This vignette shows how to transform the deterministic Markov model presented in `vignette("c-homogeneous", "heemod")`

in a probabilistic model.

We will start by re-specifying the deterministic model of HIV therapy described previously (a monotherapy strategy `mono`

and combined therapy strategy `comb`

).

But instead of defining transition probabilities and state values directly in `define_transition()`

or `define_state()`

(as in the previous vignette), parameters will be defined first in a `define_parameters()`

step. This is because only parameters defined this way can be resampled in a probabilistic analysis.

```
param <- define_parameters(
rr = .509,
p_AA_mono = .721,
p_AB_mono = .202,
p_AC_mono = .067,
p_AD_mono = .010,
p_BC_mono = .407,
p_BD_mono = .012,
p_CD_mono = .250,
p_AB_comb = p_AB_mono * rr,
p_AC_comb = p_AC_mono * rr,
p_AD_comb = p_AD_mono * rr,
p_BC_comb = p_BC_mono * rr,
p_BD_comb = p_BD_mono * rr,
p_CD_comb = p_CD_mono * rr,
p_AA_comb = 1 - (p_AB_comb + p_AC_comb + p_AD_comb),
cost_zido = 2278,
cost_lami = 2086,
cost_A = 2756,
cost_B = 3052,
cost_C = 9007
)
```

We need to define `p_AA_mono`

and `p_AA_comb`

in `define_parameters()`

because we will need to resample that value. Only values defined with `define_parameters()`

can be resampled. So we cannot use the complement alias `C`

to specify `p_AA_comb`

in `define_transition()`

, as we did before.

```
mat_trans_mono <- define_transition(
p_AA_mono, p_AB_mono, p_AC_mono, p_AD_mono,
0, C, p_BC_mono, p_BD_mono,
0, 0, C, p_CD_mono,
0, 0, 0, 1
)
```

`## No named state -> generating names.`

```
mat_trans_comb <- define_transition(
p_AA_comb, p_AB_comb, p_AC_comb, p_AD_comb,
0, C, p_BC_comb, p_BD_comb,
0, 0, C, p_CD_comb,
0, 0, 0, 1
)
```

`## No named state -> generating names.`

State definition remains the same in this example.

```
state_A <- define_state(
cost_health = 2756,
cost_drugs = dispatch_strategy(
mono = cost_zido,
comb = cost_zido + cost_lami
),
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
state_B <- define_state(
cost_health = 3052,
cost_drugs = dispatch_strategy(
mono = cost_zido,
comb = cost_zido + cost_lami
),
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
state_C <- define_state(
cost_health = 9007,
cost_drugs = dispatch_strategy(
mono = cost_zido,
comb = cost_zido + cost_lami
),
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 1
)
state_D <- define_state(
cost_health = 0,
cost_drugs = 0,
cost_total = discount(cost_health + cost_drugs, .06),
life_year = 0
)
```

Strategies must be first defined and run as in a standard deterministic analysis.

```
strat_mono <- define_strategy(
transition = mat_trans_mono,
state_A,
state_B,
state_C,
state_D
)
```

`## No named state -> generating names.`

```
strat_comb <- define_strategy(
transition = mat_trans_comb,
state_A,
state_B,
state_C,
state_D
)
```

`## No named state -> generating names.`

```
res_mod <- run_model(
mono = strat_mono,
comb = strat_comb,
parameters = param,
cycles = 50,
cost = cost_total,
effect = life_year
)
```

Now we can define the resampling distributions. The following parameters will be resampled:

- Relative risk.
- Costs (such that cost are always positive).
- Transition probability from AIDS to death.
- The transition probabilities from state A.

Since the log of a relative risk follows a lognormal distribution, relative risk follows a lognormal distribution whose mean is `rr`

and standard deviation on the log scale can be deduced from the relative risk confidence interval.

\[rr \sim lognormal(\mu = .509, \sigma = .173)\]

Programmed as:

`rr ~ lognormal(mean = .509, sdlog = .173)`

Usually costs are resampled on a gamma distribution, which has the property of being always positive. Shape and scale parameters of the gamma distribution can be calculated from the mean and standard deviation desired in the distribution. Here we assume that *mean = variance*.

\[cost_A \sim \Gamma(\mu = 2756, \sigma = \sqrt{2756})\]

This can be programmed as:

`cost_A ~ make_gamma(mean = 2756, sd = sqrt(2756))`

Proportions follow a binomial distribution that can be estimated by giving the mean proportion and the size of the sample used to estimate that proportion with `p_CD ~ prop(prob = .25, size = 40)`

.

