Decision Tree representations

A decision tree is a decision model that represents all possible pathways through sequences of events (nodes), which can be under the experimenter’s control (decisions) or not (chances). A decision tree can be represented visually according to a standardised grammar:

• Decision nodes (represented graphically by a square \(\square\)): these represent alternative paths that the model should compare, for example different treatment plans. Each Decision node must be the source of two or more Actions. A decision tree can have one or more Decision nodes, which determine the possible strategies that the model compares.
• Chance nodes (represented graphically by a circle \(\bigcirc\)): these represent alternative paths that are out of the experiment’s control, for example the probability of developing a certain side effect. Each Chance node must be the source of one or more Reactions, each with a specified probability. The probability of Reactions originating from a single Chance node must sum to 1.
• Leaf nodes (represented graphically by a triangle \(\lhd\)): these represent the final outcomes of a path. No further Actions or Reactions can occur after a Leaf node. A Leaf node can have a utility value (to a maximum of 1, indicating perfect utility) and an interval over which the utility applies.

Nodes are linked by edges:

• Actions arise from Decision nodes, and
• Reactions arise from Chance nodes.

rdecision builds a Decision Tree model by defining these elements and their relationships. For example, consider the fictitious and idealized decision problem, introduced in the package README file, of choosing between providing two forms of lifestyle advice, offered to people with vascular disease, which reduce the risk of needing an interventional procedure. The cost to a healthcare provider of the interventional procedure (e.g., inserting a stent) is 5000 GBP; the cost of providing the current form of lifestyle advice, an appointment with a dietician (“diet”), is 50 GBP and the cost of providing an alternative form, attendance at an exercise programme (“exercise”), is 750 GBP. If an advice programme is successful, there is no need for an interventional procedure. These costs can be defined as scalar variables, as follows:

cost_diet <- 50.0
cost_exercise <- 750.0
cost_stent <- 5000.0

The model for this fictional scenario can be defined by the following elements:

• Decision node: which programme to enrol the patient in.

decision_node <- DecisionNode$new("Programme")

• Chance nodes: the chance that the patient will need an interventional procedure. This is different for the two programmes, so two chance nodes must be defined.

chance_node_diet <- ChanceNode$new("Outcome")
chance_node_exercise <- ChanceNode$new("Outcome")

• Leaf nodes: the possible final states of the model, depending both on the decision (which programme) and the chance of needing an intervention. Here, we assume that the model has a time horizon of 1 year, and that the utility is the same for all patients (the default values).

leaf_node_diet_no_stent <- LeafNode$new("No intervention")
leaf_node_diet_stent <- LeafNode$new("Intervention")
leaf_node_exercise_no_stent <- LeafNode$new("No intervention")
leaf_node_exercise_stent <- LeafNode$new("Intervention")

These nodes can then be wired into a decision tree graph by defining the edges that link pairs of nodes as actions or reactions.

action_diet <- Action$new(
  decision_node, chance_node_diet, cost = cost_diet, label = "Diet"
)
action_exercise <- Action$new(
  decision_node, chance_node_exercise, cost = cost_exercise, label = "Exercise"
)

s_diet <- 12L
f_diet <- 56L
s_exercise <- 18L
f_exercise <- 40L

ip_diet <- f_diet / (s_diet + f_diet)
ip_exercise <- f_exercise / (s_exercise + f_exercise)
nnt <- 1.0 / (ip_diet - ip_exercise)

p_diet <- 1.0 - ip_diet
p_exercise <- 1.0 - ip_exercise
q_diet <- 1.0 - p_diet
q_exercise <- 1.0 - p_exercise

reaction_diet_success <- Reaction$new(
  chance_node_diet, leaf_node_diet_no_stent,
  p = p_diet, cost = 0.0, label = "Success"
)

reaction_diet_failure <- Reaction$new(
  chance_node_diet, leaf_node_diet_stent,
  p = q_diet, cost = cost_stent, label = "Failure"
)

reaction_exercise_success <- Reaction$new(
  chance_node_exercise, leaf_node_exercise_no_stent,
  p = p_exercise, cost = 0.0, label = "Success"
)

reaction_exercise_failure <- Reaction$new(
  chance_node_exercise, leaf_node_exercise_stent,
  p = q_exercise, cost = cost_stent, label = "Failure"
)

dt <- DecisionTree$new(
  V = list(
    decision_node,
    chance_node_diet,
    chance_node_exercise,
    leaf_node_diet_no_stent,
    leaf_node_diet_stent,
    leaf_node_exercise_no_stent,
    leaf_node_exercise_stent
  ),
  E = list(
    action_diet,
    action_exercise,
    reaction_diet_success,
    reaction_diet_failure,
    reaction_exercise_success,
    reaction_exercise_failure
  )
)

dt$draw()

Evaluating a decision tree

rs <- dt$evaluate()

rp <- dt$evaluate(by = "path")

du_stent <- 0.05
leaf_node_diet_stent$set_utility(1.0 - du_stent)
leaf_node_exercise_stent$set_utility(1.0 - du_stent)
rs <- dt$evaluate()

delta_c <- rs[[which(rs[, "Programme"] == "Exercise"), "Cost"]] -
  rs[[which(rs[, "Programme"] == "Diet"), "Cost"]]
delta_u <- rs[[which(rs[, "Programme"] == "Exercise"), "Utility"]] -
  rs[[which(rs[, "Programme"] == "Diet"), "Utility"]]
icer <- delta_c / delta_u

Introducing probabilistic elements

The model shown above uses a fixed value for each parameter, resulting in a single point estimate for each model result. However, parameters may be affected by uncertainty: for example, the success probability of each strategy is extracted from a small trial of few patients. This uncertainty can be incorporated into the Decision Tree model by representing individual parameters with a statistical distribution, then repeating the evaluation of the model multiple times with each run randomly drawing parameters from these defined distributions.

