---
title: "Decision tree with PSA (Jenks 2016)"
subtitle: "Tegaderm CHG IV Securement Dressing"
author: "Andrew J. Sims"
date: "July 2020"
bibliography: "REFERENCES.bib"
csl: "nature-no-et-al.csl"
output: 
  rmarkdown::html_vignette:
    fig_width: 7
    fig_height: 5
    fig_caption: true
    df_print: kable
vignette: >
  %\VignetteIndexEntry{Decision tree with PSA (Jenks 2016)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r}
#| include = FALSE,
#| purl = FALSE
knitr::opts_chunk$set(
  collapse = TRUE,
  echo = FALSE,
  comment = "#>"
)
```

```{r}
#| purl = FALSE
#nolint start
```

```{r}
library(rdecision)
```

```{r}
#| purl = FALSE
#nolint end
```

# Introduction
This vignette is an example of modelling a decision tree using the `rdecision`
package, with probabilistic sensitivity analysis (PSA). It is based on the model 
reported by Jenks *et al* [-@jenks2016] in which a transparent dressing used 
to secure vascular catheters (Tegaderm CHG) was compared with a 
standard dressing.

# Model variables

## Source variables
Eleven source variables were used in the model. The choice of
variables, their distributions and their parameters are taken from Table 4 
of Jenks *et al* [-@jenks2016], with the following additional information:

* The baseline catheter-related blood stream infection (CRBSI) rate was modelled
  as a Gamma distribution fitted by the method of moments to a mean of 1.48
  (per 1000 catheter days) and a standard deviation of 0.12 (per 1000 catheter
  days).
* The baseline local site infection (LSI) rate was modelled as a Gamma
  distribution fitted by the method of moments to a mean of 0.14 (per 1000
  catheter days) and an assumed standard deviation of 0.5 (/1000 catheter days).
* The baseline rate of dermatitis was modelled as a Beta distribution, based on
  one observed case in a trial of 476 catheter uses.
* The effect size of Tegaderm, expressed as the hazard ratio of Tegaderm
  compared with standard dressings for CRBSI and LSI, and the relative risk of
  Tegaderm compared with standard dressings for dermatitis, was modelled in
  each case using a log normal distribution. This was fitted to a sample mean
  and sample standard deviation on the natural scale, by using the "LN7"
  parametrization of `LogNormModVar`.
* The probabilities of CRBSI and LSI for standard dressings ($p$) were 
  modified by the hazard ratio $r$ for Tegaderm using the form $p * r$. This
  is an approximation which holds only for very small rates.
* Relative risks were also applied as multipliers. This is an approximation
  which holds only for very small rates.

The model variables were constructed as follows:

```{r}
#| echo = TRUE
# baseline risk
r.CRBSI <- GammaModVar$new(
  "Baseline CRBSI rate",  "/1000 catheter days",
  shape = (1.48 ^ 2L) / (0.12 ^ 2L),
  scale = (0.12 ^ 2L) / 1.48
)
r.LSI <- GammaModVar$new(
  "Baseline LSI rate", "/1000 catheter days",
  shape = (0.14 ^ 2L) / (0.5 ^ 2L),
  scale = (0.5 ^ 2L) / 0.14
)
r.Dermatitis <- BetaModVar$new(
  "Baseline dermatitis risk", "/catheter", alpha = 1L, beta = 475L
)
# relative effectiveness
hr.CRBSI <- LogNormModVar$new(
  "Tegaderm CRBSI HR", "HR",
  p1 = 0.402, p2 = (0.868 - 0.186) / (2L * 1.96), param = "LN7"
)
hr.LSI <- LogNormModVar$new(
  "Tegaderm LSI HR", "HR",
  p1 = 0.402, p2 = (0.868 - 0.186) / (2L * 1.96), param = "LN7"
)
rr.Dermatitis <- LogNormModVar$new(
  "Tegaderm Dermatitis RR", "RR", p1 = 1.0, p2 = 0.5, param = "LN7"
)
# cost variables
c.CRBSI <- GammaModVar$new(
  "CRBSI cost", "GBP",
  shape = (9900.0 ^ 2L) / (3000.0 ^ 2L),
  scale = (3000.0 ^ 2L) / 9900.0
)
c.LSI <- GammaModVar$new(
  "LSI cost", "GBP",
  shape = (100.0 ^ 2L) / (30.0 ^ 2L),
  scale = (30.0 ^ 2L) / 100.0
)
c.Dermatitis <- GammaModVar$new(
  "Dermatitis cost", "GBP",
  shape = (6.0 ^ 2L) / (3.0 ^ 2L),
  scale = (3.0 ^ 2L) / 6.0
)
# number of dressings and days with catheter
n.dressings <- GammaModVar$new(
  "No. dressings", "dressings",
  shape = (3.0 ^ 2L) / (2.0 ^ 2L),
  scale = (2.0 ^ 2L) / 3.0
)
n.cathdays <- GammaModVar$new(
  "No. days with catheter", "days",
  shape = (10.0 ^ 2L) / (5.0 ^ 2L),
  scale = (5.0 ^ 2L) / 10.0
)
```

