Elementary Semi-Markov Model (Chancellor 1997)

In rdecision: Decision Analytic Modelling in Health Economics

#| purl = FALSE,
#| include = FALSE
# read vignette source chunks from corresponding testthat script
knitr::read_chunk(file.path("..", "tests", "testthat", "test-model-AZT.R"))
# read vignette build utility functions
knitr::read_chunk(file.path("vutils.R"))

#| purl = FALSE,
#| include = FALSE
knitr::opts_chunk$set(
  echo = FALSE,
  fig.keep = "last",
  fig.align = "center",
  collapse = TRUE,
  comment = "#>"
)

#| gbp

#| purl = FALSE
#nolint start

library(rdecision)

#| purl = FALSE
#nolint end

Introduction

This vignette is an example of an elementary semi-Markov model using the rdecision package. It is based on the example given by Briggs et al [-@briggs2006] (Exercise 2.5) which itself is based on a model described by Chancellor et al [-@chancellor1997]. The model compares a combination therapy of Lamivudine/Zidovudine versus Zidovudine monotherapy in people with HIV infection.

Creating the model

Model structure

The model is constructed by forming a graph, with each state as a node and each transition as an edge. Nodes of class MarkovState and edges of class Transition have various properties whose values reflect the variables of the model (costs, rates etc.). Because the model is intended to evaluate survival, the utility of states A, B and C are set to 1 (by default) and state D to zero. Thus the incremental quality adjusted life years gained per cycle is equivalent to the survival function. Because the structure of the model is identical for monotherapy and combination therapy, we will use the same model throughout. For this reason, the costs of occupancy of each state and the costs of making transitions between states are set to zero when the model is created, and will be changed each time the model is run.

#| model,
#| echo = TRUE

Costs and discounts

The costs and discount rates used in the model (1995 rates) are numerical constants, and are defined as follows.

#| costs-det,
#| echo = TRUE

Treatment effect

The treatment effect was estimated by Chancellor et al [-@chancellor1997] via a meta-analysis, and is defined as follows:

#| txeffect-det,
#| echo = TRUE

Transition rates and probabilities

Briggs et al [-@briggs2006] interpreted the observed transition counts in 1 year as transition probabilities by dividing counts by the total transitions observed from each state. With this assumption, the annual (per-cycle) transition probabilities are calculated as follows and applied to the model via the set_probabilities function.

#| pt-mono-det,
#| echo = TRUE

More usually, fully observed transition counts are converted into transition rates, rather than probabilities, as described by Welton and Ades [-@welton2005]. This is because counting events and measuring total time at risk includes individuals who make more than one transition during the observation time, and can lead to rates with values which exceed 1. In contrast, the difference between a census of the number of individuals in each state at the start of the interval and a census at the end is directly related to the per-cycle probability. As Miller and Homan [-@miller1994], Welton and Ades [-@welton2005], Jones et al [-@jones2017] and others note, conversion between rates and probabilities for multi-state Markov models is non-trivial [@jones2017] and care is needed when modellers calculate probabilities from published rates for use in SemiMarkoModels.

Checking the model

Diagram

A representation of the model in DOT format (Graphviz) can be created using the as_DOT function of SemiMarkovModel. The function returns a character vector which can be saved in a file (.gv extension) for visualization with the dot tool of Graphviz, or plotted directly in R via the DiagrammeR package. The Markov model is shown in the figure below.

#| fig.cap = "Markov model for comparison of HIV therapy.
#|            A: 200 < cd4 < 500, B: cd4 < 200, C: AIDS, D: Death.",
#| purl = FALSE,
#| eval = FALSE
DiagrammeR::grViz(m$as_DOT())

#| gv2png

#| fig.cap = "Markov model for comparison of HIV therapy.
#|            A: 200 < cd4 < 500, B: cd4 < 200, C: AIDS, D: Death.",
# images created with dot are more compact than DiagrammeR.
pngfile <- gv2png(
  dot = m$as_DOT(rankdir = "TB", width = 7.0, height = 7.0)
)
knitr::include_graphics(pngfile)

Per-cycle transition probabilities

The per-cycle transition probabilities are the cells of the Markov transition matrix. For the monotherapy model, the transition matrix is shown below. This is consistent with the Table 1 of Chancellor et al [-@chancellor1997].

#| purl = FALSE
pander::pander(Ptm, emphasize.rownames = FALSE, justify = "lcccc")

Running the model

Model function cycle applies one cycle of a Markov model to a defined starting population in each state. It returns a table with one row per state, and each row containing several columns, including the population at the end of the state and the cost of occupancy of states, normalized by the number of patients in the cohort, with discounting applied.

