This document reports a preliminary Eastern Bering Sea walleye pollock bridge assessment implemented with the SPoRC R package (Cheng et al. 2026). The run uses the 2024 ADMB assessment data stream conditioned into SPoRC input objects, with a single-area, single-sex, age-structured configuration spanning 1964-2024 and 15 modeled ages.
The current SPoRC dev fit estimates 18,988 for spawning stock biomass, 68,827 for total biomass, 166,750 recruits, and a fishery fishing mortality of 0.043 in 2024 (Table 2). These quantities should be read as bridge-model results rather than management advice. The optimizer returned convergence code 1 with message “false convergence (8)”; the Hessian was not positive definite and the maximum absolute fixed-effect gradient was 1.92^{-4} (Table 3).
The bridge comparison with the saved 2024 ADMB result shows broad agreement in some time-series shapes, but important scale differences remain, especially for recruitment and fishing mortality (Figure 20, Table 6). This indicates that the current SPoRC setup is useful for repository development and diagnostics, but not yet a replacement for the accepted ADMB assessment.
2 Data
The SPoRC input file is generated by data-raw/build_ebs_pollock_data.R from the 2024 ADMB data file and saved as data/ebs_pollock.rds. The modeled data include catch, fishery CPUE, fishery age compositions, bottom trawl survey (BTS), acoustic trawl survey (ATS), aerial survey (AVO) indices, and survey age compositions where available. The survey time series are retained as separate information sources because earlier EBS pollock assessment work has shown that multiple surveys can carry distinct signals about abundance, composition, and distribution (O’Leary et al. 2020). Data availability by source is summarized in Table 1 and shown by year in Figure 1.
Table 1: Summary of data series included in the SPoRC input object.
Data series
Years used
Year range
Catch Fishery
61
1964-2024
Fishery age composition Fishery
60
1964-2023
Fishery index Fishery CPUE
12
1965-1976
Survey age composition ATS
19
1994-2024
Survey age composition BTS
42
1982-2024
Survey index ATS
19
1994-2024
Survey index AVO
18
2006-2024
Survey index BTS
42
1982-2024
Show code
ggplot(data_availability_df, aes(x = year, y = series)) +geom_point(aes(fill = type), shape =21, size =2.6, color ="grey20", alpha =0.9) +labs(x ="Year", y =NULL, fill ="Data type") + ggthemes::theme_few()
Figure 1: Data availability used in the SPoRC dev model.
3 Model Configuration
The dev model is configured in config/base.yml and fitted by analysis/run_scenarios.R. The model uses one region, one sex, one fishery fleet, and three survey fleets, following the single-region EBS pollock SPoRC case-study configuration (Cheng 2026). Recruitment is modeled with a Beverton-Holt stock-recruit relationship, with fixed steepness and fixed recruitment standard deviations. Initial age-composition deviations are estimated for ages 2 and older and are not penalized as equilibrium recruitment deviations in the current configuration. In this setup, equil_init_age_strc = 0 means the starting age structure is estimated more freely through ln_InitDevs, while equil_init_age_strc = 2 retains the vignette-style geometric/equilibrium initialization and associated initial recruitment penalty. SPoRC’s one-sex spawning biomass calculation applies a 0.5 multiplier internally, so the reported Aggregated_SSB is already on a female spawning-biomass basis for comparison with the ADMB assessment. Natural mortality is fixed at age-specific values, catch uncertainty is fixed, and the current dev configuration estimates penalized time-varying fishery and survey selectivity. Fishery selectivity uses a dome-shaped dbnrml fixed selectivity curve with 2dar1 deviations, correlations fixed at zero, and deviations mapped from 1964 onward; the process-error standard deviation is fixed at 0.30 to allow more interannual variability. BTS survey selectivity uses iid deviations with the underlying logistic age-50 fixed near age 3, and ATS/AVO survey selectivity uses 2dar1 deviations with fleet 3 sharing fleet 2 selectivity parameters. Selectivity is an important bridge-model diagnostic because alternative state-space and time-varying selectivity treatments can materially change catch-at-age assessment behavior (Nielsen and Berg 2014; Nielsen et al. 2021).