```{r}
#| purl = FALSE
# test that variables have expected values
local({
  # baseline CRBSI
  q <- r.CRBSI$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(r.CRBSI$mean(), 1.48, tolerance = 0.02, scale = 1.0),
    all.equal(q[[1L]], 1.28, tolerance = 0.05, scale = 1.0),
    all.equal(q[[2L]], 1.75, tolerance = 0.05, scale = 1.0)
  )
  # baseline LSI
  q <- r.LSI$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(r.LSI$mean(), 0.14, tolerance = 0.01, scale = 1.0),
    all.equal(q[[1L]], 0.0, tolerance = 0.05, scale = 1.0)
  )
  # baseline dermatitis
  q <- r.Dermatitis$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(r.Dermatitis$mean(), 1L / 476L, tolerance = 0.0001, scale = 1.0),
    all.equal(q[[1L]], 0.000, tolerance = 0.005, scale = 1.0),
    all.equal(q[[2L]], 0.010, tolerance = 0.005, scale = 1.0)
  )
  # HR of CRBSI for Tegaderm
  all.equal(hr.CRBSI$mean(), 0.402, 0.010)
  q <- hr.CRBSI$quantile(probs = c(0.025, 0.975))
  all.equal(q[[1L]], 0.186, 0.05)
  all.equal(q[[2L]], 0.868, 0.05)
  # HR of LSI for Tegaderm
  q <- hr.LSI$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(hr.LSI$mean(), 0.402, tolerance = 0.010, scale = 1.0),
    all.equal(q[[1L]], 0.186, tolerance = 0.05, scale = 1.0),
    all.equal(q[[2L]], 0.868, tolerance = 0.05, scale = 1.0)
  )
  # RR of dermatitis
  q <- rr.Dermatitis$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(rr.Dermatitis$mean(), 1.0, tolerance = 0.010, scale = 1.0),
    all.equal(q[[1L]], 0.35, tolerance = 0.05, scale = 1.0),
    all.equal(q[[2L]], 2.26, tolerance = 0.05, scale = 1.0)
  )
  # cost of CRBSI
  q <- c.CRBSI$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(c.CRBSI$mean(), 9900.0, tolerance = 10.0, scale = 1.0),
    all.equal(q[[1L]], 4921.0, tolerance = 10.0, scale = 1.0),
    all.equal(q[[2L]], 16589.0, tolerance = 10.0, scale = 1.0)
  )
  # cost of LSI
  q <- c.LSI$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(c.LSI$mean(), 100.0, tolerance = 10.0, scale = 1.0),
    all.equal(q[[1L]], 50.1, tolerance = 1.0, scale = 1.0),
    all.equal(q[[2L]], 166.8, tolerance = 1.0, scale = 1.0)
  )
  # cost of dermatitis
  q <- c.Dermatitis$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(c.Dermatitis$mean(), 6.0, tolerance = 0.1, scale = 1.0),
    all.equal(q[[1L]], 1.64, tolerance = 0.1, scale = 1.0),
    all.equal(q[[2L]], 13.1, tolerance = 0.1, scale = 1.0)
  )
  # number of dressings
  q <- n.dressings$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(n.dressings$mean(), 3.0, tolerance = 0.1, scale = 1.0),
    all.equal(q[[1L]], 0.4, tolerance = 0.1, scale = 1.0),
    all.equal(q[[2L]], 8.0, tolerance = 0.1, scale = 1.0)
  )
  # number of catheter days
  q <- n.cathdays$quantile(probs = c(0.025, 0.975))
  stopifnot(
    all.equal(n.cathdays$mean(), 10.0, tolerance = 0.1, scale = 1.0),
    all.equal(q[[1L]], 2.7, tolerance = 1.0, scale = 1.0),
    all.equal(q[[2L]], 21.9, tolerance = 1.0, scale = 1.0)
  )
})
```

## Model variable expressions
Variables in the model may be included in the decision tree via mathematical 
expressions, which involve model variables and are themselves
model variables. Forms of expression involving R functions and
multiple model variables are supported, provided they conform to R syntax.
The following code creates the model variable expressions to be used as values
in the decision tree edges. 

