Elementary decision tree (Evans 1997)

#| include = FALSE,
#| purl = FALSE
knitr::opts_chunk$set(
  echo = FALSE,
  collapse = TRUE,
  comment = "#>"
)
#| purl = FALSE
#nolint start
library(rdecision)
#| purl = FALSE
#nolint end

Introduction

This vignette is an example of modelling a decision tree using the rdecision package. It is based on the example given by Briggs [-@briggs2006] (Box 2.3) which itself is based on a decision tree which compared oral Sumatriptan versus oral caffeine/Ergotamine for migraine [@evans1997]. In this vignette, we consider the problem from the perspective of a provincial health department.

Creating the model

Model variables

The following code defines the variables for cost, utility and effect that will be used in the model. There are 14 variables in total; 4 costs, 4 utilities and 6 probabilities.

#| echo = TRUE
# Time horizon
th <- as.difftime(24L, units = "hours")

# model variables for cost
c_sumatriptan <- 16.10
c_caffeine <- 1.32
c_ed <- 63.16
c_admission <- 1093.0

# model variables for utility
u_relief_norecurrence <- 1.0
u_relief_recurrence <- 0.9
u_norelief_endures <- -0.30
u_norelief_er <- 0.1

# model variables for effect
p_sumatriptan_recurrence <- 0.594
p_caffeine_recurrence <- 0.703
p_sumatriptan_relief <- 0.558
p_caffeine_relief <- 0.379
p_er <- 0.08
p_admitted <- 0.002

Constructing the tree

The following code constructs the decision tree. In the formulation used by rdecision, a decision tree is a form of arborescence, a directed graph of nodes and edges, with a single root and a unique path from the root to each leaf node. Decision trees comprise three types of node: decision, chance and leaf nodes and two types of edge: actions (whose sources are decision nodes) and reactions (whose sources are chance nodes), see Figure 1. If the probability of traversing one reaction edge from any chance node is set to NA_real_, it will be calculated as 1 minus the sum of probabilities of the other reaction edges from that node when the tree is evaluated.

#| echo = TRUE
# Sumatriptan branch
ta <- LeafNode$new("A", utility = u_relief_norecurrence, interval = th)
tb <- LeafNode$new("B", utility = u_relief_recurrence, interval = th)
c3 <- ChanceNode$new()
e1 <- Reaction$new(
  c3, ta, p = p_sumatriptan_recurrence, label = "No recurrence"
)
e2 <- Reaction$new(
  c3, tb, p = NA_real_, cost = c_sumatriptan, label = "Relieved 2nd dose"
)
td <- LeafNode$new("D", utility = u_norelief_er, interval = th)
te <- LeafNode$new("E", utility = u_norelief_endures, interval = th)
c7 <- ChanceNode$new()
e3 <- Reaction$new(c7, td, p = NA_real_, label = "Relief")
e4 <- Reaction$new(
  c7, te, p = p_admitted, cost = c_admission, label = "Hospitalization"
)

tc <- LeafNode$new("C", utility = u_norelief_endures, interval = th)
c4 <- ChanceNode$new()
e5 <- Reaction$new(c4, tc, p = NA_real_, label = "Endures attack")
e6 <- Reaction$new(c4, c7, p = p_er, cost = c_ed, label = "ER")

c1 <- ChanceNode$new()
e7 <- Reaction$new(c1, c3, p = p_sumatriptan_relief, label = "Relief")
e8 <- Reaction$new(c1, c4, p = NA_real_, label = "No relief")

