North American Journal of Fisheries Management 35, 649-658, 2015 American Fisheries Society 2015 DOI: 10.1080/02755947.2015.1032452 Supplement: Equations for Estimating Effort and Harvest in Boat- and Shore-Based Fisheries Basic Notation The following notation is used in defining the various estimation methods: j denotes the stratum being considered (j = 1,, J); J denotes the total number of strata; i denotes the sampling day unit within the stratum (i = 1,, Nj); Nj is the total population size (all possible sampling days) in stratum j; nj is the sample size in stratum j; xij denotes the value of the ith unit of stratum j; j is the sample mean for stratum j; and = is the sample variance for stratum j. Effort Estimation for Boat- and Shore-Based Fisheries Step 1. The progressive counts of boat- and shore-based fishers were calculated separately to estimate effort for each day of sampling, = T, (S.1) where is the estimate of fishing effort for the ith sample day; is the progressive count for the ith sample day (the number of fishers counted was used for both boatand shore-based fisheries); and T is the length of the fishing day (we used the mean day length period in hours). Step 2. The daily effort estimates were then expanded for each day-type stratum (weekend or weekday) by multiplying the number of possible sample days in each stratum by the mean of the daily estimates of effort, =, (S.2)
where is the estimated mean daily fishing effort (anglers per day) for the jth daytype stratum within the season; is the estimate of fishing effort for the ith sample day in the jth day-type stratum within the season; and is the number of days sampled in the jth day-type stratum within the season. =, (S.3) where is the estimate of total effort (angler-hours) for the jth day-type stratum within the season. Step 3. We calculated the precision of effort estimates by estimating variances and SEs for each stratum: =, (S.4) where is the estimated variance of the mean daily fishing effort for the jth day-type stratum within the season; is the sample variance of the daily estimates of fishing effort for the jth day-type stratum within the season; and is the sample size as described for equation (S.2). The SE of mean daily fishing effort was then estimated as SE =, (S.5) where SE is the estimated SE of the mean daily fishing effort; and is as described for equation (S.4). The variance for total effort was determined as =, (S.6) where is the estimated variance of total effort for a stratum and was calculated separately for each day type (weekday or weekend). Finally, the SE of total effort was calculated:
SE =, (S.7) where SE is the estimated SE of total effort for a stratum; and is as described for equation (S.5). Step 4. We calculated total fishing effort separately for boat- and shore-based fisheries. This was done by adding the effort estimates for the strata to obtain seasonal totals, =, (S.8) where is the total seasonal effort, obtained by combining the effort estimates for day-type strata; and is the estimate of total effort for the jth stratum as defined for equation (S.3). Step 5. The precision of effort estimates obtained by adding stratum totals was calculated by summing the estimated variances for all strata and calculating the SE, = ( ), (S.9) where is the estimated total seasonal variance, calculated by combining the estimated effort variances for day-type strata. The SE was determined as SE=, (S.10) where SE is the estimated SE for seasonal effort totals when adding day-type strata; and is as described for equation (S.9). Harvest Rate Estimator for the Boat-Based Fishery The ratio of means was used to estimate total harvest, as the boat fishery data were based on completed trips (Jones et al. 1995; Pollock et al. 1997). This estimator is the ratio of mean harvest to mean effort; when applied to access-point surveys, it
has a statistical expectation equal to total harvest divided by total effort for the population of fishers (Pollock et al. 1997), =, (S.11) where is the ratio of means, the estimated daily harvest rate (fish/angler-hour) based on completed trips; is the complete harvest for the kth fishing party; is the complete trip length for the kth fishing party; and n is the number of fishing parties in the daily sample. The seasonal mean daily harvest rates ( ) for each day-type stratum were calculated. The estimated variances of the mean daily harvest rates ( ) were calculated by using the general form of equation (S.4), and the estimated SEs of the mean daily harvest rates (SE ) were calculated by using the general form of equation (S.5). Harvest Rate Estimator for the Shore-Based Fishery The mean of ratios was used to estimate total harvest from the shore-based fishery, as the data were based on incomplete trips (Jones et al. 1995; Hoenig et al. 1997; Pollock et al. 1997). This estimator is the mean of the individual harvest rates for all fishers interviewed on a given day. When applied to incomplete trip interviews taken by roving through the fishery, the mean-of-ratios estimator has a statistical expectation of total harvest divided by total effort for the population of fishing units provided that short-duration trips are excluded (Hoenig et al. 1997). Given this expectation, the mean-of-ratios estimator used for incomplete trips (with shorttrip exclusion) provides an equivalent measure of fishing success to the ratio-ofmeans estimator used on completed trips (Hoenig et al. 1997; Pollock et al. 1997), =, (S.12) where is the mean of ratios, the estimated shore-based fishery harvest rate (fish/angler-hour); is the harvest recorded at the time of the interview for the
