Example for a Sick-Sicker-Dead model'

In WARDEN: Workflows for Health Technology Assessments in R using Discrete EveNts

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  echo = TRUE
)

Introduction

This document runs a discrete event simulation model in the context of a late oncology model to show how the functions can be used to generate a model in only a few steps.

When running a DES, it's important to consider speed. Simulation based models can be computationally expensive, which means that using efficient coding can have a substantial impact on performance.

Main options

library(WARDEN)

library(dplyr)
library(ggplot2)
library(kableExtra)
library(purrr)

options(scipen = 999)
options(digits=3)
options(tibble.print_max = 50)

General inputs with delayed execution

Initial inputs and flags that will be used in the model can be defined below. We can define inputs that will only change across scenarios (sensitivity_inputs), inputs which are common to all patients (common_all_inputs) within a simulation, inputs that are unique to a patient independently of the treatment (e.g. natural time to death, defined in common_pt_inputs), and inputs that are unique to that patient and that treatment (unique_pt_inputs). Items can be included through the add_item function, and can be used in subsequent items. All these inputs are generated before the events and the reaction to events are executed. Furthermore, the program first executes common_all_inputs, then common_pt_inputs and then unique_pt_inputs. So one could use the items generated in common_all_inputs in unique_pt_inputs. Note that inputs are "reset" after each patient, so if patient 1 arm "noint" changes util.sick to be = 2, even if it's a common parameter for everyone, it would be reset to 1 for patient 1 arm "int".

Note that time to death is set in the common_pt_inputs, but it could also just be set in the add_tte function explained below. The user has full flexibility on how to implement this type of inputs.

There are some auxiliary functions to help setting up inputs, like pick_val_v and pick_val (and pick_psa, see below the section on Sensitivity Analysis). Note that pick_val_v and pick_val can be directly loaded as parameters (in fact, a named list will be loaded directly by R). A small tweak is needed if it's the first item added, in which the item list must be initiated by using add_item() (see below).

#We don't need to use sensitivity_inputs here, so we don't add that object

#Put objects here that do not change on any patient or intervention loop
common_all_inputs <-add_item(
                      util.sick = 0.8,
                      util.sicker = 0.5,
                      cost.sick = 3000,
                      cost.sicker = 7000,
                      cost.int = 1000,
                      coef_noint = log(0.2),
                      HR_int = 0.8,
                      drc = 0.035, #different values than what's assumed by default
                      drq = 0.035,
                      random_seed_sicker_i = sample.int(100000,1000,replace = FALSE)
)  #to be used as seeds to draw the time to event for sicker, to ensure same luck for the same patient independently of the arm


#Put objects here that do not change as we loop through treatments for a patient
common_pt_inputs <- add_item(death= max(0.0000001,rnorm(n=1, mean=12, sd=3))) 

#Put objects here that change as we loop through treatments for each patient (e.g. events can affect fl.tx, but events do not affect nat.os.s)
unique_pt_inputs <- add_item(fl.sick = 1,
                             q_default = util.sick,
                             c_default = cost.sick + if(arm=="int"){cost.int}else{0}) 

Events

Add Initial Events

Events are added below through the add_tte function. We use this function once applying to both interventions. We must define several arguments: one to indicate the intervention, one to define the names of the events used, one to define the names of other objects created that we would like to store (optional, maybe we generate an intermediate input which is not an event but that we want to save) and the actual input in which we generate the time to event. Events and other objects will be automatically initialized to Inf. We draw the times to event for the patients. Note: the order of the evts argument that appears first will be used as a reference of the order in which to process events in the case of ties (so "sick" would be processed before "sicker" if there is a tie in time to event.)

Note that the model will use the evnets defined in evts argument to look for the objects both defined in the input list and in this expression to allocate time to events. If an event is declared in evts but not defined elsewhere, then they would be assumed TTE of Inf by default.

This chunk is a bit more complex, so it's worth spending a bit of time explaining it.

The init_event_list object is populated by using the add_tte function which applies to both arms, "int" strategy and "noint" strategy. We first declare the start time to be 0. Note this could also be separated by arm if the user wants to have more clarity using two add_tte functions (i.e., add_tte(arm="noint"...) %>% add_tte(arm="int"...)).

