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 al1 in which a transparent dressing
used to secure vascular catheters (Tegaderm CHG) was compared with a
standard dressing.
Thirteen source variables were used in the model. The choice of variables, their distributions and their parameters are taken from Table 3 of Jenks et al1, with the following additional information:
LN1
parametrization of the LogNormModVar
provided in rdecision
.The model variables were constructed as follows:
# clinical variables
<- NormModVar$new(
r.CRBSI 'Baseline CRBSI rate', '/1000 catheter days', mu=1.48, sigma=0.074
)<- LogNormModVar$new(
hr.CRBSI "Tegaderm CRBSI HR", "HR", p1 = -0.911, p2 = 0.393
)<- LogNormModVar$new(
hr.LSI "Tegaderm LSI HR", "HR", p1 = -0.911, p2 = 0.393
)<- NormModVar$new(
r.Dermatitis 'Baseline dermatitis risk', '/catheter', mu=0.0026, sigma=0.00026
)<- LogNormModVar$new(
rr.Dermatitis "Tegaderm Dermatitis RR", "RR", p1=1.482, p2=0.490
)# cost variables
<- GammaModVar$new(
c.CRBSI 'CRBSI cost', 'GBP', shape=198.0, scale=50
)<- GammaModVar$new(
c.Dermatitis 'Dermatitis cost', 'GBP', shape=30, scale=5
)<- GammaModVar$new(
c.LSI 'LSI cost', 'GBP', shape=50, scale=5
)<- NormModVar$new(
n.catheters 'No. catheters', 'catheters', mu=3, sigma=0.3
)<- ExprModVar$new(
c.Tegaderm "Tegaderm CHG cost", "GBP", rlang::quo(6.21*n.catheters)
)<- ExprModVar$new(
c.Standard "Standard dressing cost", "GBP", rlang::quo(1.34*n.catheters)
)<- NormModVar$new(
n.cathdays 'No. days with catheter', 'days', mu=10, sigma=2
)
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. For probabilities, the convention \(q = 1-p\) is used to ensure that the sum of probabilities leaving each chance node is one.
# probabilities
<- ExprModVar$new(
p.Dermatitis.S 'P(dermatitis|standard dressing)', 'P',
::quo(r.Dermatitis*n.catheters)
rlang
)<- ExprModVar$new(
q.Dermatitis.S 'Q(dermatitis|standard dressing)', '1-P',
::quo(1-p.Dermatitis.S)
rlang
)<- ExprModVar$new(
p.Dermatitis.T 'P(dermatitis|Tegaderm)', 'P',
::quo(r.Dermatitis*rr.Dermatitis*n.catheters)
rlang
)<- ExprModVar$new(
q.Dermatitis.T 'Q(dermatitis|Tegaderm)', '1-P',
::quo(1-p.Dermatitis.T)
rlang
)
<- NormModVar$new(
p.LSI.S 'P(LSI|Standard)', '/patient', mu=0.1, sigma=0.01
)<- ExprModVar$new(
q.LSI.S 'Q(LSI|Standard)', '1-P', rlang::quo(1-p.LSI.S)
)<- ExprModVar$new(
p.LSI.T 'P(LSI|Tegaderm)', 'P', rlang::quo(1-(1-p.LSI.S)^hr.LSI)
)<- ExprModVar$new(
q.LSI.T 'Q(LSI|Tegaderm)', '1-P', rlang::quo(1-p.LSI.T)
)
<- ExprModVar$new(
p.CRBSI.S 'P(CRBSI|standard dressing)', 'P', rlang::quo(r.CRBSI*n.cathdays/1000)
)<- ExprModVar$new(
q.CRBSI.S 'Q(CRBSI|standard dressing)', '1-P', rlang::quo(1-p.CRBSI.S)
)<- ExprModVar$new(
p.CRBSI.T 'P(CRBSI|Tegaderm)', 'P', rlang::quo(1-(1-r.CRBSI*n.cathdays/1000)^hr.CRBSI)
)<- ExprModVar$new(
q.CRBSI.T 'Q(CRBSI|Tegaderm)', '1-P', rlang::quo(1-p.CRBSI.T)
)
The following code constructs the decision tree based on Figure 2 of
Jenks et al1. 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’ terminology2. 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, see figure.