Biological inputs used in the terminal model year are shown in Figure 2. The natural mortality curve follows the fixed age-specific values in the configuration, while maturity and spawning weight at age are taken from the conditioned pollock input data.
Show code
ggplot(bio_df, aes(x = age, y = value)) +geom_line(linewidth =0.8, color ="#1b6ca8") +geom_point(size =1.8, color ="#1b6ca8") +facet_wrap(~ quantity, scales ="free_y", ncol =1) +labs(x ="Age", y ="Value") + ggthemes::theme_few()
Figure 2: Terminal-year biological inputs and natural mortality used in the SPoRC dev model.
The estimated terminal-year selectivity patterns are shown in Figure 3. The fitted selectivity trajectories are shown as ridge plots for the fishery, BTS, and ATS in Figure 4, Figure 5, and Figure 6, following the assessment plotting approach in plot-sel.R. Each fleet-year curve is normalized to its maximum value before plotting so the figures emphasize changes in the age pattern of vulnerability through time.
Show code
ggplot(selectivity_df, aes(x = age, y = value, color = fleet_name)) +geom_line(linewidth =0.9) +facet_wrap(~ fleet_type, scales ="free_y", ncol =1) +labs(x ="Age", y ="Selectivity", color ="Fleet") + ggthemes::theme_few()
Figure 3: Terminal-year fishery and survey selectivity from the SPoRC dev model.
Show code
plot_selectivity_ridges("Fishery", fill ="#7b3294", compare_models =TRUE)
Figure 4: Relative fishery selectivity over time from the SPoRC dev and vign models. Ridgeline heights are normalized within each model and year.
Show code
plot_selectivity_ridges("BTS", fill ="#1b6ca8")
Figure 5: Relative BTS selectivity over time from the SPoRC dev model. Ridgeline heights are normalized within each year.
Show code
plot_selectivity_ridges("ATS", fill ="#4daf4a")
Figure 6: Relative ATS selectivity over time from the SPoRC dev model. Ridgeline heights are normalized within each year.
4 Results
The dev run estimates increasing spawning biomass over the terminal portion of the time series and a strong terminal recruitment estimate (Figure 7). In 2024, estimated SSB is 18,988, total biomass is 68,827, and recruitment is 166,750 (Table 2).
Figure 8: Stock-recruit relationship from the SPoRC dev model. The line is the Beverton-Holt curve using fixed steepness, estimated R0, and the median dynamic unfished spawning biomass; labels show annual model estimates by year.
The catch fit reproduces the observed catch series closely in most years by construction of the catch likelihood and associated fixed catch uncertainty (Figure 9). Index fits show larger departures for some survey and CPUE observations, which is expected at this bridge stage because the model configuration has not yet been tuned to reproduce the accepted ADMB assessment (Figure 10).
Show code
ggplot(catch_df, aes(x = year)) +geom_line(aes(y = fitted), linewidth =0.8, color ="#1b6ca8") +geom_point(aes(y = observed), size =1.8, color ="grey15") +facet_wrap(~ fleet_name, scales ="free_y") +labs(x ="Year", y ="Catch") + ggthemes::theme_few()
Figure 9: Observed and fitted catch for the SPoRC dev model. Points are observations and the line is fitted catch.
Figure 10: Observed and fitted fishery and survey indices for the SPoRC dev model. Points are observations and lines are fitted values.
Age-composition fits are shown as predicted proportions at age with observed proportions overlaid for years included in the likelihood. The fishery composition fit is shown in Figure 11, and the survey composition fits are shown separately for BTS and ATS in Figure 12 and Figure 13. The AVO survey does not contribute an age-composition series in the current conditioned input object. Following the assessment diagnostic workflow in plot_osa_comps.R, the composition fits are also summarized with residual bubble plots, normal QQ plots, and aggregate observed-versus-expected fits for the Fishery, BTS, and ATS series (Figure 14, Figure 15, Figure 16, Figure 17, and Figure 18). Age 1 is excluded from the BTS and ATS residual diagnostics because it is not used in those survey composition likelihoods. The residual diagnostic method used in this render is Pearson residual fallback.