We then proceed to generate the actual time to event. We use the draw_tte() function to generate the time to event, though one can set this up in any other way (e.g., using rexp). One should always be aware of how the competing risks interact with each other. While we have abstracted from these type of corrections here, it is recommended to have an understanding about how these affect the results and have a look at the competing risks/semi-competing risks literature.

init_event_list <- 
  add_tte(arm=c("noint","int"), evts = c("sick","sicker","death") ,input={
    sick <- 0
    sicker <- draw_tte(1,dist="exp", coef1=coef_noint, beta_tx = ifelse(arm=="int",HR_int,1), seed = random_seed_sicker_i[i]) #this way the value would be the same if it wasn't for the HR, effectively "cloning" patients luck

  })

Add Reaction to Those Events

Once the initial times of the events have been defined, we also need to declare how events react and affect each other. To do so, we use the evt_react_list object and the add_reactevt function. This function just needs to state which event is affected, and the actual reaction (usually setting flags to 1 or 0, or creating new/adjusting events).

There are a series of objects that can be used in this context to help with the reactions. Apart from the global objects and flags defined above, we can also use curtime for the current event time, prevtime for the time of the previous event, cur_evtlist for the named vector of events that is yet to happen for that patient, arm for the current treatment in the loop, evt for the current event being processed, i expresses the patient iteration, and simulation the specific simulation (relevant when the number of simulations is greater than 1). Furthermore, one can also call any other input/item that has been created before or create new ones. For example, we could even modify a cost/utility item by changing it directly, e.g. through modify_item(list(cost.idfs.tx=500)).

Item | What does it do ------------ | ------------- curtime | Current event time (numeric) prevtime | Time of the previous event (numeric) cur_evtlist | Named vector of events that is yet to happen for that patient (named numeric vector) evt | Current event being processed (character) i | Patient being iterated (numeric) arm | Intervention being iterated (character) simulation | Simulation being iterated (numeric) sens | Sensitivity analysis being iterated (numeric)

The functions to add/modify events and inputs use lists. Whenever several inputs/events are added or modified, it's recommended to group them within one function, as it reduces the computation cost. So rather than use two modify_item with a list of one element, it's better to group them into a single modify_item with a list of two elements.

new_eventallows to generate events and add them to the vector of events. It accepts more than one event. modify_event allows to modify events (e.g. delay death). When adding an event, the name of the events and the time of the events must be defined. When using modify_event, one must indicate which events are affected and what are the new times of the events. If the event specified does not exist or has already occurred, it will be ignored. modify_event with create_if_null = TRUE argument will also generate events if they don't exist. Note that one could potentially omit part of the modeling set in init_event_list and actually define new events dynamically through the reactions (we do that below for the "ae" event). However, this can have an impact in computation time, so if possible it's always better to use init_event_list.

modify_item and modify_item_seqallow to modify and add items. Elements defined within this function are not evaluated sequentially in modify_item (i.e. defining modify_item(list(fl.new = 1, var1 = fl.new * 5))) will give an error if fl.new was not defined outside this function), while modify_item_seq will do it sequentially at a slightly bigger computational cost, so both are left as choices for the user. Note that one can modify costs/utilities by using the construction type_name_category, where type is either "qaly" or "cost", name is the name (e.g., "default") and category is the category used (e.g., "instant"), so one could pass cost_default_instant and modify the cost.

The list of relevant functions to be used within add_reactevt are:

Function | What does it do | How to use it ------------ | ------------- | ------------- modify_item() | Adds & Modifies items/flags/variables for future events | modify_item(list("fl.idfs.ontx"=0,"fl.tx.beva"=0)) modify_item_seq() | Adds & Modifies items/flags/variables for future events sequentially | modify_item_seq(list("fl.idfs.ontx"=0,"fl.tx.beva"=0)) new_event() | Adds events to the vector of events for that patient | new_event(rep(list("ae"=curtime + 0.001),5)) modify_event() | Modifies (or creates, with create_if_null = TRUE) existing events by changing their time | modify_event(list("os"=curtime +5, "ttot"=curtime+0.0001))

The model will run until curtime is set to Inf, so the event that terminates the model (in this case, os), should modify curtime and set it to Inf.

