4.1.1 Disease model

We begin by constructing the model for health state transitions, which is a function of input data (i.e., covariates) and a fitted multi-state model (or a parameter object). When separate multi-state models are fit by transition, the input data consists of one observation for each treatment strategy and patient combination (joint models consist of one observation for each treatment strategy, patient, and transition combination). It can be created easily by using the expand() function to expand the hesim_data() object created above.

transmod_data <- expand(hesim_dat, 
                        by = c("strategies", "patients"))
head(transmod_data)

##    strategy_id patient_id      age female
## 1:           1          1 43.14766      0
## 2:           1          2 45.34244      1
## 3:           1          3 46.70210      0
## 4:           1          4 43.52231      0
## 5:           1          5 43.11598      0
## 6:           1          6 42.37085      1

“Clock reset” and “clock forward” transition models are created by combining the fitted models and input data with the transition matrix, desired number of PSA samples, the timescale of the model, and the starting age of each patient in the simulation (by default, patients are assumed to live no longer than age 100 in the individual-level simulation).

transmod_cr <- create_IndivCtstmTrans(wei_fits_cr, transmod_data,
                                      trans_mat = tmat, n = n_samples,
                                      clock = "reset",
                                      start_age = patients$age)
transmod_cf <- create_IndivCtstmTrans(wei_fits_cf, transmod_data,
                                      trans_mat = tmat, n = n_samples,
                                      clock = "forward",
                                      start_age = patients$age)

It is a good idea to evaluate the assumptions underlying multi-state models. hesim can help facilitate these analyses since hazards ($hazard()), cumulative hazards ($cumhazard()), and state probabilities ($stateprobs()) can be easily computed. As an illustration, we will predict hazards using the maximum likelihood estimates of the Weibull model for a single patient (patient_id = 1). To do so, we create new transition models based on a subset of the dataset transmod_data used above.

# Predict hazard
transmod_data_pat1 <- transmod_data[patient_id == 1]
predict_haz <- function(fits, clock){
 transmod_cr_pat1 <- create_IndivCtstmTrans(fits, transmod_data_pat1,
                                            trans_mat = tmat, 
                                            clock = clock,
                                            point_estimate = TRUE)
  haz <- transmod_cr_pat1$hazard(t = seq(0, 20, 1))
  title_clock <- paste(toupper(substr(clock, 1, 1)), 
                       substr(clock, 2, nchar(clock)), sep="")
  haz[, clock := title_clock]
  return(haz[, ])
}

We then plot the predicted hazard by treatment strategy and timescale.

# Plot hazards
library("ggplot2")
haz <- rbind(predict_haz(wei_fits_cr, "reset"),
             predict_haz(wei_fits_cf, "forward"))
haz[, trans_name := factor(trans,
                           levels = 1:4,
                           labels = c("Healthy-> Sick", 
                                      "Healthy -> Dead",
                                      "Sick -> Healthy",
                                      "Sick -> Dead"))]
ggplot(haz[t > 0], 
       aes(x = t, y = hazard, col = clock, linetype = factor(strategy_id))) + 
  geom_line() + 
  facet_wrap(~trans_name) +
  xlab("Years") + ylab("Hazard") +
  scale_linetype_discrete(name = "Strategy") +
  scale_color_discrete(name = "Clock") + theme_bw()

While the hazards from the healthy state are similar between the “clock forward” and “clock reset” approaches, they differ significantly in the sick state. Treatment effects (i.e., the hazard ratios between treatment strategies 1 and 2) are also largest in the sick state.

Additional analyses should be conducted as well. For instance, the hazards for treatment strategy 1 (the reference treatment strategy) could be assessed by comparing the Weibull model’s predictions with predictions from non-parametric (i.e., the Kaplan-Meier estimator) or semi-parametric (i.e., Cox) models. This can be performed using mstate, which can predict cumulative hazards and state probabilities in non-parametric and semi-parametric models. Furthermore, the Weibull model’s proportional hazards assumption should be tested using standard techniques such as plots of log time vs. the log cumulative hazard, inclusion of time-dependent covariates, and tests of the Schoenfeld residuals.

4.1.2 Utility and cost models

Mean only models (see params_mean()) can be created directly from the utility and cost tables using since they do not include covariates and therefore do not require input data.

# Utility
utilitymod <- create_StateVals(utility_tbl, n = n_samples)

# Costs
drugcostmod <- create_StateVals(drugcost_tbl, n = n_samples)
medcostmod <- create_StateVals(medcost_tbl, n = n_samples)
costmods <- list(Drug = drugcostmod,
                 Medical = medcostmod)

4.1.3 Combining the disease progression, cost, and utility models

Now that the necessary transition, utility,and cost models have been created, we combine them to create separate economic models based on the “clock reset” and “clock forward” transition models, respectively.

econmod_cr <- IndivCtstm$new(trans_model = transmod_cr,
                             utility_model = utilitymod,
                             cost_models = costmods)
econmod_cf <- IndivCtstm$new(trans_model = transmod_cf,
                             utility_model = utilitymod,
                             cost_models = costmods)

4.2.1 Disease progression

Disease progression can be simulated using the $sim_disease() method. In the individual-level simulation, unique trajectories through the multi-state model are simulated for each patient, treatment strategy, and PSA sample. Patients transition from an old health state that was entered at time time_start to a new health state at time time_stop.