```{r}
#| echo = TRUE
p.CRBSI.S <- ExprModVar$new(
  "P(CRBSI | standard dressing)", "P",
  rlang::quo(r.CRBSI * n.cathdays / 1000.0)
)
p.CRBSI.T <- ExprModVar$new(
  "P(CRBSI|Tegaderm)", "P",
  rlang::quo(p.CRBSI.S * hr.CRBSI)
)
p.LSI.S <- ExprModVar$new(
  "P(LSI | Standard)", "/patient",
  rlang::quo(r.LSI * n.cathdays / 1000.0)
)
p.LSI.T <- ExprModVar$new(
  "P(LSI | Tegaderm)", "P", rlang::quo(p.LSI.S * hr.LSI)
)
p.Dermatitis.S <- ExprModVar$new(
  "P(dermatitis | standard dressing)", "P",
  rlang::quo(r.Dermatitis)
)
p.Dermatitis.T <- ExprModVar$new(
  "P(dermatitis | Tegaderm)", "P",
  rlang::quo(p.Dermatitis.S * rr.Dermatitis)
)
c.Tegaderm <- ExprModVar$new(
  "Tegaderm CHG cost", "GBP", rlang::quo(6.26 * n.dressings)
)
c.Standard <- ExprModVar$new(
  "Standard dressing cost", "GBP", rlang::quo(1.54 * n.dressings)
)
```

# The decision tree

## Constructing the tree
The following code constructs the decision tree based on Figure 2
of Jenks *et al* [-@jenks2016]. In the formulation used by `rdecision`, 
the decision tree is constructed from sets of decision, chance and 
leaf nodes and from edges (actions and reactions).
Leaf nodes are synonymous with
pathways in Briggs' terminology [-@briggs2006]. The time horizon is
not stated explicitly in the model, and is assumed to be 7 days. It was implied
that the time horizon was ICU stay plus some follow-up, and the costs reflect
those incurred in that period, so the assumption of 7 days does not affect
the `rdecision` implementation of the model.

The tree is somewhat more complex than Figure 2 of Jenks *et al* because it
allows for patients to have more than one adverse event (AE) during their stay
(whereas their Figure 2 implies that only one event per patient is possible). 
The rates of AE were estimated independently, and allow for multiple events,
(figure 1).

In `rdecision`, if the
probability associated with one of the reactions from any chance node is set
to missing (`NA`), it will be computed before each evaluation of the tree to
ensure that the probabilities sum to unity.

