An Introduction to heemod

Transition probabilities

The probability to transition from one state to another during a time period is called a transition probability. Said time period is called a cycle.

Transition probabilities between states can be described through a 2-way table where the lines correspond to the states at the beginning of a cycle and the columns to the states at the end of a cycle. Consider a model with 2 states A and B:

When starting a cycle in state A (row A), the probability to still be in state A at the end of the cycle is found in colunm A (cell 1) and the probability to change to state B is found in column B (cell 2).

Similarly, when starting a cycle from state B (row B), the probability to be in state A or B at the end of the cycle are found in cells 3 or 4 respectively.

In the context of Markov models, this 2-way table is called a transition matrix. A transition matrix can be defined easily in heemod with the define_transition() function. If we consider the previous example, where cell values have been replaced by actual probabilities:

That transition matrix can be defined with the following command:

mat_trans <- define_transition(
  .9, .1,
  .2, .8
)

## No named state -> generating names.

mat_trans

## A transition matrix, 2 states.
## 
##   A   B  
## A 0.9 0.1
## B 0.2 0.8

Attach values to states

Values are attached to states. Cost and utility are classical examples of such values. To continue with the previous example, the following values can be attachd to state A and B:

• State A has a cost of 1234 per cycle and an utility of 0.85.
• State B has a cost of 4321 per cycle and an utility of 0.50.

A state and its values can be defined with define_state():

state_A <- define_state(
  cost = 1234,
  utility = 0.85
)
state_A

## A state with 2 values.
## 
## cost = 1234
## utility = 0.85

state_B <- define_state(
  cost = 4321,
  utility = 0.50
)
state_B

## A state with 2 values.
## 
## cost = 4321
## utility = 0.5

Combine information in a model

Now that the transition matrix and the state values are defined for a given strategy, we can combine them with define_strategy():

strat_1 <- define_strategy(
  transition_matrix = mat_trans,
  state_A,
  state_B
)

## Warning in define_strategy(transition_matrix = mat_trans, state_A,
## state_B): Argument 'transition_matrix' is deprecated, use 'transition'
## instead.

## No named state -> generating names.

strat_1

## A Markov model strategy:
## 
##     2 states,
##     2 state values

Run the model

A model is a made of one or more strategy, run with run_model() for a given number of cycles. Here we run only one strategy, so no comparison of the cost-effectiveness of the different strategies will be made. The variables corresponding to valuation of cost and effect must be given at that point.

res_mod_1 <- run_model(
  strat_1,
  cycles = 10,
  cost = cost,
  effect = utility
)

## No named model -> generating names.

res_mod_1

## 1 strategy run for 10 cycles.
## 
## Initial state counts:
## 
##      N
## A 1000
## B    0
## 
## Counting method: 'life-table'.
## 
##       cost  utility
## I 19796856 7654.552

By default the model is run for 1000 persons starting in the first state (here state A).

Analyse results

We can plot the state membership counts over time. Other plot types are available.

plot(res_mod_1)

Plots can be modified using ggplot2 syntax.

library(ggplot2)

plot(res_mod_1) +
  xlab("Time") +
  ylab("N") +
  theme_minimal() +
  scale_color_brewer(
    name = "State",
    palette = "Set1"
  )

The state membership counts and the values can be accessed with get_counts() and get_values() respectively.

get_counts(res_mod_1)

##           A        B
## 1  950.0000  50.0000
## 2  865.0000 135.0000
## 3  805.5000 194.5000
## 4  763.8500 236.1500
## 5  734.6950 265.3050
## 6  714.2865 285.7135
## 7  700.0005 299.9995
## 8  690.0004 309.9996
## 9  683.0003 316.9997
## 10 678.1002 321.8998

get_values(res_mod_1)

##    markov_cycle    cost  utility
## 1             1 1388350 832.5000
## 2             2 1650745 802.7500
## 3             3 1834422 781.9250
## 4             4 1962995 767.3475
## 5             5 2052997 757.1432
## 6             6 2115998 750.0003
## 7             7 2160098 745.0002
## 8             8 2190969 741.5001
## 9             9 2212578 739.0501
## 10           10 2227705 737.3351

Convenience functions

Convenience functions are available to easily compute transition probabilities from indidence rates, OR, RR, or probabilities estimated on a different timeframe.

Example : convert an incidence rate of 162 cases per 1,000 person-years to a 5-year probability.

rate_to_prob(r = 162, per = 1000, to = 5)

## [1] 0.5551419

See ?probability to see a list of the convenience functions available.

External data

Mortality rates by age and sex (often used as transition probabilities) can be downloaded from the WHO online database with the function get_who_mr().

External data contained in user-defined data frames can be referenced in a model with the function look_up().

Compare strategies

In order to compare different strategies it is possible to run a model with multiple strategies in parallel, examples are provided in vignette("c-homogeneous", "heemod") or vignette("d-non-homogeneous", "heemod").

Time-variability

Time varying transition probabilities and state values are available through the markov_cycle and state_cycle variables. Time-varying probabilities and values are explained in vignette("b-time-dependency", "heemod").

Transition costs

Transition costs between states can be defined by adding an additional state called a transition state. This is a special case of a tunnel state. A situation where the transtion from A to B (noted A->B) costs $100 and reduces utility by 0.1 points compared to the usual values of B can be modelled by adding a state B_trans.

The cost and utility of B_trans are the same as those of B +$100 and -0.1 QALY respectively. A->B_trans is equal to the former value of A->B. A->B is set to 0 (all transitions from A to B must pass through B_trans from now on). The probability to stay in B_trans is 0, B_trans->B is equal to B->B and similarly all B_trans->* are equal to B->*.

Parallel computing

Some operation such as PSA can be significantly sped up using parallel computing. This can be done in the following way:

• Define a cluster with the use_cluster() functions (i.e. use_cluster(4) to use 4 cores).
• Run the analysis as usual.
• To stop using parallel computing use the close_cluster() function.

Results may vary depending on the machine, but we found speed gains to be quite limited beyond 4 cores.

Going further

Probabilistic uncertainty analysis vignette("e-probabilistic", "heemod") and deterministic sensitivity analysis vignette("f-sensitivity", "heemod") can be performed. Population-level estimates and heterogeneity can be computed : vignette("g-heterogeneity", "heemod").