Oldemeyer et al.: A multiyear Bayesian model for incorporating sparse or missing salmonid data 
255 
abilities are homogenous throughout the sampling 
period. Changes in the environment, species behav¬ 
ior, or sampling equipment can alter capture efficien¬ 
cies, violating the assumption of homogenous capture 
probabilities and lead to biased abundance estimates. 
Sampling periods are often stratified temporally into 
smaller, more homogenous stratum for computation 
and later summed to minimize the violation of the 
capture probability assumption (Otis et ah, 1978; 
Schwarz and Taylor, 1998). This stratification ap¬ 
proach can be effective when sufficient data are avail¬ 
able, but stratifying sparse data can lead to wide con¬ 
fidence intervals or estimation failure resulting from 
division by zero when individuals are not recaptured 
during a stratum (Seber, 2002). 
When mark-recapture data are sparse or miss¬ 
ing, hierarchical Bayesian models have proven to be 
an effective approach for obtaining abundance esti¬ 
mates (Mackey et ag, 2008; Royle et ah, 2011; Sethi 
and Tanner, 2013). The Bayesian paradigm provides 
a framework to incorporate prior biological knowledge 
into models by using hierarchal structures between 
parameters and by specifying informative prior dis¬ 
tributions (Ellison, 2004). When prior biological infor¬ 
mation is available, structuring models to incorporate 
this information can produce more accurate and pre¬ 
cise estimates (Royle and Dorazio, 2008). Mantyniemi 
and Romakkaniemi (2002) implemented a hierarchical 
Bayesian model to estimate Atlantic salmon (Salmo 
salar) smolt abundances in the Conne River, Canada, 
and River Tornionjoki, in northern Scandinavia, while 
accounting for overdispersion associated with the spe¬ 
cies schooling behavior. Bonner and Schwarz (2011) 
increased the precision and accuracy of abundance 
estimates of Conne River Atlantic salmon smolts by 
parameterizing the expected abundances of smolts 
as a smooth function of time, using penalized Bayes¬ 
ian splines (P-splines) to address sparse data. These 
single-year models are effective with sparse data and 
short periods of missing data but fail to use or incorpo¬ 
rate all the information available in long term monitor¬ 
ing data sets. 
The goal of our study was to illustrate how a time- 
stratified hierarchical Bayesian model framework can 
incorporate prior years of information to increase the 
accuracy and precision of estimates made using sparse 
and missing data. To achieve this goal, we took a 
2-step approach. First, we compared the performance 
of 3 Bayesian models that had within-year structures 
and 1 Bayesian model with a between-year hierarchi¬ 
cal structure by using a simplified data set exhibit¬ 
ing various degrees of sparse and missing information 
roughly similar to real data. By using a simplified data 
set constructed from known parameters, we were able 
to quantify model performance by comparing the ap¬ 
proximate posterior distributions produced by each 
model to the known parameters used to create the sim¬ 
plified data set. To illustrate how the concept works 
with real data, we then compare abundance estimates 
of the 4 models to 2 data sets, one set with complete 
and abundant data and another set with sparse and 
missing data. Covariates that were suspected to influ¬ 
ence migration characteristics and juvenile abundances 
between years, e.g., seasonal hydrographic fluctuations 
and previous year counts of redds (nests dug by salmon 
in river beds), were excluded from both the data sets 
and model formulas in order to illustrate the basic con¬ 
cept and functionality of the competing models. After 
we compared the use and functionality of competing 
models, we discuss the various ways covariates could 
be included and how models could be extended to ad¬ 
dress more specific scenarios. 
The model that used multiple years of data via the 
between-year hierarchical structure was able to bridge 
large periods of missing data (upwards of several weeks 
in some years) and sparse data by using the annually 
recurring emigration characteristics expressed by the 
species in the study, juvenile Chinook salmon (On- 
corhynchus tshawytscha) in Idaho, to produce the most 
accurate and precise estimates. To our knowledge, this 
is the first time multiple years of data have been used 
to increase the robustness of abundance estimates cal¬ 
culated from sparse and missing mark-recapture data 
based on annually recurring behavioral characteristics. 
Materials and methods 
There are 2 primary components to our study. First, we 
compare estimates produced from 3 single-year Bayes¬ 
ian models and 1 hierarchical multiyear Bayesian mod¬ 
el, using simulated scenarios reflective of missing and 
sparse data typical for monitoring with rotary screw 
traps (RSTs) in Idaho. To illustrate a proof of concept 
on how the structures of the Bayesian models function, 
models where simple (e.g., did not include environ¬ 
mental covariates or individual movement parameters) 
and data for simulated scenarios were not stochastic. 
Parameter estimates produced for the simplistic simu¬ 
lated data scenarios were compared with the known 
parameters used to create the simulated data to evalu¬ 
ate bias and precision for each of the models. Next, we 
demonstrate how models performed with real juvenile 
Chinook salmon data collected at Marsh Creek and Big 
Creek, Idaho, which reflect good and poor quality data 
sets. We used the full data record from the initial year 
of trap operation to 2014 to inform the estimates for 
the 2014 emigration. 
Field sampling and data collections 
Mark-recapture studies have been widely implemented 
to calculate anadromous juvenile salmonid abundances 
at RSTs (Zabel et ah, 2005; Venditti et al. 1 ; Copeland 
1 Venditti, D. A., J. Flinders, R. Kinzer, C. Bretz, M. Corsi, 
B. Barnett, K. A. Apperson, and A. Teton. 2012. Idaho 
supplementation studies: brood year 2009 synthesis report, 
August 1, 2009—July 31, 2011. Idaho Dep. Fish Game Rep. 
12-13, 24 p. [Available from website.] 
