The most simple Markov models in health economic evaluation are models were transition probabilities between states do not change with time. Those are called homogeneous or time-homogeneous Markov models.
In this example we will model the cost effectiveness of lamivudine/zidovudine combination therapy in HIV infection (Chancellor, 1997 further described in Decision Modelling for Health Economic Evaluation, page 32. For the sake of simplicity we will not reproduce exactly the analysis from the book. See vignette vignette("i-reproduction", "heemod")
for an exact reproduction of the analysis.
This model aims to compare costs and utilities of two treatment strategies, monotherapy and combined therapy.
Four states are described, from best to worst health-wise:
Transition probabilities for the monotherapy study group are rather simple to implement with define_transition()
:
<- define_transition(
mat_mono 721, .202, .067, .010,
.0, .581, .407, .012,
0, 0, .750, .250,
0, 0, 0, 1
)
## No named state -> generating names.
mat_mono
## A transition matrix, 4 states.
##
## A B C D
## A 0.721 0.202 0.067 0.01
## B 0.581 0.407 0.012
## C 0.75 0.25
## D 1
The combined therapy group has its transition probabilities multiplied by \(rr = 0.509\), the relative risk of event for the population treated by combined therapy. Since \(rr < 1\), the combined therapy group has less chance to transition to worst health states.
The probabilities to stay in the same state are equal to \(1 - \sum P_{trans}\) where \(P_{trans}\) are the probabilities to change to another state (because all transition probabilities from a given state must sum to 1).
We use the alias C
as a convenient way to specify the probability complement, equal to \(1 - \sum P_{trans}\).
<- .509
rr
<- define_transition(
mat_comb 202*rr, .067*rr, .010*rr,
C, .0, C, .407*rr, .012*rr,
0, 0, C, .250*rr,
0, 0, 0, 1
)
## No named state -> generating names.
mat_comb
## A transition matrix, 4 states.
##
## A B C D
## A C 0.202 * rr 0.067 * rr 0.01 * rr
## B C 0.407 * rr 0.012 * rr
## C C 0.25 * rr
## D 1
We can plot the transition matrix for the monotherapy group:
plot(mat_mono)
## Le chargement a nécessité le package : diagram
And the combined therapy group:
plot(mat_comb)
The costs of lamivudine and zidovudine are defined:
<- 2278
cost_zido <- 2086 cost_lami
In addition to drugs costs (called cost_drugs
in the model), each state is associated to healthcare costs (called cost_health
). Cost are discounted at a 6% rate with the discount
function.
Efficacy in this study is measured in terms of life expectancy (called life_year
in the model). Each state thus has a value of 1 life year per year, except death who has a value of 0. Life-years are not discounted in this example.
Only cost_drug
differs between the monotherapy and the combined therapy treatment groups, the function dispatch_strategy()
can be used to account for that. For example state A can be defined with define_state()
:
<- define_state(
state_A cost_health = discount(2756, .06),
cost_drugs = discount(dispatch_strategy(
mono = cost_zido,
comb = cost_zido + cost_lami
06),
), .cost_total = cost_health + cost_drugs,
life_year = 1
) state_A
## A state with 4 values.
##
## cost_health = discount(2756, 0.06)
## cost_drugs = discount(dispatch_strategy(mono = cost_zido, comb = cost_zido +
## cost_lami), 0.06)
## cost_total = cost_health + cost_drugs
## life_year = 1
The other states for the monotherapy treatment group can be specified in the same way:
<- define_state(
state_B cost_health = discount(3052, .06),
cost_drugs = discount(dispatch_strategy(
mono = cost_zido,
comb = cost_zido + cost_lami
06),
), .cost_total = cost_health + cost_drugs,
life_year = 1
)<- define_state(
state_C cost_health = discount(9007, .06),
cost_drugs = discount(dispatch_strategy(
mono = cost_zido,
comb = cost_zido + cost_lami
06),
), .cost_total = cost_health + cost_drugs,
life_year = 1
)<- define_state(
state_D cost_health = 0,
cost_drugs = 0,
cost_total = 0,
life_year = 0
)
Strategies can now be defined by combining a transition matrix and a state list with define_strategy()
:
<- define_strategy(
strat_mono transition = mat_mono,
state_A,
state_B,
state_C,
state_D )
## No named state -> generating names.
strat_mono
## A Markov model strategy:
##
## 4 states,
## 4 state values
For the combined therapy model:
<- define_strategy(
strat_comb transition = mat_comb,
state_A,
state_B,
state_C,
state_D )
## No named state -> generating names.
Both strategies can then be combined in a model and run for 50 years with run_model()
. Strategies are given names (mono
and comb
) in order to facilitate result interpretation.
<- run_model(
res_mod mono = strat_mono,
comb = strat_comb,
cycles = 50,
cost = cost_total,
effect = life_year
)
By default models are run for 1000 persons starting in the first state (here state A).
Strategy values can then be compared with summary()
(optionally net monetary benefits can be calculated with the threshold
option):
summary(res_mod,
threshold = c(1000, 5000, 6000, 1e4))
## 2 strategies run for 50 cycles.
##
## Initial state counts:
##
## A = 1000L
## B = 0L
## C = 0L
## D = 0L
##
## Counting method: 'life-table'.
##
##
##
## Counting method: 'beginning'.
##
##
##
## Counting method: 'end'.
##
## Values:
##
## cost_health cost_drugs cost_total life_year
## mono 33891136 14870957 48762093 8585.843
## comb 48739757 44245091 92984848 17256.937
##
## Net monetary benefit difference:
##
## 1000 5000 6000 10000
## mono 35551.66 867.2847 0.000 0.00
## comb 0.00 0.0000 7803.809 42488.19
##
## Efficiency frontier:
##
## mono -> comb
##
## Differences:
##
## Cost Diff. Effect Diff. ICER Ref.
## comb 44222.75 8.671094 5100.02 mono
The incremental cost-effectiveness ratio of the combined therapy strategy is thus £5100 per life-year gained.
The counts per state can be plotted by model:
plot(res_mod, type = "counts", panel = "by_strategy") +
xlab("Time") +
theme_bw() +
scale_color_brewer(
name = "State",
palette = "Set1"
)
Or by state:
plot(res_mod, type = "counts", panel = "by_state") +
xlab("Time") +
theme_bw() +
scale_color_brewer(
name = "Strategy",
palette = "Set1"
)
The values can also be represented:
plot(res_mod, type = "values", panel = "by_value",
free_y = TRUE) +
xlab("Time") +
theme_bw() +
scale_color_brewer(
name = "Strategy",
palette = "Set1"
)
Note that classic ggplot2
syntax can be used to modifiy plot appearance.