Estimating fish mortality rates from catch curves: A plea for the abandonment of Ricker (1975)'s linear regression method

Abstract

Fisheries management often relies on the monitoring of key life-history traits to assess, for instance, any potential harvest-related detrimental effects on a given fish stock (Rochet, 1998). Mortality, which can be estimated through different approaches (Lees et al., 2021; Miranda & Bettoli, 2007), constitutes a pivotal biodemographic component for stock status assessment and population dynamics modeling (Johnson & Zúñiga-Vega, 2009; Wikström et al., 2016). Accurately estimating mortality rates is thus of prime importance to avoid any misevaluation of a given fish stock status or potentially producing overoptimistic projections from relying on downwardly biased estimates (Goto et al., 2022). Here, we advocate that this situation may arise when the linear regression (LR) method of Ricker (1975) is used to estimate mortality from catch-curve data. Our main objective with this comment is to dissuade fisheries biologists from applying this more bias-prone method by reiterating the same recommendation that was previously made by Smith et al.(2012) stating that is should no longer be used. To achieve this, we highlight several analytical disadvantages associated with this method and then reanalyse age-frequency datasets taken from four frequently cited fisheries books and four others from the Ministère de l'Environnement, de la Lutte contre les changements climatiques, de la Faune et des Parcs (MELCCFP) to compare the estimates obtained with those of more adequate, recent methods. Even if the poor performance of the LR method has already been well documented, we feel that this comment is warranted given its frequent continued application nowadays in fisheries management. When achievable, total annual mortality (A) should ideally be estimated from capture-mark-recapture (CMR) studies (Bradshaw et al., 2007; Caza-Allard et al., 2021) in which apparent total annual survival (S) is obtained, and therefore also A = 1−S. This is because such an approach can sometimes discriminate mortality from other different demographic processes (i.e., emigration) and even further partition this life-history trait into its natural (M) and fishing (F) components (Lees et al., 2021; Mucientes et al., 2023). As a less resource-demanding alternative, the age-frequency data of randomly sampled fish may be used instead to estimate the instantaneous mortality (Z) that reflects the additive but indistinguishable effects of M and F, and from which A can then be obtained (Allen & Hightower, 2010; Miranda & Bettoli, 2007; Ogle, 2016; Pauly, 1984; Simpfendorfer et al., 2005). As opposed to the CMR approach, catch-curve analyses are performed under strict assumptions of a steady-state population, which makes them more subject to biases (Murphy, 1997; Nelson, 2019). This type of data-poor analysis nevertheless provides an approximative quantification of the rate at which counts decrease with age to estimate A and is thus commonly used in fisheries management (Miranda & Bettoli, 2007; Ogle, 2016). The currently available methods to estimate Z from catch-curve analyses can be roughly summarized into three groups: (i) the log transformation of counts in an LR framework (the LR method and its weighted version: the WLR method of Maceina & Bettoli, 1998), (ii) the use of a maximum-likelihood estimator of S as proposed by Chapman & Robson (1960; the CR method) and to which correction factors have been later applied for the estimation of Z and its standard error (SE) (Hoenig et al., 1983; Smith et al., 2012) and referred to as the CR method with correction for bias (CRCB method), and (iii) a generalized linear modeling (GLM) framework (Millar, 2011; Zuur et al., 2009) in which the age frequencies are expected to be approximately Poisson distributed, as proposed by Millar (2015). The GLMPoisson approach has since been further developed into a “Poisson Model” (PM; Nelson, 2019) and a more flexible GLM-based (GB) method (Mainguy & Moral, 2021). All these different methods differ greatly in their analytical concepts, especially regarding their modeling capacity for statistical estimations and comparisons across multiple fish samples (Mainguy & Moral, 2021). More than a decade ago, Smith et al. (2012) used simulations to compare different methods for catch-curve analyses that led to four recommendations, one of which states: “Unweighted linear regression should not be used for catch-curve analysis because it is inferior to other methods.” Previous investigations of catch-curve analyses using simulations (Dunn et al., 2002; Jensen, 1985; Murphy, 1997) have unanimously shown that in many instances, the CR method was at least equal but more often superior to the LR method regarding the accuracy of Z estimates. In these studies, as for those that followed (Mainguy & Moral, 2021; Nelson, 2019), using the LR method frequently led to more pronounced biases in the estimation of A, often but not always attributable to the underestimation