1 Overview

When discrete time data is collected at evenly spaced intervals, cohort discrete time state transition models (cDTSTMs)—often referred to as Markov cohort models—can be parameterized using multinomial logistic regression. Separate multinomial logit model are estimated for each health state and predict the probability of transitioning from that state to all other states. Mathematically, the probability of a transition from state \(r\) at model cycle \(t\) to state \(s\) at model cycle \(t+1\) is given by,

\[ Pr(y_{t+1} = s| y_t = r) = \frac{e^{x_r\beta_{rs}}}{\sum_{h=1}^{H}e^{x_r\beta_{rh}}} \]

2 An example 3-state model

We illustrate by considering an illness-death model with 3 generic health states: (1) Healthy, (2) Sick, and (3) Death We will assume that patients can only transition to a more severe health state:

1. Healthy to Sick
2. Healthy to Death
3. Sick to Death

The transitions of this model can be characterized with a transition matrix with the number of rows and columns equal to the number of health states. In the multinomial logistic regression case, the reference category in each multinomial logit fit is assigned a value of zero. Elements representing transitions that are not possible are NA. All other transitions are represented with integer values from 1 to \(K_r -1\) where \(K_r\) is the number of states in the multinomial logit model for state \(r\).

tmat <- rbind(c(0, 1, 2),
              c(NA, 0, 1),
              c(NA, NA, NA))
colnames(tmat) <- rownames(tmat) <- c("Healthy", "Sick", "Death")
print(tmat)

##         Healthy Sick Death
## Healthy       0    1     2
## Sick         NA    0     1
## Death        NA   NA    NA

The cost-effectiveness analysis will compare two treatment strategies—the “Intervention” and the “Reference” treatment. Separate multinomial logistic regressions are fit for patients starting in the healthy state and the sick state using the multinom3_exdata example dataset, which tracks patient health states at yearly time intervals in a longitudinal fashion.

library("hesim")
library("data.table")
transitions_data <- data.table(multinom3_exdata$transitions)
head(transitions_data)

##    patient_id strategy_id strategy_name      age        age_cat female  year
##         <int>       <int>        <fctr>    <num>         <fctr>  <int> <num>
## 1:          1           1     Reference 37.68882       Age < 40      0     1
## 2:          2           1     Reference 20.17811       Age < 40      0     1
## 3:          3           1     Reference 62.81725      Age >= 60      0     1
## 4:          4           2  Intervention 58.12408 40 <= Age < 60      0     1
## 5:          5           2  Intervention 30.72299       Age < 40      1     1
## 6:          6           1     Reference 33.49259       Age < 40      0     1
##    state_from state_to year_cat
##        <fctr>   <fctr>   <fctr>
## 1:    Healthy     Sick Year < 3
## 2:    Healthy  Healthy Year < 3
## 3:    Healthy  Healthy Year < 3
## 4:    Healthy     Dead Year < 3
## 5:    Healthy     Sick Year < 3
## 6:    Healthy  Healthy Year < 3

In a typical Markov cohort model, either a single cohort or a small number of cohorts is simulated. However, one advantage of using an explicit statistical model to predict transition probabilities is that a large heterogeneous cohort of patients can be easily simulated. Here, we will simulate 100 representative patients (i.e., cohorts), which arguably better captures the underlying heterogeneity in the population. To do so, we will randomly sample 100 patients from the multinom3_exdata dataset. In our data, we observe differences in age and gender.

n_patients <- 100
patients <- transitions_data[year == 1, .(patient_id, age, female)][
  sample.int(nrow(transitions_data[year == 1]), n_patients)][
  , grp_id := 1:n_patients]
head(patients)

##    patient_id      age female grp_id
##         <int>    <num>  <int>  <int>
## 1:       6892 85.66808      1      1
## 2:       7340 25.41347      1      2
## 3:       1829 67.57968      0      3
## 4:       9574 23.10851      1      4
## 5:       9586 23.24127      1      5
## 6:       2788 88.16852      1      6

To perform the cost-effectiveness analysis, each patient is simulated twice, once with the reference treatment and once with the intervention. The non-death states use for simulating costs and quality-adjusted life-years (QALYs) are the healthy and sick states.

