This vignette contains an in-depth example on how to use the
`sim_discrete_time()`

function to generate complex
time-to-event data. Unlike the other vignettes, we will not rely on a
overly simple example here. Instead, our aim is to generate somewhat
realistic data about the Covid-19 pandemic. In particular, we are
interested in generating a longitudinal data set containing Covid-19
infections, vaccines and adverse side effects of those vaccines. The aim
of the real study was to create a data set that is reasonably close to
real data, including measurement problems as described below. This data
generation algorithm was then used to identify a suitable data analysis
strategy for the real data. We only describe a slightly simplified
version of the data generation part here to give a detailed example on
how the `sim_discrete_time()`

function may be used
effectively.

Note that the algorithm described here is not completely the same as the one we originally used, because that would require us to include confidential data into this vignette, which is sadly impossible.

We also strongly recommend reading the other two vignettes of this package first.

Simulating data that is reasonably close to a complex real system is not a trivial task, even when using this package. Dividing the big task of obtaining a valid data generation model into multiple sub-tasks is a great first step in the right direction. We suggest following the 7 steps below:

If you haven’t done this yet, now is the time. Try to make your goal as explicit as possible. This will help you in deciding which aspects of the system are important to you and which can be safely ignored.

This entails reading up on relevant literature and writing down any assumptions you may already have about the system. Perhaps (and usually most likely) other researchers have tried to build a simulation model for the same system (or a very similar system). A great way to encode your causal assumptions of the system is time-dependent DAG, as discussed in the other vignettes.

Real systems are incredibly complex. Any simulation will have to make some simplifying assumptions. After building a moderately detailed version of the theoretical model, you will have to decide which aspects are of interest to your research project and which aren’t.

If the simulated data should correspond to real data, it is crucial to base the input of the model on actual empirical data. Using the empirical data, you may be able to derive appropriate distributions for the root nodes and appropriate functional forms of the relationship between the considered variables.

After having specified suitable distributions and relationships,
generate the initial data used in the simulation process. It is
important to check this data thoroughly, as it will be used as a basis
for all subsequent steps of the simulation. The
`sim_from_dag()`

function might be very helpful for this
step.

Each time-varying node requires a user written function that
transforms the data at \(t\) to the
data at \(t + 1\). It might be helpful
to add one time-varying node at a time and proceeding to step
**7.)** before adding other variables, if possible.

Validating your simulation code is important to ensure that no mistakes slipped through. This may be done by fitting models relying on your modeling assumptions to the generated data and checking whether those models produce the expected results.

Since this vignette is mostly concerned with the practical implementation of the discrete-time simulation approach using this R-package, we will not spend too much time on the first 4 steps of the process mentioned in the previous section. We only briefly present the most important points.

Our actual research goal was to identify a suitable data analysis
strategy for the assessment of Covid-19 vaccine side-effects for a
particular real-life data set. To do this we decided to simulate data
that is as close to the real data as possible. Using this data we could
then try out different analysis strategies and see which one performed
adequately. The goal for this vignette is to use parts of this model to
showcase the capabilities of the `simDAG`

package. More
information on the actual simulation can be found in the first related
publication (Denz et al. 2023).

There is a seemingly endless amount of literature describing models for the Covid-19 pandemic and the associated Covid-19 vaccines. Trying to include all relevant aspects would be an unfeasible task. After deliberating on this literature we decided to include only a few key variables. To make this vignette readable we further limited the variables to only include the most essential:

`vaccination`

: The time at which the person received the first vaccine.`covid`

: Whether a Covid-19 infection occurred.`sickness`

: Whether the person has developed the sickness of interest.

We assume that these three variables vary over time and cause each other.

Instead of diving in deep from the start and trying to include all relevant variables with all relevant relationships at once, it is often better to build a very simplified version first and to start adding more and more stuff to it as we continue.

The most important variables for us are the `vaccination`

,
the `covid`

-19 infection and the `sickness`

. All
of these are variables that have a certain probability of occurrence at
each point in time. Once they occur, they last for some duration
(e.g. someone being sick for two weeks or something similar). After the
event is over, there is usually some duration where the person is
“immune” to receiving the event again. This is a perfect case for using
a time-dependent node of type `"time_to_event"`

.

