The catch taken by a fishing gear depends on the amount of fish that is available to the gear, their catchability, the selectivity and fishing power of the gear, and the effort deployed. These factors are related by the selection equation:

*C _{l,g}*=

where subscript *g* refers to gear and l refers to fish size. In principle, it is possible to
observe the catch (*C*) and the effort (*E*) directly. The stock size (*N*) can be estimated
from stock assessments. The other parameters, catchability (*q*), fishing power (*P*) and
fishing selectivity (*S*) must be estimated through designed experiments. This chapter
discusses estimation methods to obtain these parameters.

The selection equation requires normalisation: the fishing power is measured relative to a standard gear with fishing power = 1, and the selectivity is restricted to the interval [0,1] and is assumed to be 1 for one or more length classes. The model is multiplicative and it is not possible to estimate all components from catch and effort data. So, it is impossible to distinguish a gear-dependent catchability effect (more fish are available to the gear) from a difference in fishing power (the gear is more effective in retaining the fish). For example, a particular longline bait may attract more fish or alternatively ensure more successful hooking. Both mechanisms will be observed as an increase in the catch.

*C _{l,g}*: Catch

The catch by length class taken by a fishing gear is an observed variable when conducting an experiment.

*q _{l,g}*: Catchability

The catchability measures the proportion of the population that is available to the
gear. The selection equation shows that the catchability is obtained directly for the
length group with maximum selectivity (=1) when fishing with one unit of effort. The
catchability depends, among other things, on the reactions of the fish to the gear. For
gears of the same type (e.g. two different gillnets) it is commonly assumed that *q* only
depends on length, i.e. *q _{l,g}≡*

*N _{l}*: Number of fish available

*q _{l,g} N_{l}* is the number of fish encountering the gear. As noted by Hamley (1975),

*P _{g}*: Fishing power of the gear

The fishing power is a measure of the efficiency of that particular gear in retaining
fish. Many methods developed for estimating selectivity include the assumption that
the gears being compared have identical fishing power, i.e. *P _{g}* =1. The model ignores
gear saturation; this would make the fishing power a function of the amount of fish
available to the gear.

*S _{l,g}*: Selectivity of the gear

Most gears are only effective at catching certain fish and shellfish species and even then usually only over a limited size range. Traditionally this size dependence has attracted much scientific attention while other parameters in the selection equation have largely been ignored.

*E _{g}*: The effort deployed

Effort is measured by the duration of fishing and the number of gears (number of nets or hooks). Experiments with several gears of the same type deployed concurrently often use the same effort for each gear (i.e. same numbers of hooks or number of nets of similar size used for the same duration).

Methods for estimating selectivity are usually classified into two main groups based on whether information on the length distribution of the fish available to the gear exists or not.

**Direct methods**: Methods that require an external estimate of the fish abundance per size class, i.e. that*N*is known._{l}**Indirect Methods:**Methods that do not require such information, i.e. that*N*is unknown._{l}

In most cases, *N _{l}* is unknown and hence indirect methods are those used most often.
An indirect method estimates size selection and the length distribution of the fish
encountering the gear concurrently based on the selection equation. This requires
additional assumptions on the form of the selection curve.

The catch size distribution is sometimes assumed to directly present the selection curve, ignoring the stock size distribution. Hamley (1975) briefly discussed this case and noted that the result is a rough proxy as gillnets are usually very size selective. Baranov (1948) suggested as a simple rule of thumb that fish smaller or larger than 20% of the optimal selection size would only occur rarely in the catches.

**Figure 5.1 Using observed length frequencies of the catches as a proxy for
selection. The upper panel shows a stock length distribution and the middle
panel a selection curve. The catch size distribution, which is the product of
stock and selection, is shown in the lower panel. When the selection range is
narrower than the stock size distribution the size distribution of the catches
may be used as a proxy for selection. Redrawn from Baranov (1948).**

The selection equation stipulates that when the length frequency distribution of the
stock (*q _{l}N_{l}*) is relatively uniform then the length frequency distribution of the catch will
provide a proxy of the selection (Fig. 5.1).

Using the catch length distribution directly to represent the selection curve may be
useful when the gear is only effective over a very narrow size range, but the
approach has only limited use. The assumption is sometimes useful for exploratory
purposes but final selection estimates should be obtained by one of the methods that
explicitly account for the length composition of the population, *N _{l}*.