Show code
ggplot(fish_agecomp_df, aes(x = age, y = predicted)) +geom_line(linewidth =0.5, color ="#4daf4a") +geom_point(aes(y = observed), color ="#1f78b4", size =0.9, alpha =0.8) +facet_wrap(~ year, ncol =5, dir ="v") +scale_y_continuous(limits =c(0, NA)) +labs(x ="Age", y ="Proportion") + ggthemes::theme_few()
Figure 11: Observed and predicted fishery age-composition patterns for years included in the SPoRC likelihood.
Show code
ggplot(srv_agecomp_df[srv_agecomp_df$fleet_name =="BTS", ], aes(x = age, y = predicted)) +geom_line(linewidth =0.5, color ="#4daf4a") +geom_point(aes(y = observed), color ="#1f78b4", size =0.9, alpha =0.8) +facet_wrap(~ year, ncol =4, dir ="v") +scale_y_continuous(limits =c(0, NA)) +labs(x ="Age", y ="Proportion") + ggthemes::theme_few()
Figure 12: Observed and predicted BTS age-composition patterns for years included in the SPoRC likelihood.
Show code
ggplot(srv_agecomp_df[srv_agecomp_df$fleet_name =="ATS", ], aes(x = age, y = predicted)) +geom_line(linewidth =0.5, color ="#4daf4a") +geom_point(aes(y = observed), color ="#1f78b4", size =0.9, alpha =0.8) +facet_wrap(~ year, ncol =3, dir ="v") +scale_y_continuous(limits =c(0, NA)) +labs(x ="Age", y ="Proportion") + ggthemes::theme_few()
Figure 13: Observed and predicted ATS age-composition patterns for years included in the SPoRC likelihood.
Show code
plot_composition_residual_bubbles("Fishery")
Figure 14: Composition residual bubble diagnostics for the Fishery age-composition fit. Circle size is proportional to the absolute residual; triangles identify absolute residuals greater than 3.
Show code
plot_composition_residual_bubbles("BTS")
Figure 15: Composition residual bubble diagnostics for the BTS age-composition fit. Circle size is proportional to the absolute residual; triangles identify absolute residuals greater than 3.
Show code
plot_composition_residual_bubbles("ATS")
Figure 16: Composition residual bubble diagnostics for the ATS age-composition fit. Circle size is proportional to the absolute residual; triangles identify absolute residuals greater than 3.
Figure 17: Normal QQ diagnostics for the Fishery, BTS, and ATS composition residuals. Labels show the standard deviation of normalized residuals.
Show code
plot_aggregate_composition_fits()
Figure 18: Aggregate observed and expected age-composition fits for Fishery, BTS, and ATS. Bars are aggregate observed proportions and lines are aggregate expected proportions.
5 Diagnostics
The optimization diagnostics are mixed. The maximum absolute gradient is small, but the optimizer reports singular convergence and the Hessian status should be reviewed directly in Table 3. This is a clear flag that the dev fit should be treated as provisional until additional parameter mapping, starting values, bounds, and likelihood settings are reviewed.
Catchability parameters are estimated freely in the current configuration and do not have q priors (Table 4). The very large standard errors for fishery CPUE, BTS, and ATS catchability indicate weak scale identification for those index catchabilities in this provisional bridge fit. This is consistent with the false-convergence optimizer message and should be reviewed against the intended SPoRC case-study specification before using the q estimates interpretively.
Table 4: Catchability estimates, standard errors, and current q specification in the SPoRC dev model.
Fleet
Parameter
Instance
Estimate
Standard error
q specification
q prior
Fishery CPUE
ln_fish_q
1
0.2388
NaN
est_all
none
BTS
ln_srv_q
1
−1.5015
NaN
est_all
none
Likelihood contributions are dominated by the configured composition and recruitment components. The current development case is labeled dev; the earlier SPoRC vignette-style case is labeled vign when analysis/outputs/vign.rds is available. The component totals are shown with models in columns in Table 5, and the relative contribution of each component is shown in Figure 19. These values are useful for debugging model weighting and for identifying which data sources are controlling the current fit. The vign case has a nonzero initial recruitment penalty because it keeps the vignette initialization setting equil_init_age_strc = 2; the dev case frees the initial age-composition deviations and sets equil_init_age_strc = 0, so Init_Rec_nLL is zero.
