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.
If you are not familiar with heemod
, first consult the introduction vignette vignette("introduction", package = "heemod")
.
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.
This model aims to compare costs and utilities of two treatment strategies, monotherapy and combined therapy.
Four states are described, from best to worst healtwise:
Transition probabilities for the monotherapy study group are rather simple to implement:
mat_mono <-
define_matrix(
.721, .202, .067, .010,
.000, .581, .407, .012,
.000, .000, .750, .250,
.000, .000, .000, 1.00
)
mat_mono
## An unevaluated matrix, 4 states.
##
## A B C D
## A 0.721 0.202 0.067 0.01
## B 0 0.581 0.407 0.012
## C 0 0 0.75 0.25
## D 0 0 0 1
The combined therapy group has its transition probabilities multiplied by rr
, 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).
rr <- .509
mat_comb <-
define_matrix(
C, .202*rr, .067*rr, .010*rr,
.000, C, .407*rr, .012*rr,
.000, .000, C, .250*rr,
.000, .000, .000, 1.00
)
mat_comb
## An unevaluated matrix, 4 states.
##
## A B C D
## A C 0.202 * rr 0.067 * rr 0.01 * rr
## B 0 C 0.407 * rr 0.012 * rr
## C 0 0 C 0.25 * rr
## D 0 0 0 1
We can plot the transition matrix for the monotherapy group:
plot(mat_mono)
And the combined therapy group:
plot(mat_comb)
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.
For example state A can be defined with define_state
:
A_mono <-
define_state(
cost_health = 2756,
cost_drugs = cost_zido,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
A_mono
## An unevaluated state with 4 values.
##
## cost_health = 2756
## cost_drugs = cost_zido
## cost_total = discount(cost_health + cost_drugs, 0.06)
## life_year = 1
The other states for the monotherapy treatment group can be specified in the same way:
B_mono <-
define_state(
cost_health = 3052,
cost_drugs = cost_zido,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
C_mono <-
define_state(
cost_health = 9007,
cost_drugs = cost_zido,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
D_mono <-
define_state(
cost_health = 0,
cost_drugs = 0,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 0
)
Similarly, for the the combined therapy treatment group, only cost_drug
differs from the monotherapy treatment group:
A_comb <-
define_state(
cost_health = 3052,
cost_drugs = cost_zido + cost_lami,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
B_comb <-
define_state(
cost_health = 3052 + cost_lami,
cost_drugs = cost_zido,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
C_comb <-
define_state(
cost_health = 9007 + cost_lami,
cost_drugs = cost_zido,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
D_comb <-
define_state(
cost_health = 0,
cost_drugs = 0,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 0
)
Models can now be defined by combining a transition matrix and a state list with define_model
:
mod_mono <- define_model(
transition_matrix = mat_mono,
A_mono,
B_mono,
C_mono,
D_mono
)
## No named state -> generating names.
mod_mono
## An unevaluated Markov model:
##
## 0 parameter,
## 4 states,
## 4 state values,
## No starting values defined.
For the combined therapy model:
mod_comb <- define_model(
transition_matrix = mat_comb,
A_comb,
B_comb,
C_comb,
D_comb
)
## No named state -> generating names.
Both models can then be run for 20 years with run_model
. Models are given simple names (mono
and comb
) in order to facilitate result interpretation:
res_mod <- run_models(
mono = mod_mono,
comb = mod_comb,
cycles = 20,
cost = cost_total,
effect = life_year
)
By default models are run for one person starting in the first state (here state A).
Model values can then be compared with summary
:
summary(res_mod)
## 2 Markov models run for 20 cycles.
##
## Initial states:
##
## N
## A 1
## B 0
## C 0
## D 0
## cost_health cost_drugs cost_total life_year .cost .effect
## mono 45479.45 18176.56 44613.85 7.979173 0.00 0.000000
## comb 89433.47 43596.75 81026.56 13.864239 36412.71 5.885066
## .icer
## mono -Inf
## comb 6187.307
##
## Efficiency frontier:
##
## mono comb
The incremental cost-effectiveness ratio of the combiend therapy strategy is thus £6187 per life-year gained.
The counts per state can be plotted for the monotherapy group:
plot(res_mod, model = "mono", type = "counts")
And the combined therapy group:
plot(res_mod, model = "comb", type = "counts")
We can plot the strategies on the cost-efficiency plane:
plot(res_mod, type = "ce")