The `simstudy`

package is a collection of functions that
allows users to generate simulated data sets to explore modeling
techniques or better understand data generating processes. The user
defines the distributions of individual variables, specifies
relationships between covariates and outcomes, and generates data based
on these specifications. The final data sets can represent randomized
control trials, repeated measure designs, cluster-randomized trials, or
naturally observed data processes. Other complexities that can be added
include survival data, correlated data, factorial study designs, step
wedge designs, and missing data processes.

Simulation using `simstudy`

has two fundamental steps. The
user (1) **defines** the data elements of a data set and
(2) **generates** the data based on these definitions.
Additional functionality exists to simulate observed or randomized
**treatment assignment/exposures**, to create
**longitudinal/panel** data, to create
**multi-level/hierarchical** data, to create data sets with
**correlated variables** based on a specified covariance
structure, to **merge** data sets, to create data sets with
**missing** data, and to create non-linear relationships
with underlying **spline** curves.

The overarching philosophy of `simstudy`

is to create data
generating processes that mimic the typical models used to fit those
types of data. So, the parameterization of some of the data generating
processes may not follow the standard parameterizations for the specific
distributions. For example, in `simstudy`

*gamma*-distributed data are generated based on the specification
of a mean \(\mu\) (or \(\log(\mu)\)) and a dispersion \(d\), rather than shape \(\alpha\) and rate \(\beta\) parameters that more typically
characterize the *gamma* distribution. When we estimate the
parameters, we are modeling \(\mu\) (or
some function of \((\mu)\)), so we
should explicitly recover the `simstudy`

parameters used to
generate the model - illuminating the relationship between the
underlying data generating processes and the models.

This introduction provides a brief overview to the basics of defining and generating data, including treatment or exposure variables. Subsequent sections in this vignette provide more details on these processes. For information on more elaborate data generating mechanisms, please refer to other vignettes in this package that provide more detailed descriptions.

The key to simulating data in `simstudy`

is the creation
of a series of data definition tables that look like this:

varname | formula | variance | dist | link |
---|---|---|---|---|

age | 10 | 2 | normal | identity |

female | -2 + age * 0.1 | 0 | binary | logit |

visits | 1.5 - 0.2 * age + 0.5 * female | 0 | poisson | log |

These definition tables can be generated in two ways. One option is
to use any external editor that allows the creation of .csv files, which
can be read in with a call to `defRead`

. An alternative is to
make repeated calls to the function `defData`

. This script
builds a definition table internally:

```
def <- defData(varname = "age", dist = "normal", formula = 10,
variance = 2)
def <- defData(def, varname = "female", dist = "binary",
formula = "-2 + age * 0.1", link = "logit")
def <- defData(def, varname = "visits", dist = "poisson",
formula = "1.5 - 0.2 * age + 0.5 * female", link = "log")
```

The data definition table includes a row for each variable that is to
be generated, and has the following fields: `varname*`

,
`formula`

, `variance`

, `dist`

, and
`link`

. `varname`

provides the name of the
variable to be generated. `formula`

is either a value or
string representing any valid R formula (which can include function
calls) that in most cases defines the mean of the distribution.
`variance`

is a value or string that specifies either the
variance or other parameter that characterizes the distribution; the
default is 0. `dist`

is defines the distribution of the
variable to be generated; the default is *normal*. The
`link`

is a function that defines the relationship of the
formula with the mean value, and can either *identity*,
*log*, or *logit*, depending on the distribution; the
default is *identity*.

If using `defData`

to create the definition table, the
first call to `defData`

without specifying a definition name
(in this example the definition name is *def*) creates a
**new** data.table with a single row. An additional row is
added to the table `def`

each time the function
`defData`

is called. Each of these calls is the definition of
a new field in the data set that will be generated.