# Caffeine/Ergotamine branch
tf <- LeafNode$new("F", utility = u_relief_norecurrence, interval = th)
tg <- LeafNode$new("G", utility = u_relief_recurrence, interval = th)
c5 <- ChanceNode$new()
e9 <- Reaction$new(c5, tf, p = p_caffeine_recurrence, label = "No recurrence")
e10 <- Reaction$new(
  c5, tg, p = NA_real_, cost = c_caffeine, label = "Relieved 2nd dose"
)
ti <- LeafNode$new("I", utility = u_norelief_er, interval = th)
tj <- LeafNode$new("J", utility = u_norelief_endures, interval = th)
c8 <- ChanceNode$new()
e11 <- Reaction$new(c8, ti, p = NA_real_, label = "Relief")
e12 <- Reaction$new(
  c8, tj, p = p_admitted, cost = c_admission, label = "Hospitalization"
)

th <- LeafNode$new("H", utility = u_norelief_endures, interval = th)
c6 <- ChanceNode$new()
e13 <- Reaction$new(c6, th, p = NA_real_, label = "Endures attack")
e14 <- Reaction$new(c6, c8, p = p_er, cost = c_ed, label = "ER")

c2 <- ChanceNode$new()
e15 <- Reaction$new(c2, c5, p = p_caffeine_relief, label = "Relief")
e16 <- Reaction$new(c2, c6, p = NA_real_, label = "No relief")

# decision node
d1 <- DecisionNode$new("d1")
e17 <- Action$new(d1, c1, cost = c_sumatriptan, label = "Sumatriptan")
e18 <- Action$new(d1, c2, cost = c_caffeine, label = "Caffeine-Ergotamine")

# create lists of nodes and edges
V <- list(
  d1, c1, c2, c3, c4, c5, c6, c7, c8,
  ta, tb, tc, td, te, tf, tg, th, ti, tj
)
E <- list(
  e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12, e13, e14, e15, e16,
  e17, e18
)

# tree
dt <- DecisionTree$new(V, E)
#| purl = FALSE
# test that decision tree structure is as per Evans et al
stopifnot(
  all.equal(d1$label(), "d1")
)
#| results = "hide",
#| fig.keep = "all",
#| fig.align = "center",
#| fig.cap = "Figure 1. Decision tree for the Sumatriptan model"
dt$draw(border = TRUE)

Running the model

The method evaluate of decision tree objects computes the probability, cost and utility of each strategy for the model. A strategy is a unanimous prescription of the actions at each decision node. In this example there is a single decision node with two actions, and the strategies are simply the two forms of treatment to be compared. More complex decision trees are also possible.

The paths traversed in each strategy can be evaluated individually using the method evaluate(by = "path"). In rdecision a strategy is defined as a set of action edges with one action edge per decision node. It is necessary to use the option by = "path" only if information about each pathway is required; normally it is sufficient to call evaluate which will automatically aggregate the evaluation by strategy.

Model results

Base case

The evaluation of each pathway, for each strategy, is done as follows:

#| echo = TRUE
ep <- dt$evaluate(by = "path")
#| purl = FALSE
# test that evaluation by path is as per Box 2.3 of Briggs
local({
  stopifnot(
    all.equal(nrow(ep), 10L),
    setequal(
      colnames(ep),
      c(
        "Leaf", "d1", "Probability", "Cost", "Benefit", "Utility", "QALY",
        "Run"
      )
    ),
    setequal(ep[, "Leaf"], LETTERS[1L : 10L]),
    all.equal(sum(ep[, "Probability"]), 2.0, tolerace = 0.01, scale = 1.0)
  )
  ia <- which(ep[, "Leaf"] == "A")
  stopifnot(
    all.equal(ep[[ia, "d1"]], "Sumatriptan"),
    all.equal(ep[[ia, "Probability"]], 0.331, tolerance = 0.001, scale = 1.0),
    all.equal(ep[[ia, "Cost"]], 5.34, tolerance = 0.01, scale = 1.0),
    all.equal(ep[[ia, "Utility"]], 0.33, tolerance = 0.01, scale = 1.0)
  )
  ih <- which(ep[, "Leaf"] == "H")
  stopifnot(
    all.equal(ep[[ih, "d1"]], "Caffeine-Ergotamine"),
    all.equal(ep[[ih, "Probability"]], 0.571, tolerance = 0.001, scale = 1.0),
    all.equal(ep[[ih, "Cost"]], 0.75, tolerance = 0.01, scale = 1.0),
    all.equal(ep[[ih, "Utility"]], -0.17, tolerance = 0.01, scale = 1.0)
  )
})

and yields the following table:

#| echo = FALSE
with(data = ep, expr = {
  data.frame(
    Leaf = Leaf,
    Probability = round(Probability, digits = 4L),
    Cost = round(Cost, digits = 2L),
    Utility = round(Utility, digits = 5L),
    stringsAsFactors = FALSE
  )
})

There are, as expected, ten pathways (5 per strategy). The expected cost, utility and QALY (utility multiplied by the time horizon of the model) for each choice can be calculated from the table above, or by invoking the evaluate method of a decision tree object with the default parameter by = "strategy".