When the catches are obtained from fisheries covering different areas and periods,
and are of some magnitude, the observed catch distributions may provide a fairly
accurate estimate of the selection curve (see for instance Hovgård 1996a). Length
frequency information from gillnet catches may serve as guidance when planning
gillnet experiments. In recent years more elaborate methods based on interpretations
of the length frequencies have been suggested (e.g. Henderson and Wong 1991,
Helser *et al.* 1991).

Direct estimation has been used infrequently for pure selectivity studies as the
population size for each length group is rarely known. When the population length
distribution (*N _{l}*) is known, selection can be directly estimated from the selection
equation as

*C _{l.g}/ (N_{l}E_{g,})= q_{l.g}S_{l.g}P_{g}*

Only the product of the catchability, the fishing power and selection can be
estimated. Fujimori *et al.* (1996) notes in direct experiments that more shrimps were
taken in the larger mesh-sizes and interpreted this as being due to an increased
fishing power (a *P _{g}* effect) of the larger mesh-sizes. However, the observation may
equally well be due to an increase in swimming distances with increasing shrimp size
changing the catchability (a

The population size structure may be known if fishing takes place on a stocked
population in an experimental enclosure (e.g. Fujimori *et al.* 1996) or by fishing on a
tagged population in a natural environment (e.g. Hamley and Regier 1973). Both
procedures require large resources, and for tagging experiments, extensive fisheries
are needed to ensure a sufficient number of recaptures (Ricker 1975).

The population size structure has been inferred using various stock assessment
techniques such as mark-recapture experiments (e.g. Hamley and Regier 1973,
Borgstrøm 1989, 1992), acoustic surveys (e.g. Rudstram *et al.* 1987) or from fishing
using non-mesh selective gears (e.g. Borgstrøm 1989, Winters and Wheeler 1990).
However, all such assessment methods possess various weaknesses or limitations.
For example, no gear may be assumed to be ‘non-selective’ towards the stock even
when no-mesh selection is assumed (e.g. for fish above a certain size the mesh
selection is 100% in trawls and purse seines). Comparing the catches of such sizes
from a gillnet to the catches from a purse seine leads to:

where *k* is a constant absorbing the effects of effort and fishing power. The
expression shows that the catch ratio between gears only reflects selection if *q _{l.GN}* =

The uncertainty in the stock estimates influences the selection estimates. The relative uncertainty of the selectivity estimates may be written as

assuming that catch per unit effort (*C/E*) and stock (*N*) are estimated independently.
This implies that when the population size distribution is imprecisely known this will
directly be carried over into the selection estimates. The uncertainty in stock size
estimates is usually difficult to assess and is rarely reported in direct selection
studies. Borgstrøm (1989) lists confidence intervals for mark-recapture based stock
estimates.

Information on the size distribution of the population being fished is not available in
most gillnet selectivity experiments and the selectivity must be estimated indirectly.
Therefore, the size frequency distribution of the population and the selectivity
parameters are estimated simultaneously. This is possible if a) fishes of a given size
are equally available to nets of different mesh size fishing concurrently and, b) the
selectivity depends on the fish size and mesh size only. The first assumption
stipulates the catchabilities are equal between the gears compared (*q _{l.G1}=q_{l.G2}*), the
second may, for instance, be met using Baranov's principle of geometric similarities.

Three different approaches have been used to obtain the selection curve using various manipulations of the selection equation.

- Methods that use the ratio in catches for a given fish size taken in different gears reflect the difference in selection. For instance, when comparing the catches of two gears and assuming equal fishing power and effort, then . This approach is used by the various catch-relative-to-best-catch methods (e.g. Gulland and Harding 1961, Jensen 1973) and by Holt's (1963) method.
- Methods that use the ratio in catches between gears where selection is equal
reflect the difference in available fish (again assuming equal fishing power and
effort), i.e. when
*S*then . This approach is used in the graphical methods of McCombie and Fry (1960) and Kitahara (1971)._{l.1}=S_{L.2} - Methods where
*q*and_{l}N_{l}*S*are estimated simultaneously. This approach is used in the statistics-based methods proposed in recent years (e.g. Kirkwood and Walker 1986, Hovgård 1996a, Millar and Holst 1997, Hovgård_{l.g}*et al.*1999).