After the data set definitions have been created, a new data set with
\(n\) observations can be created with
a call to function ** genData**. In this
example, 1,000 observations are generated using the data set definitions
in

`def`

`dd`

```
## Key: <id>
## id age female visits
## <int> <num> <int> <int>
## 1: 1 9.78 0 0
## 2: 2 10.81 0 0
## 3: 3 8.86 0 1
## 4: 4 9.83 1 1
## 5: 5 10.58 0 0
## ---
## 996: 996 8.87 1 2
## 997: 997 10.27 0 0
## 998: 998 6.84 0 1
## 999: 999 9.28 0 2
## 1000: 1000 10.80 1 2
```

If no data definition is provided, a simple data set is created with just id’s.

```
## Key: <id>
## id
## <int>
## 1: 1
## 2: 2
## 3: 3
## 4: 4
## 5: 5
## ---
## 996: 996
## 997: 997
## 998: 998
## 999: 999
## 1000: 1000
```

In many instances, the data generation process will involve a
treatment or exposure. While it is possible to generate a treatment or
exposure variable directly using the data definition process,
`trtAssign`

and `trtObserve`

offer the ability to
generate more involved types of study designs. In particular, with
`trtAssign`

, balanced and stratified designs are
possible.

```
## Key: <id>
## id age female visits rx
## <int> <num> <int> <int> <int>
## 1: 1 9.78 0 0 3
## 2: 2 10.81 0 0 1
## 3: 3 8.86 0 1 3
## 4: 4 9.83 1 1 3
## 5: 5 10.58 0 0 3
## ---
## 996: 996 8.87 1 2 2
## 997: 997 10.27 0 0 3
## 998: 998 6.84 0 1 1
## 999: 999 9.28 0 2 1
## 1000: 1000 10.80 1 2 3
```

```
## Key: <female, rx>
## female rx N
## <int> <int> <int>
## 1: 0 1 249
## 2: 0 2 248
## 3: 0 3 248
## 4: 1 1 85
## 5: 1 2 85
## 6: 1 3 85
```

This section elaborates on the data definition process to provide more details on how to create data sets.

The data definition table for a new data set is constructed
sequentially. As each new row or variable is added, the formula (and in
some cases the variance) can refer back to a previously defined
variable. The first row by necessity cannot refer to another variable,
so the formula must be a specific value (i.e. not a string formula).
Starting with the second row, the formula can either be a value or any
valid `R`

equation with quotes and can include any variables
previously defined.

In the definition we created above, the probability being
**female** is a function of **age**, which was
previously defined. Likewise, the number of **visits** is a
function of both **age** and **female**. Since
**age** is the first row in the table, we had to use a
scalar to define the mean.

```
def <- defData(varname = "age", dist = "normal", formula = 10,
variance = 2)
def <- defData(def, varname = "female", dist = "binary",
formula = "-2 + age * 0.1", link = "logit")
def <- defData(def, varname = "visits", dist = "poisson",
formula = "1.5 - 0.2 * age + 0.5 * female", link = "log")
```

Formulas can include `R`

functions or user-defined
functions. Here is an example with a user-defined function
`myinv`

and the `log`

function from base
`R`

:

```
myinv <- function(x) {
1/x
}
def <- defData(varname = "age", formula = 10, variance = 2,
dist = "normal")
def <- defData(def, varname = "loginvage", formula = "log(myinv(age))",
variance = 0.1, dist = "normal")
genData(5, def)
```

```
## Key: <id>
## id age loginvage
## <int> <num> <num>
## 1: 1 10.31 -2.58
## 2: 2 7.90 -1.94
## 3: 3 9.83 -1.93
## 4: 4 9.10 -2.42
## 5: 5 10.18 -2.21
```

Replication is an important aspect of data simulation - it is often
very useful to generate data under different sets of assumptions.
`simstudy`

facilitates this in at least two different ways.
There is function `updateDef`

which allows row by row changes
of a data definition table. In this case, we are changing the formula of
**loginvage** in **def** so that it uses the
function `log10`

instead of `log`

:

```
## varname formula variance dist link
## <char> <char> <char> <char> <char>
## 1: age 10 2 normal identity
## 2: loginvage log10(myinv(age)) 0.1 normal identity
```

```
## Key: <id>
## id age loginvage
## <int> <num> <num>
## 1: 1 9.82 -0.338
## 2: 2 10.97 -0.633
## 3: 3 11.79 -1.267
## 4: 4 9.74 -0.882
## 5: 5 10.11 -1.519
```