#| echo = TRUE
es <- dt$evaluate()

This gives the following result, consistent with that reported by Evans et al [-@evans1997].

#| echo = FALSE
with(data = es, expr = {
  data.frame(
    d1 = d1,
    Cost = round(Cost, digits = 2L),
    Utility = round(Utility, digits = 4L),
    QALY = round(QALY, digits = 4L),
    stringsAsFactors = FALSE
  )
})
#| echo = FALSE
is <- which(es[, "d1"] == "Sumatriptan")
cost_s <- es[[is, "Cost"]]
utility_s <- es[[is, "Utility"]]
qaly_s <- es[[is, "QALY"]]

ic <- which(es[, "d1"] == "Caffeine-Ergotamine")
cost_c <- es[[ic, "Cost"]]
utility_c <- es[[ic, "Utility"]]
qaly_c <- es[[ic, "QALY"]]

delta_c <- cost_s - cost_c
delta_u <- utility_s - utility_c
delta_q <- qaly_s - qaly_c
icer <- delta_c / delta_q
#| purl = FALSE
# test that evaluation by strategy is as per Evans et al
stopifnot(
  all.equal(nrow(es), 2L),
  setequal(
    colnames(es),
    c("d1", "Run", "Probability", "Cost", "Benefit", "Utility", "QALY")
  ),
  setequal(es[, "d1"], c("Sumatriptan", "Caffeine-Ergotamine")),
  all.equal(sum(es["Probability"]), 2.0, tolerance = 0.01, scale = 1.0),
  all.equal(cost_s, 22.06, tolerance = 0.01, scale = 1.0),
  all.equal(utility_s, 0.41, tolerance = 0.01, scale = 1.0),
  all.equal(cost_c, 4.73, tolerance = 0.02, scale = 1.0),
  all.equal(utility_c, 0.20, tolerance = 0.01, scale = 1.0),
  icer / 29366.0 >= 0.95,
  icer / 29366.0 <= 1.05
)

The incremental cost was $Can r gbp(x = delta_c, p = TRUE) (r gbp(x = cost_s, p = TRUE) - r gbp(x = cost_c, p = TRUE)) and the incremental utility was r round(delta_u, 2L) (r round(utility_s, 2L) - r round(utility_c, 2L)). Because the time horizon of the model was 1 day, the incremental QALYs was the incremental annual utility divided by 365, and the ICER was therefore equal to r gbp(icer) \$Can/QALY, within 5% of the published estimate (29,366 \$Can/QALY).

Univariate sensitivity analysis

Evans et al [-@evans1997] reported the ICER for various alternative values of input variables. For example (their Table VIII), they reported that the ICER was 60,839 $Can/QALY for a relative increase in effectiveness of 9.1% (i.e., when the relief from Sumatriptan was 9.1 percentage points greater than that of Caffeine-Ergotamine) and 18,950 $Can/QALY for a relative increase in effectiveness of 26.8% (these being the lower and upper confidence intervals of the estimate of effectiveness from meta-analysis).

