# 1 Overview

An N-state partitioned survival model (PSM) simulates the probability that a patient is in each of N distinct health states at a given point of time when treated by a particular therapy. State membership is estimated from a set of non-mutually exclusive survival curves; for an N-state model, N-1 survival curves are required.

The cumulative survival function, $$S(t)_n$$, represents the probability that a patient survives to health state $$n$$ or to a lower indexed state beyond time $$t$$. The probability that a patient is in health state 1 is always just $$S(t)_1$$. State membership in health states $$2,\ldots, n-1$$ is calculated as $$S(t)_{n} - S(t)_{n-1}$$. Finally, the probability of being in the final health state $$n$$ (i.e., the dead state) is $$1-S(t)_{n-1}$$, or one minus the overall survival curve.

In hesim, an N-state PSM consists of three types of separate statistical models: a set of N-1 survival models, a model of the utility values to assign to each health state, and a set of models for costs to assign to each health state. Multiple cost models can be used to categorize costs into different categories (e.g., costs of medical care, drug costs). All models can either be fit using R or constructed with external data.

Separate survival curves are predicted for each treatment strategy and patient. That is, the probability of survival given a survival model with $$L$$ parameters, $$\alpha_1, \ldots, \alpha_L$$, for patient $$i$$ at time $$t$$ with treatment strategy $$k$$ is given by,

\begin{aligned} S_{ik}(t) &= 1 - F(y_{ik} | \alpha_1(x_{1, ik}), \ldots, \alpha_L(x_{L, ik})),\\ \alpha_l &= g^{-1}(x_{l,ik}^T \beta_l), \end{aligned}

where $$F(\cdot)$$ is the cumulative distribution function for a non-negative continuous variable, $$y$$, reflecting time to an event, $$g^{-1}(\cdot)$$ is an inverse transformation function, $$\beta_l$$, is a vector of coefficients and $$x_l$$ is an input vector of covariates. Likewise, we predict utility and cost values, $$v$$, in health state $$h$$ for patient $$i$$ with treatment strategy $$k$$,

\begin{aligned} \hat{v} &= E(v_{hik} | \alpha_1(x_{1, ikl}), \ldots, \alpha_L(x_{L, hik})),\\ \alpha_l &= g^{-1}(x_{l, hik}^T \beta_l). \end{aligned} Quality-adjusted life-years (QALYs) and total costs associated with a given health state for a given treatment strategy and patient are calculated by integrating the “weighted” probability of survival, where weights are a function of the discount factor and predicted state values. Mathematically, QALYs and total costs for the $$m$$th cost category are calculated as,

\begin{aligned} \rm{QALYs}_{hik} &= \int_{0}^{T} q_{hik} e^{-rt} p_{hik}(t)dt, \\ \rm{Costs}_{m,hik} &= \int_{0}^{T} c_{m,hik} e^{-rt} p_{hik}(t)dt, \end{aligned} where $$t$$ denotes time (typically a year), $$q$$ is a quality-of-life weight, $$c$$ represents (assuming that time is measured in years) annualized costs, $$r$$ is the discount rate, and $$p(t)$$ is the probability of being in a given health state at a given time, and QALYs and costs are calculated over $$T$$ time periods.

# 2 An example 4-state PSM

To illustrate a partitioned survival analysis, consider the evaluation of a two-line sequential treatment strategy in oncology using a 4-state PSM. The 4 health states might be:

• Progression free on first line treatment
• Progression free on second line treatment
• Progressed on second line treatment
• Death

We begin by defining the population, treatment strategies, and model structure. In this example, we model three treatment strategies, three distinct patients (that differ in age and gender), and four health states (death and three non-death states). This information is combined using hesim_data(), which creates a general object of class hesim_data.

library("data.table")
library("hesim")
set.seed(101)
strategies_dt <- data.table(strategy_id = c(1, 2, 3))
patients_dt <- data.table(patient_id = seq(1, 3),
age = c(45, 50, 60),
female = c(0, 0, 1))
states_dt <- data.frame(state_id =  seq(1, 3),
state_name = c("Progression free (1st line)",
"Progression free (2nd line)",
"Progressed (2nd line)"),
stringsAsFactors = FALSE)
hesim_dat <- hesim_data(strategies = strategies_dt,
patients = patients_dt,
states = states_dt)
print(hesim_dat)
## $strategies ## strategy_id ## 1: 1 ## 2: 2 ## 3: 3 ## ##$patients
##    patient_id age female
## 1:          1  45      0
## 2:          2  50      0
## 3:          3  60      1
##
## $states ## state_id state_name ## 1 1 Progression free (1st line) ## 2 2 Progression free (2nd line) ## 3 3 Progressed (2nd line) ## ## attr(,"class") ## [1] "hesim_data" # 3 Parameter estimation ## 3.1 Survival models The parameters of the survival models would typically be estimated in one of two ways. First, if access to patient level data from a clinical trial were available, survival models would be fit using available R packages. Second, a formal evidence synthesis—such as a network meta-analysis—might be conducted. Here, we provide an example of an analysis with trial level data. We consider an estimation dataset (surv_est_data) with three survival endpoints with endpoint 1 denoting the lowest indexed state, endpoint 2 the next indexed state, and endpoint 3 representing overall survival. In our example, endpoint 1 represents progression on 1st line treatment, endpoint 2 represents progression on 2nd line treatment, and endpoint 3 represents death. Survival models in hesim can be fit using either flexsurvreg or flexsurvspline from the flexsurv package. hesim currently supports parametric (exponential, Weibull, Gompertz, gamma, log-logistic, lognormal, and generalized gamma), splines, and fractional polynomial survival models (see params_surv). In this example, we fit a Weibull model to each of the three survival endpoints. library("flexsurv") surv_est_data <- psm4_exdata$survival
fit1 <- flexsurv::flexsurvreg(Surv(endpoint1_time, endpoint1_status) ~ age + female +
factor(strategy_id),
data = surv_est_data,
dist = "weibull")
fit2 <- flexsurv::flexsurvreg(Surv(endpoint2_time, endpoint2_status) ~ age + female +
factor(strategy_id),
data = surv_est_data,
dist = "weibull")
fit3 <- flexsurv::flexsurvreg(Surv(endpoint3_time, endpoint3_status) ~ age + female +
factor(strategy_id),
data = surv_est_data,
dist = "weibull")
psfit_wei <- flexsurvreg_list(fit1, fit2, fit3)

