5. Interpret
5.3.3 Interpret data -- associated techniques
Common tools for analysis of data
1. Area under the curve
Sometimes the only practical way to measure escapement is to measure a proxy. Very commonly biologists collect periodic visual impressions on the number present on the spawning grounds. Several authors have attempted to explain the process by which these periodic measurements can be converted into a logically sound estimate of total escapement magnitude (English et al. 1992; Bue et al. 1998; Hilborn et al. 1999; Parken et al. 2003). In essence, the process can be broken down into three parts or three ideas. The first idea is that each observer incorrectly perceives the number of fish present because of random error and because of that particular individual's bias or scaling error. In principle, this scaling error can be measured by having each observer repeatedly view a different known number of fish over time. Naturally, it is important that the scaling error is the same no matter how many fish are present--which is an assumption that is usually false. Although incorrect, sometimes this assumption is reasonable in the sense that the analyst can still produce adequate estimates. The second idea is that there is a curve that tracks the actual number of fish present on the spawning grounds at each point in time. Over time, some new fish will move into the spawning population and other fish will die leave the population, so that the curve will go up and go down. In principle, this curve can be approximated by adjusting the basic observations for the bias or scaling error and then simply connecting the dots produced by these adjusted values when graphed as a function of time. The third idea is based on the notion that the true, underlying curve can be integrated with calculus (which is also called finding the area under the curve). It turns out that this calculus result will produce a parameter that when divided by the average stream life of the fish on the spawning grounds will result in the actual total number of fish that moved into the spawning ground--or the total escapement that the analyst wants to know. Unfortunately, this process can lead to misleading results if the average stream life was not accurately estimated for each year, if the individual scaling conversions are not found by looking at the whole range of possible numbers of fish that are present, or by one of many other problems.
Bue, B. G., and coauthors. 1998. Estimating salmon escapement using area-under-the-curve, aerial observer efficiency, and stream-life estimates: the Prince William Sound pink salmon example. North Pacific Anadromous Fish Commission Bulletin 1:240-250.
English, K. K., R. C. Bocking, and J. R. Irvine. 1992. A robust procedure for estimating salmon escapement based on the area-under-the-curve method. Canadian Journal of Fisheries and Aquatic Science 49:1982-1989.
Hilborn, R., B. G. Bue, and S. Sharr. 1999. Estimating spawning escapements from periodic counts: a comparison of methods. Canadian Journal of Fisheries and Aquatic Science 56:888-896.
Parken, C.K., J.R. Irvine, and R.E. Bailey. 2003. Incorporating uncertainty into area-under-the-curve and peak count salmon escapement estimation. North American Journal of Fisheries Management 23:78-90.
2. Escapement goal methods
One of the most conventional and important components of a salmon stock assessment is the recommendation of a harvest policy based on an analysis of the stock's recent performance. In recent years some management agencies, especially in regions with highly eroded freshwater habitat, have recommended highly restricted harvest policies, often recommending against all directed harvests for some important stocks. In contrast, in Alaska, Canada, and other areas with pristine freshwater habitat, managers have recommended harvest policies with high harvest rates intended to produce the greatest possible sustained yield. The theory and practical recommendation for carrying out this kind of analysis can be found in many modern fisheries textbooks, such as the ones by Quinn and Deriso (1999), the one by Hilborn and Walters (1992), or one of several others.
One common way to produce escapement goals is with a "Ricker analysis," using linear regression to fit the parameters of a Ricker curve. The analysis begins by constructing a table of historic stock size estimates for each cohort (group of fish that were born in a single year). Next the analyst adds to the table estimates of the total return (all fish that were either harvested or else those that escaped the fishery and joined the breeding pool) over the life of the cohort. An estimate of the escapement level that will produce maximum sustained yield can be found in a computer spreadsheet or found with more complex analyses involving specialized software (PSC 1999). In comparison, many salmon stocks do not have sufficient stock-specific spawner and recruit data to directly estimate fish production relationships with a Ricker analysis. For these data limited stocks, habitat-based methods have been used to estimate fish production reference points and escapement goals to evaluate stock status using a meta-analysis of stock-recruitment data and habitat area observed across numerous stocks (Parken et al. 2006, Liermann et al. 2010).