To calculate these ICERs, we set the value of the model variable p_sumatriptan_relief, and re-evaluate the model. The lower range of ICER (with the greater relative increase in effectiveness) is calculated as follows:

#| echo = TRUE
p_sumatriptan_relief <- p_caffeine_relief + 0.268
e7$set_probability(p_sumatriptan_relief)
es <- dt$evaluate()
#| echo = FALSE
is <- which(es[, "d1"] == "Sumatriptan")
cost_s_upper <- es[[is, "Cost"]]
utility_s_upper <- es[[is, "Utility"]]
qaly_s_upper <- es[[is, "QALY"]]

ic <- which(es[, "d1"] == "Caffeine-Ergotamine")
cost_c_upper <- es[[ic, "Cost"]]
utility_c_upper <- es[[ic, "Utility"]]
qaly_c_upper <- es[[ic, "QALY"]]

delta_c_upper <- cost_s_upper - cost_c_upper
delta_u_upper <- utility_s_upper - utility_c_upper
delta_q_upper <- qaly_s_upper - qaly_c_upper
icer_upper <- delta_c_upper / delta_q_upper
#| purl = FALSE
# test that upper relief threshold ICER agrees with Evans et al
stopifnot(
  icer_upper / 18950.0 >= 0.95,
  icer_upper / 18950.0 <= 1.05
)

This yields the following table, from which the ICER is calculated as r gbp(icer_upper) \$Can/QALY, close to the published estimate of 18,950 \$Can/QALY.

#| echo = FALSE
with(data = es, expr = {
  data.frame(
    d1 = d1,
    Cost = round(Cost, digits = 2L),
    Utility = round(Utility, digits = 4L),
    QALY = round(QALY, digits = 4L),
    stringsAsFactors = FALSE
  )
})

The upper range of ICER (with the smaller relative increase in effectiveness) is calculated as follows:

#| echo = TRUE
p_sumatriptan_relief <- p_caffeine_relief + 0.091
e7$set_probability(p_sumatriptan_relief)
es <- dt$evaluate()
#| echo = FALSE
is <- which(es[, "d1"] == "Sumatriptan")
cost_s_lower <- es[[is, "Cost"]]
utility_s_lower <- es[[is, "Utility"]]
qaly_s_lower <- es[[is, "QALY"]]

ic <- which(es[, "d1"] == "Caffeine-Ergotamine")
cost_c_lower <- es[[ic, "Cost"]]
utility_c_lower <- es[[ic, "Utility"]]
qaly_c_lower <- es[[ic, "QALY"]]

delta_c_lower <- cost_s_lower - cost_c_lower
delta_u_lower <- utility_s_lower - utility_c_lower
delta_q_lower <- qaly_s_lower - qaly_c_lower
icer_lower <- delta_c_lower / delta_q_lower
#| purl = FALSE
# test that lower relief threshold ICER agrees with Evans et al
stopifnot(
  icer_lower / 60839.0 >= 0.95,
  icer_lower / 60839.0 <= 1.05
)

This yields the following table, from which the ICER is calculated as r gbp(icer_lower) \$Can/QALY, close to the published estimate of 60,839 \$Can/QALY.

#| echo = FALSE
with(data = es, expr = {
  data.frame(
    d1 = d1,
    Cost = round(Cost, digits = 2L),
    Utility = round(Utility, digits = 4L),
    QALY = round(QALY, digits = 4L),
    stringsAsFactors = FALSE
  )
})
#| purl = FALSE
# test that upper and lower ICER thresholds can be replicatd with thresholding
local({

  # model variables with uncertainty
  p_sumatriptan_relief <- ConstModVar$new(
    "P(relief|sumatriptan)", "P", 0.558
  )

  # set probabilities for edges associated with model variables
  e7$set_probability(p_sumatriptan_relief)
  e15$set_probability(p_caffeine_relief)

  # upper 95% relief rate threshold for ICER (Table VIII)
  p_relief_upper <- dt$threshold(
    index = list(e17), ref = list(e18), outcome = "ICER",
    mvd = p_sumatriptan_relief$description(),
    a = 0.6, b = 0.7,
    lambda = 18950.0, tol = 0.0001
  )
  # lower 95% relief rate threshold for ICER (Table VIII)
  p_relief_lower <- dt$threshold(
    index = list(e17), ref = list(e18), outcome = "ICER",
    mvd = p_sumatriptan_relief$description(),
    a = 0.4, b = 0.5,
    lambda = 60839.0, tol = 0.0001
  )