of Z. One can easily foresee that such a situation is not without consequences from a fisheries management perspective. Numerous fisheries biologists have nonetheless kept applying the LR method to estimate mortality rates in recently published studies as indicated by this non-exhaustive list: Chestnut-Faull et al. (2021), Lowe et al. (2021), Pacicco et al. (2021), Sanchez and Rooker (2021), Santander-Neto et al. (2021), Scarnecchia et al. (2021), Turner et al. (2021), Anderson et al. (2022), Tsai and Huang (2022), Yamamoto and Katayama (2022), Andrade et al. (2023), He et al. (2023), Klein and McCormick (2023), and Madenjian et al. (2023). We believe the LR method is still frequently used nowadays as (i) it is commonly described in fisheries books and as a result often used in published catch-curve studies, (ii) it can be easily applied using readily available software (e.g., Ogle, 2016; Slipke & Maceina, 2014), (iii) it has a more attractive analytical simplicity when compared to alternatives, and (iv) it has sometimes been established as a “default” approach after decades of use. Nevertheless, we describe here why we believe these reasons are no longer sufficient to justify its continued use. Modeling count data should primarily be done using an appropriate error structure that was designed for the discrete case, that is, the Poisson distribution (Hilbe, 2014). Extensions of this distribution have also been developed with an additional scaling parameter to allow the analysis of counts that are not equidispersed (i.e., variance ≈ mean), a prerequisite for the GLMPoisson (Hilbe, 2014). Counts are frequently overdispersed in ecological studies (Richards, 2008) but can sometimes be underdispersed too, thus often requiring better-adapted Poisson extensions (Brooks et al., 2019; Lindén & Mäntyniemi, 2011). The availability of appropriate error structures to analyse counts led Steel et al. (2013) to state that “there is no longer a need to transform count data” especially since “Gaussian models based on log-transformed data can perform poorly under many circumstances such as zero-inflation, small means, or over-dispersion.” This statement is mostly based on the work of O'Hara and Kotze (2010) unambiguously entitled: “Do not log-transform count data.” It is therefore not surprising that error structures that were designed for the analysis of counts are increasingly being adopted in ecological studies to the detriment of data transformation (St-Pierre et al., 2018). The main problem with data transformation is that it changes the relationship between a response (Y) and a predictor (X), which also complicates effect size interpretation (Zuur et al., 2009). Regarding catch curves, counts (Y) are modeled based on age (X) to estimate the absolute value of the slope parameter (Z). As such, modeling Y and X on the raw scale within a GLMPoisson (Millar, 2015) or based on one of its extensions when required (Mainguy & Moral, 2021) will necessarily yield a different outcome for the estimation of Z and its SE than regressing loge(Y) on X in a linear model (Ricker, 1975). Moreover, the use of a distribution that accommodates counts in a GLM also allows us to produce a 95% CI lower limit for A that cannot be negative as the lower bound is innately set to zero with a count variable (Hilbe, 2014). Conversely, the LR method can sometimes generate unrealistic, negative values when the counts are highly dispersed (see the brook trout [Salvelinus fontinalis] example in Table 1). For all these analytical considerations, we believe that catch-curve analyses should not be performed on log-transformed counts in linear models but rather directly on counts in log-linear models (Hilbe, 2014; O'Hara & Kotze, 2010; Steel et al., 2013). Zero counts are frequently ignored in the descending limb of the catch curve when employing the LR method as loge(0) is not calculable as a real number. As a result, at least two problems will arise: (i) the full information contained in a fish sample will not be utilized because of omission of true zeros and (ii) by being likely discarded, the old age groups with zero counts will have a diminished influence on the regression curve and, thus, the estimated slope. To partly circumvent such disadvantages, the FAMS software (Slipke & Maceina, 2014) sometimes used for catch-curve analyses (e.g., Chestnut-Faull et al., 2021) will instead add a constant (+1) to each count when ≥ 1 age group(s) were not sampled in the descending limb. Although such practice has also been criticized (O'Hara & Kotze, 2010), this offers a better solution as loge(0 + 1) = 0, therefore, replacing a zero with a zero instead of discarding it. Doing so in most instances should reduce bias for the estimation of Z and its SE. However, as log-linear models can adequately handle zeros when present (Hilbe, 2014), they offer a better analytical solution for catch-curves analyses (Millar, 2015). As indicated by Miranda and Bettoli (2007) and Allen and Hightower (2010), it is customary when using the LR method to truncate the older age