hesim_dat <- hesim_data(
  patients = patients,
  strategies = data.table(strategy_id = 1:2,
                          strategy_name = c("Reference", "Intervention")),
  states = data.table(state_id = c(1, 2),
                     state_name = rownames(tmat)[1:2]) # Non-death health states
)

We can use get_labels() to create nice labels for the plots and summary tables.

labs <- get_labels(hesim_dat)
print(labs)

## $strategy_id
##    Reference Intervention 
##            1            2 
## 
## $state_id
## Healthy    Sick   Death 
##       1       2       3

3 Parameter estimation

library("nnet")
library("splines")

# Transitions from healthy state
data_healthy <- transitions_data[state_from == "Healthy"]
fit_healthy <- multinom(state_to ~ strategy_name + female + 
                          ns(age, df = 5) + year_cat, 
                        data = data_healthy, trace = FALSE)

# Transitions from sick state
data_sick <- droplevels(transitions_data[state_from == "Sick"])
fit_sick <- multinom(state_to ~ strategy_name + female + 
                       ns(age, df = 5) + year_cat, 
                     data = data_sick, trace = FALSE)

transfits <- multinom_list(healthy = fit_healthy, sick = fit_sick)

utility_tbl <- stateval_tbl(multinom3_exdata$utility,
                            dist = "beta")
head(utility_tbl)

##    state_id state_name  mean        se
##       <num>     <char> <num>     <num>
## 1:        1    Healthy  0.90 0.1732051
## 2:        2       Sick  0.65 0.2000000

drugcost_tbl <- stateval_tbl(multinom3_exdata$costs$drugs,
                            dist = "fixed")
medcost_tbl <- stateval_tbl(multinom3_exdata$costs$medical,
                            dist = "gamma")

4 Simulation

The economic model consists of a model for disease progression and models for assigning utility and cost values to health states. Since we are performing a PSA, we must specify the number of times to sample the parameters.

n_samples <- 100

tintervals <- time_intervals(unique(transitions_data[, .(year_cat)])
                             [, time_start := c(0, 2, 6)])
transmod_data <- expand(hesim_dat, times = tintervals)
transmod <- create_CohortDtstmTrans(transfits,
                                    input_data = transmod_data,
                                    trans_mat = tmat,
                                    n = n_samples,
                                    uncertainty = "normal")

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

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

econmod <- CohortDtstm$new(trans_model = transmod,
                           utility_model = utilitymod,
                           cost_models = costmods)

econmod$sim_stateprobs(n_cycles = 20)
autoplot(econmod$stateprobs_, labels = labs,
         ci = TRUE, ci_style = "ribbon")

econmod$sim_qalys(dr = .03)
econmod$sim_costs(dr = .03)

5 Decision analysis

Now that costs and QALYs have been simulated with the PSA, a decision analysis can be performed. We will focus on the pairwise comparison between the “Intervention” and the “Reference” treatment.

ce_sim <- econmod$summarize(by_grp = TRUE)
cea_pw_out <- cea_pw(ce_sim, comparator = 1, dr_qalys = .03, dr_costs = .03)

Since we modeled the health state transitions as a function of covariates, transition rates vary by patient characteristics. The impact of gender and the (nonlinear) impact of age on transitions have a similar effect on the incremental cost-effectiveness ratio (ICER). The ICERs were lower for females and tended to rise with age, suggesting that the “Intervention” was more effective for patients with a lower baseline risk.

icers <- merge(cea_pw_out$summary,
               hesim_dat$patients[, .(grp_id, age, female)],
               by = "grp_id")
icers[, gender := factor(female, 
                         levels = c(0, 1),
                         labels = c("Male", "Female"))]

# Plot of ICER by demographics
library("ggplot2")
ggplot(icers, aes(x = age, y = as.numeric(gsub(",", "", icer)), col = gender)) +
  geom_point() +
  xlab("Age") + ylab("Incremental cost-effectiveness ratio") +
  scale_y_continuous(label = scales::dollar_format()) +
  scale_colour_discrete(name = "") + 
  theme_bw()

Of note, the heterogeneous ICER arises completely through differences in baseline risk since the effect of the “Intervention” relative to the “Reference” on transitions was assumed to be constant across patients (i.e., the treatment strategy covariate did not have any effect modifiers). This example highlights an advantage of modeling disease progression with a statistical model in that it can facilitate analyses that assess the consequences of patient heterogeneity.