# create decision tree
<- as.difftime(7, units="days")
th # standard dressing
<- LeafNode$new("t01", interval=th)
t01 <- LeafNode$new("t02", interval=th)
t02 <- ChanceNode$new()
c01 <- Reaction$new(c01,t01,p=p.Dermatitis.S,cost=c.Dermatitis,
e01 label="Dermatitis")
<- Reaction$new(c01,t02,p=q.Dermatitis.S,cost=0,
e02 label="No dermatitis")
#
<- LeafNode$new("t03", interval=th)
t03 <- LeafNode$new("t04", interval=th)
t04 <- ChanceNode$new()
c02 <- Reaction$new(c02,t03,p=p.Dermatitis.S,cost=c.Dermatitis,
e03 label="Dermatitis")
<- Reaction$new(c02,t04,p=q.Dermatitis.S,cost=0,
e04 label="No dermatitis")
#
<- ChanceNode$new()
c03 <- Reaction$new(c03,c01,p=p.LSI.S,cost=c.LSI,label="LSI")
e05 <- Reaction$new(c03,c02,p=q.LSI.S,cost=0,label="No LSI")
e06 #
<- LeafNode$new("t11", interval=th)
t11 <- LeafNode$new("t12", interval=th)
t12 <- ChanceNode$new()
c11 <- Reaction$new(c11,t11,p=p.Dermatitis.S,cost=c.Dermatitis,
e11 label="Dermatitis")
<- Reaction$new(c11,t12,p=q.Dermatitis.S,cost=0,label="No Dermatitis")
e12 #
<- LeafNode$new("t13", interval=th)
t13 <- LeafNode$new("t14", interval=th)
t14 <- ChanceNode$new()
c12 <- Reaction$new(c12,t13,p=p.Dermatitis.S,cost=c.Dermatitis,
e13 label="Dermatitis")
<- Reaction$new(c12,t14,p=q.Dermatitis.S,cost=0,label="No dermatitis")
e14 #
<- ChanceNode$new()
c13 <- Reaction$new(c13,c11,p=p.LSI.S,cost=c.LSI,label="LSI")
e15 <- Reaction$new(c13,c12,p=q.LSI.S,cost=0,label="No LSI")
e16 #
<- ChanceNode$new()
c23 <- Reaction$new(c23,c03,p=p.CRBSI.S,cost=c.CRBSI,label="CRBSI")
e21 <- Reaction$new(c23,c13,p=q.CRBSI.S,cost=0,label="No CRBSI")
e22 #
# Tegaderm branch
<- LeafNode$new("t31", interval=th)
t31 <- LeafNode$new("t32", interval=th)
t32 <- ChanceNode$new()
c31 <- Reaction$new(c31,t31,p=p.Dermatitis.T,cost=c.Dermatitis,
e31 label="Dermatitis")
<- Reaction$new(c31,t32,p=q.Dermatitis.T,cost=0,label="no dermatitis")
e32 #
<- LeafNode$new("t33", interval=th)
t33 <- LeafNode$new("t34", interval=th)
t34 <- ChanceNode$new()
c32 <- Reaction$new(c32,t33,p=p.Dermatitis.T,cost=c.Dermatitis,
e33 label="Dermatitis")
<- Reaction$new(c32,t34,p=q.Dermatitis.T,cost=0,label="No dermatitis")
e34 #
<- ChanceNode$new()
c33 <- Reaction$new(c33,c31,p=p.LSI.T,cost=c.LSI,label="LSI")
e35 <- Reaction$new(c33,c32,p=q.LSI.T,cost=0,label="No LSI")
e36 #
<- LeafNode$new("t41", interval=th)
t41 <- LeafNode$new("t42", interval=th)
t42 <- ChanceNode$new()
c41 <- Reaction$new(c41,t41,p=p.Dermatitis.T,cost=c.Dermatitis,
e41 label="Dermatitis")
<- Reaction$new(c41,t42,p=q.Dermatitis.T,cost=0,label="No dermatitis")
e42 #
<- LeafNode$new("t43", interval=th)
t43 <- LeafNode$new("t44", interval=th)
t44 <- ChanceNode$new()
c42 <- Reaction$new(c42,t43,p=p.Dermatitis.T,cost=c.Dermatitis,
e43 label="Dermatitis")
<- Reaction$new(c42,t44,p=q.Dermatitis.T,cost=0,label="No dermatitis")
e44 #
<- ChanceNode$new()
c43 <- Reaction$new(c43,c41,p=p.LSI.T,cost=c.LSI,label="LSI")
e45 <- Reaction$new(c43,c42,p=q.LSI.T,cost=0,label="No LSI")
e46 #
<- ChanceNode$new()
c53 <- Reaction$new(c53,c43,p=p.CRBSI.T,cost=c.CRBSI,label="CRBSI")
e51 <- Reaction$new(c53,c33,p=q.CRBSI.T,cost=0,label="no CRBSI")
e52 #
# decision node
<- DecisionNode$new("d1")
d1 <- Action$new(d1,c23,label="Standard",cost=c.Standard)
e9 <- Action$new(d1,c53,label="Tegaderm",cost=c.Tegaderm)
e10 #
# create decision tree
<- list(d1,c01,c02,c03,c11,c12,c13,c23,c31,c32,c33,c41,c42,c43,c53,
V
t01,t02,t03,t04,t11,t12,t13,t14,t31,t32,t33,t34,t41,t42,t43,t44)<- list(e01,e02,e03,e04,e05,e06,e11,e12,e13,e14,e15,e16,e21,e22,
E
e31,e32,e33,e34,e35,e36,e41,e42,e43,e44,e45,e46,e51,e52,e9,e10)<- DecisionTree$new(V,E) DT
In the company’s model, the uncertainties in the probabilities associated with the polytomous chance nodes were modelled as independent variables. This is not recommended because there is a chance that a particular run of the PSA will yield probabilities that are outside the range [0,1].
