A simple delay-difference assessment model using a
time-series of catches and a relative abundance index and coded in TMB. The model
can be conditioned on either (1) effort and estimates predicted catch or (2) catch and estimates a predicted index.
In the state-space version `DD_SS`

, recruitment deviations from the stock-recruit relationship are estimated.

```
DD_TMB(
x = 1,
Data,
condition = c("catch", "effort"),
AddInd = "B",
SR = c("BH", "Ricker"),
rescale = "mean1",
MW = FALSE,
start = NULL,
prior = list(),
fix_h = TRUE,
dep = 1,
LWT = list(),
n_itF = 3L,
silent = TRUE,
opt_hess = FALSE,
n_restart = ifelse(opt_hess, 0, 1),
control = list(iter.max = 5000, eval.max = 10000),
...
)
DD_SS(
x = 1,
Data,
condition = c("catch", "effort"),
AddInd = "B",
SR = c("BH", "Ricker"),
rescale = "mean1",
MW = FALSE,
start = NULL,
prior = list(),
fix_h = TRUE,
fix_sd = FALSE,
fix_tau = TRUE,
dep = 1,
LWT = list(),
n_itF = 3L,
integrate = FALSE,
silent = TRUE,
opt_hess = FALSE,
n_restart = ifelse(opt_hess, 0, 1),
control = list(iter.max = 5000, eval.max = 10000),
inner.control = list(),
...
)
```

- x
An index for the objects in

`Data`

when running in closed loop simulation. Otherwise, equals to 1 when running an assessment.- Data
An object of class Data.

- condition
A string to indicate whether to condition the model on catch or effort (ratio of catch and index).

- AddInd
A vector of integers or character strings indicating the indices to be used in the model. Integers assign the index to the corresponding index in Data@AddInd, "B" (or 0) represents total biomass in Data@Ind, "VB" represents vulnerable biomass in Data@VInd, and "SSB" represents spawning stock biomass in Data@SpInd.

- SR
Stock-recruit function (either

`"BH"`

for Beverton-Holt or`"Ricker"`

).- rescale
A multiplicative factor that rescales the catch in the assessment model, which can improve convergence. By default,

`"mean1"`

scales the catch so that time series mean is 1, otherwise a numeric. Output is re-converted back to original units.- MW
Logical, whether to fit to mean weight. In closed-loop simulation, mean weight will be grabbed from

`Data@Misc[[x]]$MW`

, otherwise calculated from`Data@CAL`

.- start
Optional list of starting values. Entries can be expressions that are evaluated in the function. See details.

- prior
A named list for the parameters of any priors to be added to the model. See below.

- fix_h
Logical, whether to fix steepness to value in

`Data@steep`

in the assessment model. Automatically false if a prior is used.- dep
The initial depletion in the first year of the model. A tight prior is placed on the model objective function to estimate the equilibrium fishing mortality rate that corresponds to the initial depletion. Due to this tight prior, this F should not be considered to be an independent model parameter. Set to zero to eliminate this prior.

- LWT
A named list of likelihood weights. For

`LWT$Index`

, a vector of likelihood weights for each survey, while for`LWT$MW`

a numeric.- n_itF
Integer, the number of iterations to solve F within an annual time step when conditioning on catch.

- silent
Logical, passed to

`MakeADFun`

, whether TMB will print trace information during optimization. Used for diagnostics for model convergence.- opt_hess
Logical, whether the hessian function will be passed to

`nlminb`

during optimization (this generally reduces the number of iterations to convergence, but is memory and time intensive and does not guarantee an increase in convergence rate). Ignored if`integrate = TRUE`

.- n_restart
The number of restarts (calls to

`nlminb`

) in the optimization procedure, so long as the model hasn't converged. The optimization continues from the parameters from the previous (re)start.- control
A named list of parameters regarding optimization to be passed to

`nlminb`

.- ...
Additional arguments (not currently used).

- fix_sd
Logical, whether the standard deviation of the data in the likelihood (index for conditioning on catch or catch for conditioning on effort). If

`TRUE`

, the SD is fixed to value provided in`start`

(if provided), otherwise, value based on either`Data@CV_Cat`

or`Data@CV_Ind`

.- fix_tau
Logical, the standard deviation of the recruitment deviations is fixed. If

`TRUE`

, tau is fixed to value provided in`start`

(if provided), otherwise, equal to 1.- integrate
Logical, whether the likelihood of the model integrates over the likelihood of the recruitment deviations (thus, treating it as a random effects/state-space variable). Otherwise, recruitment deviations are penalized parameters.

