Probabilistic Uncertainty Analysis

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

Model definition

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
)

Resampling distributions

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)
)

Run probabilistic model

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    98284639   78519552   93492006 17992.565
## mono    48433870   19741658   48864658  8666.224
## 
## Net monetary benefit difference:
## 
##          1000     5000    6000    10000
## mono 35301.01    0.000     0.0     0.00
## comb     0.00 2004.357 11330.7 48636.06
## 
## Efficiency frontier:
## 
## mono -> comb
## 
## Differences:
## 
##      Cost Diff. Effect Diff.     ICER Ref.
## comb   44627.35     9.326341 4785.087 mono

Result interpretation

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)

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()

Parallel computing

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.

Interface with BCEA and SAVI

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<4800 and mono for k>=4800
## 
## 
## Analysis for willingness to pay parameter k = 10000
## 
##      Expected utility
## comb         37797579
## mono         86433640
## 
##                    EIB CEAC   ICER
## comb vs mono -48636061    0 4785.1
## 
## Optimal intervention (max expected utility) for k=10000: mono
##                      
## EVPI -0.0000000050678