```{r}
#| echo = TRUE
# create decision tree
th <- as.difftime(7L, units = "days")
# standard dressing
t01 <- LeafNode$new("t01", interval = th)
t02 <- LeafNode$new("t02", interval = th)
c01 <- ChanceNode$new()
e01 <- Reaction$new(
  c01, t01, p = p.Dermatitis.S, cost = c.Dermatitis, label = "Dermatitis"
)
e02 <- Reaction$new(
  c01, t02, p = NA_real_, cost = 0.0, label = "No dermatitis"
)
t03 <- LeafNode$new("t03", interval = th)
t04 <- LeafNode$new("t04", interval = th)
c02 <- ChanceNode$new()
e03 <- Reaction$new(
  c02, t03, p = p.Dermatitis.S, cost = c.Dermatitis, label = "Dermatitis"
)
e04 <- Reaction$new(
  c02, t04, p = NA_real_, cost = 0.0, label = "No dermatitis"
)
c03 <- ChanceNode$new()
e05 <- Reaction$new(c03, c01, p = p.LSI.S, cost = c.LSI, label = "LSI")
e06 <- Reaction$new(c03, c02, p = NA_real_, cost = 0.0, label = "No LSI")
t11 <- LeafNode$new("t11", interval = th)
t12 <- LeafNode$new("t12", interval = th)
c11 <- ChanceNode$new()
e11 <- Reaction$new(
  c11, t11, p = p.Dermatitis.S, cost = c.Dermatitis, label = "Dermatitis"
)
e12 <- Reaction$new(
  c11, t12, p = NA_real_, cost = 0.0, label = "No Dermatitis"
)
t13 <- LeafNode$new("t13", interval = th)
t14 <- LeafNode$new("t14", interval = th)
c12 <- ChanceNode$new()
e13 <- Reaction$new(
  c12, t13, p = p.Dermatitis.S, cost = c.Dermatitis, label = "Dermatitis"
)
e14 <- Reaction$new(
  c12, t14, p = NA_real_, cost = 0.0, label = "No dermatitis"
)
c13 <- ChanceNode$new()
e15 <- Reaction$new(c13, c11, p = p.LSI.S, cost = c.LSI, label = "LSI")
e16 <- Reaction$new(c13, c12, p = NA_real_, cost = 0.0, label = "No LSI")
c23 <- ChanceNode$new()
e21 <- Reaction$new(c23, c03, p = p.CRBSI.S, cost = c.CRBSI, label = "CRBSI")
e22 <- Reaction$new(c23, c13, p = NA_real_, cost = 0.0, label = "No CRBSI")

# Tegaderm branch
t31 <- LeafNode$new("t31", interval = th)
t32 <- LeafNode$new("t32", interval = th)
c31 <- ChanceNode$new()
e31 <- Reaction$new(
  c31, t31, p = p.Dermatitis.T, cost = c.Dermatitis, label = "Dermatitis"
)
e32 <- Reaction$new(
  c31, t32, p = NA_real_, cost = 0.0, label = "no dermatitis"
)
t33 <- LeafNode$new("t33", interval = th)
t34 <- LeafNode$new("t34", interval = th)
c32 <- ChanceNode$new()
e33 <- Reaction$new(
  c32, t33, p = p.Dermatitis.T, cost = c.Dermatitis, label = "Dermatitis"
)
e34 <- Reaction$new(
  c32, t34, p = NA_real_, cost = 0.0, label = "No dermatitis"
)
c33 <- ChanceNode$new()
e35 <- Reaction$new(c33, c31, p = p.LSI.T, cost = c.LSI, label = "LSI")
e36 <- Reaction$new(c33, c32, p = NA_real_, cost = 0.0, label = "No LSI")
t41 <- LeafNode$new("t41", interval = th)
t42 <- LeafNode$new("t42", interval = th)
c41 <- ChanceNode$new()
e41 <- Reaction$new(
  c41, t41, p = p.Dermatitis.T, cost = c.Dermatitis, label = "Dermatitis"
)
e42 <- Reaction$new(
  c41, t42, p = NA_real_, cost = 0.0, label = "No dermatitis"
)
t43 <- LeafNode$new("t43", interval = th)
t44 <- LeafNode$new("t44", interval = th)
c42 <- ChanceNode$new()
e43 <- Reaction$new(
  c42, t43, p = p.Dermatitis.T, cost = c.Dermatitis, label = "Dermatitis"
)
e44 <- Reaction$new(
  c42, t44, p = NA_real_, cost = 0.0, label = "No dermatitis"
)
c43 <- ChanceNode$new()
e45 <- Reaction$new(c43, c41, p = p.LSI.T, cost = c.LSI, label = "LSI")
e46 <- Reaction$new(c43, c42, p = NA_real_, cost = 0.0, label = "No LSI")
c53 <- ChanceNode$new()
e51 <- Reaction$new(c53, c43, p = p.CRBSI.T, cost = c.CRBSI, label = "CRBSI")
e52 <- Reaction$new(c53, c33, p = NA_real_, cost = 0.0, label = "no CRBSI")