incomplete trip made by the kth fisher; is the length of the incomplete trip at the time of the interview for the kth fisher; and n is the number of anglers in the daily sample. The seasonal mean daily harvest rates ( ) for each day-type stratum were calculated. The estimated variances of the mean daily harvest rates ( ) were determined by using the general form of equation (S.4), and the estimated SEs of the mean daily harvest rates (SE ) were calculated using the general form of equation (S.5). Harvest Rate Estimation for Boat- and Shore-Based Fisheries The contribution of each day-type stratum to the estimated seasonal harvest rate was apportioned by the relative size of each day-type stratum within the season (Pollock et al. 1994). Essentially, greater weighting was given to the weekday stratum because there are more weekdays than weekend days in a season, = +, (S.13) where is the stratified mean daily harvest rate (fish/angler-hour) for a season (the estimator [equation S.11] was used for the boat-based fishery; the estimator [equation S.12] was used for the shore-based fishery); is the number of weekdays in the season; is the number of weekend days (includes public holidays) in the season; is the total number of days in the season; is the mean daily harvest rate for the weekday stratum; and is the mean daily harvest rate for the weekend day stratum. Estimates of variance for the stratified mean daily harvest rates were calculated as follows: ( ) = ( ) + ( ), (S.14) where ( ) is the estimated variance of the stratified mean daily harvest rate for the season; ( ) is the estimated variance of the mean daily harvest rates for
the weekday stratum in the season; and ( ) is the estimated variance of the mean daily harvest rates for the weekend day stratum in the season. Both ( ) and ( ) can be calculated by using the general form of equation (S.4). Estimates of SE for the stratified mean daily harvest rates were obtained as SE( ) = ( ), (S.15) where SE( ) is the SE of the stratified mean daily harvest rate; and ( ) is as described for equation (S.14). Harvest Estimation for Boat- and Shore-Based Fisheries To estimate total harvest for boat- and shore-based fisheries, the independent effort estimate was multiplied by the appropriate harvest rate estimate (Pollock et al. 1994, 1997; Hoenig et al. 1997): =, (S.16) where is the estimate of harvest (numbers of fish) for the boat-based fishery (estimation was for each day-type stratum within the season); is the estimate of effort (angler-hours) for the boat-based fishery; and is the estimate of mean daily harvest rate (fish/angler-hour) as described for equation (S.11). Likewise, for the shore-based fishery, =, (S.17) where is the estimate of harvest (numbers of fish) for the shore-based fishery (estimation was for each day-type stratum within the season); is the estimate of effort (angler-hours) for the shore-based fishery; and is the estimate of mean daily harvest rate (fish/angler-hour) as described for equation (S.12). Estimates of variance in harvest for the boat- and shore-based fisheries were calculated, and the base level of estimation was for a day-type stratum within the season:
= ()+ (), (S.18) where is the estimated variance for the boat-based fishery when using equivalent terms from equation (S.16) and is the estimated variance for the shorebased fishery when using equivalent terms from equation (S.17); is the estimate of mean daily harvest rate for a stratum (the estimator [equation S.11] was used for the boat-based fishery; the estimator [equation S.12] was used for the shore-based fishery); () is the estimated variance of the mean daily harvest rate for a stratum and is calculated by using the general form of equation (S.4); is the total effort for a stratum and is described in equation (S.3); and is the estimated variance of the total effort for a stratum, as described in equation (S.6). The estimated SE of the harvest was calculated as SE =, (S.19) where SE is the estimated SE of the harvest for the boat-based fishery when using equivalent terms from equation (S.16) and is the estimated SE of the harvest for the shore-based fishery when using equivalent terms from equation (S.17); and is the estimated variance of the harvest, as described in equation (S.18). Harvest estimates for weekday and weekend day strata were combined to obtain seasonal totals. The general forms of the equations used in the effort and harvest estimations and the associated variances and SEs were used for catch estimation. SUPPLEMENTAL REFERENCES Hoenig, J. M., C. M. Jones, K. H. Pollock, D. S. Robson, and D. L. Wade. 1997. Calculation of catch rate and total catch in roving surveys of anglers. Biometrics 53:306 317. Jones, C. M., D. S. Robson, H. D. Lakkis, and J. Kressel. 1995. Properties of catch rates used in analysis of angler surveys. Transactions of the American Fisheries Society 124:911 928.
Pollock, K. H., J. M. Hoenig, C. M. Jones, D. S. Robson, and C. J. Greene. 1997. Catch rate estimation for roving and access point surveys. North American Journal of Fisheries Management 17:11 19. Pollock, K. H., C. M. Jones, and T. L. Brown. 1994. Angler survey methods and their applications in fisheries management. American Fisheries Society, Special Publication 25, Bethesda, Maryland.