Qldemeyer et al.: A multiyear Bayesian model for incorporating sparse or missing salmonid data 
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Model set 
We compared 4 models by using the constructed like¬ 
lihood components outlined above with different pa¬ 
rameter structures. The most basic model, M PS (pooled 
capture probability and simple abundance parameters), 
analyzed 1 year of data with a simple prior for U 
where log(?7 i ) shared identical normal priors with fixed 
mean and variance and where capture probability (p) 
was pooled, pp-p for i=l,...,s across all strata. The M PS 
model structures are functionally similar to a pooled 
Lincoln-Petersen model in that p are structured to be 
pooled and constant across strata within the year and 
U is independent between strata. 
The second model was similar to models used by 
Mantyniemi and Romakkaniemi (2002), which al¬ 
lowed information about U and p to be shared among 
strata within the year, and is denoted M H w (hierarchi¬ 
cal within-year parameter structure). This model con¬ 
tained a hierarchical structure that assigned both U 
and p, log(Uj) and logit(pj), normal priors with common 
mean and variance within the year. The M HW hierar¬ 
chical structures allow information about U and p to be 
shared across all strata within a single year. 
The third model is structured from Bonner and 
Schwarz (2011) and uses the P-spline hierarchical pri¬ 
or for U to smooth estimates between adjacent strata, 
localizing the sharing of information between adjacent 
strata within the year, and is denoted Mg PL iNE- A hier¬ 
archical structure for p assigned logit(pj) normal priors, 
with common but unknown mean and variance within 
the year, was used for the M SPLINE model. The M SPL ine 
hierarchical structures allows information about p to 
be shared among strata throughout the year but uses 
the temporal ordering of strata within the year to give 
greater weight to adjacent strata for sharing informa¬ 
tion in regards to U. Following the recommendations 
of Lang and Brezger (2004) and Schwarz and Bonner, 3 
we placed knot points evenly across strata at'4-week 
intervals. These authors found that this spacing was 
a suitable compromise between spline overfitting and 
sharing information among strata. We chose to exclude 
the additional parameter used to model the probability 
that an individual is available for recapture in proceed¬ 
ing strata as explained above. 
The hierarchical multiyear (between-year) model, 
Mjjg, allows information from the same temporal peri¬ 
od among years to be shared in regard to p and U. M H b 
contains a hierarchical structure for U and p, where 
log(C7y) and logitlpy) have a common mean within stra¬ 
ta of the same ordinal time period between years. By 
structuring the hierarchy between years, recurring run 
characteristics specific to each stratum (ordinal week 
for this study) were integrated into the model. 
Selection of prior distributions for the highest level 
of the model hierarchies was chosen to be vague and 
weakly informative in regard to their parameters and 
to be identical to prior distributions used in previous re¬ 
search when applicable. Prior distributions selected also 
aligned with expert knowledge of and experience with 
RSTs. For instance, capture probabilities at RSTs rarely 
exceed 0.5 and often average 0.1-0.2 depending on the 
trap location and time of year. The prior distributions 
selected for the hyperparameters of each capture prob¬ 
ability for each model had a median of roughly 0.1 and 
95 th percentiles from 0.0 to 0.5. Additionally, prior dis¬ 
tributions for the hyperparameters of each abundance 
parameter have a median abundance of roughly 22,000 
with 95 th percentiles from 814 to 583,381, with the ex¬ 
ception of the Mspline model that uses hyperparameters 
implemented by Bonner and Schwarz (2011). Complete 
structures for all models can be found in Supplementary 
Table 1 and Supplementary Figures 1-4 (online only). 
Test data sets 
A simplified data set roughly similar to RST data found 
in Idaho was created with known U and p parameters. 
The simulated data set used for the scenarios spanned 
10 years (j=10) with 35 strata per year (i= 35). This 
scheme roughly corresponds to the early spring transi¬ 
tion in mid-March and the late fall transition in mid- 
November. The migration of juvenile Chinook salmon 
in Idaho typically has 2 pulses, one in the spring and 
one in the fall; therefore, parameters U V) followed a 
smooth bimodal run with a small peak at strata 6 
(mid-May) and the majority of the individuals centered 
on a peak at stratum 29 (late September). Parameters 
Py were constant at 0.333 for strata 1-4 and 13-35, 
and constant at 0.111 for strata 5-12. As discharge 
increases in the spring, RSTs are often relocated out 
of the thalweg to slower portions of the river transect 
to avoid woody debris and decrease mechanical stress 
on the sampling equipment. In addition, as discharge 
increases, the relative amount of water sampled by 
RSTs decreases. These conditions often decrease trap 
efficiencies and are the justification for the decreased 
capture efficiencies for strata 5-12 in our simulated 
data. The number of marked individuals, n iy released 
in each stratum was equivalent to the number of un¬ 
marked individuals captured, u iy up to 50 individuals. 
The restriction to 50 individual was implemented to 
mimic tag and handling permit constraints that are 
common when dealing with threatened or fragile spe¬ 
cies at RSTs. Total yearly abundance summed over all 
35 stratum was 23,477. 
This data set was modified to simulate realistic oc¬ 
currence of sparse and missing data. The scenario us¬ 
ing data produced from the parameters listed above 
was denoted as “full” owing to the completeness across 
strata and years and was the initial scenario in our 
simulation study used for subsequent modification. 
The second scenario is identical to the full scenario 
with the exception of information from strata 5-8 in 
the first year being removed. This 4-stratum exclusion 
mimicked RST conditions when spring flows halt RST 
operations for several weeks. The third scenario had 
information from strata 5-8 removed from the first 
year in addition to reducing the number of marked 
and recaptured individuals by 60% across all strata 