# decision node
d1 <- DecisionNode$new("d1")
e9 <- Action$new(d1, c23, label = "Standard", cost = c.Standard)
e10 <- Action$new(d1, c53, label = "Tegaderm", cost = c.Tegaderm)

# create decision tree
V <- list(
  d1,
  c01, c02, c03, c11, c12, c13, c23, c31, c32, c33, c41, c42, c43, c53,
  t01, t02, t03, t04, t11, t12, t13, t14, t31, t32, t33, t34,
  t41, t42, t43, t44
)
E <- list(
  e01, e02, e03, e04, e05, e06, e11, e12, e13, e14, e15, e16, e21, e22,
  e31, e32, e33, e34, e35, e36, e41, e42, e43, e44, e45, e46, e51, e52,
  e9, e10
)
DT <- DecisionTree$new(V, E)
```

```{r}
#| purl = FALSE
# test that model variables are as expected
local({
  mv <- DT$modvars()
  stopifnot(
    all.equal(length(mv), 19L)
  )
  MVT <- DT$modvar_table()
  stopifnot(
    all.equal(nrow(MVT), 19L),
    all.equal(sum(MVT$Est), 8L)
  )
})
```

## Tree diagram
The `draw` method of a `DecisionTree` object creates a graphical representation
of the tree, as follows.
```{r}
#| results = "hide",
#| fig.keep = "all",
#| fig.align = "center",
#| fig.cap = "Figure 1. Decision tree for the Tegaderm model",
#| echo = TRUE
DT$draw(border = TRUE)
```

## Summary of model variables in the tree
The model variables which will be associated with actions, reactions and leaf 
nodes
can be tabulated using the method `modvar_table`. This returns a data
frame describing each variable, its description, units and uncertainty
distribution. Variables inheriting from type `ModVar` will be included in the
tabulation unless explicitly excluded, regular numeric values will not be
listed. In the Tegaderm model, the input model variables are in the following
table, including those constructed from expressions. 

```{r}
with(data = DT$modvar_table(), expr = {
  data.frame(
    Description = Description,
    Distribution = Distribution,
    stringsAsFactors = FALSE
  )
})
```

The units, point estimates, lower 95% and upper 9% confidence intervals are
are obtained from the same call, in the remaining columns. 

```{r}
with(data = DT$modvar_table(), expr = {
  data.frame(
    Variable = paste(Description, Units, sep = ", "),
    Mean = round(E, digits = 3L),
    LowerCI = round(Q2.5, digits = 3L),
    UpperCI = round(Q97.5, digits = 3L),
    stringsAsFactors = FALSE
  )
})
```

# Running the model

## Base case
The following code runs a single model scenario, using the `evaluate`
method of a decision node to evaluate each pathway from the decision node,
shown in the table. This model did not consider utility, and the columns 
associated with utility are removed. 

```{r}
#| echo = TRUE
RES <- DT$evaluate()
```

```{r}
#| purl = FALSE
# test that EAC base case agrees with direct calculation
local({
  # values from Table 4
  r_crbsi <- 1.48
  r_lsi <- 0.14
  r_derm <- 0.0021
  hr_crbsi_teg <- 0.402
  hr_lsi_teg <- 0.402
  rr_derm_teg <- 1.0
  c_crbsi <- 9900.0
  c_derm <- 6.0
  c_lsi <- 100.0
  n_cdays <- 10.0
  n_dress <- 3L
  c_teg <- 6.26
  c_std <- 1.54
  # probabilities
  p_crbsi_std <- r_crbsi * (n_cdays / 1000.0)
  p_lsi_std <- r_lsi * (n_cdays / 1000.0)
  p_derm_std <- r_derm
  p_crbsi_teg <- p_crbsi_std * hr_crbsi_teg
  p_lsi_teg <- p_lsi_std * hr_lsi_teg
  p_derm_teg <- rr_derm_teg * p_derm_std
  # component costs
  cdress_std <- c_std * n_dress
  cdress_teg <- c_teg * n_dress
  ccrbsi_std <- c_crbsi * p_crbsi_std
  ccrbsi_teg <- c_crbsi * p_crbsi_teg
  clsi_std <- c_lsi * p_lsi_std
  clsi_teg <- c_lsi * p_lsi_teg
  cderm_std <- c_derm * p_derm_std
  cderm_teg <- c_derm * p_derm_teg
  # total costs
  c_std <- cdress_std + ccrbsi_std + clsi_std + cderm_std
  c_teg <- cdress_teg + ccrbsi_teg + clsi_teg + cderm_teg
  with(data = RES, expr = {
    stopifnot(
      # check against the model
      all.equal(
        Cost[[which(d1 == "Standard")]], c_std, tolerance = 2.0, scale = 1.0
      ),
      all.equal(
        Cost[[which(d1 == "Tegaderm")]], c_teg, tolerance = 2.0, scale = 1.0
      ),
      # check against the Excel model
      all.equal(
        Cost[[which(d1 == "Standard")]], 151.29, tolerance = 2.0, scale = 1.0
      ),
      all.equal(
        Cost[[which(d1 == "Tegaderm")]], 77.75, tolerance = 2.0, scale = 1.0
      )
    )
  })
})
```