Multiple cycles are run by feeding the state populations at the end of one cycle into the next. Function cycles does this and returns a data frame with one row per cycle, and each row containing the state populations and the aggregated cost of occupancy for all states, with discounting applied. This is done below for the first 20 cycles of the model for monotherapy, with discount. For convenience, and future use with probabilistic sensitivity analysis, a function, run_mono is used to wrap up the steps needed to run 20 cycles of the model for monotherapy. The arguments to the function are the transition probability matrix, the occupancy costs for states A, B, and C, and logical variables which determine whether to apply half-cycle correction to the state populations, costs and QALYs returned in the Markov trace.

#| run-mono,
#| echo = TRUE

Coding note: In function run_mono, the occupancy costs for states A, B and C are set via calls to function set_cost() which is associated with a MarkovState object. Although these are set after the state objects sA, sB and sC have been added to model m, the updated costs are used when the model is cycled. This is because R's R6 objects, such as Markov states and transitions, are passed by reference. That is, if an R6 object such as a MarkovState changes, any other object that refers to it, such as a SemiMarkovModel will see the changes. This behaviour is different from regular R variable types, such as numeric variables, which are passed by value; that is, a copy of them is created within the function to which they are passed, and any change to the original would not apply to the copy.

The model is run by calling the new function, with appropriate arguments. The cumulative cost and life years are calculated by summing the appropriate columns from the Markov trace, as follows:

#| mono-det-results,
#| echo = TRUE

The populations and discounted costs are consistent with Briggs et al, Table 2.3 [-@briggs2006], and the QALY column is consistent with Table 2.4 (without half cycle correction). No discount was applied to the utilities.

#| purl = FALSE
keep <- c("Years", "A", "B", "C", "D", "Cost", "QALY")
pander::pander(MT.mono[, keep], row.names = FALSE, justify = "rrrrrrr",
               round = c(2L, 0L, 0L, 0L, 0L, 0L, 3L))

Model results

Monotherapy

The estimated life years is approximated by summing the proportions of patients left alive at each cycle (Briggs et al [@briggs2006], Exercise 2.5). This is an approximation because it ignores the population who remain alive after 21 years, and assumes all deaths occurred at the start of each cycle. For monotherapy the expected life gained is r round(el.mono, 3L) years at a cost of r gbp(cost.mono) GBP.

Combination therapy

For combination therapy, a similar model was created, with adjusted costs and transition rates. Following Briggs et al [@briggs2006] the treatment effect was applied to the probabilities, and these were used as inputs to the model. More usually, treatment effects are applied to rates, rather than probabilities.

#| pt-comb-det,
#| echo = TRUE

The resulting per-cycle transition matrix for the combination therapy is as follows:

#| purl = FALSE
pander::pander(Ptc, emphasize.rownames = FALSE, justify = "lcccc")

In this model, lamivudine is given for the first 2 years, with the treatment effect assumed to persist for the same period. The state populations and cycle numbers are retained by the model between calls to cycle or cycles and can be retrieved by calling get_populations. In this example, the combination therapy model is run for 2 cycles, then the population is used to continue with the monotherapy model for the remaining 18 years. The reset function is used to set the cycle number and elapsed time of the new run of the mono model. As before, function run_comb is created to wrap up these steps, so they can be used repeatedly for different values of the model variables.

#| run-combo,
#| echo = TRUE

The model is run by calling the new function, with appropriate arguments, as follows. The incremental cost effectiveness ratio (ICER) is also calculated, as the ratio of the incremental cost to the incremental life years of the combination therapy compared with monotherapy.

#| comb-det-results,
#| echo = TRUE

The Markov trace for combination therapy is as follows:

#| purl = FALSE
keep <- c("Years", "A", "B", "C", "D", "Cost", "QALY")
pander::pander(MT.comb[, keep], row.names = FALSE, justify = "rrrrrrr",
               round = c(2L, 0L, 0L, 0L, 0L, 0L, 3L))

Comparison of treatments

Over the 20 year time horizon, the expected life years gained for monotherapy was r round(el.mono,3) years at a total cost per patient of r gbp(cost.mono) GBP. The expected life years gained with combination therapy for two years was r round(el.comb, 3L) at a total cost per patient of r gbp(cost.comb) GBP. The incremental change in life years was r round(el.comb - el.mono, 3L) years at an incremental cost of r gbp(cost.comb - cost.mono) GBP, giving an ICER of r gbp(icer) GBP/QALY. This is consistent with the result obtained by Briggs et al [-@briggs2006] (6276 GBP/QALY), within rounding error.

