In pbs-assess/gfsynopsis: Data Synopsis Reports for PBS Groundfish

GROWTH AND MATURITY {#app:growth-and-maturity}

MATURITY OGIVES {#sec:maturity-models}

We fit maturity ogives as logistic regressions of maturity (mature vs.\ not mature) against length or age:

\begin{align} y_i &\sim \mathrm{Binomial}(\pi_i)\ \mathrm{logit} \left( \pi_i \right) &= \beta_0 + \beta_1 x_i + \beta_2 F_i \end{align} where $y_i$ represents a 1 if fish $i$ is considered mature and a 0 if fish $i$ is considered immature. The $\beta$ parameters represent estimated coefficients, $x_i$ represents either the length or age of fish $i$, and $F_i$ represents a binary predictor that is 1 if the fish is female and 0 if the fish is male. The variable $\pi_i$ represents the expected probability of fish $i$ being mature. We only fit these models if there are at least 20 mature males, 20 immature males, 20 mature females, and 20 immature females to ensure reasonably representative sampling and sufficient sample sizes.

LENGTH-AGE MODELS {#sec:length-age-models}

We fit von Bertalanffy length-age growth models [@vonbertalanffy1938] as:

\begin{equation} L_i \sim \operatorname{Log-normal} \left( \log(l_\mathrm{inf} (1 - \exp(-k (A_i - t_0)))) - \sigma^2 / 2, \sigma \right), \end{equation}

where $L_i$ and $A_i$ represent the length and age of fish $i$, $l_\mathrm{inf}$, $k$, and $t_0$ represent the von Bertalanffy growth parameters, and $\sigma$ represents the log standard deviation or scale parameter. The term $- \sigma^2 /2$ represents a lognormal bias adjustment term so we model the mean length rather than the median. We fit the models with Template Model Builder (TMB) [@kristensen2016] with starting values of $k = 0.2$, $l_\mathrm{inf} = 40$, $\ln (\sigma) = \ln (0.1)$, and $t_0 = -1$.

LENGTH-WEIGHT MODELS {#sec:length-weight-models}

We fit the length-weight models as robust linear regressions of log(length) on log(weight) with Student-t error and a degrees of freedom parameter fixed to 3. By using Student-t error instead of Gaussian error we down-weight the influence of outlying values [e.g. @anderson2017c] and help generate reasonable model fits across all species without handpicking outlying measurements to discard. The underlying growth model can be written as:

\begin{equation} W_i = a \cdot L_i^b \cdot e_i, \end{equation}

with $W_i$ and $L_i$ representing the weight and length for fish $i$ and $e_i$ representing error. The variables $a$ and $b$ represent the estimated length-weight parameters. We fit the model as:

\mathchardef\mhyphen="2D

\begin{equation} \log (W_i) \sim \mathrm{Student\mhyphen t} (df = 3, \log(a) + b \log(L_i), \sigma) \end{equation}

using Template Model Builder [@kristensen2016], where $a$ and $b$ have the same meaning, $df$ represents the degrees of freedom, and $\sigma$ represents the scale parameter.

pbs-assess/gfsynopsis documentation built on March 26, 2024, 7:30 p.m.