These methods are based on the catch ratio of two gears being fished with the same effort:

under the assumption that the fishing power of the two gears is equal (e.g. two
different mesh sizes of otherwise identical gillnets). There are two unknowns, *S _{l.g1}* and

The simple procedure suggested by Jensen (1973) has been used as an example of this class of methods. The calculation scheme is illustrated in Example 5.1 where Baranov's principle of geometric similarity (see Section 3) has been used to combine the information from the different mesh sizes.

**Example 5.1 Approximated selection estimates from comparing catch to best
catch.**

Table 5.1 shows the gillnet catches of sockeye salmon reported by Peterson (1954). The catch amounts to 6 333 salmon caught in eight different mesh sizes between 13.5 and 19 cm.

**Table 5.1 Catches of sockeye salmon taken in a gillnet experiment reported by
mesh size and fish size by Peterson, 1954.**

Length | Mesh size | |||||||

13.5 | 14.0 | 14.8 | 15.4 | 15.9 | 16.6 | 17.8 | 19.0 | |

52.5 | 52 | 11 | 1 | 1 | 0 | 0 | 0 | 0 |

54.5 | 102 | 91 | 16 | 4 | 4 | 2 | 0 | 3 |

56.5 | 295 | 232 | 131 | 61 | 17 | 13 | 3 | 1 |

58.5 | 309 | 318 | 362 | 243 | 95 | 26 | 4 | 3 |

60.5 | 118 | 173 | 326 | 342 | 199 | 100 | 10 | 11 |

62.5 | 79 | 87 | 191 | 239 | 202 | 201 | 39 | 15 |

64.5 | 27 | 48 | 111 | 143 | 133 | 185 | 72 | 25 |

66.5 | 14 | 17 | 44 | 51 | 52 | 122 | 74 | 41 |

68.5 | 8 | 6 | 14 | 23 | 25 | 59 | 65 | 76 |

70.5 | 7 | 3 | 8 | 14 | 15 | 16 | 34 | 33 |

72.5 | 0 | 3 | 1 | 2 | 5 | 4 | 6 | 15 |

The best catches for each size group are compiled in Table 5.2, first column. These
are used in the calculation of the selection proxy which is derived as *S _{l.g} = C_{l.g} / C_{l.best}*
(Table 5.2.). For fish below 56.5cm the smallest mesh size always provides the best
catches, as do the largest mesh for sizes above 68.5cm. For these extreme size
groups the best catch can hardly be expected to be 100% selected and therefore it is
reasonable to restrict the data to be used to the interval 56.5 to 68.5cm.

**Table 5.2 Jensen's selection proxies (catch divided by best catch) using the
data provided in Table 5.1. Only the selection proxies (grey shaded area) are
considered useful.**

Length Midpoint Cm | Mesh size | ||||||||

Best catch, see Table 6.1. | 13.5 | 14.0 | 14.8 | 15.4 | 15.9 | 16.6 | 17.8 | 19.0 | |

52.5 | 52 | 1.00 | 0.21 | 0.02 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 |

54.5 | 102 | 1.00 | 0.89 | 0.16 | 0.04 | 0.04 | 0.02 | 0.00 | 0.03 |

56.5 | 295 | 1.00 | 0.79 | 0.44 | 0.21 | 0.06 | 0.04 | 0.01 | 0.00 |

58.5 | 362 | 0.85 | 0.88 | 1.00 | 0.67 | 0.26 | 0.07 | 0.01 | 0.01 |

60.5 | 342 | 0.35 | 0.51 | 0.95 | 1.00 | 0.58 | 0.29 | 0.03 | 0.03 |

62.5 | 239 | 0.33 | 0.36 | 0.80 | 1.00 | 0.85 | 0.84 | 0.16 | 0.06 |

64.5 | 185 | 0.15 | 0.26 | 0.60 | 0.77 | 0.72 | 1.00 | 0.39 | 0.14 |

66.5 | 122 | 0.11 | 0.14 | 0.36 | 0.42 | 0.43 | 1.00 | 0.61 | 0.34 |

68.5 | 76 | 0.11 | 0.08 | 0.18 | 0.30 | 0.33 | 0.78 | 0.86 | 1.00 |

70.5 | 34 | 0.21 | 0.09 | 0.24 | 0.41 | 0.44 | 0.47 | 1.00 | 0.97 |

72.5 | 15 | 0.00 | 0.20 | 0.07 | 0.13 | 0.33 | 0.27 | 0.40 | 1.00 |

**Figure 5.2 A selection curve for the size range 56.5–68.5cm describing
Peterson's (1954) sockeye salmon data using the catch / best catch approach
suggested by Jensen (1973). The selection is plotted versus length/mesh-size.
Curve fitted by eye.**