Finally, note that there could be two different ways of accumulating continuous outcomes, backwards (i.e., in the example below, we would set q_default = util.sick at the sicker event, and modify the q_default value in the death event) and forwards (as in the example below). This option can be modified in the run_sim function using the accum_backwards argument, which assumes forwards by default.

evt_react_list <-
  add_reactevt(name_evt = "sick",
               input = {}) %>%
  add_reactevt(name_evt = "sicker",
               input = {
                 modify_item(list(q_default = util.sicker,
                                  c_default = cost.sicker + if(arm=="int"){cost.int}else{0},
                                  fl.sick = 0)) 
               }) %>%
  add_reactevt(name_evt = "death",
               input = {
                 modify_item(list(q_default = 0,
                                  c_default = 0, 
                                  curtime = Inf)) 
               }) 

Extract Interactions Between Events

As an additional optional step, to easily see the interactions between the reactions of the events, we can also now use the extract_from_reactions function to obtain a data.frame with all the relationships defined in the reactions in the model. This functions looks at all modify_item, modify_item_seq, modify_event and new_event and checks which elements are being defined there, their definition, and whether they are triggered conditionally (e.g., "if(a==1){modify_item(list(b=2))}"). Note it would be straightforward to build a network graph showcasing all the interactions between events in terms of events affecting other events, or to show which (and how) events affect specific items.

df_interactions <- extract_from_reactions(evt_react_list)

kable(df_interactions)

Costs and Utilities

Costs and utilities are introduced below. However, it's worth noting that the model is able to run without costs or utilities.

Utilities/Costs/Other outputs are defined by declaring which object belongs to utilities/costs/other outputs, and whether they need to be discounted continuously or discretely (instantaneous). These will be passed to the run_sim function.

Utilities

util_ongoing <- "q_default"

Costs

cost_ongoing <- "c_default"

Model

Model Execution

The model can be run using the function run_sim below. We must define the number of patients to be simulated, the number of simulations, whether we want to run a PSA or not, the strategy list, the inputs, events and reactions defined above, utilities, costs and also if we want any extra output and the level of ipd data desired to be exported.

It is worth noting that the psa_bool argument does not run a PSA automatically, but is rather an additional input/flag of the model that we use as a reference to determine whether we want to use a deterministic or stochastic input. As such, it could also be defined in common_all_inputs as the first item to be defined, and the result would be the same. However, we recommend it to be defined in run_sim.

Note that the distribution chosen, the number of events and the interaction between events can have a substantial impact on the running time of the model.

Debugging can be implemented using the argument debug in the run_sim function.

#Logic is: per patient, per intervention, per event, react to that event.
results <- run_sim(  
  npats=1000,                               # number of patients to be simulated
  n_sim=1,                                  # number of simulations to run
  psa_bool = FALSE,                         # use PSA or not. If n_sim > 1 and psa_bool = FALSE, then difference in outcomes is due to sampling (number of pats simulated)  
  arm_list = c("int", "noint"),             # intervention list
  common_all_inputs = common_all_inputs,    # inputs common that do not change within a simulation
  common_pt_inputs = common_pt_inputs,      # inputs that change within a simulation but are not affected by the intervention
  unique_pt_inputs = unique_pt_inputs,      # inputs that change within a simulation between interventions
  init_event_list = init_event_list,        # initial event list
  evt_react_list = evt_react_list,          # reaction of events
  util_ongoing_list = util_ongoing,
  cost_ongoing_list = cost_ongoing,
  ipd = 1
)

Post-processing of Model Outputs

Summary of Results

Once the model has been run, we can use the results and summarize them using the summary_results_det to print the results of the last simulation (if nsim = 1, it's the deterministic case), and summary_results_sim to show the PSA results (with the confidence intervals). We can also use the individual patient data generated by the simulation, which we collect here to plot in the psa_ipd object.

summary_results_det(results[[1]][[1]]) #print first simulation

summary_results_sim(results[[1]])

summary_results_sens(results)

psa_ipd <- bind_rows(map(results[[1]], "merged_df")) 

psa_ipd[1:10,] %>%
  kable() %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"))

We can also check what has been the absolute number of events per strategy.

psa_ipd %>% group_by(arm,evtname) %>% summarise(n=n()) %>% arrange(arm,-n)%>%
  kable() %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"))

Plots

We now use the data output to plot the histograms/densities of the simulation.

data_plot <- results[[1]][[1]]$merged_df %>%
  filter(evtname != "sick") %>%
  group_by(arm,evtname,simulation) %>%
  mutate(median = median(evttime)) %>%
  ungroup()

ggplot(data_plot) +
  geom_density(aes(fill = arm, x = evttime),
               alpha = 0.7) +
  geom_vline(aes(xintercept=median,col=arm)) +
  facet_wrap( ~ evtname, scales = "free") +
  scale_y_continuous(expand = c(0, 0)) +
  scale_x_continuous(expand = c(0, 0)) +
  theme_bw()

We can also plot the patient level incremental QALY/costs.