```{r}
with(data = RES, expr = {
  data.frame(
    Run = Run,
    d1 = d1,
    Cost = gbp(Cost, p = TRUE, char = FALSE),
    stringsAsFactors = FALSE
  )
})
```

## Univariate sensitivity analysis
The sensitivity of the decision tree results to each source model variable,
varied independently of the others, is demonstrated by a tornado diagram. The
method `tornado` can be used to generate such a plot (and also provides a
tabulated version of the values used in the plot).

Tornado diagrams compare
outcomes for two interventions, labelled as `index` and `ref`. In a decision
tree, an intervention is defined as a strategy for traversing the tree,
expressed as a list of the action edges emanating from each decision node. In
trees with a single decision node, the `index` and `ref` parameters may be
expressed as a single action edge. Source variables are
varied over their 95% confidence limits (figure 2).

```{r}
#| results = "hide",
#| fig.keep = "all",
#| fig.align = "center",
#| fig.cap = "Figure 2. Tornado diagram for the Tegaderm model",
#| echo = TRUE
to <- DT$tornado(index = e10, ref = e9, draw = TRUE)
```

The object returned from method `tornado` (`to`) is a data frame which includes
the values of the cost difference when each model variable is univariately at
the limits of its 95% confidence interval, as follows:

```{r}
with(data = to, expr = {
  data.frame(
    Variable = paste(Description, Units, sep = ", "),
    LL = round(x = LL, digits = 2L),
    UL = round(x = UL, digits = 2L),
    Min.CostDiff = round(x = outcome.min, digits = 2L),
    Max.CostDiff = round(x = outcome.max, digits = 2L),
    stringsAsFactors = FALSE
  )
})
```

## Probabilistic sensitivity analysis
Multivariate probabilistic sensitivity analysis is supported through the use of
sampling model variables. The same call, with extra parameters, is used to run 
the PSA and save the results in a data frame. Additionally, the cost difference
is computed for each run of the model, as follows:

```{r}
#| echo = TRUE
N <- 1000L
psa <- DT$evaluate(setvars = "random", by = "run", N = N)
psa[, "Difference"] <- psa[, "Cost.Standard"] - psa[, "Cost.Tegaderm"]
```

The first few runs of PSA are as follows; the `by = "run"` option reshapes the
table to give one row per simulation, rather than one row per run, per strategy.

```{r}
with(data = head(psa, n = 10L), expr = {
  data.frame(
    Run = Run,
    Cost.Tegaderm = gbp(Cost.Tegaderm, p = TRUE, char = FALSE),
    Cost.Standard = gbp(Cost.Standard, p = TRUE, char = FALSE),
    Cost.Difference = gbp(Difference, p = TRUE, char = FALSE),
    stringsAsFactors = FALSE
  )
})
```

From PSA (`r N` runs), the mean cost of treatment with Tegaderm 
was `r gbp(mean(psa[, "Cost.Tegaderm"]), p = TRUE)` GBP,
the mean cost of treatment with standard dressings was
`r gbp(mean(psa[, "Cost.Standard"]), p = TRUE)` GBP
and the mean cost saving was `r gbp(mean(psa[, "Difference"]), p = TRUE)` GBP.
The 95% confidence interval for cost saving was 
`r gbp(quantile(psa[, "Difference"], probs = 0.025), p = TRUE)` GBP to 
`r gbp(quantile(psa[, "Difference"], probs = 0.975), p = TRUE)` GBP; the
standard deviation of the cost saving was 
`r gbp(sd(psa[, "Difference"]), p = TRUE)` GBP.
Overall, `r round(100.0 * sum(psa[, "Difference"] > 0.0) / nrow(psa), 2L)`% of
runs found that Tegaderm was cost saving. These results replicate those reported
by Jenks *et al* (saving of 72.90 GBP, 97.8% cases cost saving; mean cost of
standard dressing 151.29 GBP, mean cost of Tegaderm 77.75 GBP).

