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+
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+
+ + +Content not found. Please use links in the navbar. + +
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+

Site built with pkgdown 1.5.1.

+
+ +
+
+ + + + + + + + diff --git a/docs/articles/expected-values.html b/docs/articles/expected-values.html new file mode 100644 index 00000000..55523635 --- /dev/null +++ b/docs/articles/expected-values.html @@ -0,0 +1,183 @@ + + + + + + + +Simulating costs and quality-adjusted life-years • hesim + + + + + + + + + + +
+
+ + + + +
+
+ + + + +
+

+Overview

+

Before performing a decision analysis, (discounted) costs and quality-adjusted life-years (QALYs) must be simulated. In cohort models, they are simulated as a function of previously simulated state occupancy probabilities. In individual-level models, they are simulated as a function of simulated trajectories characterizing disease progression. In hesim, the discrete time state transition and partitioned survival models are cohort models and the continuous time state transition model is an individual-level model.

+
+
+

+Cohort Models

+

Costs and QALYs in cohort models are computed by integrating the “weighted” probability of being in each state. Weights are a function of the discount factor and state values (e.g., annualized costs and utility) predicted using either the cost or utility model. Mathematically, for a time horizon \(T\), discounted costs and QALYs in health state \(h\) are computed as,

+

\[ +\int_0^{T} z_h(t) e^{-rt} P_h(t) dt, +\]

+

where \(z_h(t)\) is the predicted cost or utility value at time \(t\), \(r\) is the discount rate, and \(P_h(t)\) is the probability of being in a given health state. Note that the state values, \(z_h(t)\), can depend on time since the start of the model but not on time since entering a new health state.

+
+

+Integration methods

+

Three types of approaches are currently available for numerical integration given values of state probabilities at distinct discrete times.

+
    +
  • +Left Riemann sum: The function is approximated by its value at the left most point (i.e., start) of each time interval.
  • +
  • +Right Riemann sum: The function is approximated by its value at the right most point (i.e., end) of each time interval.
  • +
  • +Trapezoid rule: The function is approximated by the average of its values at the left and right endpoints. Specifically, for an interval [\(t_1\), \(t_2\)] with value \(y_1\) at the left endpoint and \(y_2\) at the right endpoint, the function is approximated as \(\frac{t_2 - t_1}{2}(y_1 + y_2)\).
  • +
+

The Riemann sum rules approximate the area under the curve using rectangles in each time interval whiles the trapezoid rule approximates the area under the curve using a trapezoid. In general, the left Riemann sum will underestimate costs and QALYs whereas the right Riemann sum will overestimate them.

+
+
+
+

+Individual-level models

+

In individual-level models, costs and QALYs are computed using the continuous time present value given a flow of state values, which change as patients transition between health states or as costs vary as a function of time. The state values can be partitioned into \(M\) time intervals indexed by \(m = 1,\ldots, M\) where interval \(m\) contains times \(t\) such that \(t_m\leq t \leq t_{m+1}\) and values for state \(h\) are equal to \(z_{hm}\) during interval \(m\). \(z_{hm}\) will equal zero during time intervals in which a patient is not in state \(h\). Discounted costs and QALYs for health state \(h\) are then given by,

+

\[ +\sum_{m = 1}^M \int_{t_m}^{t_m+1} z_{hm}e^{-rt}dt = \sum_{m = 1}^M z_{hm} \left(\frac{e^{-r{t_{m}}} - e^{-r{t_{m+1}}}}{r}\right), +\]

+

where \(r > 0\) is the discount rate. If \(r = 0\), then the present value simplifies to \(\sum_{m = 1}^M z_{hm}(t_{m+1} - t_{m})\).

+

Note that while state values in cohort models can depend on time since the start of the model, state values in individual-level models can depend on either time since the start of the model or time since entering the most recent health state. Individual-level models consequently not only afford more flexibility than cohort models when simulating disease progression, but when simulating costs and/or QALYs as well.

+
+
+ + + +
+ + + +
+ +
+

Site built with pkgdown 1.5.1.