data_qaly_cost<- psa_ipd[,.SD[1],by=.(pat_id,arm,simulation)][,.(arm,qaly=total_qalys,cost=total_costs,pat_id,simulation)]
data_qaly_cost[,ps_id:=paste(pat_id,simulation,sep="_")]


mean_data_qaly_cost <- data_qaly_cost %>% group_by(arm) %>% summarise(across(where(is.numeric),mean))

ggplot(data_qaly_cost,aes(x=qaly, y = cost, col = arm)) + 
  geom_point(alpha=0.15,shape = 21) +
  geom_point(data=mean_data_qaly_cost, aes(x=qaly, y = cost, fill = arm), shape = 21,col="black",size=3) +
  scale_y_continuous(expand = c(0, 0)) +
  scale_x_continuous(expand = c(0, 0)) +
  theme_bw()+
  theme(axis.text.x = element_text(angle = 90, vjust = .5))

Sensitivity Analysis

Inputs

In this case, inputs must be created first to change across sensitivity analysis. To do so, the item list sensitivity_inputs can be used. In this case, we also use pick_val_v which allows the model to automatically pick the relevant value (no PSA, PSA or sensitivity analysis) based on the corresponding boolean flags of psa_bool and sensitivity_bool. In this case we also use the sens iterator for each sensitivity analysis and the n_sensitivity which is an argument in run_sim.

Note that we have then just changed how the inputs are handled in common_all_inputs, but the same could be done with unique_pt_inputs, but in those cases, as the inputs change per patient, the pick_val_v or pick_val functions should be applied within unique_pt_inputs to make sure they are evaluated when it correspond.

Note that for the psa we are directly calling the distributions and passing the parameters.Note also that the sens_name_used is automatically computed by the engine and is accesible to the user (it's the name of the sensitivity analysis, e.g., "scenario 1").

The indicator parameter in pick_val_v and pick_val is used to determine which parameters are left "as is" and which ones are to be substituted with the sensitivity value. There are two ways to do this, either by setting it in a binary way (1 or 0), or by using the indicator as the number of the parameter values to be varied (useful when several parameters are varied at the same time, or only specific values of a vector are varied). This can be set by using indicator_sens_binary argument.

Note that pick_val_v and pick_val can be directly loaded as parameters (in fact, a named list will be loaded directly by R). A small tweak is needed if it's the first item added, in which the item list must be initiated by using add_item() (see below). Note that one can use a list of lists in the case where the base_value or any of the other parameters are vectors instead of elements of length 1. In this case, we showcase a list but it could also use a data.frame.

pick_psa can be used to select the correct PSA distributions.

#Load some data
list_par <- list(parameter_name = list("util.sick","util.sicker","cost.sick","cost.sicker","cost.int","coef_noint","HR_int"),
                              base_value = list(0.8,0.5,3000,7000,1000,log(0.2),0.8),
                              DSA_min = list(0.6,0.3,1000,5000,800,log(0.1),0.5),
                              DSA_max = list(0.9,0.7,5000,9000,2000,log(0.4),0.9),
                              PSA_dist = list("rnorm","rbeta_mse","rgamma_mse","rgamma_mse","rgamma_mse","rnorm","rlnorm"),
                              a=list(0.8,0.5,3000,7000,1000,log(0.2),log(0.8)),
                              b=lapply(list(0.8,0.5,3000,7000,1000,log(0.2),log(0.8)), function(x) abs(x/5)),
                              scenario_1=list(0.6,0.3,1000,5000,800,log(0.1),0.5),
                              scenario_2=list(0.9,0.7,5000,9000,2000,log(0.4),0.9)
                              )

sensitivity_inputs <-add_item(
            indicators = if(sensitivity_bool){ create_indicators(sens,n_sensitivity*length(sensitivity_names),rep(1,length(list_par[[1]])))}else{
                                rep(1,length(list_par[[1]]))} #vector of indicators, value 0 everywhere except at sens, where it takes value 1 (for dsa_min and dsa_max, if not sensitivity analysis, then we activate all of them, i.e., in a PSA)
                              )

common_all_inputs <-add_item() %>% 
  add_item(
            pick_val_v(base        = list_par[["base_value"]],
                       psa         = pick_psa(list_par[["PSA_dist"]],rep(1,length(list_par[["PSA_dist"]])),list_par[["a"]],list_par[["b"]]),
                       sens        = list_par[[sens_name_used]],
                       psa_ind     = psa_bool,
                       sens_ind    = sensitivity_bool,
                       indicator   = indicators,
                       names_out   = list_par[["parameter_name"]]
                       )
            ) %>%
  add_item(random_seed_sicker_i = sample(1:1000,1000,replace = FALSE)) #we don't add this variable ot the sensitivity analysis

