This vignette demonstrates basic features of `RTMB`

by
implementing a *random regression* model from scratch and fit it
to a built-in R dataset.

We’ll take as starting point the built-in data set ‘ChickWeight’:

The data contains `weight`

of 50 chicks by
`Time`

and individual identifier `Chick`

. A useful
plot of growth by individual is available from the help page by running
`example(ChickWeight)`

.

Although not the most natural based on the plot, we’ll setup a
*random regression model*. It is defined by first drawing slopes
and intercepts by individual from an underlying distribution

\[ a_1,...,a_{50} \sim N(\mu_a , \sigma_a^2) \]

\[ b_1,...,b_{50} \sim N(\mu_b , \sigma_b^2) \]

and then state a normal regression given these random coefficients:

\[ \text{weight}_{i} \sim N(a_{\text{Chick}_i} * \text{Time}_i + b_{\text{Chick}_i}, \sigma^2) \]

To implement the model in RTMB we have to set up the objective function and define the parameters and random effects.

Parameter objects are gathered in a list that also serves as initial guess when fitting the model:

```
parameters <- list(
mua=0, ## Mean slope
sda=1, ## Std of slopes
mub=0, ## Mean intercept
sdb=1, ## Std of intercepts
sdeps=1, ## Residual Std
a=rep(0, 50), ## Random slope by chick
b=rep(0, 50) ## Random intercept by chick
)
```

The objective function takes as input such a parameter list, and is defined by

```
f <- function(parms) {
getAll(ChickWeight, parms, warn=FALSE)
## Optional (enables extra RTMB features)
weight <- OBS(weight)
## Initialize joint negative log likelihood
nll <- 0
## Random slopes
nll <- nll - sum(dnorm(a, mean=mua, sd=sda, log=TRUE))
## Random intercepts
nll <- nll - sum(dnorm(b, mean=mub, sd=sdb, log=TRUE))
## Data
predWeight <- a[Chick] * Time + b[Chick]
nll <- nll - sum(dnorm(weight, predWeight, sd=sdeps, log=TRUE))
## Get predicted weight uncertainties
ADREPORT(predWeight)
## Return
nll
}
```

This function calculates the negative log likelihood `nll`

using straight forward R operations corresponding exactly to the model
definition. In addition, some RTMB specific statements are used:

- The
`getAll`

function makes all the list elements of data and parameters visible inside the function, so that one can write e.g.`weight`

rather than`ChickWeight$weight`

. - The
`weight <- OBS(weight)`

statement tells RTMB that that`weight`

is the response. This is needed to enable automatic simulation and residual calculations from the model object. - The
`ADREPORT(predWeight)`

statement tells RTMB that we want uncertainties for this intermediate calculation.

The objective function `f`

is processed by RTMB using the
call

where we also specify that `a`

and `b`

are
random effects.

We optimize the model using `nlminb`

(or any other
suitable gradient based optimizer in R)

Uncertainties are now calculated using

```
## sdreport(.) result
## Estimate Std. Error
## mua 8.467355 0.5010482
## sda 3.463551 0.3630097
## mub 29.044188 1.8040325
## sdb 10.541116 1.5588779
## sdeps 12.890654 0.4237485
## Maximum gradient component: 1.117405e-05
```

By default, the shown output is very brief, containing only the model
parameters. The `sdr`

object contains much more. It is often
convenient to inspect parameter estimates and other output as a list
similar to the one containing the parameters. For instance, to get
parameters estimates and standard errors as separate lists use:

Pass `report=TRUE`

to get `ADREPORT`

ed
quantities:

New datasets can be generated from the estimated model using
`obj$simulate()`

assuming that the model is implemented in
accordance with the principles defined in the help page
`?Simulation`

.

When building random effect models from scratch, many mistakes can be
made leading to wrong results. We always recommend to run the model
through the completely automatic consistency check (this requires that
`obj$simulate()`

works for the implementation). By default,
the check simulates 100 datasets and calculates gradients for each
replicate. A standard output tells you whether the implementation is
consistent with simulation (message ‘simulation appears to be correct’).
It also gives an idea of the parameter bias usually caused by the
Laplace approximation (not an issue for the random regression model for
which the Laplace approximation is exact). We run the standard check
by:

```
## Parameters used for simulation:
## mua sda mub sdb sdeps
## 8.467355 3.463551 29.044188 10.541116 12.890654
##
## Test correct simulation (p.value):
## [1] 0.916118
## Simulation appears to be correct
##
## Estimated parameter bias:
## mua sda mub sdb sdeps
## 0.041037447 -0.001069032 -0.092780844 0.057738402 0.027244614
```

As expected for this case, everything looks fine. A complete simulation study, that re-estimates parameters for each replicate, can also be performed although that takes longer to run:

For more details we refer to the help page
`?TMB::checkConsistency`

.

Quantile residuals can be generated automatically using the
`oneStepPredict`

function. These residuals are conceptually
superior to other methods (e.g. Pearson residuals), but much trickier to
calculate. It is very important to specify an appropriate
`method`

(see `?TMB::oneStepPredict`

) because
using an inappropriate method can give wrong residuals. For the random
regression model the ‘fullGaussian’ method is computationally exact.

The argument `discrete=FALSE`

is necessary in this case
because the data has duplicates, which is a zero-probability event for
continuous distributions. If the model is correct, the residuals are
standard, independent normally distributed, which is obviously not the
case here.

For the random regression model, we could run `MakeADFun`

without problems. However, model implementation might not always work as
smoothly. In case of errors it is useful to test the implementation
step-wise.

Going back to our objective function

`f`

, first step is to check that you can evaluate the function as a normal R function:An error at this point is obviously not due to

`RTMB`

.Next, it is useful to check that

`MakeADFun`

can be run*without*random effects:Should an error occur at this point, you can enable standard R debugging

`debug(f)`

and run`MakeADFun`

again to figure out which code line caused the error. A common cause is to that an unsupported (e.g. non-differentiable) operation has been used.Once

`obj`

has been constructed successfully, you should evaluate itand verify that it gives the same result as

`f(parameters)`

.

The random regression model could alternatively have been written
using the RTMB ‘tilde operator’ (`%~%`

):

```
f2 <- function(parms) {
getAll(ChickWeight, parms, warn=FALSE)
## Optional (enables extra RTMB features)
weight <- OBS(weight)
## Random slopes
a %~% dnorm(mean=mua, sd=sda)
## Random intercepts
b %~% dnorm(mean=mub, sd=sdb)
## Data
predWeight <- a[Chick] * Time + b[Chick]
weight %~% dnorm(predWeight, sd=sdeps)
## Get predicted weight uncertainties
ADREPORT(predWeight)
}
```

This syntax is closer to other probabilistic languages (e.g. BUGS,
JAGS and Stan). But more importantly, it prevents some very common TMB
mistakes, by passing `log=TRUE`

automatically and making sure
the sign of the objective function is correct.

Otherwise, `f2`

is identical to `f`

, and the
model object can be constructed and fitted in the same way: