In more complex Markov models in health economic evaluation transition probabilities between states can vary with time. These models are called non-homogeneous or time-inhomogeneous Markov models. A further distinction can be made depending on whether:

  1. transition probabilities depend on how long the entire model has been running (model-time variation): this situation can be modelled with a non-homogeneous Markov model;
  2. transition probabilities depend on how long an individual has been in a state (individual-time variation): this situation usually needs to be modelled with a microsimulation.

However in some special cases non-homogeneous Markov models can be used in situation 2, when the state with transition probabilities depending on individual-time is the starting state and it is not possible to go back to that state after having left it. In this situation individual-time is equivalent to model-time. This is the case in the following example.

If you are not familiar with heemod, first consult the introduction vignette vignette("introduction", package = "heemod").

Model description

This example is an implementation of the assessment of a new total hip replacement (THR) technology described in chapter 3.5 of Decision Modelling for Health Economic Evaluation. A more detailed report is available at this location, event though this reports goes a bit further in the analysis.

This model has 5 states:

Two transition probabilities are time-varying in this model:

Other-cause death probabilities (mortality rate mr) vary with age and gender with the following values:

death_prob <- data.frame(
  age = rep(seq(35, 85, 10), each = 2),
  sex = rep(0:1, 6),
  mr = c(
    1.51e-3, .99e-3, 3.93e-3,
    2.6e-3, 10.9e-3, 6.7e-3,
    31.6e-3, 19.3e-3, 80.1e-3,
    53.5e-3, 187.9e-3, 154.8e-3
  )
)
death_prob
##    age sex      mr
## 1   35   0 0.00151
## 2   35   1 0.00099
## 3   45   0 0.00393
## 4   45   1 0.00260
## 5   55   0 0.01090
## 6   55   1 0.00670
## 7   65   0 0.03160
## 8   65   1 0.01930
## 9   75   0 0.08010
## 10  75   1 0.05350
## 11  85   0 0.18790
## 12  85   1 0.15480

Primary THR revision probability for the standard THR, called standardRR increases with time with the following formula (a Weibull distribution):

\[ P_{revision} = 1 - \exp(\lambda \times ((t-1)^\gamma-t^\gamma)) \]

Where \(t\) is the time since revision, \(\gamma = 1.45367786\) and:

\[ \lambda = exp(cons + ageC \times age + maleC \times sex) \]

Where age and sex (female = 0, male = 1) are individual characteristics, cons = -5.49094, ageC = -0.0367 and maleC = 0.768536.

For the NP1 procedure the starndard procedure revision probability standardRR is modified by the relative risk rrNP1 = 0.260677.

\[ P_{revision} = 1 - \exp(\lambda \times rrNP1 \times ((t-1)^\gamma-t^\gamma)) \]

Revision THR re-revision (rrr) probability is set to be constant at 0.04 per year.

Parameters definition

The key element to specify time-varying elements in heemod is through the use of a package-defined variable, markov_model. This variable takes increasing values with each cycles, starting from 1. For example the age of individuals at any moment can be defined as Initial age + markov_cycle.

In order to build this more complex Markov model, parameters need to be defined through define_parameters(). The equations decribed in the previous section can be written easily, here for a female population (sex = 0) starting at 60 years old (age_init = 60):

# a function to return age-related mortality rate
# given age and sex
mr_func <- function(age, sex) {
  age  <- floor(age/10-.5)*10+5
  age <- ifelse(age > 85, 85, age)
  merge(data.frame(age = age, sex = sex), death_prob)$mr
}

param_standard <- define_parameters(
    age_init = 60,
    sex = 0,
    # age increases with cycles
    age = age_init + markov_cycle,
    
    # operative mortality rates
    omrPTHR = .02,
    omrRTHR = .02,
    
    # re-revision mortality rate
    rrr = .04,
    
    # parameters for calculating primary revision rate
    cons = -5.49094,
    ageC = -.0367,
    maleC = .768536,
    lambda = exp(cons + ageC * age_init + maleC * sex),
    gamma = 1.45367786,
    
    rrNP1 = .260677,
    
    # revision probability of primary procedure
    standardRR = 1 - exp(lambda * ((markov_cycle - 1) ^ gamma -
                                     markov_cycle ^ gamma)),
    
    # age-related mortality rate
    mr = mr_func(age, sex)
)
param_standard
## 14 unevaluated parameters.
## 
## age_init = 60
## sex = 0
## age = age_init + markov_cycle
## omrPTHR = 0.02
## omrRTHR = 0.02
## rrr = 0.04
## cons = -5.49094
## ageC = -0.0367
## maleC = 0.768536
## lambda = exp(cons + ageC * age_init + maleC * sex)
## gamma = 1.45367786
## rrNP1 = 0.260677
## standardRR = 1 - exp(lambda * ((markov_cycle - 1)^gamma - markov_cycle^gamma))
## mr = mr_func(age, sex)