+
+ +
+
+ + + + + + diff --git a/docs/articles/getting-started.html b/docs/articles/getting-started.html new file mode 100644 index 00000000..5d6268b0 --- /dev/null +++ b/docs/articles/getting-started.html @@ -0,0 +1,365 @@ + + + + + + + +Getting started with a Markov cohort model • hesim + + + + + + + + + + +
+
+ + + + +
+
+ + + + +
+

+Overview

+

The simplest models in hesim are cohort discrete time state transitions models (cDTSTMs), which encompass both time-homogeneous and time-inhomogeneous Markov models. cDTSTMs (that can easily incorporate patient heterogeneity and parameter uncertainty) are most conveniently simulated with a four step process specifying:

+
    +
  1. +Data consisting of the treatment strategies and patient population
  2. +
  3. +Parameter objects storing the values of all parameters used in the model
  4. +
  5. +Random number generation expressions that randomly generate probability distributions for the parameters for probabilistic sensitivity analysis (PSA)
  6. +
  7. +Transformed parameter expressions that transform the parameter estimates into values used for simulation
  8. +
+

We illustrate by replicating the HIV model developed by Chancellor et al. and used for didactic purposes in the Decision Modeling for Health Economic Evaluation textbook. The analysis compares two treatment strategies, zidovudine monotherapy and zidovudine + lamivudine combination therapy. The model is time-inhomogeneous because patients are assumed to only use lamivudine for 2 years before switching back to monotherapy.

+

The model consists 4 health states. Ordered from worst to best to worst, they are: State A (CD4 cells > 200 and < 500 cells/mm3), State B (CD4 < 200 cells/mm3), State C (AIDS), and State D (death). The model assumes that patients can transition to any state that is more severe (i.e., progress) but cannot transition back to a more severe state.

+

+
+
+

+Define the model

+
+

+Data

+

We set up the model for two treatment strategies and one representative patient.

+
library("hesim")
+library("data.table")
+strategies <- data.table(strategy_id = 1:2,
+                         strategy_name = c("Monotherapy", "Combination therapy"))
+patients <- data.table(patient_id = 1)
+hesim_dat <- hesim_data(strategies = strategies,
+                        patients = patients)
+print(hesim_dat)
+
## $strategies
+##    strategy_id       strategy_name
+## 1:           1         Monotherapy
+## 2:           2 Combination therapy
+## 
+## $patients
+##    patient_id
+## 1:          1
+## 
+## attr(,"class")
+## [1] "hesim_data"
+

The model will be simulated for each treatment strategy and patient combination. Such data containing treatment strategy and patient characteristics can be generated using expand().

+
data <- expand(hesim_dat, by = c("strategies", "patients"))
+head(data)
+
##    strategy_id patient_id       strategy_name
+## 1:           1          1         Monotherapy
+## 2:           2          1 Combination therapy
+
+
+

+Parameters

+

Counts of transitions at yearly intervals between each of the 4 health states were available from a longitudinal dataset from Chelsea and Westminister hospital.

+
trans_mono <- matrix(c(1251, 350, 116, 17,
+                       0, 731, 512, 15,
+                       0, 0, 1312, 437,
+                       0, 0, 0, 469),
+                      ncol = 4, nrow = 4, byrow = TRUE)
+colnames(trans_mono) <- rownames(trans_mono) <-  c("A", "B", "C", "D")
+print(trans_mono)
+
##      A   B    C   D
+## A 1251 350  116  17
+## B    0 731  512  15
+## C    0   0 1312 437
+## D    0   0    0 469
+

The transition probabilities from state \(r\) to each of the 4 states (State A, State B, State C, and State D) can be modeled using Dirichlet distributions, which are parameterized by a vector of concentration parameters \(\alpha = (\alpha_{r1}, \alpha_{r2}, \alpha_{r3}, \alpha_{r4})\). The Dirichlet distribution is particularly useful for modeling uncertainty in the transition probabilities because it is a conjugate prior for the multinomial distribution. The posterior distribution of \(\alpha\) consequently follows a Dirichlet distribution which can be thought of as the “effective sample size” for each transition (see Appendix).

+

Treatment effects are defined in terms of the relative risk, which is assumed to reduce the probability of all transitions to a more severe health state by an equal amount. We work with the log relative risk since it is approximately normally distributed. The estimated relative risk of disease progression is 0.509 (95% CI 0.365 to 0.710). We can then calculate the mean and standard error on the log scale.

+

The cost-effectiveness analysis is based on life-years, meaning that each health state is assigned a utility value of 1. The costs of zidovudine and lamivudine are $2,278 and $2,086.50, respectively. Medical costs are split into direct medical and community medical costs.

+

The parameter estimates are stored in a list.

+
params <- list(
+  alpha_mono = trans_mono,
+  lrr_mean = log(.509),
+  lrr_lower <- log(.365),
+  lrr_upper = log(.710),
+  c_dmed_mean = c(A = 1701, B = 1774, C = 6948),
+  c_cmed_mean = c(A = 1055, B = 1278, C = 2059),
+  c_zido = 2278,
+  c_lam = 2086.50,
+  u = 1
+)
+
+
+

+Random number generation

+

To perform a PSA, it is necessary to draw samples of the parameters from suitable probability distributions. While the probability distributions could have been drawn manually and stored in the params object above, it can be more easily done by using random number generation functions with define_rng().