```{r}
rm(psa)
```

## Scenario - low baseline rate of CRBSI
Jenks *et al* modelled an additional scenario, in which the baseline rate
of CRBSI was 0.3 per 1000 catheter days (modelled as a Gamma distribution fitted
to a sample mean of 0.3 and a sample 95% confidence interval of 0.2 to 0.6). A
way to achieve this in `rdecision` is to replace the model variable for the
baseline rate of CRBSI, and any other model variables that depend on it via
expressions, and then reconstruct the model, as follows.

```{r}
#| echo = TRUE
r.CRBSI <- GammaModVar$new(
  "Baseline CRBSI rate",  "/1000 catheter days",
  shape = (0.30 ^ 2L) / (0.102 ^ 2L),
  scale = (0.102 ^ 2L) / 0.30
)
p.CRBSI.S <- ExprModVar$new(
  "P(CRBSI | standard dressing)", "P",
  rlang::quo(r.CRBSI * n.cathdays / 1000.0)
)
p.CRBSI.T <- ExprModVar$new(
  "P(CRBSI|Tegaderm)", "P",
  rlang::quo(p.CRBSI.S * hr.CRBSI)
)
e21 <- Reaction$new(c23, c03, p = p.CRBSI.S, cost = c.CRBSI, label = "CRBSI")
e22 <- Reaction$new(c23, c13, p = NA_real_, cost = 0.0, label = "No CRBSI")
e51 <- Reaction$new(c53, c43, p = p.CRBSI.T, cost = c.CRBSI, label = "CRBSI")
e52 <- Reaction$new(c53, c33, p = NA_real_, cost = 0.0, label = "no CRBSI")
E <- list(
  e01, e02, e03, e04, e05, e06, e11, e12, e13, e14, e15, e16, e21, e22,
  e31, e32, e33, e34, e35, e36, e41, e42, e43, e44, e45, e46, e51, e52,
  e9, e10
)
DT <- DecisionTree$new(V, E)
```

```{r}
#| purl = FALSE
# test that scenario case agrees with direct calculation
local({
  # evaluate the scenario as a point estimate
  s_sco <- DT$evaluate()
  # values from Table 4
  r_crbsi <- 0.30
  r_lsi <- 0.14
  r_derm <- 0.0021
  hr_crbsi_teg <- 0.402
  hr_lsi_teg <- 0.402
  rr_derm_teg <- 1.0
  c_crbsi <- 9900.0
  c_derm <- 6.0
  c_lsi <- 100.0
  n_cdays <- 10.0
  n_dress <- 3L
  c_teg <- 6.26
  c_std <- 1.54
  # probabilities
  p_crbsi_std <- r_crbsi * (n_cdays / 1000.0)
  p_lsi_std <- r_lsi * (n_cdays / 1000.0)
  p_derm_std <- r_derm
  p_crbsi_teg <- p_crbsi_std * hr_crbsi_teg
  p_lsi_teg <- p_lsi_std * hr_lsi_teg
  p_derm_teg <- rr_derm_teg * p_derm_std
  # component costs
  cdress_std <- c_std * n_dress
  cdress_teg <- c_teg * n_dress
  ccrbsi_std <- c_crbsi * p_crbsi_std
  ccrbsi_teg <- c_crbsi * p_crbsi_teg
  clsi_std <- c_lsi * p_lsi_std
  clsi_teg <- c_lsi * p_lsi_teg
  cderm_std <- c_derm * p_derm_std
  cderm_teg <- c_derm * p_derm_teg
  # total costs
  c_std <- cdress_std + ccrbsi_std + clsi_std + cderm_std
  c_teg <- cdress_teg + ccrbsi_teg + clsi_teg + cderm_teg
  with(data = s_sco, expr = {
    stopifnot(
      # check against the model
      all.equal(
        Cost[[which(d1 == "Standard")]], c_std, tolerance = 2.0, scale = 1.0
      ),
      all.equal(
        Cost[[which(d1 == "Tegaderm")]], c_teg, tolerance = 2.0, scale = 1.0
      ),
      # check against the excel model
      all.equal(
        Cost[[which(d1 == "Standard")]], 34.47, tolerance = 2.0, scale = 1.0
      ),
      all.equal(
        Cost[[which(d1 == "Tegaderm")]], 30.79, tolerance = 2.0, scale = 1.0
      )
    )
  })
})
```

The model for this scenario was run under PSA, as for the base case:

