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. This reports goes a bit further in the analysis. 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.

This model has 5 states:

- Primary THR: starting state, individuals receive either the standard
or the new THR (called NP1), outcomes are either success of primary THR
or death (the operative mortality rate of primary THR will be called
`omrPTHR`

); - Success of primary THR: state after receiving primary THR if the surgery is successfull, individuals can stay in that state, need a THR revision, or die of other causes;
- Revision of primary THR: for individuals whose primary THR needed
revision, outcomes are either success of revision THR or death (the
operative mortality rate of revision THR will be called
`omrRTHR`

); - Success of revision THR: state after receiving revision THR if the surgery is successfull, individuals can stay in that state, need a THR re-revision, or die of other causes;
- Death (either caused by THR or another cause).

Two transition probabilities are time-varying in this model:

- Probability of death by another cause increases with age;
- Probability of primary THR revision changes with time.

Other-cause death probabilities (mortality rate `mr`

) for
the United Kingdom is taken from WHO databases using the
`get_who_mr()`

function. The variable `sex`

,
taking values `0`

and `1`

, must be recoded in
`"FMLE"`

and `"MLE"`

before being passed to this
function.

- 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 standard procedure revision probability
`standardRR`

is modified by the relative risk \(rrNP1 = 0.260677\).

\[ P_{revision} = 1 - \exp(\lambda \times rr \times NP1 \times ((t-1)^\gamma-t^\gamma)) \]

- Revision THR re-revision (
`rrr`

) probability is set to be constant at*0.04*per year.

The key element to specify time-varying elements in
`heemod`

is through the use of the **package-defined
variables** `model_time`

and `state_time`

.
See vignette `vignette("b-time-dependency", "heemod")`

for
more details.

In order to build this more complex Markov model, parameters need to
be defined through `define_parameters()`

(for 2 reasons: to
keep the transition matrix readable, and to avoid repetition by re-using
parameters between strategies).

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*).

```
param <- define_parameters(
age_init = 60,
sex = 0,
# age increases with cycles
age = age_init + model_time,
# 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 * ((model_time - 1) ^ gamma -
model_time ^ gamma)),
np1RR = 1 - exp(lambda * rrNP1 * ((model_time - 1) ^ gamma -
model_time ^ gamma)),
# age-related mortality rate
sex_cat = ifelse(sex == 0, "FMLE", "MLE"),
mr = get_who_mr(age, sex_cat, local = TRUE),
# state values
u_SuccessP = .85,
u_RevisionTHR = .30,
u_SuccessR = .75,
c_RevisionTHR = 5294
)
param
```

```
## 20 unevaluated parameters.
##
## age_init = 60
## sex = 0
## age = age_init + model_time
## 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 - ...
## np1RR = 1 - ...
## sex_cat = ifelse(sex == 0, "FMLE", "MLE")
## mr = get_who_mr(age, sex_cat, local = TRUE)
## u_SuccessP = 0.85
## u_RevisionTHR = 0.3
## u_SuccessR = 0.75
## c_RevisionTHR = 5294
```

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

```
mat_standard <- define_transition(
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_standard
```

```
## A transition matrix, 5 states.
##
## PrimaryTHR SuccessP RevisionTHR SuccessR Death
## PrimaryTHR C omrPTHR
## SuccessP C standardRR mr
## RevisionTHR C omrRTHR + mr
## SuccessR rrr C mr
## Death 1
```

```
mat_np1 <- define_transition(
state_names = c(
"PrimaryTHR",
"SuccessP",
"RevisionTHR",
"SuccessR",
"Death"
),
0, C, 0, 0, omrPTHR,
0, C, np1RR, 0, mr,
0, 0, 0, C, omrRTHR+mr,
0, 0, rrr, C, mr,
0, 0, 0, 0, 1
)
mat_np1
```

```
## A transition matrix, 5 states.
##
## PrimaryTHR SuccessP RevisionTHR SuccessR Death
## PrimaryTHR C omrPTHR
## SuccessP C np1RR mr
## RevisionTHR C omrRTHR + mr
## SuccessR rrr C mr
## Death 1
```

While it is possible to plot the matrix thanks to the
`diagram`

package, the results may not always be easy to
read.

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 strategies for the control group and the NP1 treatment.

We use `define_starting_values()`

to take into account the
cost of surgery.

```
strat_standard <- define_strategy(
transition = mat_standard,
PrimaryTHR = define_state(
utility = 0,
cost = 0
),
SuccessP = define_state(
utility = discount(u_SuccessP, .015),
cost = 0
),
RevisionTHR = define_state(
utility = discount(u_RevisionTHR, .015),
cost = discount(c_RevisionTHR, .06)
),
SuccessR = define_state(
utility = discount(u_SuccessR, .015),
cost = 0
),
Death = define_state(
utility = 0,
cost = 0
),
starting_values = define_starting_values(
cost = 394
)
)
strat_standard
```

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

```
strat_np1 <- define_strategy(
transition = mat_np1,
PrimaryTHR = define_state(
utility = 0,
cost = 0
),
SuccessP = define_state(
utility = discount(u_SuccessP, .015),
cost = 0
),
RevisionTHR = define_state(
utility = discount(u_RevisionTHR, .015),
cost = discount(c_RevisionTHR, .06)
),
SuccessR = define_state(
utility = discount(u_SuccessR, .015),
cost = 0
),
Death = define_state(
utility = 0,
cost = 0
),
starting_values = define_starting_values(
cost = 579
)
)
strat_np1
```

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

Both strategies can now be run for 60 years. By default models are
computed for 1000 person starting in `PrimaryTHR`

.

```
res_mod <- run_model(
standard = strat_standard,
np1 = strat_np1,
parameters = param,
cycles = 60,
cost = cost,
effect = utility
)
```

`## Fetching mortality data from package cached data.`

`## Using cached data from year 2019.`

A comparison of both strategies can be done with
`summary()`

. The incremental cost and effect are displayed in
columns `Cost`

and `Effect`

.

```
## 2 strategies run for 60 cycles.
##
## Initial state counts:
##
## PrimaryTHR = 1000L
## SuccessP = 0L
## RevisionTHR = 0L
## SuccessR = 0L
## Death = 0L
##
## Counting method: 'life-table'.
##
##
##
## Counting method: 'beginning'.
##
##
##
## Counting method: 'end'.
##
## Values:
##
## utility cost
## standard 16949.57 535002.2
## np1 17009.10 616348.5
##
## Efficiency frontier:
##
## standard -> np1
##
## Differences:
##
## Cost Diff. Effect Diff. ICER Ref.
## np1 81.3463 0.05953445 1366.374 standard
```

The new treatment costs £1366 more per QALY gained.

It should be noted that this result differs from the original study.
This difference is explained by higher population-level all-causes
mortality rates in the original study than in the WHO database (used
here). See vignette `vignette("i-reproduction", "heemod")`

for an exact reproduction of the analysis.

We can plot the counts per state:

```
plot(res_mod, type = "counts", panel = "by_state", free_y = TRUE) +
theme_bw() +
scale_color_brewer(
name = "Strategy",
palette = "Set1"
)
```

```
## Scale for colour is already present.
## Adding another scale for colour, which will replace the existing scale.
```