Update Overview

In this update, I provide an alternative stratification and allocation scheme that is based on finding the best stratification and allocation. In this case, “best” is defined as the minimum sample size that satisfies the specified CV of the mean trap catch for each species. This is accomplished by repeatedly applying the Bethel Algorithm to different stratification options and using a genetic optimization algorithm to approach a near-optimal solution. I specified the CV as 0.10 for each species and I chose to allow the optimization to operate on the covariates in continuous space rather than group each covariate into factor levels prior to determining the optimal stratification.

Longitude, latitude, depth, and sample sites for each stratum as determined by the optimum stratification and allocation method.
Stratum lon.min lon.max lat.min lat.max depth.min depth.max Sample Sites
1 -80.44677 -80.06440 27.33250 28.75534 16 22 207
2 -80.94271 -80.02306 27.22535 29.42881 18 39 133
3 -81.06300 -80.00793 27.25772 29.66942 19 64 204
4 -81.22112 -79.72498 29.66429 31.49537 13 66 668
5 -80.81340 -78.55297 29.66430 32.68385 14 68 1055
6 -80.26138 -76.69541 29.66306 33.72164 20 94 961
7 -79.04702 -75.86499 32.25075 34.50163 16 96 495
8 -76.94690 -75.80977 34.51068 34.57761 15 63 63
9 -79.04079 -75.45364 32.25172 35.01521 14 115 180

I also specified the abundance reduction time series to be a decline of approximately 75% which is less than in previous runs so that computation of the CV metrics were not heavily influenced by very low abundance of the species with lowest catches.

Sampling Process

Gray triggerfish, black sea bass, red porgy, gag grouper, red snapper, scamp grouper, vermillion snapper, and red grouper were annually sampled in each of 12 years among 1,500 samples during the first 6 years and 750 samples during the last 6 years. Annual samples were chosen from the sampling universe by simple random samples, stratified samples with equal allocation,stratified samples with Bethel Allocation, and the optimal stratification and allocation method.

Estimation Results

I computed indices for each replicate sample (n=1000), species, observation method (trap and video), and sample type (Figures 1-8). Each model used latitude, longitude, and depth for both the negative binomial and binomial portions of the zero inflated error structure.

Figure 1 Grey triggerfish true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 1 Grey triggerfish true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 2. Black sea bass true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 2. Black sea bass true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 3. Red porgy true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 3. Red porgy true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 4. Gag grouper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 4. Gag grouper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 5. Red snapper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 5. Red snapper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 6. Scamp grouper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 6. Scamp grouper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 7. Vermillion snapper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 7. Vermillion snapper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 8. Red grouper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.
Figure 8. Red grouper true abundance trend (red line) and 1000 replicate index trends from chevron trap sampling (gray lines) and video sampling using Sumcount (blue lines) and for simple random sampling (Top Left), stratified sampling with Bethel allocation (Bottom Left), stratified sampling with equal allocation (Top Right), and optimal stratification and allocation.

Evaluation Process

I computed annual coefficients of variation for each species, observation method (trap and video), and sampling method (Figures 9-16). Note the increase in the CV during the second half of the timeseries owing to the reduction in annual sample size. Additionally, there is not a consistent or notable apparent shifting in the relative magnitude of CV among methods between the first and second halves of the timeseries. This suggests that CV under smaller sample size is not generally improved for one sampling method versus the others.

Figure 9. Grey triggerfish coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 9. Grey triggerfish coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 10. Black sea bass coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 10. Black sea bass coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 11. Red porgy coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 11. Red porgy coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 12. Gag grouper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red). Note different scales on the y-axes.
Figure 12. Gag grouper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red). Note different scales on the y-axes.
Figure 13. Red snapper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 13. Red snapper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 14. Scamp grouper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red). Note different scales on the y-axes.
Figure 14. Scamp grouper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red). Note different scales on the y-axes.
Figure 15. Vermillion snapper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 15. Vermillion snapper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red).
Figure 16. Red grouper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red). Note different scales on the y-axes.
Figure 16. Red grouper coefficients of variation for simple random sampling (Blue), stratified sampling with Bethel Allocation (Black), and stratified sampling with Equal Allocation (Red). Note different scales on the y-axes.


I computed performance scores based on average annual CV for each species and summed across all species for the chevron trap index and the video index (Figure 17-18). The relative degree of difference among sampling methods for the chevron trap indices is larger than for the video indices. For the species that have relatively low CV in the chevron trap indices (grey triggerfish, black sea bass, red porgy, red snapper, and vermillion snapper) there is really very little difference among sampling methods. Similarly and considering the video sampling indices, there is very little difference in total CV scores and individual species CV among the different sampling methods. The largest difference is for red grouper.

Figure 17. Chevron trap index CV scoring.
Figure 17. Chevron trap index CV scoring.


Figure 18. Video index CV scoring.
Figure 18. Video index CV scoring.


Following up on Erik’s suggestion, I also computed the relative magnitude of CV in the stratified sampling designs versus the simple random sample (figure 19-20). Considering the optimal stratification and allocation design, we again see the biggest differences among the grouper species and red snapper compared to the most abundance species. The overall scores are the sum of the species CV proportions so a score of 8 in aggregate would be no difference compared to simple random sampling.

Figure 19. Relative chevron trap index CV scoring.
Figure 19. Relative chevron trap index CV scoring.


Figure 20. Relative video index CV scoring.
Figure 20. Relative video index CV scoring.

Preliminary Conclusion and Possible Next Steps

Considering these results, stratified sampling is perhaps marginally beneficial in improving the precision of indices for important assessed species.

I think the next steps will be to run a set of simulations structured by different values of sampling intensity and abundance decline. One thought is to specify constant sampling intensities throughout a 12 years sampling duration where N= 1500, 1000, or 500. Alternatively, I could run three sets of sampling intensity where the first 6 years is N= 1500 and the last 6 years is either N=1500, 1000, or 500. I also thought I could have three levels of abundance decline either 0%, 30%, or 60%. These levels are arbitrary but would seemingly be a good start.

I would like to hear your thoughts on the current set of simulations including the use of the optimal stratification and allocation methodology. I would also like to hear your thoughts on different performance scoring metrics and this proposed set of structured simulations to examine the performance of stratification among depletion level and sample size.