```{r}
#| echo = TRUE
N <- 1000L
psa <- DT$evaluate(setvars = "random", by = "run", N = N)
psa[, "Difference"] <- psa[, "Cost.Standard"] - psa[, "Cost.Tegaderm"]
```

From PSA (`r N` runs), the mean cost of treatment with Tegaderm 
was `r gbp(mean(psa[, "Cost.Tegaderm"]), p = TRUE)` GBP,
the mean cost of treatment with standard dressings was 
`r gbp(mean(psa[, "Cost.Standard"]), p = TRUE)` GBP
and the mean cost saving was `r gbp(mean(psa[, "Difference"]), p = TRUE)` GBP.
The 95% confidence interval for cost saving was 
`r gbp(quantile(psa[, "Difference"], probs = 0.025), p = TRUE)` GBP to 
`r gbp(quantile(psa[, "Difference"], probs = 0.975), p = TRUE)` GBP; the
standard deviation
of the cost saving was `r gbp(sd(psa[, "Difference"]), p = TRUE)` GBP.
Overall, `r round(100.0 * sum(psa[, "Difference"] > 0.0) / nrow(psa), 2L)`% of
runs found that Tegaderm was cost saving. These results replicate those reported
by Jenks *et al* (saving of 3.56 GBP, 57.9% cases cost saving; mean cost of
standard dressing 34.47 GBP, mean cost of Tegaderm 30.79 GBP).

Two threshold analyses were reported for this scenario. This can be achieved
in `rdecision` by using the `threshold` method of the decision tree. Firstly,
the threshold hazard ratio of a CRBSI with Tegaderm versus a CRBSI with a
standard dressing was varied in the range 0.1 to 0.9, as follows:

```{r}
#| echo = TRUE
hr_threshold <- DT$threshold(
  index = list(e10),
  ref = list(e9),
  outcome = "saving",
  mvd = "Tegaderm CRBSI HR",
  a = 0.1,
  b = 0.9,
  tol = 0.01
)
```

```{r}
#| purl = FALSE
# test that scenario hazard rate threshold agrees with that reported
local({
  stopifnot(
    all.equal(hr_threshold, 0.53, tolerance = 0.05, scale = 1.0)
  )
})
```

This gave a threshold value of `r round(hr_threshold, 2L)`, above which
Tegaderm became cost incurring (the reported threshold was 0.53).
Secondly, the cost of each CRBSI was varied between 0 GBP and 9900 GBP to find
the threshold of cost saving, as follows:

```{r}
#| echo = TRUE
c_crbsi_threshold <- DT$threshold(
  index = list(e10),
  ref = list(e9),
  outcome = "saving",
  mvd = "CRBSI cost",
  a = 0.0,
  b = 9900.0,
  tol = 10.0
)
```

```{r}
#| purl = FALSE
# test_that scenario CRBSI cost threshold agrees with reported value
local({
  stopifnot(
    all.equal(c_crbsi_threshold, 8000.0, tolerance = 300.0, scale = 1.0)
  )
})
```

This gave a threshold value of `r gbp(c_crbsi_threshold, p = TRUE)` GBP, below
which Tegaderm became cost incurring (the reported threshold was 8000 GBP).

# References