# "Clock reset"
econmod_cr$sim_disease()
head(econmod_cr$disprog_)

##    sample strategy_id patient_id from to final time_start  time_stop
## 1:      1           1          1    1  3     1   0.000000 4.13418530
## 2:      1           1          2    1  3     1   0.000000 1.46575756
## 3:      1           1          3    1  2     0   0.000000 1.13072166
## 4:      1           1          3    2  1     0   1.130722 7.43575179
## 5:      1           1          3    1  3     1   7.435752 8.26788036
## 6:      1           1          4    1  2     0   0.000000 0.05154252

# "Clock forward"
econmod_cf$sim_disease()

State occupancy probabilities at different time points are computed using $sim_stateprobs(). First, we simulate state probabilities for the “clock reset” model.

econmod_cr$sim_stateprobs(t = seq(0, 20 , 1/12)) 

We can then compare state probabilities between the competing treatment strategies.

# Short funtion add create state name variable to data.tabale
add_state_name <- function(x){
  x[, state_name := factor(state_id,
                           levels = 1:nrow(tmat),
                           labels = colnames(tmat))] 
}

# Short function to create state probability "dataset" for plotting
summarize_stprobs <- function(stateprobs){
  x <- stateprobs[, .(prob_mean = mean(prob)),
                  by = c("strategy_id", "state_id", "t")]
  add_state_name(x)
}

# Plot of state probabilities
stprobs_cr <- summarize_stprobs(econmod_cr$stateprobs_)
ggplot(stprobs_cr, aes(x = t, y = prob_mean, col = factor(strategy_id))) +
  geom_line() + facet_wrap(~state_name) + 
  xlab("Years") + ylab("Probability in health state") +
  scale_color_discrete(name = "Strategy") +
  theme(legend.position = "bottom") +
  theme_bw()

Next, we compare the state probabilities from the “clock reset” and “clock forward” models.

econmod_cf$sim_stateprobs(t = seq(0, 20 , 1/12)) 
stprobs_cf <- summarize_stprobs(econmod_cf$stateprobs_)

# Compare "clock forward" and "clock reset" cases
stprobs <- rbind(data.table(stprobs_cf, clock = "Forward"),
                 data.table(stprobs_cr, clock = "Reset"))
ggplot(stprobs[strategy_id == 1], 
       aes(x = t, y = prob_mean, col = clock)) +
  geom_line() + facet_wrap(~state_name) + 
  xlab("Years") + ylab("Probability in health state") +
  scale_color_discrete(name = "Clock") +
  theme(legend.position = "bottom") +
  theme_bw()

The probabilities are generally quite similar, implying that the choice of timescale has a small impact on the results. This is not unexpected given that patients spend considerably more time in the healthy state and the predicted hazard rates are very similar in the healthy state.

4.2.2 QALYs

QALYs (and life-years) are simulated using $sim_qalys(). By default, mean QALYs are computed by treatment strategy, health state, and PSA sample (the by_patient option can be used to compute aggregated QALYs at the patient level). Here, we used the “clock reset” model to compute both undiscounted QALYs (dr = 0) and QALYs discounted at 3%.

econmod_cr$sim_qalys(dr = c(0,.03))
head(econmod_cr$qalys_)

##    sample strategy_id state_id dr     qalys       lys
## 1:      1           1        1  0 3.4835421 5.4570542
## 2:      1           1        2  0 0.4112720 1.3869591
## 3:      1           2        1  0 3.9301615 6.1566943
## 4:      1           2        2  0 0.2502997 0.8441018
## 5:      2           1        1  0 4.2161973 5.6976188
## 6:      2           1        2  0 1.2866570 1.6265598

We summarize the simulated QALYs by computing means by treatment strategy and health state across the PSA samples.

qalys_summary <- econmod_cr$qalys_[, .(mean = mean(qalys)),
                                    by = c("strategy_id", "state_id", "dr")]
add_state_name(qalys_summary)
ggplot(qalys_summary[dr == .03],
       aes(x = factor(strategy_id), y = mean, fill = state_name)) + 
  geom_bar(stat = "identity") +
  scale_fill_discrete(name = "") +
  xlab("Strategy") + ylab("Mean QALYs") +
  theme_bw()

4.2.3 Costs

Costs are computed in the same way as QALYs, except that they are computed by category. We use the “clock reset” model and a 3% discount rate.

econmod_cr$sim_costs(dr = 0.03)
head(econmod_cr$costs_)

##    sample strategy_id state_id   dr category     costs
## 1:      1           1        1 0.03     Drug 22422.281
## 2:      1           1        2 0.03     Drug  5725.925
## 3:      1           2        1 0.03     Drug 49847.146
## 4:      1           2        2 0.03     Drug  7152.902
## 5:      2           1        1 0.03     Drug 23237.139
## 6:      2           1        2 0.03     Drug  6568.404

As with QALYs, we summarize costs by computing means (now by treatment strategy and category) across the PSA samples.

library("scales")
costs_summary <- econmod_cr$costs_[dr == .03 , .(mean = mean(costs)),
                                   by = c("strategy_id", "category")]
ggplot(costs_summary,
       aes(x = factor(strategy_id), y = mean, fill = category)) + 
  geom_bar(stat = "identity") +
  scale_fill_discrete(name = "Category") +
  scale_y_continuous(label = scales::dollar_format()) +
  xlab("Strategy") + ylab("Mean costs") +
  theme_bw()