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, with expression model variables excluded.
Description | Distribution |
---|---|
Dermatitis cost | Ga(30,5) |
Baseline dermatitis risk | N(0.0026,0.00026) |
No. catheters | N(3,0.3) |
LSI cost | Ga(50,5) |
P(LSI|Standard) | N(0.1,0.01) |
CRBSI cost | Ga(198,50) |
Baseline CRBSI rate | N(1.48,0.074) |
No. days with catheter | N(10,2) |
Tegaderm Dermatitis RR | LN(1.482,0.49) |
Tegaderm LSI HR | LN(-0.911,0.393) |
Tegaderm CRBSI HR | LN(-0.911,0.393) |
The point estimates, units and distributional properties are obtained from the same call, in the remaining columns.
Variable | Mean | Q2.5 | Q97.5 |
---|---|---|---|
Dermatitis cost, GBP | 150 | 101 | 208 |
Baseline dermatitis risk, /catheter | 0.0026 | 0.00209 | 0.00311 |
No. catheters, catheters | 3 | 2.41 | 3.59 |
LSI cost, GBP | 250 | 186 | 324 |
P(LSI|Standard), /patient | 0.1 | 0.0804 | 0.12 |
CRBSI cost, GBP | 9900 | 8569 | 11326 |
Baseline CRBSI rate, /1000 catheter days | 1.48 | 1.33 | 1.63 |
No. days with catheter, days | 10 | 6.08 | 13.9 |
Tegaderm Dermatitis RR, RR | 4.96 | 1.68 | 11.5 |
Tegaderm LSI HR, HR | 0.434 | 0.186 | 0.869 |
Tegaderm CRBSI HR, HR | 0.434 | 0.186 | 0.869 |
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.
Run | d1 | Cost |
---|---|---|
1 | Standard | 176.7 |
1 | Tegaderm | 99.54 |
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). Source variables are varied over their 95% confidence limits,
see figure.
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:
<- 1000
N <- DT$evaluate(setvars="random", by="run", N=N) PSA
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.
Run | Cost.Tegaderm | Cost.Standard | Difference |
---|---|---|---|
1 | 76.34 | 205.2 | -128.8 |
2 | 110.8 | 164.4 | -53.65 |
3 | 117.7 | 215.6 | -97.92 |
4 | 107 | 204.1 | -97.14 |
5 | 108.3 | 181.8 | -73.54 |
6 | 84.23 | 229.9 | -145.7 |
7 | 95.24 | 194.7 | -99.49 |
8 | 91.57 | 161.9 | -70.35 |
9 | 158.3 | 145.7 | 12.62 |
10 | 138.1 | 157.7 | -19.65 |
From PSA (1000 runs), the mean cost of treatment with Tegaderm was 99.06, the mean cost of treatment with standard dressings was 175.7 and the mean cost saving was -76.64. The 95% confidence interval for cost saving was -141.14 to -13.15; the standard deviation of the cost saving was 32.38. Overall, 98.9% of runs found that Tegaderm was cost saving. These results replicate those reported by the manufacturer (saving of 77.76, 98.5% cases cost saving; mean cost of standard dressing 176.89, mean cost of Tegaderm 99.63).