Table 5: Negative log-likelihood components by model case.
Component
dev
vign
Joint objective
1,779
526
Fishery age composition
396
1,153
Survey age composition
876
846
Survey index
436
−77
Recruitment
86
111
Fishery index
93
−4
Fishing mortality penalty
88
74
Initial recruitment
0
38
Catch
−66
−61
Selectivity penalty
−130
−1,553
Show code
ggplot(nll_df, aes(x = component, y = value, fill = model)) +geom_col(position ="dodge", width =0.7) +coord_flip() +labs(x =NULL, y ="Negative log-likelihood", fill ="Model") + ggthemes::scale_fill_colorblind() + ggthemes::theme_few()
Figure 19: Negative log-likelihood components by SPoRC model case.
6 Bridge Comparison With ADMB
The bridge comparison uses /Users/jim/_mymods/pollock/results/derived/admb_2024.rds as the ADMB reference object. The comparison is limited to shared annual quantities: fishing mortality, spawning biomass, and recruitment. The maximum absolute relative differences are shown in Table 6, and the time series are plotted in Figure 20.
Show code
if (nrow(comparison_summary) >0) {gt(comparison_summary) |>fmt_number(columns = max_abs_relative_difference, decimals =4) |>cols_label(quantity ="Quantity",max_abs_relative_difference ="Max. absolute relative difference" )} else {gt(data.frame(note ="ADMB comparison object was not available."))}
Table 6: Maximum absolute relative difference between the SPoRC dev model and the 2024 ADMB reference object.
Quantity
Max. absolute relative difference
F
1.9808
Recruit
8.1190
SSB
5.9496
Show code
if (!is.null(bridge_ts)) {ggplot(bridge_ts, aes(x = year, y = value, color = model)) +geom_line(linewidth =0.8) +facet_wrap(~ quantity, scales ="free_y", ncol =1) +labs(x ="Year", y ="Value", color ="Model") + ggthemes::theme_few()}
Figure 20: Shared SPoRC and ADMB time series for the dev-model bridge comparison.
The current bridge run does not yet reproduce the ADMB model closely enough for operational use. The largest differences occur in recruitment and fishing mortality, and those differences are large enough to affect interpretation of terminal stock status and future harvest advice. The next development step is to inspect the mapping of recruitment deviations, selectivity forms, catch likelihood treatment, index scaling, and composition likelihood settings against the ADMB model.
7 Reproducibility
The assessment artifacts used by this document are:
SPoRC data object: data/ebs_pollock.rds
SPoRC dev model: analysis/outputs/base.rds
SPoRC vignette-style model: analysis/outputs/vign.rds, regenerated by analysis/run_vignette_case.R
Cheng, Matthew, Dan Goethel, Pete Hulson, and Curry Cunningham. 2026. SPoRC: A Generalized Stochastic Population Model over Regional Components. https://chengmatt.github.io/SPoRC/.
Nielsen, Anders, and Casper W. Berg. 2014. “Estimation of Time-Varying Selectivity in Stock Assessments Using State-Space Models.”Fisheries Research 158: 96–101. https://doi.org/10.1016/j.fishres.2014.01.014.
Nielsen, Anders, Niels T Hintzen, Henrik Mosegaard, Vanessa Trijoulet, and Casper W Berg. 2021. “Multi-Fleet State-Space Assessment Model Strengthens Confidence in Single-Fleet SAM and Provides Fleet-Specific Forecast Options.”ICES Journal of Marine Science 78 (6): 2043–52. https://doi.org/10.1093/icesjms/fsab078.