+

As noted above, we draw the transition probabilities with monotherapy from a dirichlet distribution and the relative risk (of a transition to a worse state with combination therapy relative to monotherapy) from a lognormal distribution. Drug costs and utility are assumed to be fixed while medical costs are drawn from gamma distributions.

+

Sampled parameters values are stored in a list, meaning that samples for each parameter can be stored as either a vector or matrix. For instance, we sample medical costs for each (non-death) health state from gamma distributions and they a stored in a matrix (one row for each sample and one column for each health state). Likewise, the parameters of each cell in the transition probability matrix are store in a matrix, with parameters of the matrix ordered rowwise across the columns. On the other hand, samples of the relative risk are stored in a vector.

+
rng_def <- define_rng({
+  lrr_se <- (lrr_upper - lrr_lower)/(2 * qnorm(.975)) # Local object 
+                                                      # not returned
+  list( # Parameters to return
+    p_mono = dirichlet_rng(alpha_mono),
+    rr_comb = lognormal_rng(lrr_mean, lrr_se),
+    c_zido = c_zido,
+    c_lam = c_lam,
+    c_dmed = gamma_rng(mean = c_dmed_mean, sd = c_dmed_mean),
+    c_cmed = gamma_rng(mean = c_cmed_mean, sd = c_cmed_mean),
+    u = u
+  )
+}, n = 1000)
+
+
+

+Transformed parameters

+

The next step is to “transform” the parameters for use in the economic model using define_tparams(). This typically involves modeling them as a function of the treatment strategies and patient (i.e., the data). To maximize computational efficiency, all operations are vectorized across the rows in the data. Parameters not include in a transformed parameter function are assumed constant across patients and treatment strategies.

+

An important helper function is tpmatrix(), which allows users to define transition matrices as a function of other parameters. In this example “vectorized” transition probabilities are defined in terms of the transition matrix for monotherapy (p_mono) and the relative risk (rr). The relative risk is simply 1 for monotherapy (since the corresponding transition matrix is just p_mono) and equal to rr_comb for combination therapy.

+

The only costs in the model that vary across treatment strategies (rather than just states) are drug costs. The monotherapy treatment strategy include the cost of zidovudine only whereas the combination therapy includes the costs of both zidovudine and lamivudine.

+

Parameters may depend on time (since the start of the model) by using the package defined time variable. The time variable corresponds to the stopping time of time intervals, which are specified using the times argument in define_tparams(). Since patients are only assumed to use lamivudine for two years, rr_comb and the costs of lamivudine are not applied beyond year 2.

+
tparams_def <- define_tparams({
+  ## The treatment effect (relative risk) is transformed so that it varies by 
+  ## strategies and only applies for the first 2 years (Monotherapy is 
+  ## the reference strategy). 
+  rr <- ifelse(strategy_name == "Monotherapy" | time > 2, 1, rr_comb)
+
+  list(
+    tpmatrix = tpmatrix(
+      C, p_mono$A_B * rr, p_mono$A_C * rr, p_mono$A_D * rr,
+      0, C, p_mono$B_C * rr, p_mono$B_D * rr,
+      0, 0, C, p_mono$C_D * rr,
+      0, 0, 0, 1
+    ),
+    utility = u,
+    costs = list(
+        drug = ifelse(strategy_name == "Monotherapy" | time > 2,
+                    c_zido, c_zido + c_lam),
+        community_medical = c_cmed,
+        direct_medical = c_dmed
+    )
+  )
+}, times = c(2, Inf))
+
+
+

+The model

+

The model is defined using define_model() function. The names of transition probability matrix, utility, and cost objects are specified, as are the names of the health states.

+
mod_def <- define_model(tparams_def = tparams_def,
+                        rng_def = rng_def,
+                        params = params)
+
+
+
+

+Simulation

+

A economic model (of class CohortDtstm) can be created from a defined model (of class model_def) and data using the generic function create_CohortDtstm(). The economic model is an R6 object consisting of a transition model (trans_model) for simulating transition probabilities with sim_stateprobs(), a utility model for simulating quality-adjusted life-years with sim_qalys(), and a set of cost models (for each cost category) for simulating costs with sim_costs().

+
econmod <- create_CohortDtstm(mod_def, data)
+
+

+Health state probabilities

+

At the start of the model, all patients are assumed to be in State A (the default setting of the transition model of class CohortDtstmTrans). State occupancy probabilities are generated by simulating the discrete time Markov chain; as expected, patients survive longer with combination therapy.