As noted the use of the catch / best catch approach includes uncertainty on whether the best catch corresponds to 100% selection. Normally the fitted selection curve actually negates this assumption and Gulland and Harding (1961) therefore suggested an iterative procedure. Their procedure includes the assumption that the observed catches can be corrected by the estimated selection, i.e.

where *S _{l.g}* is the selection read from the selection curve. The analysis is then repeated
with the new corrected set of catches (

Holt's method (Holt 1963) is one of the most commonly used methods for estimating
gillnet selectivity. The method has also been used to estimate longline selection (e.g.
Cortes-Zaragoza *et al*. 1989). The method is based upon standard linear regression
and can be carried out using a pocket calculator. However, it is restrictive as it
assumes the normal curve as the selection model. This selection model does not
conform to the principle of geometric similarity.

The method compares the catch in the same length group taken by two gillnets that have nearly the same mesh size. The selection model is the normal curve, i.e.

where *k* is the selection factor and σ^{2} a measure of the width of the selection curve.
Holt (1963) proposed to use linear regression on the logarithmic ratio between
catches from the same fish length group.

If the same population is fished with two gillnets with the same effort (e.g. two nets of identical power but having different mesh sizes) the relative selection is:

and

with regression constants (α, β) derived from the normal distribution:

The selection can therefore be estimated by linear regression giving α and β. Solving
for the selection parameters *k* and σ^{2} gives

and

Re-introduction of the estimated selectivity in the selection model allows estimation of the relative population size:

**Example 5.2 Estimation of selection using Holt's method**

The example is based on data in Peterson (1954), see Table 5.1. The logarithmic catch ratios are given in Table 5.3.

**Table 5.3 Logarithmic catch ratios, In ( C _{l,m1}/C_{l,m2}) for neighbouring combinations
of mesh sizes. The data have been curtailed, as only ratios where at least 5 fish in
both meshes are caught have been included in the calculations**.

**Range of meshes being compared**

Mid length of fish (cm) | 13.5 | 14.0 | 14.8 | 15.4 | 15.9 | 16.8 | 17.8 |

14.0 | 14.8 | 15.4 | 15.9 | 16.8 | 17.8 | 19.0 | |

52.5 | 1.5533 | ||||||

54.5 | 0.1141 | 1.7383 | |||||

56.5 | 0.2402 | 0.5715 | 0.7643 | 1.2777 | 0.2683 | ||

58.5 | -0.0287 | -0.1296 | 0.3986 | 0.9392 | 1.2958 | ||

60.5 | -0.3826 | -0.6336 | -0.0479 | 0.5415 | 0.6881 | 2.3026 | -0.0953 |

62.5 | -0.0965 | -0.7864 | -0.2242 | 0.1682 | 0.0050 | 1.6397 | 0.9555 |

64.5 | -0.5754 | -0.8383 | -0.2533 | 0.0725 | -0.3300 | 0.9437 | 1.0578 |

66.5 | -0.1942 | -0.9510 | -0.1476 | -0.0194 | -0.8528 | 0.5000 | 0.5905 |

68.5 | 0.2877 | -0.8473 | -0.4964 | -0.0834 | -0.8587 | -0.0969 | -0.1564 |

70.5 | -0.5596 | -0.0690 | -0.0645 | -0.7538 | 0.0299 | ||

72.5 | -0.9163 |

Regressing the data in Table 5.3 linearly with the fish length as the independent variable gives:

Smallest mesh | 13.5 | 14 | 14.8 | 15.4 | 15.9 | 16.8 | 17.8 |

Largest mesh | 14 | 14.8 | 15.4 | 15.9 | 16.8 | 17.8 | 19 |

Intercept α | 3.977 | 10.012 | 5.248 | 6.521 | 6.845 | 20.345 | 6.7744 |

Slope β | -0.06405 | -0.16661 | -0.08376 | -0.09713 | -0.1075 | -0.29908 | -0.09872 |

Which leads to the estimates of the selection parameters of the different mesh-sizes

Mesh difference | 0.5 | 0.8 | 0.6 | 0.5 | 0.9 | 1 | 1.2 | Overall mean |

Mean Mesh | 13.75 | 14.4 | 15.1 | 15.65 | 16.35 | 17.3 | 18.4 | |

K | 4.52 | 4.17 | 4.15 | 4.29 | 3.89 | 3.93 | 3.73 | 4.10 |

σ^{2} | 35.25 | 20.04 | 29.72 | 22.08 | 32.61 | 13.15 | 45.33 | 28.31 |

σ | 5.9 | 4.5 | 5.5 | 4.7 | 5.7 | 3.6 | 6.7 | 5.3 |

The model formulation implies that the mode in the selection curve is proportional to
mesh-size (i.e. the mode equals *K* multiplied by mesh-size), but that the spread of the
selection curve (σ) is constant for all mesh-sizes. Using the overall mean values
allows the selection curves to be drawn (Fig. 5.3).

