Custom distributions can be specified in `defData`

and
`defDataAdd`

by setting the argument *dist* to
“custom”. When defining a custom distribution, you provide the name of
the user-defined function as a string in the *formula* argument.
The arguments of the custom function are listed in the *variance*
argument, separated by commas and formatted as “**arg_1 =
val_form_1, arg_2 = val_form_2, \(\dots\), arg_K = val_form_K**”.

Here, the *arg_k’s* represent the names of the arguments
passed to the customized function, where \(k\) ranges from \(1\) to \(K\). You can use values or formulas for
each *val_form_k*. If formulas are used, ensure that the
variables have been previously generated. Double dot notation is
available in specifying *value_formula_k*. One important
requirement of the custom function is that the parameter list used to
define the function must include an argument”**n = n**”,
but do not include \(n\) in the
definition as part of `defData`

or
`defDataAdd`

.

Here is an example where we would like to generate data from a
zero-inflated beta distribution. In this case, there is a user-defined
function `zeroBeta`

that takes on shape parameters \(a\) and \(b\), as well as \(p_0\), the proportion of the sample that is
zero. Note that the function also takes an argument \(n\) that will not to be be specified in the
data definition; \(n\) will represent
the number of observations being generated:

```
zeroBeta <- function(n, a, b, p0) {
betas <- rbeta(n, a, b)
is.zero <- rbinom(n, 1, p0)
betas*!(is.zero)
}
```

The data definition specifies a new variable \(zb\) that sets \(a\) and \(b\) to 0.75, and \(p_0 = 0.02\):

```
def <- defData(
varname = "zb",
formula = "zeroBeta",
variance = "a = 0.75, b = 0.75, p0 = 0.02",
dist = "custom"
)
```

The data are generated:

```
## Key: <id>
## id zb
## <int> <num>
## 1: 1 0.93922887
## 2: 2 0.35609519
## 3: 3 0.08087245
## 4: 4 0.99796758
## 5: 5 0.28481522
## ---
## 99996: 99996 0.81740836
## 99997: 99997 0.98586333
## 99998: 99998 0.68770216
## 99999: 99999 0.45096868
## 100000: 100000 0.74101272
```

A plot of the data reveals dis-proportion of zero’s:

In this second example, we are generating sets of truncated Gaussian
distributions with means ranging from \(-1\) to \(1\). The limits of the truncation vary
across three different groups. `rnormt`

is a customized
(user-defined) function that generates the truncated Gaussiandata. The
function requires four arguments (the left truncation value, the right
truncation value, the distribution average and the standard
deviation).

```
rnormt <- function(n, min, max, mu, s) {
F.a <- pnorm(min, mean = mu, sd = s)
F.b <- pnorm(max, mean = mu, sd = s)
u <- runif(n, min = F.a, max = F.b)
qnorm(u, mean = mu, sd = s)
}
```

In this example, truncation limits vary based on group membership. Initially, three groups are created, followed by the generation of truncated values. For Group 1, truncation occurs within the range of \(-1\) to \(1\), for Group 2, it’s \(-2\) to \(2\) and for Group 3, it’s \(-3\) to \(3\). We’ll generate three data sets, each with a distinct mean denoted by M, using the double-dot notation to implement these different means.

```
def <-
defData(
varname = "limit",
formula = "1/4;1/2;1/4",
dist = "categorical"
) |>
defData(
varname = "tn",
formula = "rnormt",
variance = "min = -limit, max = limit, mu = ..M, s = 1.5",
dist = "custom"
)
```

The data generation requires three calls to `genData`

. The
output is a list of three data sets:

Here are the first six observations from each of the three data sets:

```
## [[1]]
## Key: <id>
## id limit tn
## <int> <int> <num>
## 1: 1 2 0.6949619
## 2: 2 2 -0.3641963
## 3: 3 2 -0.4721632
## 4: 4 3 -2.6083796
## 5: 5 2 -0.6800441
## 6: 6 3 -0.5813880
##
## [[2]]
## Key: <id>
## id limit tn
## <int> <int> <num>
## 1: 1 1 0.4853614
## 2: 2 2 -0.5690811
## 3: 3 2 0.5282246
## 4: 4 2 0.1107778
## 5: 5 2 -0.3504309
## 6: 6 2 1.9439890
##
## [[3]]
## Key: <id>
## id limit tn
## <int> <int> <num>
## 1: 1 2 1.3560628
## 2: 2 2 1.4543616
## 3: 3 3 1.4491010
## 4: 4 2 0.7328855
## 5: 5 2 -0.1254556
## 6: 6 2 -0.7455908
```

A plot highlights the group differences.