Often, we’d like to explore data generation and modeling under different scenarios. For example, we might want to understand the operating characteristics of a model given different variance or other parametric assumptions. There is functionality built into `simstudy`

to facilitate this type of dynamic exploration. First, the functions `updateDef`

and `updateDefAdd`

essentially allow us to edit lines of existing data definition tables. Second, there is a built-in mechanism - called *double-dot* reference - to access external variables that do not exist in a defined data set or data definition.

The `updateDef`

function updates a row in a definition table created by functions `defData`

or `defRead`

. Analogously, `updateDefAdd`

function updates a row in a definition table created by functions `defDataAdd`

or `defReadAdd`

.

The original data set definition includes three variables `x`

, `y`

, and `z`

, all normally distributed:

```
defs <- defData(varname = "x", formula = 0, variance = 3, dist = "normal")
defs <- defData(defs, varname = "y", formula = "2 + 3*x", variance = 1, dist = "normal")
defs <- defData(defs, varname = "z", formula = "4 + 3*x - 2*y", variance = 1, dist = "normal")
defs
```

```
## varname formula variance dist link
## 1: x 0 3 normal identity
## 2: y 2 + 3*x 1 normal identity
## 3: z 4 + 3*x - 2*y 1 normal identity
```

In the first case, we are changing the relationship of `y`

with `x`

as well as the variance:

```
defs <- updateDef(dtDefs = defs, changevar = "y", newformula = "x + 5", newvariance = 2)
defs
```

```
## varname formula variance dist link
## 1: x 0 3 normal identity
## 2: y x + 5 2 normal identity
## 3: z 4 + 3*x - 2*y 1 normal identity
```

In this second case, we are changing the distribution of `z`

to *Poisson* and updating the link function to *log*:

```
defs <- updateDef(dtDefs = defs, changevar = "z", newdist = "poisson", newlink = "log")
defs
```

```
## varname formula variance dist link
## 1: x 0 3 normal identity
## 2: y x + 5 2 normal identity
## 3: z 4 + 3*x - 2*y 1 poisson log
```

And in the last case, we remove a variable from a data set definition. Note in the case of a definition created by `defData`

that it is not possible to remove a variable that is a predictor of a subsequent variable, such as `x`

or `y`

in this case.

```
defs <- updateDef(dtDefs = defs, changevar = "z", remove = TRUE)
defs
```

```
## varname formula variance dist link
## 1: x 0 3 normal identity
## 2: y x + 5 2 normal identity
```

For a truly dynamic data definition process, `simstudy`

(as of `version 0.2.0`

) allows users to reference variables that exist outside of data generation. These can be thought of as a type of hyperparameter of the data generation process. The reference is made directly in the formula itself, using a double-dot (“..”) notation before the variable name. Here is a simple example:

```
def <- defData(varname = "x", formula = 0,
variance = 5, dist = "normal")
def <- defData(def, varname = "y", formula = "..B0 + ..B1 * x",
variance = "..sigma2", dist = "normal")
def
```

```
## varname formula variance dist link
## 1: x 0 5 normal identity
## 2: y ..B0 + ..B1 * x ..sigma2 normal identity
```

```
B0 <- 4;
B1 <- 2;
sigma2 <- 9
set.seed(716251)
dd <- genData(100, def)
fit <- summary(lm(y ~ x, data = dd))
coef(fit)
```

```
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.00 0.284 14.1 2.56e-25
## x 2.01 0.130 15.4 5.90e-28
```

`fit$sigma`

`## [1] 2.83`

It is easy to create a new data set on the fly with a difference variance assumption without having to go to the trouble of updating the data definitions.

```
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.35 0.427 10.19 4.57e-17
## x 2.12 0.218 9.75 4.32e-16
```