O’Leary, C. A., J. T. Thorson, J. N. Ianelli, and S. Kotwicki. 2020. “Adapting to Climate‐driven Distribution Shifts Using Model‐based Indices and Age Composition from Multiple Surveys in the Walleye Pollock (Gadus Chalcogrammus) Stock Assessment.”Fisheries Oceanography 29 (6): 541–57. https://doi.org/10.1111/fog.12494.
8 Appendix
8.1 SparseNUTS MCMC
The SPoRC dev model is RTMB-based, so the fitted objective can be passed directly to SparseNUTS. Following the Rceattle assessment workflow, I ran SparseNUTS::sample_snuts() outside the Quarto render using the package defaults for chains, samples, warmup selection, metric selection, initial values, and control settings, with cores = 1 for serial RTMB execution. The sampler output and diagnostic figures are saved as assessment artifacts.
This output is a computational diagnostic rather than accepted posterior inference. The default run reported divergent transitions and () and effective sample size diagnostics indicate non-convergence for catchability parameters, so these posterior summaries should not be used for management interpretation without a longer and retuned MCMC run.
Show code
if (file.exists(sparsenuts_path)) { sparsenuts <-readRDS(sparsenuts_path) snuts <- sparsenuts$snuts_fit snuts_diag <- sparsenuts$diagnostics settings <- sparsenuts$sampler_settings postwarmup_per_chain <-dim(snuts$samples)[1] - snuts$warmup total_postwarmup <- postwarmup_per_chain *dim(snuts$samples)[2] max_rhat <-max(snuts$monitor$rhat, na.rm =TRUE) min_ess <-min(snuts$monitor$ess_bulk, na.rm =TRUE) worst_rhat <- snuts$monitor$variable[which.max(snuts$monitor$rhat)] worst_ess <- snuts$monitor$variable[which.min(snuts$monitor$ess_bulk)] divergent_transitions <-round(snuts_diag$perc_divergent /100* total_postwarmup) snuts_status <-data.frame(metric =c("Algorithm","Metric","Chains","Warmup iterations per chain","Post-warmup samples per chain","Total post-warmup draws","Minimum bulk ESS","Minimum bulk ESS parameter","Maximum Rhat","Maximum Rhat parameter","Divergent transitions","Percent divergent","Interpretation" ),value =c( snuts$algorithm, snuts$metric,dim(snuts$samples)[2], snuts$warmup, postwarmup_per_chain, total_postwarmup,sprintf("%.1f", min_ess), worst_ess,sprintf("%.3f", max_rhat), worst_rhat, divergent_transitions,sprintf("%.2f%%", snuts_diag$perc_divergent),ifelse( snuts_diag$perc_divergent ==0&& max_rhat <1.01,"No obvious warning by divergence and Rhat checks.","Diagnostic only; do not use for inference without retuning." ) ) )gt(snuts_status) |>cols_label(metric ="Metric", value ="Value") |>tab_header(title ="SparseNUTS Diagnostic Run")} else {gt(data.frame(note ="SparseNUTS output not available. Run analysis/run_sporc_sparsenuts.R."))}
Table 7: SparseNUTS diagnostic-run summary for the SPoRC dev model.
SparseNUTS Diagnostic Run
Metric
Value
Algorithm
SNUTS
Metric
dense
Chains
4
Warmup iterations per chain
150
Post-warmup samples per chain
1000
Total post-warmup draws
4000
Minimum bulk ESS
6.3
Minimum bulk ESS parameter
ln_srv_q[1]
Maximum Rhat
1.694
Maximum Rhat parameter
ln_srv_q[1]
Divergent transitions
8
Percent divergent
0.20%
Interpretation
Diagnostic only; do not use for inference without retuning.
Figure 27: SparseNUTS::plot_uncertainties output for the SPoRC diagnostic run.
8.2 Retrospective Analysis
A five-year retrospective was run by peeling terminal years from the conditioned SPoRC input data and refitting the same dev configuration. The resulting spawning biomass and recruitment trajectories are shown in Figure 28, with relative differences from the full fit in Figure 29. Mohn’s rho values are calculated from terminal-year relative differences across the five peels and reported in Table 8.
