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("reproduction", package = "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 healtwise:

• 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:

mat_mono <-
  define_matrix(
    .721, .202, .067, .010,
    .000, .581, .407, .012,
    .000, .000, .750, .250,
    .000, .000, .000, 1.00
  )

## No named state -> generating names.

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).

We use the alias C as a convenient way to specify the probability complement, equal to 1 - sum(row probabilities)

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
  )

## No named state -> generating names.

mat_comb

## An unevaluated matrix, 4 states.
## 
##   A B          C          D         
## A   0.202 * rr 0.067 * rr 0.01 * rr 
## B 0            0.407 * rr 0.012 * rr
## C 0 0                     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)

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.

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 = 2756,
    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
  )

Model definition

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:
## 
##     4 states,
##     4 state values

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.

Running models

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 1000
## B    0
## C    0
## D    0
##      cost_health cost_drugs cost_total life_year
## mono    45479452   18176555   47290682  7979.173
## comb    87728706   43596748   84574956 13864.239
## 
## Efficiency frontier:
## 
## mono comb
## 
## Model difference:
## 
##          Cost   Effect     ICER
## comb 37284.27 5.885066 6335.405

The incremental cost-effectiveness ratio of the combiend therapy strategy is thus £6335 per life-year gained.

The counts per state can be plotted for the monotherapy group:

plot(res_mod, model = "mono", type = "counts") +
  xlab("Time") +
  theme_minimal() +
  scale_color_brewer(
    name = "State",
    palette = "Set1"
  )

And the combined therapy group:

plot(res_mod, model = "comb", type = "counts") +
  xlab("Time") +
  theme_minimal() +
  scale_color_brewer(
    name = "State",
    palette = "Set1"
  )

Note that classic ggplot2 syntax can be used to modifiy plot appearance.