**Figure 5.3 Selection curves from Peterson's data using the Holt (1963) method**.

The methods described in the earlier sections compared catches in the same length group taken by different gears. In the class of methods discussed in this section we compare catches in different length classes that are exposed to the same size selectivity. This class of methods is typically used for gillnets as the methods are based on Baranov's principle of geometric similarity (selection is described as a function of length/mesh-size) and assume that the fishing power is the same for all mesh-sizes.

The selection equation is now used to compare catches of two gillnets with different
mesh sizes *m _{1}* and

According to Baranov's principle of geometric similarity, equal selection is found for
constant values of fish-size/mesh-size (i.e. *l _{1}/m_{1}*=

For each particular length class, *q _{l}N_{l}* is the same across all mesh sizes used. This
implies that

Estimation of the selection curve is done as follows:

Plot

*C*against (_{l,m}/E_{m}*l/m*) by length class.Choose an appropriate model for describing the scaled points (e.g. choose a normal distribution if the points appear as a bell-shaped distribution).

Scale the points for the individual length classes to a common magnitude. This implies ‘guessing an appropriate scaling factor’ for each length class. These scaling factors are implicit measures of 1/

*q*._{l}N_{l}

When effort is identical between mesh sizes, as when fishing with groups of gillnets with different mesh sizes, the catches can be used directly instead of catch per effort for the analyses. This was done in the graphical estimations presented by McCombie and Fry (1969) and Kitahara (1971) whose methods are essential similar. Example 5.3 illustrates the graphical approach.

**Example 5.3 Estimation of the selection curve by the McCombie and Fry (1960)
method**

Figure 5.4 shows Peterson's gillnet salmon data (Table 5.1) where the catches per size class are plotted on the abscissa: length class/mesh-size. The curves for the individual length classes are of the same shape except for a scaling factor. Table 5.4 shows scaling values that bring the various length-classes to common amplitudes.

Figure 5.5 shows the catches scaled. The scaling factors are equivalent to 1/ *q _{l}N_{l}*, i.e.
the reciprocal of the numbers of fish encountering the gear.

**Table 5.4 Factors used for scaling the individual size classes shown on
Figure 5.5 to common amplitude.**

Length | 52.5 | 54.5 | 56.5 | 58.5 | 60.5 | 62.5 | 64.5 | 66.5 | 68.5 | 70.5 | 72.5 |

Guess | 52 | 101 | 260 | 381 | 320 | 238 | 169 | 95 | 63 | 33 | 5 |

Graphical methods are simple means to show the catch information split into
selection and stock size. However, this split is conditioned by the use of some
predetermined selection curve. The graphical approaches have been relatively little
used in recent years (e.g. Spangler and Collins 1992, Fujimori *et al.* 1996).

**Figure 5.4 Peterson's (1954) sockeye salmon data plotted per size class vs.
transformed length. For three size classes the points have been joined to
indicate that the different size classes follow similarly shaped curves differing
only in amplitude.**

**Figure 5.5 Gillnet selection curve derived by McCombie and Fry's method, by
scaling the catch data shown in Figure 5.4 to equal height.**

Several rigorous statistical methods to estimate the selection curve from indirect selection experiments have been proposed starting with Kirkwood and Walker (1987). These methods use established statistical methods for optimising a fit between a specified model and the observed catches.

The selection equation is typically simplified as most methods ignore the effort and fishing power terms (both assumed to be the same for all mesh sizes). The selection equation is, on the other hand, extended to include a specific model for the noise in the observation. The selection equation now reads:

*C _{l,m}*=

The noise term includes several mechanisms such as patchy distribution of fish, variability of fish behaviour to the gear, gear performance variability and sampling variance. Note that representing noise as an additive random term is very natural for some probability distributions, such as the normal distribution, but not for others, such as the Poisson distribution. Nevertheless, here we use it to represent all random factors for consistency.