  # check parameters of threshold function
  # mean relief rate threshold for ICER
  pt <- dt$threshold(
    index = list(e17), ref = list(e18), outcome = "ICER",
    mvd = p_sumatriptan_relief$description(),
    a = 0.5, b = 0.6,
    lambda = 29366.0, tol = 0.0001
  )
  # check values against Table VIII
  stopifnot(
    all.equal(pt, p_caffeine_relief + 0.179, tolerance = 0.02, scale = 1.0),
    all.equal(
      p_relief_upper, p_caffeine_relief + 0.268, tolerance = 0.02, scale = 1.0
    ),
    all.equal(
      p_relief_lower, p_caffeine_relief + 0.091, tolerance = 0.02, scale = 1.0
    )
  )
})
#| purl = FALSE
# test that ICERs computed by tornado function are as expected
local({

  # probability variables with uncertainty
  p_sumatriptan_relief <- ConstModVar$new(
    "P(relief|sumatriptan)", "P", 0.558
  )
  e7$set_probability(p_sumatriptan_relief)
  e15$set_probability(p_caffeine_relief)

  # cost variables with uncertainty
  c_sumatriptan <- GammaModVar$new(
    "Sumatriptan", "CAD", shape = 16.10, scale = 1.0
  )
  c_caffeine <- GammaModVar$new(
    "Caffeine", "CAD", shape = 1.32, scale = 1.0
  )
  e2$set_cost(c_sumatriptan)
  e10$set_cost(c_caffeine)
  e17$set_cost(c_sumatriptan)
  e18$set_cost(c_caffeine)

  # check ICER ranges in tornado diagram (branches B and G get 2nd dose)
  TO <- dt$tornado(index = e17, ref = e18, outcome = "ICER", draw = FALSE)
  c_sumatriptan$set("expected")
  c_caffeine$set("expected")
  x <- qgamma(p = 0.025, shape = 16.10, rate = 1.0)
  deltac <- (x - c_sumatriptan$get()) * 1.227
  stopifnot(
    all.equal(
      TO[[which(TO$Description == "Sumatriptan"), "LL"]],
      x,
      tolerance = 0.01,
      scale = 1.0
    ),
    all.equal(
      TO[[which(TO$Description == "Sumatriptan"), "outcome.min"]],
      (cost_s - cost_c + deltac) / delta_q,
      tolerance = 100.0,
      scale = 1.0
    )
  )
  x <- qgamma(p = 0.975, shape = 16.10, rate = 1.0)
  deltac <- (x - c_sumatriptan$get()) * 1.227
  stopifnot(
    all.equal(
      TO[[which(TO$Description == "Sumatriptan"), "UL"]],
      x,
      tolerance = 0.01,
      scale = 1.0
    ),
    all.equal(
      TO[[which(TO$Description == "Sumatriptan"), "outcome.max"]],
      (cost_s - cost_c + deltac) / delta_q,
      tolerance = 100.0,
      scale = 1.0
    )
  )
  x <- qgamma(p = 0.025, shape = 1.32, rate = 1.0)
  deltac <- (c_caffeine$get() - x) * 1.113
  stopifnot(
    all.equal(
      TO[[which(TO$Description == "Caffeine"), "LL"]],
      x,
      tolerance = 0.01,
      scale = 1.0
    ),
    all.equal(
      TO[[which(TO$Description == "Caffeine"), "outcome.min"]],
      (cost_s - cost_c + deltac) / delta_q,
      tolerance = 100.0,
      scale = 1.0
    )
  )
  x <- qgamma(p = 0.975, shape = 1.32, rate = 1.0)
  deltac <- (c_caffeine$get() - x) * 1.113
  stopifnot(
    all.equal(
      TO[[which(TO$Description == "Caffeine"), "UL"]],
      x,
      tolerance = 0.01,
      scale = 1.0
    ),
    all.equal(
      TO[[which(TO$Description == "Caffeine"), "outcome.max"]],
      (cost_s - cost_c + deltac) / delta_q,
      tolerance = 100.0,
      scale = 1.0
    )
  )
})

References



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rdecision documentation built on April 3, 2025, 6:09 p.m.