Show code
if (!is.null(retro_df)) {ggplot(retro_df, aes(x = year, y = value, color = peel_label, group = peel_label)) +geom_line(linewidth =0.8) +facet_wrap(~ quantity, scales ="free_y", ncol =1) +labs(x ="Year", y ="Value", color ="Retrospective peel") + ggthemes::theme_few()}
Figure 28: Five-year retrospective trajectories for spawning biomass and recruitment from the SPoRC dev model.
Show code
if (!is.null(retro_rel)) {ggplot(retro_rel, aes(x = year, y = relative_difference, color = peel_label, group = peel_label)) +geom_hline(yintercept =0, linetype ="dashed", color ="grey45") +geom_line(linewidth =0.8) +geom_point(data = retro_terminal, size =2) +facet_wrap(~ quantity, scales ="free_y", ncol =1) +labs(x ="Year", y ="Relative difference from full fit", color ="Retrospective peel") + ggthemes::theme_few()}
Figure 29: Relative differences between each retrospective peel and the full SPoRC dev fit. Points identify the terminal year of each peel.
Show code
if (nrow(retro_rho) >0) {gt(retro_rho) |>fmt_number(columns = mohns_rho, decimals =3) |>cols_label(quantity ="Quantity",region ="Region",mohns_rho ="Mohn's rho" )} else {gt(data.frame(note ="The five-year retrospective output was not available."))}
Table 8: Mohn’s rho for the five-year SPoRC retrospective analysis.
Quantity
Region
Mohn's rho
Recruitment
1
−0.221
Spawning biomass
1
−0.235
8.3 Model Configurations
The dev model configuration is read from config/base.yml. The vign column summarizes the specification reproduced by analysis/run_vignette_case.R from the SPoRC single-region EBS pollock vignette. The flattened and summarized settings in Table 9 document the model configurations used by the rendered assessment.
The age-composition sample sizes used in the multinomial likelihood calculations are year-specific and are read directly from the conditioned SPoRC input object. The values in Table 10 are the input sample sizes used for fishery, BTS, and ATS age-composition likelihoods.
Table 10: Input sample sizes used for multinomial age-composition likelihood calculations by fleet and year.
Likelihood
Year
Input sample size
Fishery
Multinomial
1964
10.00
Multinomial
1965
10.00
Multinomial
1966
10.00
Multinomial
1967
10.00
Multinomial
1968
10.00
Multinomial
1969
10.00
Multinomial
1970
10.00
Multinomial
1971
10.00
Multinomial
1972
10.00
Multinomial
1973
10.00
Multinomial
1974
10.00
Multinomial
1975
10.00
Multinomial
1976
10.00
Multinomial
1977
10.00
Multinomial
1978
39.00
Multinomial
1979
39.00
Multinomial
1980
39.00
Multinomial
1981
39.00
Multinomial
1982
39.00
Multinomial
1983
39.00
Multinomial
1984
39.00
Multinomial
1985
39.00
Multinomial
1986
39.00
Multinomial
1987
39.00
Multinomial
1988
39.00
Multinomial
1989
39.00
Multinomial
1990
39.00
Multinomial
1991
260.30
Multinomial
1992
228.05
Multinomial
1993
344.45
Multinomial
1994
286.36
Multinomial
1995
256.95
Multinomial
1996
189.08
Multinomial
1997
318.68
Multinomial
1998
354.57
Multinomial
1999
476.14
Multinomial
2000
482.43
Multinomial
2001
328.17
Multinomial
2002
486.56
Multinomial
2003
440.91
Multinomial
2004
390.52
Multinomial
2005
494.38
Multinomial
2006
506.21
Multinomial
2007