A more powerful feature of `simstudy`

that allows for
dynamic definition tables is the external reference capability through
the *double-dot* notation. Any variable reference in a formula
that is preceded by “..” refers to an externally defined variable:

```
age_effect <- 3
def <- defData(varname = "age", formula = 10, variance = 2,
dist = "normal")
def <- defData(def, varname = "agemult", formula = "age * ..age_effect",
dist = "nonrandom")
def
```

```
## varname formula variance dist link
## <char> <char> <num> <char> <char>
## 1: age 10 2 normal identity
## 2: agemult age * ..age_effect 0 nonrandom identity
```

```
## Key: <id>
## id age agemult
## <int> <num> <num>
## 1: 1 9.69 29.1
## 2: 2 9.63 28.9
```

But the real power of dynamic definition is seen in the context of iteration over multiple values:

```
age_effects <- c(0, 5, 10)
list_of_data <- list()
for (i in seq_along(age_effects)) {
age_effect <- age_effects[i]
list_of_data[[i]] <- genData(2, def)
}
list_of_data
```

```
## [[1]]
## Key: <id>
## id age agemult
## <int> <num> <num>
## 1: 1 11.4 0
## 2: 2 10.7 0
##
## [[2]]
## Key: <id>
## id age agemult
## <int> <num> <num>
## 1: 1 11.3 56.6
## 2: 2 11.2 56.1
##
## [[3]]
## Key: <id>
## id age agemult
## <int> <num> <num>
## 1: 1 9.32 93.2
## 2: 2 10.62 106.2
```

The foundation of generating data is the assumptions we make about
the distribution of each variable. `simstudy`

provides 15
types of distributions, which are listed in the following table:

name | formula | string/value | format | variance | identity | log | logit |
---|---|---|---|---|---|---|---|

beta | mean | both | - | dispersion | X | - | X |

binary | probability | both | - | - | X | X | X |

binomial | probability | both | - | # of trials | X | X | X |

categorical | probability | string | p_1;p_2;…;p_n | a;b;c | X | - | X |

clusterSize | total N | both | - | dispersion | X | - | - |

custom | function | string | - | arguments | X | - | - |

exponential | mean | both | - | - | X | X | - |

gamma | mean | both | - | dispersion | X | X | - |

mixture | formula | string | x_1 | p_1 + … + x_n | p_n | - | X | - | - |

negBinomial | mean | both | - | dispersion | X | X | - |

nonrandom | formula | both | - | - | X | - | - |

normal | mean | both | - | variance | X | - | - |

noZeroPoisson | mean | both | - | - | X | X | - |

poisson | mean | both | - | - | X | X | - |

trtAssign | ratio | string | r_1;r_2;…;r_n | stratification | X | X | - |

uniform | range | string | from ; to | - | X | - | - |

uniformInt | range | string | from ; to | - | X | - | - |

A *beta* distribution is a continuous data distribution that
takes on values between \(0\) and \(1\). The `formula`

specifies the
mean \(\mu\) (with the ‘identity’ link)
or the log-odds of the mean (with the ‘logit’ link). The scalar value in
the ‘variance’ represents the dispersion value \(d\). The variance \(\sigma^2\) for a beta distributed variable
will be \(\mu (1- \mu)/(1 + d)\).
Typically, the beta distribution is specified using two shape parameters
\(\alpha\) and \(\beta\), where \(\mu = \alpha/(\alpha + \beta)\) and \(\sigma^2 = \alpha\beta/[(\alpha + \beta)^2 (\alpha
+ \beta + 1)]\).