Finally multinomial distributions are declared with the number of individuals in each group in the sample used to estimate the proportions. These proportions follow a Dirichlet distribution:

`p_AA + p_AB + p_AC + p_AD ~ multinomial(721, 202, 67, 10)`

```
rsp <- define_psa(
rr ~ lognormal(mean = .509, sdlog = .173),
cost_A ~ gamma(mean = 2756, sd = sqrt(2756)),
cost_B ~ gamma(mean = 3052, sd = sqrt(3052)),
cost_C ~ gamma(mean = 9007, sd = sqrt(9007)),
p_CD_mono ~ binomial(prob = .25, size = 40),
p_AA_mono + p_AB_mono + p_AC_mono + p_AD_mono ~ multinomial(721, 202, 67, 10)
)
```

Now that the distributions of parameters are set we can simply run the probabilistic model as follow:

```
pm <- run_psa(
model = res_mod,
psa = rsp,
N = 100
)
```

`## Resampling strategy 'mono'...`

`## Resampling strategy 'comb'...`

The average results are computed. In theory these values are more accurate than simple estimates because of non-linearities. An optional `threshold`

can be passed to `summary()`

to compute net monetary benefit.

```
summary(
pm,
threshold = c(1000, 5000, 6000, 1e4))
```

```
## 2 strategies run for 50 cycles.
##
## Initial state counts:
##
## A = 1000L
## B = 0L
## C = 0L
## D = 0L
##
## Counting method: 'life-table'.
##
## Values:
##
## cost_health cost_drugs cost_total life_year
## comb 105161238 82671952 96617909 18944.077
## mono 50926810 20358609 50336176 8937.054
##
## Net monetary benefit difference:
##
## 1000 5000 6000 10000
## mono 36274.71 0.000 0.00 0.0
## comb 0.00 3753.382 13760.41 53788.5
##
## Efficiency frontier:
##
## mono -> comb
##
## Differences:
##
## Cost Diff. Effect Diff. ICER Ref.
## comb 46281.73 10.00702 4624.925 mono
```

The results of the analysis can be plotted on the cost-effectiveness plane. We can see there seem to be little uncertainty on the costs compared to the uncertainty on the effects, resulting in an uncertainty cloud that looks like a line.

`plot(pm, type = "ce")`

And as cost-effectiveness acceptability curves or EVPI:

`plot(pm, type = "ac", max_wtp = 10000, log_scale = FALSE)`

`plot(pm, type = "evpi", max_wtp = 10000, log_scale = FALSE)`

A covariance analysis can be performed on strategy results:

`plot(pm, type = "cov")`

`## Loading required namespace: mgcv`

Or on the difference between strategies:

`plot(pm, type = "cov", diff = TRUE, threshold = 5000)`

```
## Warning in compute_prop_var(mgcv::gam(formula = form_effect, data = .)):
## Parameter excluded because of perfect collinearity: p_AD_mono.
```

As usual plots can be modified with the standard `ggplot2`

syntax.

```
library(ggplot2)
plot(pm, type = "ce") +
xlab("Life-years gained") +
ylab("Additional cost") +
scale_color_brewer(
name = "Strategy",
palette = "Set1"
) +
theme_minimal()
```

Resampling can be significantly sped up by 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.

To compute EVPPI the results can also be exported with `export_savi()`

in a format compatible with the SAVI software (Sheffield Accelerated Value of Information).

The results can be post-processed by the `BCEA`

package with the `run_bcea()`

function.

`bcea <- run_bcea(pm, plot = TRUE, Kmax = 10000)`

`## Loading required namespace: BCEA`

`summary(bcea)`

```
## NB: k (wtp) is defined in the interval [0 - 10000]
##
## Cost-effectiveness analysis summary
##
## Reference intervention: comb
## Comparator intervention: mono
##
## Optimal decision: choose comb for k<4640 and mono for k>=4640
##
##
## Analysis for willingness to pay parameter k = 10000
##
## Expected utility
## comb 39034362
## mono 92822860
##
## EIB CEAC ICER
## comb vs mono -53788498 0 4624.9
##
## Optimal intervention (max expected utility) for k=10000: mono
##
## EVPI -0.00000000029788
```