Model Execution

The model is executed as before, just adding the sensitivity_inputs, sensitivity_names, sensitivity_bool and n_sensitivity arguments. Note that the total number of sensitivity iterations is given not by n_sensitivity, but by n_sensitivity * length(sensitivity_names), so in this case it will be 2 x n_sensitivity, or 2 x 7 = 14. For two scenario analysis it would be 2 x 1 = 2, with the indicators variable defined in the previous section taking value 1 for all the variables altered in the scenario, and 0 otherwise.

results <- run_sim(  
  npats=100,                               # number of patients to be simulated
  n_sim=1,                                  # number of simulations to run
  psa_bool = FALSE,                         # use PSA or not. If n_sim > 1 and psa_bool = FALSE, then difference in outcomes is due to sampling (number of pats simulated)  
  arm_list = c("int", "noint"),             # intervention list
  common_all_inputs = common_all_inputs,    # inputs common that do not change within a simulation
  common_pt_inputs = common_pt_inputs,      # inputs that change within a simulation but are not affected by the intervention
  unique_pt_inputs = unique_pt_inputs,      # inputs that change within a simulation between interventions
  init_event_list = init_event_list,        # initial event list
  evt_react_list = evt_react_list,          # reaction of events
  util_ongoing_list = util_ongoing,
  cost_ongoing_list = cost_ongoing,
  sensitivity_inputs = sensitivity_inputs,
  sensitivity_names = c("DSA_min","DSA_max"),
  sensitivity_bool = TRUE,
  n_sensitivity = length(list_par[[1]]),
  input_out = unlist(list_par[["parameter_name"]])
)

Check results

We briefly check below that indeed the engine has been changing the corresponding parameter value.

data_sensitivity <- bind_rows(map_depth(results,2, "merged_df"))

#Check mean value across iterations as PSA is off
data_sensitivity %>% group_by(sensitivity) %>% summarise_at(c("util.sick","util.sicker","cost.sick","cost.sicker","cost.int","coef_noint","HR_int"),mean)

Model Execution, probabilistic DSA

The model is executed as before, just activating the psa_bool option

results <- run_sim(  
  npats=100,                               
  n_sim=6,                                  
  psa_bool = TRUE,                         
  arm_list = c("int", "noint"),             
  common_all_inputs = common_all_inputs,    
  common_pt_inputs = common_pt_inputs,      
  unique_pt_inputs = unique_pt_inputs,      
  init_event_list = init_event_list,        
  evt_react_list = evt_react_list,          
  util_ongoing_list = util_ongoing,
  cost_ongoing_list = cost_ongoing,
  sensitivity_inputs = sensitivity_inputs,
  sensitivity_names = c("DSA_min","DSA_max"),
  sensitivity_bool = TRUE,
  n_sensitivity = length(list_par[[1]]),
  input_out = unlist(list_par[["parameter_name"]])
)

Check results

We briefly check below that indeed the engine has been changing the corresponding parameter value.

data_sensitivity <- bind_rows(map_depth(results,2, "merged_df"))

#Check mean value across iterations as PSA is off
data_sensitivity %>% group_by(sensitivity) %>% summarise_at(c("util.sick","util.sicker","cost.sick","cost.sicker","cost.int","coef_noint","HR_int"),mean)

Model Execution, Simple PSA

The model is executed as before, just activating the psa_bool option and deactivating the sensitivity_bool and removing sensitivity_names and setting n_sensitivity = 1

results <- run_sim(  
  npats=100,                               
  n_sim=10,                                  
  psa_bool = TRUE,                         
  arm_list = c("int", "noint"),             
  common_all_inputs = common_all_inputs,    
  common_pt_inputs = common_pt_inputs,      
  unique_pt_inputs = unique_pt_inputs,      
  init_event_list = init_event_list,        
  evt_react_list = evt_react_list,          
  util_ongoing_list = util_ongoing,
  cost_ongoing_list = cost_ongoing,
  sensitivity_inputs = sensitivity_inputs,
  sensitivity_bool = FALSE,
  n_sensitivity = 1,
  input_out = unlist(list_par[["parameter_name"]])
)

Check results

We briefly check below that indeed the engine has been changing the corresponding parameter values.

data_simulation <- bind_rows(map_depth(results,2, "merged_df"))

#Check mean value across iterations as PSA is off
data_simulation %>% group_by(simulation) %>% summarise_at(c("util.sick","util.sicker","cost.sick","cost.sicker","cost.int","coef_noint","HR_int"),mean)

WARDEN documentation built on April 11, 2025, 5:41 p.m.