A *binary* distribution is a discrete data distribution that
takes values \(0\) or \(1\). (It is more conventionally called a
*Bernoulli* distribution, or is a *binomial* distribution
with a single trial \(n=1\).) The
`formula`

represents the probability (with the ‘identity’
link), the relative risk (with the ‘log’ link), or the log odds (with
the ‘logit’ link) that the variable takes the value of 1. The mean of
this distribution is \(p\), and
variance \(\sigma^2\) is \(p(1-p)\).

A *binomial* distribution is a discrete data distribution that
represents the count of the number of successes given a number of
trials. The formula specifies the probability of success (with the
‘identity’ link), the relative risk (with the ‘log’ link), or the log
odds (with the ‘logit’ link) that the variable takes the value of 1. and
the variance field is used to specify the number of trials \(n\). Given a value of \(p\), the mean \(\mu\) of this distribution is \(n*p\), and the variance \(\sigma^2\) is \(np(1-p)\).

A *categorical* distribution is a discrete data distribution
taking on values from \(1\) to \(K\), with each value representing a
specific category, and there are \(K\)
categories. The categories may or may not be ordered. For a categorical
variable with \(k\) categories, the
`formula`

is a string of probabilities that sum to 1, each
separated by a semi-colon: \((p_1 ; p_2 ; ...
; p_k)\). \(p_1\) is the
probability of the random variable falling in category \(1\), \(p_2\) is the probability of category \(2\), etc. The probabilities can be
specified as functions of other variables previously defined. The helper
function `genCatFormula`

is an easy way to create different
probability strings. The `link`

options are *identity*
or *logit*. The `variance`

field is optional an allows
to provide categories other than the default `1...n`

in the
same format as `formula`

: “a;b;c”. Numeric variance Strings
(e.g. “50;100;200”) will be converted to numeric when possible. All
probabilities will be rounded to 1e12 decimal points to prevent possible
rounding errors.

The *clusterSize* distribution allocates a total sample size
*N* (specified in the *formula* argument) across the
*k* rows of the data.table such that the sum of the rows equals
*N*. If the *dispersion* argument is set to 0, the
allocation to each row is *N/k*, with some rows getting an
increment of 1 to ensure that the sum is N. It is possible to relax the
assumption of balanced cluster sizes by setting *dispersion >
0*. As the dispersion increases, the variability of cluster sizes
across clusters increases.

Custom distributions can be specified in `defData`

and
`defDataAdd`

by setting the argument *dist* to
“custom”. When defining a custom distribution, 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**”. The
*arg_k’s* represent the names of the arguments passed to the
customized function, where \(k\) ranges
from \(1\) to \(K\). Values or formulas can be used 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`

.

An *exponential* distribution is a continuous data
distribution that takes on non-negative values. The `formula`

represents the mean \(\theta\) (with
the ‘identity’ link) or log of the mean (with the ‘log’ link). The
`variance`

argument does not apply to the
*exponential* distribution. The variance \(\sigma^2\) is \(\theta^2\).

A *gamma* distribution is a continuous data distribution that
takes on non-negative values. The `formula`

specifies the
mean \(\mu\) (with the ‘identity’ link)
or the log of the mean (with the ‘log’ link). The `variance`

field represents a dispersion value \(d\). The variance \(\sigma^2\) is is \(d \mu^2\).

The *mixture* distribution is a mixture of other predefined
variables, which can be defined based on any of the other available
distributions. The formula is a string structured with a sequence of
variables \(x_i\) and probabilities
\(p_i\): \(x_1 | p_1 + … + x_n | p_n\). All of the
\(x_i\)’s are required to have been
previously defined, and the probabilities must sum to \(1\) (i.e. \(\sum_1^n p_i = 1\)). The result of
generating from a mixture is the value \(x_i\) with probability \(p_i\). The `variance`

and
`link`

fields do not apply to the *mixture*
distribution.