groups, for instance, with fewer than five fish, to reduce their influence on the overall estimate of Z (e.g., He et al., 2023). Defining which age groups should be kept or removed when n is below a given threshold can also have varying impacts on the estimation of Z. As previously noted by Hoenig (1983), Z and longevity are inversely related, and, as such, discarding the oldest individuals in a catch-curve analysis aimed at estimating Z due to their yet naturally occurring low counts appears somewhat counterintuitive. When this decision rule is applied, how the remaining counts for younger age groups are distributed will then determine the direction of the potentially induced bias for Z. See, for instance, fig. 6 presented in Yamamoto and Katayama (2022) for an example of a severely right-truncated descending limb in which Z may rather be overestimated from using the LR method. Although Chapman and Robson (1960) have originally suggested such practice, Smith et al. (2012) later recommended that no data truncation should be applied. Millar (2015) later argued that all age frequencies, including zeros, are valid observed counts and recommended that the age frequencies of the descending limb should also be extended by adding zero counts beyond the oldest sampled age to avoid the underestimation of Z as obtained from a GLMPoisson. The rationale behind this analytical step is biologically sound as some older individuals than the oldest one sampled may not have been caught due to their generally low abundance. As doing so better reflects the biological reality being modeled, Nelson (2019) and Mainguy and Moral (2021) have also included this additional step in their respective PM and GB methods, although such procedure is less required when analysing the catch curves of short-lived species in which the oldest possible age has been sampled. Regardless of the retained method used to estimate mortality, the criterion behind the selection of the first age group to be used for the descending limb can sometimes have a major influence on the estimated slope parameter. As initially mentioned, we have reanalysed the age-frequency datasets presented in four fisheries books from Pauly (1984), Miranda and Bettoli (2007), Allen and Hightower (2010), and Ogle (2016) from which the results of mortality estimates are provided in Table 1. A first striking ascertainment was that the way the first considered fully recruited age was selected differed among these authors. Pauly (1984) described and applied the “Peak Plus” criterion and so did Miranda and Bettoli (2007). On the contrary, Allen and Hightower (2010) instead determined the starting point from their own catch-curve interpretation, which resulted in what could be called a “Peak Minus” criterion. Finally, Ogle (2016) referred to Smith et al. (2012) for the application of method-specific decision rules but used his own interpretation, which corresponded to the application of the “Peak” criterion. We believe that the “Peak Plus” criterion should be systematically used as it offers a better safeguard from including an age group that was nearly but not yet fully recruited by the fishing gear, as originally proposed by Pauly (1984) and later by Smith et al. (2012). However, using the stricter “Peak Plus” criterion translates into losing one degree of freedom as the age groups correspond to the sampling units, and thus not their combined sample sizes. When the “Peak” criterion is used instead, it should then produce a similar Z estimate to that obtained when “Peak Plus” is applied if the catch-curve assumptions are met (Pauly, 1984). Although it is more constraining for short-lived species, the use of the “Peak Plus” criterion appears as the best available option as a standardization effort regarding the identification of the first age group to be included in the descending limb of a catch curve. Pauly (1984) [over] 0.658 ± 0.062b 48.2 [40.0, 55.2] 0.633 ± 0.059 46.9 [38.9, 53.8] 0.597 ± 0.034 45.0 [41.1, 48.6] 0.623 ± 0.043 46.4 [41.6, 50.8] Miranda and Bettoli (2007) [under/equi] 0.511 ± 0.016 40.0 [37.7, 42.2]c 0.475 ± 0.024 37.8 [34.2, 41.3] 0.511 ± 0.016 40.0 [37.7, 42.3] 0.544 ± 0.028 42.0 [38.7, 45.1] 0.546 ± 0.030 42.1 [38.8; 45.1] Allen and Hightower (2010) [equi] 0.64d 47d 0.763 ± 0.129b 53.4 [35.0, 66.6] 0.525 ± 0.122 40.9 [19.0, 56.8] 0.704 ± 0.089 50.6 [41.2, 58.4] 0.713 ± 0.073 51.0 [44.0, 57.1] Ogle (2016) [over] 0.660 ± 0.137 48.3 [20.1, 66.6] 0.676 ± 0.236 49.1 [−40.6, 81.6]f 0.761 ± 0.337 53.3 [9.6, 75.9] 0.608 ± 1.344 45.5 [16.5, 64.5] Bélanger (2023) [over] - - 0.190 ± 0.045 17.3 [8.5, 25.2] 0.174 ± 0.036 16.0 [9.0, 22.5] 0.254 ± 0.050 22.5 [14.5, 29.7] 0.231 ± 0.045 20.6 [13.5, 27.2] Mainguy and Beaupré (2019) [equi] - - 0.674 ± 0.079 49.0 [34.4, 60.4] 0.454 ± 0.115 36.5 [8.5, 55.9] 0.708 ± 0.167 50.7 [31.7, 64.5] 0.736 ± 0.133 52.1 [39.0, 62.4] - - 1.210 ± 0.162 70.2 [50.0, 82.2] 0.951 ± 0.148 61.4 [41.7, 74.4] 1.278 ± 0.146 72.1 [62.9, 79.1] 1.302 ± 0.116 72.8 [66.5, 77.9] - - 0.435 ± 0.078 35.2 [22.1, 46.2] 0.450 ± 0.100g 36.2 [18.5, 50.1] 0.379 ± 0.121 