Normally the estimation problem is simplified by assuming that the observation in each length class is independent of the observation in another length class. Also for simplification the trivial summation over several experiments is usually ignored and the problem only formulated for a single experiment using several “gears” (e.g. panels with different mesh sizes, or section of longline using different hooks).

Estimation may be done by either maximising the log-likelihood function (Maximum
Likelihood [ML] methods, e.g. Kirkwood and Walker 1987, Millar and Holst 1997) or
by minimising least squares (Least Square methods [LSQ] e.g. Hovgård 1996a and
Hovgård *et al.* 1999). ML methods require an explicit assumption of the error
structure in the observations. Millar and Holst (1997) suggest from experience that
although the error structure is over-dispersed, Poisson distributed errors generally
provide an adequate description.

The LSQ methods rely on weaker assumptions, i.e. that the statistical noise is symmetrically distributed around the expected values with a common variance. This usually can not be assumed and therefore it is common practice to transform the data. Elliott (1983) suggests power transformation as a flexible tool useful for normalising the most commonly occurring error structures. The LSQ and the ML estimates are identical when the errors are normally distributed. That means that the LSQ estimates of logarithmic transformed observations corresponds to ML estimates assuming log-normal distributed errors. Erzini and Castro (1998) found little differences between estimates derived by ML or LSQ methods applied to the same data.

These methods are based on the standard statistical estimation technique of maximum likelihood (e.g. see Lehmann 1983) that was introduced by R.A. Fisher in the 1920s. He proposed that the best guess on the unknown parameters is to assume that the set of observations is the most probable.

Kirkwood and Walker (1987) applied the method assuming Poisson distributed errors. Millar (1992) investigated the methods based on the general family of exponential error distribution applied to trawl selection models. Millar and Holst (1997) investigated the maximum likelihood methods for gillnet selection models.

Assuming that the experiments are independent, the logarithm of the observation probability is the sum of the log-probabilities of the observed catches given a particular set of parameters. The maximum likelihood principle requires that this expression is maximised by varying the parameter values.

*5.4.1.1 Poisson Distributed Observation Variance*

The procedure using Poisson distributed errors is presented below following Kirkwood and Walker (1987). Assuming Poisson errors (i.e. that the variance is proportional to the mean) is in many ways the most simple case. The stochasticity in the observations is in this formulation:

The log-likelihood function is

and the estimation equations become for a parameter *θ* in the selection model

The estimation equations are equal to zero at the maximum, which can be used to find the maximum likelihood parameters.

The parameters are log(*qN _{l}*) and the selection parameters

The estimation equations for the selection parameters can only be specified when
the selection model has been decided upon. There are numerous selection
formulations available, see Table 3.1. The method is then to calculate log*L* as a
function of the selection parameters and maximise that sum.

**Example 5.4 Maximum likelihood estimation assuming poisson distributed
errors.**

The example uses data from Peterson (1954) presented in Table 5.1. Selection is assumed to follow the log-normal distribution.

where *k*, *Ψ* are the mode and the spread of the log-normal selection curve,
respectively.

The estimates can be obtained in EXCEL using the SOLVER add-in, which requires initial input parameters (guesses) for the two parameters in the log-normal distribution to start the fitting process. Setting the calculations up in an EXCEL worksheet requires 3 matrices with the dimensions:

columns = numbers of mesh-sizes,

rows = numbers of size classes.

**Matrix 1:** contains the catches by mesh size and length class.

**Matrix 2:** contains the calculated selection using the two input parameters. Add
three extra columns

A: summed catch,

B: summed selection and

C: the ratio A/B which express the number of fish in each length class
encountering the nets (*q _{l}N_{l}*)

**Matrix 3:** contains the log probabilities = *C**log(*q***N***S*)-*q***N***S*-log*C*!.

The log *C*! is calculated using the log-gamma function, in EXCEL called GammaLn.
Log C! = GammaLn(*C*+1). However, as this function is not dependent on the
parameters, it is not necessary to calculate the function to find the maximum of the
function.

The sum of values in this matrix is stored in a cell that is used as the target cell in
SOLVER. This facility in EXCEL is run to maximise the value stored in this cell by
changing the two parameter values of the log-normal selection model: *log k* and *Ψ ^{2}*.