In rdecision, model variables that are described by a distribution are represented by ModVar objects. Many commonly used distributions, such as the Normal, Log-Normal, Gamma and Beta distributions are included in the package, and additional distributions can be easily implemented from the generic ModVar class. Additionally, model variables that are calculated from other r probabilistic variables using an expression can be represented as ExprModVar objects.

Fixed costs can be left as numerical values, or also be represented by ModVars which ensures that they are included in variable tabulations.

cost_diet <- ConstModVar$new("Cost of diet programme", "GBP", 50.0)
cost_exercise <- ConstModVar$new("Cost of exercise programme", "GBP", 750.0)
cost_stent <- ConstModVar$new("Cost of stent intervention", "GBP", 5000.0)

In our simplified example, the probability of success of each strategy should include the uncertainty associated with the small sample that they are based on. This can be represented statistically by a Beta distribution, a probability distribution constrained to the interval [0, 1]. A Beta distribution that captures the results of the trials can be defined by the alpha (observed successes) and beta (observed failures) parameters.

p_diet <- BetaModVar$new(
  alpha = s_diet, beta = f_diet, description = "P(diet)", units = ""
)
p_exercise <- BetaModVar$new(
  alpha = s_exercise, beta = f_exercise, description = "P(exercise)", units = ""
)

q_diet <- ExprModVar$new(
  rlang::quo(1.0 - p_diet), description = "1 - P(diet)", units = ""
)
q_exercise <- ExprModVar$new(
  rlang::quo(1.0 - p_exercise), description = "1 - P(exercise)", units = ""
)

These distributions describe the probability of success of each strategy; by the constraints of a Decision Tree, the sum of all probabilities associated with a chance node must be 1, so the probability of failure should be calculated as 1 - p(Success). This can be represented by an ExprModVar.

The newly defined ModVars can be incorporated into the Decision Tree model using the same grammar as the non-probabilistic model. Because the actions and reactions are objects already included in the tree, we can change their properties using set_ calls and those new properties will be used when the tree is evaluated.

action_diet$set_cost(cost_diet)
action_exercise$set_cost(cost_exercise)

reaction_diet_success$set_probability(p_diet)

reaction_diet_failure$set_probability(q_diet)
reaction_diet_failure$set_cost(cost_stent)

reaction_exercise_success$set_probability(p_exercise)

reaction_exercise_failure$set_probability(q_exercise)
reaction_exercise_failure$set_cost(cost_stent)

All the probabilistic variables included in the model can be tabulated using the modvar_table() method, which details the distribution definition and some useful parameters, such as mean, SD and 95% CI.

A call to the evaluate() method with the default settings uses the expected (mean) value of each variable, and so replicates the point estimate above.

rs <- dt$evaluate()

However, because each variable is described by a distribution, it is now possible to explore the range of possible values consistent with the model. For example, a lower and upper bound can be estimated by setting each variable to its 2.5-th or 97.5-th percentile:

rs_025 <- dt$evaluate(setvars = "q2.5")
rs_975 <- dt$evaluate(setvars = "q97.5")

The costs for each choice when all variables are at their upper and lower confidence levels are as follows:

To sample the possible outcomes in a completely probabilistic way, the setvar = "random" option can be used, which draws a random value from the distribution of each variable. Repeating this process a sufficiently large number of times builds a collection of results compatible with the model definition, which can then be used to calculate ranges and confidence intervals of the estimated values.

N <- 1000L
rs <- dt$evaluate(setvars = "random", by = "run", N = N)

The estimates of cost for each intervention can be plotted as follows:

plot(
  rs[, "Cost.Diet"],
  rs[, "Cost.Exercise"],
  pch = 20L,
  xlab = "Cost of diet (GBP)", ylab = "Cost of exercise (GBP)",
  main = paste(N, "simulations of vascular disease prevention model")
)
abline(a = 0.0, b = 1.0, col = "red")

A tabular summary is as follows:

The variables can be further manipulated, for example calculating the difference in cost between the two strategies for each run of the randomised model:

rs[, "Difference"] <- rs[, "Cost.Diet"] - rs[, "Cost.Exercise"]
CI <- quantile(rs[, "Difference"], c(0.025, 0.975))

hist(
  rs[, "Difference"], 100L,  main = "Distribution of saving",
  xlab = "Saving (GBP)"
)

knitr::kable(
  rs[1L : 10L, c(1L, 3L, 8L, 12L)], digits = 2L,
  row.names = FALSE
)

Plotting the distribution of the difference of the two costs reveals that, in this model, the uncertainties in the input parameters are large enough that either strategy could have a lower net cost, within a 95% confidence interval [-771.19, 749.44].

cost_threshold <- dt$threshold(
  index = list(action_exercise),
  ref = list(action_diet),
  outcome = "saving",
  mvd = cost_exercise$description(),
  a = 0.0, b = 5000.0, tol = 0.1
)

success_threshold <- dt$threshold(
  index = list(action_exercise),
  ref = list(action_diet),
  outcome = "saving",
  mvd = p_exercise$description(),
  a = 0.0, b = 1.0, tol = 0.001
)