Fits the re-parameterized von Bertalanffy growth equation of Francis (1988) by using nonlinear least-squares

```
vbfr(age = NULL, L = NULL, agephi = NULL, agepsi = NULL, graph = TRUE,
gestimate = TRUE, Lphiparms = c(NA, NA, NA), Lchiparms = c(NA, NA, NA),
Lpsiparms = c(NA, NA, NA),control = list(maxiter = 10000))
```

age

Vector of ages of individual fish.

L

Vector of lengths of individual fish.

agephi

Arbitrary reference age phi

agepsi

Arbitrary reference age psi. agepsi>agephi.

graph

Logical specifiying whether observed versus predicted, and residual plots should be drawn. Default=TRUE.

gestimate

Logical specifying whether automatic generation of starting
values of *lphi*, *lchi* and *lpsi* should be used. Default=TRUE.
If gestimate=FALSE, user-specified starting, lower and upper limits of parameters
must be entered.

Lphiparms

If gestimate=FALSE, starting value, lower limit and upper limit
of *lphi* used in *nls*.

Lchiparms

If gestimate=FALSE, starting value, lower limit and upper limit
of *lchi* used in *nls*.

Lpsiparms

If gestimate=FALSE, starting value, lower limit and upper limit
of *lpsi* used in *nls*.

control

see `control`

under function *nls*.

*nls* object of model results. Use *summary* to extract results.

Francis (1988) re-parameterized the von Bertalanffy growth equation for age-length
in order to make equivalent comparison of parameters to parameters of a common model
used to estimate growth from tagging data. Three parameters, *lphi*, *lchi*
and *lpsi*, are estimated. The re-parameterization also has better
statistical properties than the original equation.

The formulae to get the conventional von Bertalanffy parameters are:

Linf = lphi + (lpsi-lphi)/(1-r^2) where r = (lpsi-lchi)/(lchi-lphi)

K = -(2*log(r))/(agepsi-agephi)

t0 = agephi + (1/K)*log((Linf-lphi)/Linf)

If gestimate=TRUE, unconstrained nonlinear least-squares (function *nls*) is used
to fit the model. If gestimate=FALSE, constrained nonlinear least-squares
is used (algorithm "port" in *nls*).

Francis, R. I. C. C. 1988. Are growth parameters estimated from tagging and age-length data comparable? Can. J. Fish. Aquat. Sci. 45: 936-942.

```
# NOT RUN {
data(pinfish)
with(pinfish,vbfr(age=age,L=sl,agephi=3,agepsi=6))
# }
```

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