The selection parameters are the obtained as

**Parameters**

*log k* 1.401307

*Ψ ^{2}* 0.00659

**Matrix 4:** contains the residuals (*C _{l,m} q_{l}N_{l}*S_{l,m}*). This matrix is not used in the
estimation but is required for inspection of the goodness of fit.

The format of the worksheet is shown in Fig. 5.6

**Figure 5.5 A example worksheet for estimating selectivity of gillnets or
longlines using the maximum likelihood procedure suggested by Kirkwood and
Walker (1987). Data input is required in the cells marked grey.**

*Choose selection qurve formulation, e.g. log-normal
which has two parameters, In k and sigma, i.e.*

Input parameters | |||

Guess | |||

S(l,m)= | exp (-0.5*(In(l/m)-Ink)^2/ sigma^2), | Ln k | |

sigma |

**Matrix 1** Observed catches (C)by mesh-size(m1-mk) and length(L1-Lj)

Mesh-size | m1 | m2 | mk | ||

Length | |||||

L1 | |||||

L2 | |||||

Lj |

**Matrix 2** Calculated selections (S) using selection curve formulation with the
parameters given as input. qN, the population encountering the gear,
derived by values in column A divided by values in column B.

A | B | A/B | ||||||

Mesh-size | m1 | m2 | mk | Sum Catch | Sum Selection | qN | ||

Length | ||||||||

L1 | ||||||||

L2 | ||||||||

Lj |

**Matrix 3** Log Probabilities=C*ln(qN)*S-qN*S-In(C!). The C values taken from
cells in Matrix 1, S and qN values taken from Matrix 2. Ln (C!)
derived by the EXCEL function GAMMALN(C+1)

Optimization cell | ||||||||

Mesh-size | m1 | m2 | mk | |||||

Length | (sum of all cells in matrix 3) | |||||||

L1 | ||||||||

L2 | ||||||||

Lj |

**Matrix 4** Residuals, r = C- qN*S, C taken from matrix1, S and qN

from matrix 2

Mesh-size | m1 | m2 | mk | ||

Length | |||||

L1 | |||||

L2 | |||||

Lj |

*5.4.1.2 Log-Normal Distributed Observation Variance*

Most authors have estimated gillnet selection curves by assuming Poisson
distributed errors, either in ML estimations (e.g. Kirkwood and Walker 1987, Millar
and Holst 1997) or implicitly by choosing a data transformation matching this
assumption (e.g. Hovgård *et al.* 1999). Poisson distributed errors correspond to fish
being caught at random. However, catches should a *priori* be expected to be
overdispersed as fish are usually patchily distributed. The log-normal error structure
where the variance is estimated independently of the mean allows for overdispersed
observations.

The stochastic noise in the observations is now:

where *Δl* is the width of the length class. The logarithmic likelihood function now
becomes

Again, as was the case for the Poisson distributed errors, it is possible to estimate the population as a function of the selection model. Since the error structure has now changed the estimator also changes (see Appendix 1 for derivation). The catch rate term can be derived directly, based on a defined selectivity model:

Again, a shape of selection curve must be chosen — the log-normal selection mode is again chosen as the example. Note that there are two “log-normal” models involved here. The first, discussed above, is the account of the noise in the observations and reflects the stochastic element of the observations. The second “log-normal” refers to the shape of the selection curve.

where *k*, *ψ*^{2} are the mode and the width of the log-normal selection curve.

**Example 5.5 Maximum likelihood estimation assuming log-normal distributed
errors. Selection is assumed to follow the log-normal distribution.**

An EXCEL worksheet may be constructed following principles similar to those used in Example 5.4, however Matrix 3 in that example is now calculated in EXCEL as:

Ln[[LOGNORMDIST(Catch+0.5,log(qNS), ψ^{2}) -

LOGNORMDIST(Catch-0.5, log(qNS), Ψ^{2})]/Catch]

This is the log-likelihood for log-normal selection with log-normal errors. The results of the estimations are:

Log k 1.4145

*ψ ^{2}* 0.01128

Compared to Example 5.4 where Poisson distributed errors were assumed, the log *k*
values are quite similar, but the width of the selection curve (*ψ ^{2}*) differs more. Figure 5.7A
shows the estimated population sizes and Figure 5.7B shows the selection curves
obtained by the two assumptions - i.e. Poisson observation noise and log-normal
observation noise. The observation variance estimated from the log-normal error
assumption (CV approximately 200 %) is substantially higher than what is assumed in the
Poisson model (CV < 20 % for those combinations of length and mesh with significant
catches).

