Results and parameterization from double_logistic_selectivity.R
Published
2026-02-02 11:46
1 Overview
This document summarizes the 3-parameter double logistic selectivity curve used for prior elicitation in the EBS pollock assessment workflow. The content and plots below are based on double_logistic_selectivity.R and are intended as a starting point for an RTMB prior demonstration.
2 Model definition
Let age be \(a\), with parameters \(p_1\), \(p_2\), and \(p_3\) and derived inflection points \(\gamma_1\) and \(\gamma_2\):
\(p_1 > 0\) controls the ascending slope and is the distance from \(\gamma_1\) (50 percent selectivity) to the 95 percent point.
\(p_2\) shifts the ascending limb through \(\gamma_1 = p_1 + p_2\).
\(p_3 > 0\) controls the descending slope and is the distance from \(\gamma_2\) (50 percent on the descending limb) to the 5 percent point.
The dome width is \(\gamma_2 - \gamma_1 = p_1 + p_3\).
The factor \(0.95^{-2}\) normalizes the curve so that when both limbs are at 0.95 the product is near 1 (then capped at 1.0).
3 Scenarios used in double_logistic_selectivity.R
scenario
p1
p2
p3
gamma1
gamma2
dome_width
Wide dome
2.0
4
2.5
6.0
10.5
4.5
Asymmetric
(fast rise, slow decline)
0.8
4
3.0
4.8
8.6
3.8
Asymmetric
(slow rise, fast decline)
2.0
3
0.8
5.0
7.8
2.8
Nearly asymptotic
1.5
4
8.0
5.5
15.0
9.5
Table 1: Scenario parameters and derived inflection points
The scenario summary in Table 1 lists the four parameter sets and the derived inflection points (\(\gamma_1\), \(\gamma_2\)) used in the selectivity scenarios.
4 Faceted scenario curves
Each scenario appears on its own panel in Figure 1 so the ascent, descent, and dome width can be compared against the 50 and 95 percent reference lines.
Figure 1: Double logistic selectivity for each parameter scenario.
5 Overlay comparison
All scenarios are overlaid in Figure 2 to highlight differences in peak age and decline shape when the curves are viewed on a common axis.
Figure 2: All scenarios overlaid for direct comparison.
6 Parameter sensitivity analysis
The three panels in Figure 3 show how each parameter changes the curve when the other two are held fixed.
Figure 3: Sensitivity to p1, p2, and p3 with other parameters fixed.
7 RTMB draft with priors
A common approach is to estimate on the log scale for the positive parameters and apply lognormal priors on \(p_1\) and \(p_3\):
Parameter \(p_2\) can remain unconstrained (normal prior) or be modeled on the log scale if you want to enforce \(p_2 > 0\).
The draft RTMB objective below is set up for a single selectivity curve with Gaussian observation error and priors on \(p_1\), \(p_2\), and \(p_3\). Replace sel_obs and sel_sd with observed selectivity or pseudo-data for a prior conditioning exercise.
Gradient evaluation took 4.5e-05 seconds
1000 transitions using 10 leapfrog steps per transition would take 0.45 seconds.
Adjust your expectations accordingly!
Elapsed Time: 0.019 seconds (Warm-up)
0.125 seconds (Sampling)
0.144 seconds (Total)
Model 'selectivity_set1' has 3 pars, and was fit using NUTS with a 'diag' metric
1 chain(s) of 1300 total iterations (150 warmup) were used
Average run time per chain was 0.14 seconds
Minimum ESS=360.7 (31.37%), and maximum Rhat=1.015
There were 0 divergences after warmup
8 MCMC evaluations
Figure 4: Marginal posterior distributions for selectivity parameters (SparseNUTS, prior set 1).
The marginal posterior densities in Figure 4 summarize \((p_1, p_2,
p_3)\) under prior set 1, highlighting the implied uncertainty in each parameter.
Figure 5: Pairwise posterior scatterplots for selectivity parameters (SparseNUTS, prior set 1).
Pairwise posterior relationships in Figure 5 show \((p_1, p_2, p_3)\) and are useful for diagnosing parameter correlations and non-linear structure.
Figure 6: MCMC selectivity draws for prior set 1 (p1 median 1.5, p3 median 2.5, p2 sd 1).
Posterior selectivity draws in Figure 6 show the spread of age-specific selectivity implied by prior set 1.
Gradient evaluation took 2.5e-05 seconds
1000 transitions using 10 leapfrog steps per transition would take 0.25 seconds.
Adjust your expectations accordingly!
Elapsed Time: 0.018 seconds (Warm-up)
0.118 seconds (Sampling)
0.136 seconds (Total)
Model 'selectivity_set2' has 3 pars, and was fit using NUTS with a 'diag' metric
1 chain(s) of 1300 total iterations (150 warmup) were used
Average run time per chain was 0.14 seconds
Minimum ESS=393.8 (34.24%), and maximum Rhat=1.003
There were 0 divergences after warmup
Gradient evaluation took 2.2e-05 seconds
1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
Adjust your expectations accordingly!
Elapsed Time: 0.017 seconds (Warm-up)
0.12 seconds (Sampling)
0.137 seconds (Total)
Model 'selectivity_set3' has 3 pars, and was fit using NUTS with a 'diag' metric
1 chain(s) of 1300 total iterations (150 warmup) were used
Average run time per chain was 0.14 seconds
Minimum ESS=427 (37.13%), and maximum Rhat=1.005
There were 0 divergences after warmup
Figure 7: MCMC selectivity draws for prior set 2 (p1 median 1.5, p3 median 8, p2 sd 1).
The selectivity draws in Figure 7 show prior set 2, which widens the dome by increasing the median of \(p_3\).
Figure 8: MCMC selectivity draws for prior set 3 (p1 median 1.5, p3 median 8, p2 sd 2).
Posterior draws in Figure 8 emphasize how a larger \(p_2\) standard deviation shifts the curves horizontally under prior set 3.
9 Parameter correlations
After fitting, use sdreport() to extract the covariance matrix and convert to correlation with cov2cor(). cov.fixed is on the estimation scale (log_p1, p2, log_p3), while cov.report is on the reported scale (p1, p2, p3) if you used ADREPORT().
The heatmap in ?@fig-dl-corr-heatmap visualizes the natural-scale correlation structure among \((p_1, p_2, p_3)\) and helps identify any strong dependencies implied by the fitted model.