31.5 [13.2, 46.0] 0.491 ± 0.141 38.8 [20.4, 53.0] Given the different analytical shortcomings of the LR method, the CR and CRCB methods have sometimes been preferred to estimate mortality rates (e.g., Clark et al., 2023; Coulson & Poad, 2021) if not the WLR method instead (e.g., Jefferson et al., 2021; Sullivan et al., 2020). Although the CRCB method constitutes an improvement over its et al., it nonetheless that Z and its SE be from S as obtained from the method to then correction factors (Hoenig et al., 1983; Smith et al., This makes the approach more but in the of overdispersed it will yield a SE estimate that better reflects the observed as for the PM that also a similar correction Even if the for A is then more adequately statistical regarding comparisons cannot be with the CRCB method and the PM as 95% may still be of statistical for a & The WLR method, on the contrary, being an of the LR method, relies on a procedure that was as by Smith et al. (2012) and thus appears to us as a Mainguy and Moral (2021) have shown with simulations that the LR and WLR methods may yield severely mortality estimates when the variance in observed counts as a of the a commonly in ecological studies & Mäntyniemi, 2011). The use of a GLM framework as a better analytical especially as the of model et al., model selection et al., and model et al., can all be applied to better describe the biological (Mainguy & Moral, 2021). To further why we fully the statement made in the of Smith et al. (2012) the LR “Unweighted linear regression should no longer be used for catch-curve we present in Table 1 the of the catch-curve to the and GB methods. For the LR method, estimates were under three different approaches in which zero counts were (i) as originally estimated and by the (ii) using the “Peak” criterion with data truncation of the older with low counts when present and (iii) using the “Peak Plus” criterion without data truncation for the LR and CRCB methods were obtained with the et al., but linear models for the were also to the assumptions of and were was used for the GB method by the models based on the et al., with as such threshold to a model that is already less likely than the one et al., 2011). information the age frequencies used as for the estimation of Z and A in Table 1 can be in Table and of the the analytical and are provided in Mainguy and Moral (2021) and a The method-specific A estimates obtained were compared with between the and the CRCB or GB methods. which could be performed as the were distributed, indicated that the method lower A estimates when compared to the CRCB method = = = with a of and the same was when compared to the GB method = = = with a of As opposed to studies (Mainguy & Moral, 2021; Smith et al., 2012) in which the is the age structure of the sampled population and thus this was not the in the analysis of these catch-curve As such, no bias could be but results are nonetheless directly in with what could have been In most instances, we believe that such in Z estimates are from different age-frequency data that are made to the use of a linear log zeros and of the descending limb. should no longer be when analysing catch-curve data. could arise when estimates obtained with the LR method need to be compared on the same to more recent estimates without to the data. such more recent methods should also be used to with the interpretation of the estimates obtained by Given the recommendation made by Smith et al. (2012) of no longer using the LR method, the of from other studies that it often and of catch-curve provided here that further its from better alternatives, we advocate that the LR method should no longer be used. This is because the and GB methods have been shown to be better for catch-curve Although additional to these methods could be we believe that should be to which the mortality estimates obtained from catch-curve analyses with those of a CMR approach for the same catch-curve analyses of a assessment of different on the one the CMR approach annual S estimates for the age groups based on rates on the the biological processes and analytical to describe them so are likely the estimates. Nevertheless, how such estimations obtained from different approaches et al., Turner et al., 2021) be to further for instance, a or bias in A estimates may be observed by using catch-curve and the made over the analytical investigations are likely still required to better the accuracy of mortality estimates obtained from a catch curve. to further reduce any estimation biases should therefore be and as such, we believe that the LR method a step in that and with by and We all the fish & and biologists have in the different monitoring from which the age-frequency data presented in this especially and Ministère de l'Environnement, de la Lutte contre les changements climatiques, de la Faune et des The that they have no or that could have to influence the work in this The is not for the or of any information by the authors. than should be to the for the

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