The difference between the two selection curves may be interpreted as an uncertainty due to the true error structure being unknown.

**Figure 5.7 Estimated population length distribution and selection curve using
Maximum Likelihood based on Poisson and Log-normal error distributions.
Considering that population length distribution estimates from indirect
methods are only relative, the shape of the two length distributions, rather than
the actual numbers, should be compared.**

The regression framework presented here is a generalisation of the approach used
by Hovgård (1996a) and Hovgård *et al.* (1999). The framework allows the researcher
to account for the error structure by using a power transformation of the catch data
and least-squares estimation. The regression minimises the difference between the
observed and the predicted catches on the transformed scale, (i.e. min Σ(*C ^{β}*,

The method estimates the population per size class as:

This is the least-squares estimate for the *qN _{l}*'s. The remaining parameters may need
to be estimated by minimising the least squares sum:

The effort term (*E _{m}*) may be ignored if the gears compared are of equal size and
operated for identical duration. Similarly, if the fishing power is assumed to be equal,
this term may be ignored.

*5.4.2.1 Formulation in an Excel Spreadsheet*

The spreadsheet (Figure 5.7) contains four matrixes and supplementary cells for parameters. Data inputs are required in the cells marked grey. These include:

The power used in the transformation; setting the power at 0.5 is approximately the same as assuming Poisson distributed errors.

The initial parameters for the particular model chosen. The parameter values must be initially guessed.

**Matrix 1:** The catch per length and mesh-size.

**Matrix 2:** The fishing power of the various nets in the first row if such estimates are
available, otherwise all power values are set at 1.0. The subsequent rows contain the
calculated selections derived using the guessed selection input parameters. The last
column in Matrix 2 contains the estimated *qN*'s taken from the *qN _{l}* equation above.

**Matrix 3:** The estimated catches, which are calculated as the Fishing-power
* Selection * estimated-*qN*.

**Matrix 4:** The transformed residual, i.e. *C ^{β}_{obs} - Ĉ ^{β}*. The residual are squared and
summed in the SSQ cell.

Estimations are carried out using Excel's Solver facility. The target cell is the SSQ
cell, which is to be minimised, by changing the values in the parameter cells. Solver
iteratively adjusts the parameter values until a minimum is found. The output then
contains the estimated parameters, the estimated selections per mesh-size, the
estimated *qN*'s and the transformed residuals.

Example 5.6 shows the analysis of Peterson's Fraser River salmon data (Table 5.1) using four different selection formulations: following Baranov's principle of geometric similarity they are evaluated as in the model of Holt, where all selection curves follow a normal distribution with identical spreads.

**Figure 5.8 Example worksheet for estimating selectivity of gillnets or longlines
using the regression framework described in section 5.4.2. Input parameters
are in gray cells.**

Choose selection qurve formulation, e.g. log-normal which has two parameters, In k and sigma, i.e.S(l,m)= exp (-0.5*(In(l/m)-Ink)^2/ sigma^2) | Input parameters | |

Guess | ||

Ln k | ||

sigma | ||

Choose the error structure (stabilise variance) assumed by setting the power of Beta,e.g. Beta=0.5 for Poisson distributed errors | ||

Beta |

**Matrix 1** Observed catches (C)by mesh-size(m1-mk) and length(L1-Lj)

Mesh-size | m1 | m2 | mk | ||

Length | |||||

L1 | |||||

L2 | |||||

Lj |

**Matrix 2** Calculated selection (S) using selection curve formulation with the

parameters given as input. qN, the population encountering the gear,

derived by expression given in text if available, fishing power (FP) may be

supplied (see section 5.5).

Mesh-size | m1 | m2 | mk | qN | ||

FP | p1 | p2 | pk | |||

Length | ||||||

L1 | ||||||

L2 | ||||||

Lj |

**Matrix 3** Predicted catch=qN*S*FP. Values are taken from

Matrix 2

Mesh-size | m1 | m2 | mk | ||

Length | |||||

L1 | |||||

L2 | |||||

Lj |

**Matrix 4** Transformed residuals= catch^beta-predicted catch^beta. Catch

taken from Matrix 1, predicted catch from Matrix 3.

Mesh-size | m1 | m2 | mk | |||||

Length | Optimization cell | |||||||

L1 | ||||||||

L2 | ||||||||

(sum of squares of all cells in Matrix 4) | ||||||||

Lj |