Making the most of survey data: Incorporating age uncertainty when fitting growth parameters

Abstract Individual growth is an important parameter and is linked to a number of other biological processes. It is commonly modeled using the von Bertalanffy growth function (VBGF), which is regularly fitted to age data where the ages of the animals are not known exactly but are binned into yearly age groups, such as fish survey data. Current methods of fitting the VBGF to these data treat all the binned ages as the actual ages. We present a new VBGF model that combines data from multiple surveys and allows the actual age of an animal to be inferred. By fitting to survey data for Atlantic herring (Clupea harengus) and Atlantic cod (Gadus morhua), we compare our model with two other ways of combining data from multiple surveys but where the ages are as reported in the survey data. We use the fitted parameters as inputs into a yield‐per‐recruit model to see what would happen to advice given to management. We found that each of the ways of combining the data leads to different parameter estimates for the VBGF and advice for policymakers. Our model fitted to the data better than either of the other models and also reduced the uncertainty in the parameter estimates and models used to inform management. Our model is a robust way of fitting the VBGF and can be used to combine data from multiple sources. The model is general enough to fit other growth curves for any taxon when the age of individuals is binned into groups.


A.3 Model III
For model III, θ III = k, l ∞ , t 0 , σ 2 , c 2 T . The prior for c 2 was The likelihood was and N (·|·, ·) being described in equation S1. Figure S1 gives an example of f (x) for the standard VBGF, Somers' version and a linear interpolator.  Figure S1: The values of f (x) for three different models. For Somers' version, t s = 0.5 and c = 1 and for the linear interpolator, m = 1, q 2 = 0.75 and c 2 = 0.63.

A.4 Model IV
For model IV, θ IV = k, l ∞ , t 0 , σ 2 , s 1 , . . . , s N T . The prior for the spawning times, s i (for i = 1 . . . N ), was with I 0 being the modified Bessel function of order 1. The likelihood was and N (·|·, ·) being described in equation S1.

A.5 Model V
For model V, θ V = k, l ∞ , t 0 , σ 2 , s 1 , . . . , s N , c 2 T . The priors for c 2 and s i (for i = 1 . . . N ) are shown in equations S2 and S4 respectively. The likelihood was f (x) being described in equation S3 and N (·|·, ·) being described in equation S1.

B Markov Chain Monte Carlo
The posterior distributions cannot be calculated analytically and therefore we sampled from the posterior distribution using a Markov Chain Monte Carlo (MCMC) algorithm, the No-U-turn-Hamiltonian Monte Carlo algorithm (see Hoffman and Gelman (2011) for details). Figure S2 shows the trace plot for model IV fitted to herring data. All four chains converge quickly to the same stationary distribution. We found similar results for all other models fitted to both the survey and simulated data (figures not included).

C Yield per recruit model
The Yield per recruit model described by Gabriel, Sissenwine and Overholtz (1989). The number of individuals aged a, N a , follows a Ricker stock recruitment model (Ricker, 1954) such that where r a is the catchability at age a, F is the fishing effort and M a is the natural mortality at age a. For both herring and cod, r 0 = r 1 = 0 and r i = 1 for i ≥ 2 (ICES, 2016a,b). The catch (numbers) at age a is The yield for age a fish is then the weight of the catch i.e.
where l t is the expected length at age a calculated by where k, l ∞ , t 0 and σ 2 are sampled parameters from the von Bertalanffy growth curve described in equation 2. The initial condition, N 0 = R, is the initial recruits aged 0 in the model. The total yield is then with the yield per recruit being Y /R.

D Simulation study D.1 Herring
We sampled herring from the age distribution (t and q) of herring caught in SWC-IBTS and spawning times, s i , from equation 3 with µ = 0.868 and τ = 0.007. The lengths were sampled from equation 2 with a i = t i − s i + q i . This means that the age distributions would represent one seen in reality and assuming that a fish grows according to the VBGF.
In Figures S3 and 2 we show that Model I and II are not consistent respectively (i.e. as the amount of data increases we get more certain about the wrong value of the parameters). In Figure S4 we show that Model IV appears to be consistent, as the amount of data increases the posterior distributions appear to concentrate around the true value of the parameters. We also investigated the effect of combining different data points from different quarters. Figure S5 shows Model IV fitted to increasing amounts of data Sample size Figure S4: Model IV fitted with data from surveys. As we get more data we get more certain and seem to be consistent with the truth.
from quarter 1 only and Figure S6 shows Model IV fitted to increasing amounts of data from quarter 4 only. It does not appear that the model is not consistent and after 500 data points the posterior distribution for quarter 1 and quarter 4 only appear to be similar. Prior to being fitted to 500 data points, the posterior distributions appear to be similar but with more uncertainty in quarter 1.

D.2 Cod
We sampled cod from the age distribution (t and q) of cod caught in SWC-IBTS and spawning times, s i , from equation 3 with µ = 0.312 and τ = 5.473. The lengths were sampled from equation 2 with a i = t i − s i + q i with k = 0.24, l ∞ = 1148.3, t 0 = −0.165 and σ 2 = 0.139 2 . This means that the age distributions would represent one seen in reality and assuming that a fish grows according to the VBGF. In Figures S7 and S8 we show that Model I and II are not consistent re- spectively. In Figure S9 we show that Model IV appears to be consistent, as the amount of data increases the posterior distributions appear to concentrate around the true value of the parameters. We also investigated the effect of combining different data points from different quarters. Figure S10 shows Model IV fitted to increasing amounts of data from quarter 1 only and Figure S11 shows Model IV fitted to increasing amounts of data from quarter 4 only. Unlike with herring, fitting cod to only quarter 1 or quarter 4 leads to the posterior distributions converging to different values. We believe this is because of the short spawning period. For herring we are able to see a larger range of ages due to it's larger spawning time and therefore there is more information in the survey data from one quarter than for cod. In future research we are going to further investigate how the gear used on the surveys and the time of the surveys are sensitive to fitting the VBGF. Sample size Figure S10: Model IV fitted with data from surveys in the first quarter only for cod.