The parameters for the NP1 group are almost the same as for the standard group, with the exception of the primary THR revision probability formula. Parameters can by modified through the function modify():

param_np1 <- modify(
  param_standard,
  standardRR = 1 - exp(
    lambda *
      rrNP1 *
      (
        (markov_cycle - 1) ^ gamma - 
          markov_cycle ^ gamma)
  )
)
param_np1
## 14 unevaluated parameters.
## 
## age_init = 60
## sex = 0
## age = age_init + markov_cycle
## omrPTHR = 0.02
## omrRTHR = 0.02
## rrr = 0.04
## cons = -5.49094
## ageC = -0.0367
## maleC = 0.768536
## lambda = exp(cons + ageC * age_init + maleC * sex)
## gamma = 1.45367786
## rrNP1 = 0.260677
## standardRR = 1 - exp(lambda * rrNP1 * ((markov_cycle - 1)^gamma - markov_cycle^gamma))
## mr = mr_func(age, sex)

Transition matrix definition

Now that parameters are defined, the probability transitions can be easily written:

mat_trans <- define_matrix(
    state_names = c(
      "PrimaryTHR",
      "SuccessP",
      "RevisionTHR",
      "SuccessR",
      "Death"
    ),
    0, C, 0,          0, omrPTHR,
    0, C, standardRR, 0, mr,
    0, 0, 0,          C, omrRTHR+mr,
    0, 0, rrr,        C, mr,
    0, 0, 0,          0, 1
)
mat_trans
## An unevaluated matrix, 5 states.
## 
##             PrimaryTHR SuccessP RevisionTHR SuccessR Death       
## PrimaryTHR  0          C        0           0        omrPTHR     
## SuccessP    0          C        standardRR  0        mr          
## RevisionTHR 0          0        0           C        omrRTHR + mr
## SuccessR    0          0        rrr         C        mr          
## Death       0          0        0           0        1

While it is possible to plot the matrix thanks to the diagram package, the results may not always be easy to read.

plot(mat_trans)

State & model definition

Utilities and costs are then associated to states. In this model costs are discounted at a rate of 6% and utilities at a rate of 1.5%.

Now that parameters, transition matrix and states are defined we can define the models for the control group and the NP1 treatment:

mod_standard <- define_model(
  parameters = param_standard,
  transition_matrix = mat_trans,
  PrimaryTHR = define_state(
    utility = 0,
    cost = 0
  ),
  SuccessP = define_state(
    utility = discount(.85, .015),
    cost = 0
  ),
  RevisionTHR = define_state(
    utility = discount(.30, .015),
    cost = discount(5294, .06)
  ),
  SuccessR = define_state(
    utility = discount(.75, .015),
    cost = 0
  ),
  Death = define_state(
    utility = 0,
    cost = 0
  ),
  starting_values = c(cost = 394)
)
mod_standard
## An unevaluated Markov model:
## 
##     14 parameters,
##     5 states,
##     2 state values,
##     Starting values defined.
mod_np1 <- define_model(
  parameters = param_np1,
  transition_matrix = mat_trans,
  PrimaryTHR = define_state(
    utility = 0,
    cost = 0
  ),
  SuccessP = define_state(
    utility = discount(.85, .015),
    cost = 0
  ),
  RevisionTHR = define_state(
    utility = discount(.30, .015),
    cost = discount(5294, .06)
  ),
  SuccessR = define_state(
    utility = discount(.75, .015),
    cost = 0
  ),
  Death = define_state(
    utility = 0,
    cost = 0
  ),
  starting_values = c(cost = 579)
)

Analyse models

Both models can now be run for 60 years, for 1 person starting in PrimaryTHR:

res_mod <- run_models(
  standard = mod_standard,
  np1 = mod_np1,
  cycles = 60,
  cost = cost,
  effect = utility
)
res_mod
## 2 Markov models, run for 60 cycles.
## 
## Model names:
## 
## standard
## np1

A comparison of both models can be done with summary(). The incremental cost and effect are displayed in columns .cost and .effect.

summary(res_mod)
## 2 Markov models run for 60 cycles.
## 
## Initial states:
## 
##             N
## PrimaryTHR  1
## SuccessP    0
## RevisionTHR 0
## SuccessR    0
## Death       0
##           utility     cost    .cost    .effect    .icer
## standard 14.67977 615.9570  0.00000 0.00000000     -Inf
## np1      14.72271 637.8967 21.93971 0.04293769 510.9662
## 
## Efficiency frontier:
## 
## standard np1

The new treatment costs £511 per QALY gained.

We can plot the counts per state for the standard model:

plot(res_mod, model = "standard", type = "counts")

And the np1 model:

plot(res_mod, model = "np1", type = "counts")

Models on the cost-effectiveness plane:

plot(res_mod, type = "ce")