Hilborn and Walters discuss some of the problems that can arise with this kind of analysis. Many of such problems stem from uncertainty due to measurement error, process error, missing data, and changing productivity. In actual practice, this simple linear regression approach is insufficient to make good recommendations. Many of the problems with this approach stem from measurement error in the estimates of stock size, missing data, and changing stock productivity. An extensive discussion of the potential pitfalls is beyond the scope of this web site and analysts should refer to the above books. Suffice it to say that this kind of analysis is one of the most important interpretive uses of salmon stock assessment indicators.
Pacific Salmon Commission. 1999. Maximum sustained yield or biologically-based escapement goals for selected Chinook salmon stock used by the Pacific Salmon Commission's Chinook Technical Committee for escapement assement. TCCHINOOK (99)-3.
Parken, C.K., R.E. McNicol, and J.R. Irvine. 2006. Habitat-based methods to estimate escapement goals for Chinook salmon stocks in British Columbia, 2004. Canadian Science Advisory Secretariat, Research Document 2006/083.
Specific examples from Alaska
Where possible, escapement goal ranges in Alaska are established with a stock-recruit analysis (Clark et al. 2009). When a stable, sustainable fishery is already in place, escapement goals have been established by selecting percentiles of historical escapement estimates or indices (e.g, Hasbrouck and Edmundsen 2007). To establish a lower-bound escapement threshold, Bernard et al. (2009) tried to balance the risks of management error by fitting a statistical model of the escapement time series. Fleischman and Borba (2009) explicitly incorporated measurement error in the estimates of stock size, missing data, and changing stock productivity into their stock recruit analyses.
Agency Reports:
Bernard, D.R., J.J. Hasbrouck, B.G. Bue, and R.A. Clark. 2009. Estimating risk of management error from precautionary reference points (PRP) for non-targeted salmon stocks. Alaska Department of Fish and Game, Special Publication No. 09-09, Anchorage. http://www.sf.adfg.state.ak.us/FedAidpdfs/SP09-09.pdf
Clark, R. A., Bernard, D. R., and S. J. Fleischman. 2009. Stock-recruitment analysis for escapement goal development: a case study of Pacific salmon in Alaska. American Fisheries Society Symposium 70:743-757.
Fleischman, S. J. and B. M. Borba 2009. Escapement estimation, spawner-recruit analysis, and escapement goal recommendation for fall chum salmon in the Yukon River drainage. Alaska Department of Fish and Game, Fishery Manuscript Series No. 09-08, Anchorage. http://www.sf.adfg.state.ak.us/FedAidpdfs/fms09-08.pdf
Hasbrouck, J. J., and J. A. Edmundson. 2007. Escapement goals for salmon stocks in Upper Cook Inlet, Alaska: report to the Alaska Board of Fisheries, January 2005. Alaska Department of Fish and Game, Special Publication No. 07-10, Anchorage. http://www.sf.adfg.state.ak.us/FedAidpdfs/SP07-10.pdf
Specific examples from Canada
- Spawning Chinook and sockeye salmon in the Lower Shuswap River, B.C.
Under Canada's Wild Salmon Policy, the biological status of a conservation unit is based on comparing indicators of abundance, trend, spatial distribution, and fishing mortality rates to benchmarks that identify when the biological status changes significantly. Abundance benchmarks can be based on criteria such as a percentage of the spawning escapement that produces the maximum sustained yield (MSY), a time frame that permits recovery, or an abundance associated with a spatial distribution that provides confidence that the conservation unit does not have a high probability of extinction. Trend benchmarcks can be based on the criteria used by the Committee on the Status of Endangered Wildlife in Canada. Spatial distribution benchmarks may identify when abundance becomes unusually concentrated or over-dispersed within a convservation unit. Benchmarks for fishing mortality rates can be based on fish production relationships, current survival conditions, and corresponding abundance reference points defined by the fish production relationship (e.g. spawning abundance producing MSY). Recent benchmark research has focussed on innovative approaches that are robust and meet multiple status evaluation criteria simultaneously.
Agency Reports:
Holt, C.A., A. Cass, B. Holtby, B. Riddell. 2009. Indicators of status and benchmarks for conservation units under Canada's wild salmon policy. Canadian Science Advisory Secretariat, Research Document 2009/058.
Bradford, M., and C. Wood. 2004. A review of biological principles and methods involved in setting minimum population sizes and recovery objectives for the September 2004 drafts of the Cultus and Sakinaw Lake sockeye salmon and Interior Fraser coho salmon recovery plans. Canadian Science Advisory Secretariat, Research Document 2004/128.