`fit$sigma`

`## [1] 4.21`

The double-dot notation can be flexibly applied using `lapply`

(or the parallel version `mclapply`

) to create a range of data sets under different assumptions:

```
sigma2s <- c(1, 2, 6, 9)
gen_data <- function(sigma2, d) {
dd <- genData(200, d)
dd$sigma2 <- sigma2
dd
}
dd_4 <- lapply(sigma2s, function(s) gen_data(s, def))
dd_4 <- rbindlist(dd_4)
ggplot(data = dd_4, aes(x = x, y = y)) +
geom_point(size = .5, color = "grey30") +
facet_wrap(sigma2 ~ .) +
theme(panel.grid = element_blank())
```

The double-dot notation is also *array-friendly*. For example if we want to create a mixture distribution from a vector of values (which we can also do using a *categorical* distribution), we can define the mixture formula in terms of the vector. In this case we are generating permuted block sizes of 2 and 4:

```
defblk <- defData(varname = "blksize",
formula = "..sizes[1] | .5 + ..sizes[2] | .5", dist = "mixture")
defblk
```

```
## varname formula variance dist link
## 1: blksize ..sizes[1] | .5 + ..sizes[2] | .5 0 mixture identity
```

```
## id blksize
## 1: 1 4
## 2: 2 4
## 3: 3 4
## 4: 4 2
## 5: 5 4
## ---
## 996: 996 2
## 997: 997 2
## 998: 998 4
## 999: 999 4
## 1000: 1000 4
```

In this second example, there is a vector variable *tau* of positive real numbers that sum to 1, and we want to calculate the weighted average of three numbers using *tau* as the weights. We could use the following code to estimate a weighted average *theta*:

`## [1] 0.319 0.550 0.132`

```
d <- defData(varname = "a", formula = 3, variance = 4)
d <- defData(d, varname = "b", formula = 8, variance = 2)
d <- defData(d, varname = "c", formula = 11, variance = 6)
d <- defData(d, varname = "theta", formula = "..tau[1]*a + ..tau[2]*b + ..tau[3]*c",
dist = "nonrandom")
set.seed(1)
genData(4, d)
```

```
## id a b c theta
## 1: 1 1.75 8.47 12.4 6.84
## 2: 2 3.37 6.84 10.3 6.18
## 3: 3 1.33 8.69 14.7 7.13
## 4: 4 6.19 9.04 12.0 8.52
```

We can simplify the calculation of *theta* by using matrix multiplication:

```
d <- updateDef(d, changevar = "theta", newformula = "t(..tau) %*% c(a, b, c)")
set.seed(1)
genData(4, d)
```

```
## id a b c theta
## 1: 1 1.75 8.47 12.4 6.84
## 2: 2 3.37 6.84 10.3 6.18
## 3: 3 1.33 8.69 14.7 7.13
## 4: 4 6.19 9.04 12.0 8.52
```

These arrays can also have **multiple dimensions**, as in a \(2 \times 2\) matrix. If we want to specify the mean outcomes for a factorial study design with two interventions \(a\) and \(b\), we can use a simple matrix and draw the means directly from the matrix, which in this example is stored in the variable *effect*:

```
## [,1] [,2]
## [1,] 0 5
## [2,] 4 7
```

Using double dot notation, it is possible to reference the matrix cell values directly:

```
d1 <- defData(varname = "a", formula = ".5;.5", variance = "1;2", dist = "categorical")
d1 <- defData(d1, varname = "b", formula = ".5;.5", variance = "1;2", dist = "categorical")
d1 <- defData(d1, varname = "outcome", formula = "..effect[a, b]", dist="nonrandom")
```

```
dx <- genData(8, d1)
dx
```

```
## id a b outcome
## 1: 1 2 2 7
## 2: 2 2 2 7
## 3: 3 2 1 4
## 4: 4 2 1 4
## 5: 5 1 1 0
## 6: 6 2 2 7
## 7: 7 2 1 4
## 8: 8 1 2 5
```

It is possible to generate normally distributed data based on these means:

The plot shows the individual values as well as the mean values by intervention arm: