#| include = FALSE, #| purl = FALSE knitr::opts_chunk$set( echo = FALSE, collapse = TRUE, comment = "#>" )
#| purl = FALSE #nolint start
library(rdecision)
#| purl = FALSE #nolint end
This vignette is an example of modelling a decision tree using the rdecision
package. It is based on the example given by Briggs [-@briggs2006] (Box 2.3)
which itself is based on a decision tree which compared oral Sumatriptan versus
oral caffeine/Ergotamine for migraine [@evans1997]. In this vignette, we
consider the problem from the perspective of a provincial health department.
The following code defines the variables for cost, utility and effect that will be used in the model. There are 14 variables in total; 4 costs, 4 utilities and 6 probabilities.
#| echo = TRUE # Time horizon th <- as.difftime(24L, units = "hours") # model variables for cost c_sumatriptan <- 16.10 c_caffeine <- 1.32 c_ed <- 63.16 c_admission <- 1093.0 # model variables for utility u_relief_norecurrence <- 1.0 u_relief_recurrence <- 0.9 u_norelief_endures <- -0.30 u_norelief_er <- 0.1 # model variables for effect p_sumatriptan_recurrence <- 0.594 p_caffeine_recurrence <- 0.703 p_sumatriptan_relief <- 0.558 p_caffeine_relief <- 0.379 p_er <- 0.08 p_admitted <- 0.002
The following code constructs the decision tree. In the
formulation used by rdecision
, a decision tree is a form of
arborescence, a directed graph of nodes and edges, with a single
root and a unique path from the root to each leaf node. Decision trees
comprise three types of node: decision, chance and leaf nodes and two
types of edge: actions (whose sources are decision nodes) and reactions
(whose sources are chance nodes), see Figure 1.
If the probability of traversing one reaction edge from any chance node is set
to NA_real_
, it will be calculated as 1 minus the sum of probabilities of the
other reaction edges from that node when the tree is evaluated.
#| echo = TRUE # Sumatriptan branch ta <- LeafNode$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$draw(border = TRUE)
The method evaluate
of decision tree objects computes
the probability, cost and utility of each strategy for the model. A strategy
is a unanimous prescription of the actions at each decision node. In this
example there is a single decision node with two actions, and the strategies
are simply the two forms of treatment to be compared. More complex decision
trees are also possible.
The paths traversed in each strategy can be evaluated individually
using the method evaluate(by = "path")
. In rdecision
a strategy is defined
as a set of action edges with one action edge per decision node. It is necessary
to use the option by = "path"
only if information about each pathway is
required; normally it is sufficient to call evaluate
which will automatically
aggregate the evaluation by strategy.
The evaluation of each pathway, for each strategy, is done as follows:
#| echo = TRUE ep <- dt$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).
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 ) ) })
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