COSEWIC's Assessment Process and Criteria
3. Trend analysis
The most obvious questions about a salmon stock have to do with whether or not it is changing substantially over time. Those questions can be refined into a question such as, is the stock's productivity increasing or decreasing substantially, has the catch been increasing or decreasing substantially, or has some other feature been increasing or decreasing? Because the most notable feature of salmon population dynamics is a large amount of random, year-to-year variation, questions about trends are sometimes not easy to answer because trends are obscured by this randomness. Analysts try to approach this question with statistical hypothesis testing, using a null hypothesis of no change at all. Because the ocean is full of short-term and long-term trends, this kind of statistical hypothesis test approach does not really make sense. With a long time series, changes that are of no practical significance are sometimes determined to be statistically "significant," while short times series -- even ones with sharp decreases that should be alarming -- are sometimes considered to be "not significant." For this reason the analyst should usually provide a graph of the trend. Also, the analyst should directly estimate the magnitude of the change as well as the uncertainty in that estimate, and the analyst's full interpretation should include a clear idea of what magnitude of change should be considered meaningful in a management context.
Another important problem with using common statistical tools to look for trends is that common linear regression and correlation analyses are highly influenced by outliers, which, unfortunately, tend to be very common in salmon stock assessment time series. With these common tools, the more atypical an observation is, the greater the influence it will have on the estimated trend. Geiger and Zhang (2002) used a technique called robust regression to get around this problem and to put a decline into some kind of meaningful context. Their suggestion was to take either the previous 15 or 21 years of observations and pair these with a year number, starting with year 1. Next, they used a robust kind of statistical regression to find a trend line. This robust regression gives outliers the same weight in the analysis as points lying close to the line. Next, they then back-cast the trend line to a statistically stable estimate of the series in "year zero," the year just before the first observation. The slope of this kind of trend line gives a statistically stable estimate of the annual decline or increase in the population. If there is a decrease, then this decrease can be expressed as a percentage of the year zero value. In the populations Geiger and Zhang considered, they determined that 5% decline was what they called "biologically meaningful," although the level of decline that should trigger an alarm needs to be determined on a stock-by-stock basis.
4. Using data from multiple sites with hierarchical models
Everyone who samples biological data with monitoring programs shares a substantial challenge -- data collected represent not only real phenomena or states but also observation error and natural variability in processes. In some cases, observation error can be large and can create considerable imprecision and bias in estimates. Analyses with such data can usually only estimate the collective effect of these and other sources of uncertainty because observation error is particularly difficult to estimate. One common feature of ecological systems, the tendency for positive correlation to exist among certain components, offers a partial solution to this problem of unknown observation error.
For instance, such positive correlations in survival rate have been documented among nearby populations of many plants and animals (e.g., Peltonen et al. 2002), including salmon (Peterman et al. 1998; Pyper et al. 2005), with the magnitude of correlation decreasing with increasing distance between the populations (Bjornstad et al. 1999). It is likely that random observation error, which causes imprecise estimates, will be independent from one population to the next, so statistical models have been developed to use information from multiple populations simultaneously while estimating parameter values of processes such as maximum reproductive rate and magnitude of depensation (Allee effect at low abundance) in fish populations (Liermann and Hilborn 1997; Mueter et al. 2002a; Myers et al. 1999; Myers 2001, Su et al. 2004).
These cited analyses have all used some form of hierarchical model (Banerjee et al. 2004), one type of which is a mixed-effects model. Such models require data for each of the separate entities, such as fish populations in this example. These hierarchical models work by assuming that some parameter, such as maximum reproductive rate for a particular population, is part of a probability distribution of rates across numerous populations. The hierarchical model uses the data from multiple populations to derive a more precise estimate of the grand mean maximum reproductive rate across all populations than could be derived by analyzing each population separately. Hierarchical models are now becoming widely used by ecologists (Clark and Gelfand 2006), but even so, there are limitations (Cressie et al. 2009).
Hierarchical statistical modelling therefore has the potential to assist with interpretation of data collected from salmon monitoring programs by taking into account the tendency for positive correlation across space in survival and growth rates. This should help estimate parameters more precisely than with the most common current methods, which use data from each population separately.
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