500.22
Multinomial
2008
523.65
Multinomial
2009
420.03
Multinomial
2010
549.41
Multinomial
2011
728.07
Multinomial
2012
607.46
Multinomial
2013
753.27
Multinomial
2014
606.34
Multinomial
2015
821.23
Multinomial
2016
704.11
Multinomial
2017
606.72
Multinomial
2018
667.84
Multinomial
2019
700.80
Multinomial
2020
565.24
Multinomial
2021
817.58
Multinomial
2022
641.24
Multinomial
2023
443.04
BTS
Multinomial
1982
162.04
Multinomial
1983
130.70
Multinomial
1984
63.27
Multinomial
1985
73.54
Multinomial
1986
151.95
Multinomial
1987
61.21
Multinomial
1988
82.23
Multinomial
1989
53.19
Multinomial
1990
48.27
Multinomial
1991
79.42
Multinomial
1992
85.22
Multinomial
1993
91.98
Multinomial
1994
86.85
Multinomial
1995
104.90
Multinomial
1996
226.47
Multinomial
1997
73.38
Multinomial
1998
65.19
Multinomial
1999
95.43
Multinomial
2000
90.69
Multinomial
2001
183.01
Multinomial
2002
186.33
Multinomial
2003
125.66
Multinomial
2004
188.66
Multinomial
2005
194.49
Multinomial
2006
329.61
Multinomial
2007
192.70
Multinomial
2008
235.41
Multinomial
2009
144.72
Multinomial
2010
107.89
Multinomial
2011
165.27
Multinomial
2012
107.12
Multinomial
2013
66.53
Multinomial
2014
117.66
Multinomial
2015
103.67
Multinomial
2016
122.84
Multinomial
2017
234.59
Multinomial
2018
158.32
Multinomial
2019
120.68
Multinomial
2021
149.48
Multinomial
2022
172.83
Multinomial
2023
126.78
Multinomial
2024
64.21
ATS
Multinomial
1994
43.00
Multinomial
1996
32.00
Multinomial
1997
49.00
Multinomial
1999
67.00
Multinomial
2000
70.00
Multinomial
2002
72.00
Multinomial
2004
51.00
Multinomial
2006
47.00
Multinomial
2007
39.00
Multinomial
2008
35.00
Multinomial
2009
26.00
Multinomial
2010
34.00
Multinomial
2012
44.00
Multinomial
2014
79.00
Multinomial
2016
61.00
Multinomial
2018
50.00
Multinomial
2020
1.00
Multinomial
2022
52.00
Multinomial
2024
79.00
8.5 Parameter Estimates
Estimated fixed-effect parameters and their standard errors from the SPoRC dev model are shown in Table 11. Repeated parameter names are distinguished by their instance number in the underlying parameter vector.
Table 11: SPoRC dev-model fixed-effect parameter estimates and standard errors.
Parameter
Instance
Estimate
Standard error
ln_global_R0
1
11.8034
NaN
ln_InitDevs
1
−2.7475
NaN
ln_InitDevs.1
1
−2.7671
NaN
ln_InitDevs.2
1
−4.0436
NaN
ln_InitDevs.3
1
−5.1608
NaN
ln_InitDevs.4
1
−3.6680
NaN
ln_InitDevs.5
1
−4.2403
NaN
ln_InitDevs.6
1
−6.0079
NaN
ln_InitDevs.7
1
−38.7926
NaN
ln_InitDevs.8
1
−36.0825
NaN
ln_InitDevs.9
1
−33.7880
NaN
ln_InitDevs.10
1
−33.7768
NaN
ln_InitDevs.11
1
−23.4553
NaN
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NaN
ln_srvsel_devs.1028
1
0.0095
NaN
ln_srvsel_devs.1029
1
−0.0013
NaN
ln_srvsel_devs.1030
1
−0.0013
NaN
ln_srvsel_devs.1031
1
−0.0079
NaN
ln_srvsel_devs.1032
1
−0.0220
NaN
ln_srvsel_devs.1033
1
−0.0201
NaN
ln_srvsel_devs.1034
1
−0.0053
NaN
ln_srvsel_devs.1035
1
−0.0047
NaN
ln_srvsel_devs.1036
1
−0.0159
NaN
8.6 Gradient Diagnostics
The fixed-effect parameters with the largest absolute gradients are shown in Table 12. These entries identify the parameters that are furthest from satisfying first-order optimality in the current dev fit.