in panel (C1A) %RB at points a and b are -33% and 92%, respectively, while in panel (C1B) %RB at points a and b are 22% and -19%, respectively. The failure to detect a true temporal trend in estimated prevalence will occur if there is insufficient mark–recapture data to support models with time-dependence in the demographic parameters that are truly time-dependent when using the Senar and Conroy (2004) approach to estimating prevalence.

LITERATURE CITED

Senar, J. C., and M. J. Conroy. 2004. Multi-state analysis of the impacts of avian pox on a population of Serins (Serinus serinus): The importance of estimating recapture rates. Animal Biodiversity and Conservation 27:133–146.

White, G. C., and K. P. Burnham. 1999. Program MARK: Survival estimation from populations of marked animals. Bird Study 46 (Supplement):120–138.