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.
Model description
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:
- A: CD4 cells > 200 and < 500 cells/mm3;
- B: CD4 < 200 cells/mm3, non-AIDS;
- C: AIDS;
- D: Death.
Transition probabilities
Transition probabilities for the monotherapy study group are rather
simple to implement with define_transition()
:
mat_mono <- define_transition(
.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 , the relative risk of event for the population treated by combined therapy. Since , the combined therapy group has less chance to transition to worst health states.
The probabilities to stay in the same state are equal to where 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
.
rr <- .509
mat_comb <- define_transition(
C, .202*rr, .067*rr, .010*rr,
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)
## Loading required namespace: diagram
And the combined therapy group:
plot(mat_comb)
State values
The costs of lamivudine and zidovudine are defined:
cost_zido <- 2278
cost_lami <- 2086
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()
:
state_A <- define_state(
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(...)
## cost_total = cost_health + cost_drugs
## life_year = 1
The other states for the monotherapy treatment group can be specified in the same way:
state_B <- define_state(
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
)
state_C <- define_state(
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
)
state_D <- define_state(
cost_health = 0,
cost_drugs = 0,
cost_total = 0,
life_year = 0
)
Strategy definitions
Strategies can now be defined by combining a transition matrix and a
state list with define_strategy()
:
strat_mono <- define_strategy(
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:
strat_comb <- define_strategy(
transition = mat_comb,
state_A,
state_B,
state_C,
state_D
)
## No named state -> generating names.
Running the model
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.
res_mod <- run_model(
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).
Result interpretation
Strategy values can then be compared with summary()
(optionally net monetary benefits can be calculated with the
threshold
option):
## 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 modify
plot appearance.