## 3.2 Utility model

Although utility and cost models can be fit using regression models, in many cases, we won’t fit a model directly, but will use estimated mean values from published sources. In this case, we can use a stateval_tbl() to store estimates. For instance, consider an example where we are only given a range of values for utility and we therefore approximate the posterior distribution of mean utility values by health state using a uniform distribution.

psm <- Psm$new(survival_models = survmods, utility_model = utilitymod, cost_models = list(medical = medcostmod, drug = drugcostmod)) # 5 Simulating outcomes ## 5.1 Simulating survival curves Survival curves are predicted by treatment strategy and patient with the function $sim_survival(). The predicted curves are returned as a tidy data.table to facilitate plotting and stored as a field in the class named survival_. Below we plot the mean survival curve for patient 2 with treatment strategy 3.

# Simulate
times <- seq(0, 10, by = .1)
psm$sim_survival(t = times) # Plot library("ggplot2") surv_means <- psm$survival_[, .(mean_surv = mean(survival)),
by = c("curve", "strategy_id", "patient_id", "t")]
theme_set(theme_bw())
p_data <- surv_means[strategy_id == 3 & patient_id == 2]
p_data[, curve := factor(curve)]
p_data[, mean_surv_min := c(rep(0, length(times)), p_data[curve !=3, mean_surv])]
ggplot(p_data, aes(x = t, y = mean_surv, fill = curve)) +
geom_line(aes(col = curve)) +
geom_ribbon(aes(ymin = mean_surv_min, ymax = mean_surv), alpha = .65) +
xlab("Years") + ylab("Proportion surviving") +
scale_color_discrete(name = "Survival curve") +
guides(fill = FALSE) +
theme(legend.position = "bottom")

## 5.2 Simulating health state probabilities

After simulating survival, we can calculate the probability of being in each of the four health states using $sim_stateprobs(). At a given point in time, the probability of being in state 1 is the survival probability from the first survival curve, the probability of being in state 2 is the difference in survival probabilities between the 2nd and 1st curves, the probability of being in state 3 is the difference in survival probabilities between the 3rd and 2nd curves, and the probability of death is 1 minus the survival probability from curve 3. We plot these state probabilities for the first patient with the 30th randomly sampled parameter set. # Simulate psm$sim_stateprobs()

# Plot
stateprobs <- psm$stateprobs_[sample == 30 & patient_id == 1] stateprobs[, state := factor(state_id, levels = rev(unique(state_id)))] stateprobs[, strategy := factor(strategy_id, labels = c("Strategy 1", "Strategy 2", "Strategy 3"))] ggplot(stateprobs[strategy_id %in% c(1, 2)], aes(x = t, y = prob, fill = state, group = state)) + geom_area(alpha = .65) + facet_wrap(~strategy) + xlab("Years") + ylab("Proportion in state") + scale_fill_manual(name = "Health state", values = c("gray92", "green4", "orange", "purple"), labels = c("Death", rev(hesim_dat$states$state_name))) + guides(fill = guide_legend(reverse = TRUE)) + theme(legend.position = "bottom") ## 5.3 Simulating costs and QALYs Finally, we can simulate discounted costs and QALYs by numerically integrating the “weighted” probabilities in $stateprobs_ as described above. Costs and QALYs are produced for each treatment strategy, patient, health state, and sampled parameter set.

# Costs
psm$sim_costs(dr = .03) head(psm$costs_)
##    sample strategy_id patient_id state_id   dr category     costs
## 1:      1           1          1        1 0.03  medical 16701.379
## 2:      1           1          1        2 0.03  medical  7633.707
## 3:      1           1          1        3 0.03  medical 17411.772
## 4:      1           1          2        1 0.03  medical 15929.483
## 5:      1           1          2        2 0.03  medical  7408.622
## 6:      1           1          2        3 0.03  medical 16593.823
# QALYs
psm$sim_qalys(dr = .03) head(psm$qalys_)
##    sample strategy_id patient_id state_id   dr category     qalys
## 1:      1           1          1        1 0.03    qalys 0.5285081
## 2:      1           1          1        2 0.03    qalys 0.1752568
## 3:      1           1          1        3 0.03    qalys 0.3472255
## 4:      1           1          2        1 0.03    qalys 0.5040817
## 5:      1           1          2        2 0.03    qalys 0.1700892
## 6:      1           1          2        3 0.03    qalys 0.3309139`