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

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 original version (Smith et al., 2012), it nonetheless requires that Z and its SE must be approximated from S as obtained from the original method to then apply correction factors (Hoenig et al., 1983; Smith et al., 2012). This makes the approach more analytically complex, but in the presence of overdispersed counts, it will yield a more-reliable SE estimate that better reflects the observed variation, as for the PM that also applies a similar overdispersion correction factor. Even if the predicted uncertainty for A is then more adequately described, reliable statistical inferences regarding between-group comparisons cannot be satisfactorily achieved with the CRCB method and the PM as partially overlapping 95% CIs may still be indicative of statistical support for a difference (Schenker & Gentleman, 2001). The WLR method, on the contrary, being an extension of the already-biased LR method, relies on a weighting procedure that was categorized as "completely ad hoc" by Smith et al. (2012) and thus appears to us as a less-optimal choice. Indeed, Mainguy and Moral (2021) have shown with simulations that both the LR and WLR methods may yield severely under-biased mortality estimates when the variance in observed counts behaves as a quadratic function of the mean, a commonly encountered pattern in ecological studies (Lindén & Mäntyniemi, 2011). The use of a GLM framework stands out as a better analytical option, especially as the steps of model adequacy (Moral et al., 2017), model selection (Burnham et al., 2011), and model averaging (Dormann et al., 2018) can all be applied to better statistically describe the investigated biological process (Mainguy & Moral, 2021). To further illustrate why we fully support the statement made in the abstract of Smith et al. (2012) about the LR method: "Unweighted linear regression should no longer be used for analyzing catch-curve data," we present in Table 1 the reanalysis of the eight catch-curve examples according to the LR, CRCB, and GB methods. For the LR method, estimates were calculated under three different approaches in which zero counts were omitted: (i) as originally estimated and reported by the authors (LRAU), (ii) using the "Peak" (P) criterion with data truncation (T) of the older ages with low counts when present (LRPT), and (iii) using the "Peak Plus" (PP) criterion without data truncation (LRPP). Estimates for the LR and CRCB methods were obtained with the FSA package (Ogle et al., 2022), but corresponding linear models for the former were also tested to assess whether the parametric assumptions of normality and homoscedasticity were respected (Supplementary Material). Model averaging was used for the GB method by retaining only the adequate models based on the hnp package (Moral et al., 2017) with ΔAICc < 4 as such threshold corresponds to a model that is already 7.4 times less likely than the top-ranking one (Burnham et al., 2011). Detailed information about the age frequencies used as examples for the estimation of Z and A reported in Table 1 can be found in Table S1 and Figures S1–S8 of the Supplementary Material, whereas the analytical steps and related R scripts are provided in Mainguy and Moral (2021) and a companion website: https://sites.google.com/view/catch-curve-analyses/accueil. The method-specific A estimates obtained were compared with paired t-tests between the LRPP and either the CRCB or GB methods. These paired comparisons, which could be performed as the differences were sufficiently normally distributed, indicated that the LRPP method mainly yielded lower A estimates when compared to the CRCB method (t = −2.23, df = 7, p = 0.06) with a mean difference of −5.1 percentage points [−10.5, 0.3], and the same pattern was found when compared to the GB method (t = −2.30, df = 7, p = 0.06) with a mean difference of −5.7 percentage points [−10.8, 0.2]. As opposed to simulation studies (Mainguy & Moral, 2021; Smith et al., 2012) in which the "truth" is known about the age structure of the sampled population and thus Z, this was obviously not the case in the analysis of these eight catch-curve examples. As such, no bias could be determined, but our results are nonetheless directly in line with what could have been reasonably predicted. In most instances, we believe that such differences in Z estimates are introduced from different unnecessary age-frequency data manipulations that are made to accommodate the use of a linear model, namely log transformation, zeros omission, and right-truncation of the descending limb. Such practices should no longer be employed when analysing catch-curve data. An exception could however arise when historical estimates obtained with the LR method would need to be compared on the same basis to more recent estimates without having access to the original data. Under such situation, more recent methods should however also be used to help with the interpretation of the estimates obtained by contrasting them. Given the recommendation made by Smith et al. (2012) of no longer using the LR method, the important body of evidence from other simulation studies indicating that it often underestimates Z, and our reanalysis of catch-curve examples provided here that further highlights its departure from better alternatives, we advocate that the LR method should no longer be used. This is mainly because the CRCB, PM, and GB methods have been shown to be better adapted for catch-curve analyses. Although additional improvements to these methods could be proposed, we believe that efforts should be directed toward investigating to which extent the mortality estimates obtained from catch-curve analyses agree with those of a CMR approach for the same studied population. Because catch-curve analyses consist of a cross-sectional assessment of different cohorts on the one hand, whereas the individual-based CMR approach yields annual S estimates for the monitored age groups based on detection rates on the other, the biological processes and analytical concepts to describe them differ—and so are likely the resulting estimates. Nevertheless, comparing how much such estimations obtained from different approaches agree (Scheffel et al., Turner et al., 2021) would be to further for instance, whether a or bias in A estimates may be observed by using catch-curve analyses. and the made over the analytical investigations are likely still required to better assess 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 would 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 authors that they have no known or that could have to influence the work reported in this The is not for the or of any information by the authors. than should be directed to the corresponding for the

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