Results with half-cycle correction

With half-cycle correction applied to the state populations, the model can be recalculated as follows.

#| hcc-det,
#| echo = TRUE

Over the 20 year time horizon, the expected life years gained for monotherapy was r round(el.mono.hcc, 3L) years at a total cost per patient of r gbp(cost.mono.hcc) GBP. The expected life years gained with combination therapy for two years was r round(el.comb.hcc, 3L) at a total cost per patient of r gbp(cost.comb.hcc) GBP. The incremental change in life years was r round(el.comb.hcc - el.mono.hcc, 3L) years at an incremental cost of r gbp(cost.comb.hcc - cost.mono.hcc) GBP, giving an ICER of r gbp(icer.hcc) GBP/QALY.

Probabilistic sensitivity analysis

In their Exercise 4.7, Briggs et al [-@briggs2006] extended the original model to account for uncertainty in the estimates of the values of the model variables. In this section, the exercise is replicated in rdecision, using the same assumptions.

Costs

Although it is possible to sample from uncertainty distributions using the functions in R standard package stats (e.g., rbeta), rdecision introduces the notion of a ModVar, which is an object that can represent a model variable with an uncertainty distribution. Many of the class methods in redecision will accept a ModVar as alternative to a numerical value as an argument, and will automatically sample from its uncertainty distribution.

The model costs are represented as ModVars of various types, as follows. The state occupancy costs for both models involve a summation of other variables. Package rdecision introduces a form of ModVar that is defined as a mathematical expression (an ExprModVar) potentially involving ModVars. The uncertainty distribution of cAm, for example, is complex, because it is a sum of two Gamma-distributed variables and a scalar, but rdecision takes care of this when cAm is sampled.

#| costs-psa,
#| echo = TRUE

Treatment effect

The treatment effect is also represented by a ModVar whose uncertainty follows a log normal distribution.

#| txeffect-psa,
#| echo = TRUE

Transition matrix

The following function generates a transition probability matrix from observed counts, using Dirichlet distributions, as described by Briggs et al. This could be achieved using the R stats function rgamma, but rdecision offers the DirichletDistribition class for convenience, which is used here.

#| pt-psa,
#| echo = TRUE

Running the PSA

The following code runs 1000 iterations of the model. At each run, the model variables are sampled from their uncertainty distributions, the transition matrix is sampled from count data, and the treatment effect is applied. Functions run_mono and run_comb are used to generate Markov traces for each form of therapy, and the incremental costs, life years and ICER for each run are saved in a matrix.

#| run-psa,
#| echo = TRUE

Coding note: The state occupancy costs cAm, cBm etc. are now ModVars, rather than numeric variables as they were in the deterministic model. However, they can still be passed as arguments to MarkovState$set_cost(), via the arguments to helper functions run_mono and run_comb, and rdecision will manage them appropriately, without changing any other code. Documentation for functions in rdecision explains where this is supported by the package.

Results

The mean (95% confidence interval) for the cost of monotherapy was r gbp(mean(psa[, "cost.mono"])) (r gbp(quantile(psa[, "cost.mono"], probs = 0.025)) to r gbp(quantile(psa[, "cost.mono"], probs = 0.975))) GBP, and the mean (95% CI) cost for combination therapy was r gbp(mean(psa[, "cost.comb"])) (r gbp(quantile(psa[, "cost.comb"], probs = 0.025)) to r gbp(quantile(psa[, "cost.comb"], probs = 0.975))) GBP. The life years gained for monotherapy was r round(mean(psa[, "el.mono"]), 3L) (r round(quantile(psa[, "el.mono"], probs = 0.025), 3L) to r round(quantile(psa[, "el.mono"], probs = 0.975), 3L)), and the life years gained for combination therapy was r round(mean(psa[, "el.comb"]), 3L) (r round(quantile(psa[, "el.comb"], probs = 0.025), 3L) to r round(quantile(psa[, "el.comb"], probs = 0.975), 3L)). The mean ICER was r gbp(mean(psa[, "icer"])) GBP/QALY with 95% confidence interval r gbp(quantile(psa[, "icer"], probs = 0.025)) to r gbp(quantile(psa[, "icer"], probs = 0.975)) GBP/QALY.

References

rdecision documentation built on June 22, 2024, 10:02 a.m.