A *negative binomial* distribution is a discrete data
distribution that represents the number of successes that occur in a
sequence of *Bernoulli* trials before a specified number of
failures occurs. It is often used to model count data more generally
when a *Poisson* distribution is not considered appropriate; the
variance of the negative binomial distribution is larger than the
*Poisson* distribution. The `formula`

specifies the
mean \(\mu\) or the log of the mean.
The variance field represents a dispersion value \(d\). The variance \(\sigma^2\) will be \(\mu + d\mu^2\).

Deterministic data can be “generated” using the *nonrandom*
distribution. The `formula`

explicitly represents the value
of the variable to be generated, without any uncertainty. The
`variance`

and `link`

fields do not apply to
*nonrandom* data generation.

A *normal* or *Gaussian* distribution is a continuous
data distribution that takes on values between \(-\infty\) and \(\infty\). The `formula`

represents the mean \(\mu\) and the
`variance`

represents \(\sigma^2\). The `link`

field is
not applied to the *normal* distribution.

The *noZeroPoisson* distribution is a discrete data
distribution that takes on positive integers. This is a truncated
*poisson* distribution that excludes \(0\). The `formula`

specifies the
parameter \(\lambda\) (link is
‘identity’) or log() (`link`

is log). The
`variance`

field does not apply to this distribution. The
mean \(\mu\) of this distribution is
\(\lambda/(1-e^{-\lambda})\) and the
variance \(\sigma^2\) is \((\lambda + \lambda^2)/(1-e^{-\lambda}) -
\lambda^2/(1-e^{-\lambda})^2\). We are not typically interested
in modeling data drawn from this distribution (except in the case of a
*hurdle model*), but it is useful to generate positive count data
where it is not desirable to have any \(0\) values.

The *poisson* distribution is a discrete data distribution
that takes on non-negative integers. The `formula`

specifies
the mean \(\lambda\) (link is
‘identity’) or log of the mean (`link`

is log). The
`variance`

field does not apply to this distribution. The
variance \(\sigma^2\) is \(\lambda\) itself.

The *trtAssign* distribution is an implementation of the
`trtAssign`

functionality in the context of the
`defData`

workflow. Sometimes, it might be convenient to
assign treatment or group membership while defining other variables. The
`formula`

specifies the relative allocation to the different
groups. For example three-arm randomization with equal allocation to
each arm would be specified as *“1;1;1”*. The
`variance`

field defines the stratification variables, and
would be specified as *“s_1;s_2”* if *s_1* and
*s_2* are the stratification variables. The `link`

field is used to indicate if the allocations should be perfectly
balanced; if nothing is specified (and link defaults to
*identity*), the allocation will be balanced; if link is
specified to be different from *identity*, then the allocation
will not be balanced.

A *uniform* distribution is a continuous data distribution
that takes on values from \(a\) to
\(b\), where \(b\) > \(a\), and they both lie anywhere on the real
number line. The `formula`

is a string with the format “a;b”,
where *a* and *b* are scalars or functions of previously
defined variables. The `variance`

and `link`

arguments do not apply to the *uniform* distribution.

A *uniform integer* distribution is a discrete data
distribution that takes on values from \(a\) to \(b\), where \(b\) > \(a\), and they both lie anywhere on the
integer number line. The `formula`

is a string with the
format “a;b”, where *a* and *b* are scalars or functions
of previously defined variables. The `variance`

and
`link`

arguments do not apply to the *uniform integer*
distribution.

`defRepeat`

allows us to specify multiple versions of a
variable based on a single set of distribution assumptions. The function
will add `nvar`

variables to the *data definition*
table, each of which will be specified with a single set of distribution
assumptions. The names of the variables will be based on the
`prefix`

argument and the distribution assumptions are
specified as they are in the `defData`