- inner.control
A named list of arguments for optimization of the random effects, which is passed on to

`newton`

via`MakeADFun`

.

An object of `Assessment`

containing objects and output from TMB.

For `start`

(optional), a named list of starting values of estimates can be provided for:

`R0`

Unfished recruitment. Otherwise, Data@OM$R0[x] is used in closed-loop, and 400% of mean catch otherwise.`h`

Steepness. Otherwise, Data@steep[x] is used, or 0.9 if empty.`M`

Natural mortality. Otherwise, Data@Mort[x] is used.`k`

Age of knife-edge maturity. By default, the age of 50% maturity calculated from the slots in the Data object.`Rho`

Delay-difference rho parameter. Otherwise, calculated from biological parameters in the Data object.`Alpha`

Delay-difference alpha parameter. Otherwise, calculated from biological parameters in the Data object.`q_effort`

Scalar coefficient when conditioning on effort (to scale to F). Otherwise, 1 is the default.`F_equilibrium`

Equilibrium fishing mortality rate leading into first year of the model (to determine initial depletion). By default, 0.`omega`

Lognormal SD of the catch (observation error) when conditioning on effort. By default, Data@CV_Cat[x].`tau`

Lognormal SD of the recruitment deviations (process error) for`DD_SS`

. By default, Data@sigmaR[x].`sigma`

Lognormal SD of the index (observation error) when conditioning on catch. By default, Data@CV_Ind[x]. Not used if multiple indices are used.`sigma_W`

Lognormal SD of the mean weight (observation error). By default, 0.1.

Multiple indices are supported in the model. Data@Ind, Data@VInd, and Data@SpInd are all assumed to be biomass-based. For Data@AddInd, Data@I_units are used to identify a biomass vs. abundance-based index.

Similar to many other assessment models, the model depends on assumptions such as stationary productivity and proportionality between the abundance index and real abundance. Unsurprisingly the extent to which these assumptions are violated tends to be the biggest driver of performance for this method.

The following priors can be added as a named list, e.g., `prior = list(M = c(0.25, 0.15), h = c(0.7, 0.1)`

.
For each parameter below, provide a vector of values as described:

`R0`

- A vector of length 3. The first value indicates the distribution of the prior:`1`

for lognormal,`2`

for uniform on`log(R0)`

,`3`

for uniform on R0. If lognormal, the second and third values are the prior mean (in normal space) and SD (in log space). Otherwise, the second and third values are the lower and upper bounds of the uniform distribution (values in normal space).`h`

- A vector of length 2 for the prior mean and SD, both in normal space. Beverton-Holt steepness uses a beta distribution, while Ricker steepness uses a normal distribution.`M`

- A vector of length 2 for the prior mean (in normal space) and SD (in log space). Lognormal prior.`q`

- A matrix for nsurvey rows and 2 columns. The first column is the prior mean (in normal space) and the second column for the SD (in log space). Use`NA`

in rows corresponding to indices without priors.

See online documentation for more details.

Model description and equations are available on the openMSE website.

`DD_TMB`

: Cat, Ind, Mort, L50, vbK, vbLinf, vbt0, wla, wlb, MaxAge`DD_SS`

: Cat, Ind, Mort, L50, vbK, vbLinf, vbt0, wla, wlb, MaxAge

`DD_TMB`

: steep`DD_SS`

: steep, CV_Cat

Carruthers, T, Walters, C.J,, and McAllister, M.K. 2012. Evaluating methods that classify fisheries stock status using only fisheries catch data. Fisheries Research 119-120:66-79.

Hilborn, R., and Walters, C., 1992. Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty. Chapman and Hall, New York.

```
# \donttest{
#### Observation-error delay difference model
res <- DD_TMB(x = 3, Data = MSEtool::SimulatedData)
# Provide starting values
start <- list(h = 0.95)
res <- DD_TMB(x = 3, Data = MSEtool::SimulatedData, start = start)
summary(res@SD) # Parameter estimates
#> Estimate Std. Error
#> R0x -1.828102e+00 2.164137e-02
#> log_sigma -1.071746e+00 1.414214e-02
#> R0 1.579920e+02 3.419164e+00
#> h 9.500000e-01 0.000000e+00
#> q 1.675884e-04 5.816162e-06
#> sigma 3.424100e-01 4.842410e-03
### State-space version
### Set recruitment variability SD = 0.3 (since fix_tau = TRUE)
res <- DD_SS(x = 3, Data = MSEtool::SimulatedData, start = list(tau = 0.3))
# }
```