+
econmod$sim_stateprobs(n_cycles = 20)
+
+# Plot
+library("ggplot2")
+theme_set(theme_bw())
+stateprob_summary <- econmod$stateprobs_[, .(prob_mean = mean(prob),
+                                              prob_lower = quantile(prob, .025),
+                                              prob_upper = quantile(prob, .975)),
+                                          by = c("strategy_id", "state_id", "t")]
+stateprob_summary[, strategy_name := factor(strategy_id,
+                                          labels = strategies$strategy_name)]
+ggplot(stateprob_summary, aes(x = t, y = prob_mean)) +
+  geom_line(aes(col = strategy_name)) +
+  geom_ribbon(aes(x = t, ymin = prob_lower, ymax = prob_upper,
+                  fill = strategy_name), alpha = .3) +
+  facet_wrap(~factor(state_id, labels = LETTERS[1:4])) +
+  xlab("Year") + ylab("Probability") +
+  scale_fill_discrete("Strategy") + scale_color_discrete("Strategy")
+

+
+
+

+Costs and QALYs

+

Costs and quality-adjusted life-years (QALYs) are computed by integrating the previously simulated state probabilities. To maintain consistency with Chancellor et al., we approximate the integral with a right Riemann sum, which is simply a sum of costs and QALYs measured at the right endpoint of each interval (e.g., costs during model cycle 1 are measured at time 1 (rather than at say, time 0)).

+
econmod$sim_qalys(dr = 0, integrate_method = "riemann_right")
+econmod$sim_costs(dr = 0.06, integrate_method = "riemann_right")
+
+
+
+

+Decision analysis

+

A decision analysis is performed by comparing combination therapy to monotherapy.

+
ce_sim <- econmod$summarize()
+wtp <- seq(0, 25000, 500)
+icea_pw_out <- icea_pw(ce_sim, comparator = 1, dr_qalys = 0, dr_costs = .06,
+                       k = wtp)
+

The incremental cost-effectiveness ratio (ICER) estimated using means from the PSA is $6,300, which is very close to the estimate from the deterministic analysis from Chancellor et al. 

+
icer_tbl(icea_pw_out)
+
##                   1   2                        
+## Incremental QALYs "-" "0.92 (0.56, 1.22)"      
+## Incremental costs "-" "5,821 (3,027, 9,548)"   
+## Incremental NMB   "-" "40,376 (22,948, 54,439)"
+## ICER              "-" "6,300"                  
+## Conclusion        "-" "Cost-effective"
+

A cost-effectiveness acceptability curve shows that the probability that combination therapy is more cost-effective than monotherapy increases quickly with the willingness to pay (WTP) for a QALY. At $10,000, the probability is 0.944 and approaches 1 at slightly higher WTP values.

+
ggplot(icea_pw_out$ceac,
+       aes(x = k, y = prob,
+           col = factor(strategy_id, labels = strategies$strategy_name[-1]))) +
+  geom_line()  + xlab("Willingness to pay") +
+  ylab("Probability cost-effective") +
+  scale_x_continuous(breaks = seq(0, max(wtp), 5000), label = scales::dollar) +
+  theme(legend.position = "bottom") + scale_colour_discrete(name = "Strategy")
+

+
+
+

+Appendix: Dirichlet distribution

+

The multinomial distribution is a discrete probability distribution for the number of successes for each of k mutually exclusive categories in n trials. The probabilities of the categories are given by \(\pi_1,\ldots, \pi_k\) with \(\sum_{j=1}^k \pi_j=1\) and each \(\pi_j\) defined on \([0,1]\). The Dirichlet distribution is parameterized by the concentration parameters \(\alpha_1,\ldots, \alpha_k\) with \(\alpha_j > 0\). Letting \(x_1,\ldots, x_k\) denote the number of successes in each category, the prior distribution and likelihood are,

+

\[ +\begin{aligned} +p(\pi_1,\ldots,\pi_k |\alpha_1,\ldots, \alpha_k) = \text{Dirichlet}(\alpha_1,\ldots,\alpha_k) \\ +p(x_1,\ldots,x_k | \pi_1,\ldots,\pi_k) = \text{Multin}(n, \pi_1,\ldots,\pi_k). +\end{aligned} +\]

+

The posterior distribution for \(\pi_1,\ldots,\pi_k\) is then given by,

+

\[ +p\left(\pi_1,\ldots,\pi_k| x_1,\ldots,x_k, \alpha_1,\ldots,\alpha_k \right) = \text{Dirichlet}\left(\alpha_1 + x_1, \ldots, \alpha_k + x_k\right). +\]

+
+
+ + + +
+ + + +
+ +
+

Site built with pkgdown 1.5.1.

+
+ +
+
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Individualized cost-effectiveness analysis (iCEA) evaluates the cost-effectiveness of treatments at the individual (or subgroup) level. This has two major implications:

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