We start out modeling every one of these variables as completely independent of each other using the following DAG:

```
library(data.table)
library(ggplot2)
#> Warning: Paket 'ggplot2' wurde unter R Version 4.2.3 erstellt
library(simDAG)
dag <- empty_dag() +
node_td("vaccination", type="time_to_event", prob_fun=0.001,
event_duration=21, immunity_duration=Inf) +
node_td("covid", type="time_to_event", prob_fun=0.001, event_duration=30,
immunity_duration=80) +
node_td("sickness", type="time_to_event", prob_fun=0.0001,
event_duration=2, immunity_duration=2)
```

In the DAG above, we supplied a constant value to each of the
`prob_fun`

arguments, indicating that regardless of time and
other variables, each event has a constant probability of occurring on
each day. We set the `event_duration`

of
`vaccination`

to 21, because we want to model the time after
vaccination in which the risk for the adverse side-effect (e.g. the
`sickness`

) is higher than usual later on. By setting the
`immunity_duration`

of the `vaccination`

to
`Inf`

, we are currently only allowing the person to get one
vaccination over the entire time. The `sickness`

is allowed
to occur again directly after it was over.

We can make this data-generation process a little more interesting by
making both the `vaccination`

and `covid`

have an
effect on the probability of developing the sickness. We will do this by
simply raising the probability of occurrence of the
`sickness`

by a constant factor whenever either a
`covid`

or `vaccination`

event is currently
happening. This can be done by formulating an appropriate
`prob_fun`

for the `sickness`

node:

```
prob_sickness <- function(data, rr_covid, rr_vacc, base_p) {
# multiply base probability by relevant RRs
p <- base_p * rr_vacc^(data$vaccination_event) * rr_covid^(data$covid_event)
return(p)
}
```

This works because any number to an exponent of 1 is itself, while
any number to an exponent of 0 is one. The
`vaccination_event`

and `covid_event`

columns are
always either `TRUE`

(when an event is currently happening)
or `FALSE`

(when no event is currently happening), which are
interpreted as 1 and 0 by R. Let’s update our DAG:

```
dag <- empty_dag() +
node_td("vaccination", type="time_to_event", prob_fun=0.001,
event_duration=21, immunity_duration=Inf) +
node_td("covid", type="time_to_event", prob_fun=0.001, event_duration=30,
immunity_duration=80) +
node_td("sickness", type="time_to_event", prob_fun=prob_sickness,
parents=c("vaccination_event", "covid_event"),
base_p=0.0001, rr_covid=3.5, rr_vacc=3.24,
event_duration=2, immunity_duration=2)
```

Instead of passing a constant value to the `prob_fun`

argument, we are now passing it the previously defined function. Because
our function has `base_p`

, `rr_covid`

and
`rr_vacc`

as arguments without defaults, we have to specify
those in the `node_td`

call as well. We keep the original
`base_p`

, and set the relative risks to 3.5 and 3.24
respectively. Additionally, we have to set both the
`vaccination_event`

and the `covid_event`

columns
as `parents`

now, because they are used in the
`prob_sickness`

function.

So far we assumed that the `covid`

infection probability
is unaffected by whether the person received the vaccine or not. We will
now change this by implementing a time-window after receiving the
vaccine in which the person cannot develop a `covid`

infection. Again, this can be done by defining an appropriate
`prob_fun`

function, this time for the `covid`

node:

```
prob_covid <- function(data, base_p, vacc_duration) {
p <- fifelse(data$vaccination_time_since_last < vacc_duration,
0, base_p, na=base_p)
return(p)
}
```

In this function we use the column
`vaccination_time_since_last`

, which is a column that can
optionally be created in time-to-event nodes by setting
`time_since_last`

to `TRUE`

. So let’s again update
our DAG accordingly:

```
dag <- empty_dag() +
node_td("vaccination", type="time_to_event", prob_fun=0.001,
event_duration=21, immunity_duration=Inf,
time_since_last=TRUE) +
node_td("covid", type="time_to_event", prob_fun=prob_covid,
parents=c("vaccination_time_since_last"),
base_p=0.001, vacc_duration=80, event_duration=30,
immunity_duration=80) +
node_td("sickness", type="time_to_event", prob_fun=prob_sickness,
parents=c("vaccination_event", "covid_event"),
base_p=0.0001, rr_covid=3.5, rr_vacc=3.24,
event_duration=2, immunity_duration=2)
```

Instead of just updating the `parents`

and
`prob_fun`

arguments of the `covid`

node, we now
also had to set the `time_since_last`

argument of the
`vaccination`

node to `TRUE`

as well to get the
required additional column. Our data-generation algorithm is getting
better now. But there is still a lot we can do.