function. Calls to
`defRepeat`

can be integrated with calls to
`defData`

.

```
def <- defRepeat(nVars = 4, prefix = "g", formula = "1/3;1/3;1/3",
variance = 0, dist = "categorical")
def <- defData(def, varname = "a", formula = "1;1", dist = "trtAssign")
def <- defRepeat(def, 3, "b", formula = "5 + a", variance = 3,
dist = "normal")
def <- defData(def, "y", formula = "0.10", dist = "binary")
def
```

```
## varname formula variance dist link
## <char> <char> <num> <char> <char>
## 1: g1 1/3;1/3;1/3 0 categorical identity
## 2: g2 1/3;1/3;1/3 0 categorical identity
## 3: g3 1/3;1/3;1/3 0 categorical identity
## 4: g4 1/3;1/3;1/3 0 categorical identity
## 5: a 1;1 0 trtAssign identity
## 6: b1 5 + a 3 normal identity
## 7: b2 5 + a 3 normal identity
## 8: b3 5 + a 3 normal identity
## 9: y 0.10 0 binary identity
```

Until this point, we have been generating new data sets, building them up from scratch. However, it is often necessary to generate the data in multiple stages so that we would need to add data as we go along. For example, we may have multi-level data with clusters that contain collections of individual observations. The data generation might begin with defining and generating cluster-level variables, followed by the definition and generation of the individual-level data; the individual-level data set would be adding to the cluster-level data set.

There are several important functions that facilitate the
augmentation of data sets. `defDataAdd`

,
`defRepeatAdd`

, and `readDataAdd`

are similar to
their counterparts `defData`

, `defRepeat`

, and
`readData`

, respectively; they create data definition tables
that will be used by the function `addColumns`

. The formulas
in these “*add*-ing” functions are permitted to refer to fields
that exist in the data set to be augmented, so all variables need not be
defined in the current definition able.

```
d1 <- defData(varname = "x1", formula = 0, variance = 1,
dist = "normal")
d1 <- defData(d1, varname = "x2", formula = 0.5, dist = "binary")
d2 <- defRepeatAdd(nVars = 2, prefix = "q", formula = "5 + 3*rx",
variance = 4, dist = "normal")
d2 <- defDataAdd(d2, varname = "y", formula = "-2 + 0.5*x1 + 0.5*x2 + 1*rx",
dist = "binary", link = "logit")
dd <- genData(5, d1)
dd <- trtAssign(dd, nTrt = 2, grpName = "rx")
dd
```

```
## Key: <id>
## id x1 x2 rx
## <int> <num> <int> <int>
## 1: 1 -1.3230 1 0
## 2: 2 -0.0494 0 1
## 3: 3 -0.4064 1 0
## 4: 4 -0.5562 1 0
## 5: 5 -0.0941 0 1
```

```
## Key: <id>
## id x1 x2 rx q1 q2 y
## <int> <num> <int> <int> <num> <num> <int>
## 1: 1 -1.3230 1 0 4.589 5.70 0
## 2: 2 -0.0494 0 1 9.829 11.74 1
## 3: 3 -0.4064 1 0 2.117 4.47 0
## 4: 4 -0.5562 1 0 0.798 3.24 0
## 5: 5 -0.0941 0 1 7.601 6.98 0
```

In certain situations, it might be useful to define a data
distribution conditional on previously generated data in a way that is
more complex than might be easily handled by a single formula.
`defCondition`

creates a special table of definitions and the
new variable is added to an existing data set by calling
`addCondition`

. `defCondition`

specifies a
condition argument that will be based on a variable that already exists
in the data set. The new variable can take on any `simstudy`

distribution specified with the appropriate `formula`

,
`variance`

, and `link`

arguments.

In this example, the slope of a regression line of \(y\) on \(x\) varies depending on the value of the predictor \(x\):

```
d <- defData(varname = "x", formula = 0, variance = 9, dist = "normal")
dc <- defCondition(condition = "x <= -2", formula = "4 + 3*x",
variance = 2, dist = "normal")
dc <- defCondition(dc, condition = "x > -2 & x <= 2", formula = "0 + 1*x",
variance = 4, dist = "normal")
dc <- defCondition(dc, condition = "x > 2", formula = "-5 + 4*x",
variance = 3, dist = "normal")
dd <- genData(1000, d)
dd <- addCondition(dc, dd, newvar = "y")
```