In reality, very little people who were currently experiencing a
Covid-19 infection went and got the vaccine. In fact, this is absolutely
discouraged by doctors world-wide. To add this circumstance to the
model, we once again simply have to update the probability of receiving
a vaccination, by defining an appropriate `prob_fun`

:

Using this function, the probability of getting vaccinated for any
individual that is currently experiencing a `covid`

infection
is 0. Let’s update our DAG one more time to include these changes:

```
dag <- empty_dag() +
node_td("vaccination", type="time_to_event",
prob_fun=prob_vaccination,
parents=c("covid_event"), base_p=0.001,
event_duration=21, immunity_duration=Inf,
time_since_last=TRUE) +
node_td("covid", type="time_to_event", prob_fun=prob_covid,
parents=c("vaccination_time_since_last"),
base_p=0.001, vacc_duration=80, event_duration=30,
immunity_duration=80) +
node_td("sickness", type="time_to_event", prob_fun=prob_sickness,
parents=c("vaccination_event", "covid_event"),
base_p=0.0001, rr_covid=3.5, rr_vacc=3.24,
event_duration=2, immunity_duration=2)
```

Again we simply changed the `prob_fun`

argument and added
the correct `parents`

to the appropriate node. Our final
“DAG” looks like this:

Note that in this plot it doesn’t look like a classic DAG anymore,
because it has a bi-directional arrow between `covid`

and
`vaccination`

due to the time-dependent nature of their
relationship.

Suppose we are now pleased with the complexity of our data-generation
algorithm and want to simulate data from it. We can do this by simply
calling the `sim_discrete_time()`

function on the specified
DAG:

```
set.seed(42)
sim <- sim_discrete_time(dag, n_sim=1000, max_t=800)
summary(sim)
#> A simDT object with:
#> - 1000 observations
#> - 800 distinct points in time
#> - 3 time-varying variables in total
#> - 3 time_to_event nodes
#> - 0 competing_events nodes
#> Only the last state of the simulation was saved.
```

For exemplary purposes, we kind of arbitrarily used 1000 individuals
and let the simulation run for 800 days. By calling the
`plot()`

method, we get a concise overview over the process
we simulated:

A more useful output of the resulting data can be obtained using the
`sim2data()`

function. For example, we could transform the
output to the start-stop format:

```
sim2data(sim, to="start_stop")
#> .id start stop vaccination covid sickness
#> <int> <int> <num> <lgcl> <lgcl> <lgcl>
#> 1: 1 1 178 FALSE FALSE FALSE
#> 2: 1 179 199 TRUE FALSE FALSE
#> 3: 1 200 800 FALSE FALSE FALSE
#> 4: 2 1 501 FALSE FALSE FALSE
#> 5: 2 502 531 FALSE TRUE FALSE
#> ---
#> 3466: 1000 1 47 FALSE FALSE FALSE
#> 3467: 1000 48 49 FALSE FALSE TRUE
#> 3468: 1000 50 131 FALSE FALSE FALSE
#> 3469: 1000 132 152 TRUE FALSE FALSE
#> 3470: 1000 153 800 FALSE FALSE FALSE
```

As can be seen, we managed to implement a fairly complex data-generation mechanism using only a few small function definitions and a few lines of code, allowing us to generate a complex dataset with three interdependent time-varying variables with only minimal effort.

There is no need to stop here. We could make this simulation model even more complex by implementing any of the following things:

- Adding time-dependent base-probabilities for
`vaccination`

,`covid`

and`sickness`

- Adding different kinds of
`vaccination`

s, perhaps with different effects on`covid`

and/or`sickness`

- Adding time-fixed variables such as
`sex`

which have an effect on any of the other variables - Allowing multiple vaccinations
- Changing the constant raising of the probabilities in the form of a relative risk to a more realistic non-linear time-dependent relative risk

There are of course many more possible extensions, all of which can
be implemented by augmenting the respective `prob_fun`

arguments and updating the `dag`

accordingly. In fact, in the
real monte-carlo simulation we conducted, that is exactly what we did.
We used empirical data to model time-dependent base-probabilities and
more. How much complexity you really need is completely up to you. We
hope that the `simDAG`

package can help you with whatever you
need.

Banks, Jerry, John S. Carson II, Barry L. Nelson, and David M. Nicol (2014). Discrete-Event System Simulation. Vol. 5. Edinburgh Gate: Pearson Education Limited.

Denz, Robin, Katharina Meiszl, Peter Ihle, Doris F. Oberle, Ursula Drechsel-Bäuerle, Katrin Scholz, Ingo Meyer and Nina Timmesfeld (2023). “Impact of Record-Linkage Errors in Covid-19 Vaccine-Safety Analyses using German Health-Care Data: A Simulation Study”. In: arXiv:2310.15016