# Survival models

## Distributions

The heemod package provides a number of ways to estimate transition probabilities from survival distributions. Survival distributions can come from at least three different sources:

• User-defined parametric distributions with define_surv_dist() or define_surv_spline().
• Fitted survival models with a Kaplan-Meier estimator or parametric distributions with define_surv_fit()
• Survival Tables with define_surv_table()

Once defined, each of these types of distributions can be combined and modified using a standard set of operations.

User-defined parametric distributions are created using the define_surv_dist() and define_surv_spline() functions:

surv_dist_1 <- define_surv_dist(
distribution = "exp",
rate = .5
)

surv_dist_2 <- define_surv_spline(
scale = "odds",
gamma = c(-11.643, 1.843, 0.208),
knots = c(4.077537, 5.883183, 6.458338)
)
## Le chargement a nécessité le package : flexsurv

Fitted Kaplan-Meier curves are created using survival::survfit() wrapped into define_surv_fit()

library(flexsurv)
## Le chargement a nécessité le package : survival
fit_w <- flexsurvreg(
formula = Surv(futime, fustat) ~ 1,
data = ovarian, dist = "weibull"
) |>
define_surv_fit()

plot(fit_w)

fit_spl <- flexsurvspline(
formula = Surv(futime, fustat) ~ 1,
data = ovarian,
scale = "odds",
k=1
) |>
define_surv_fit()

plot(fit_spl)

Fitted models can include covariates. In order to use a model with covariates in heemod, you can use the set_covariates() function on the fitted model and provide as additional arguments the covariate values you want to model. You can also provide a data frame of covariate levels to aggregate survival probabilities over different groups. By default, heemod will aggregate over predicted survival probabilities for each subject in the dataset to which the model was fit.

fit_cov <- flexsurvreg(
formula = Surv(rectime, censrec) ~ group,
dist = "weibull",
data = bc
)|>
define_surv_fit()
plot(fit_cov)
## No covariates provided, returning aggregate survival across all subjects.

fitcov_good   <- set_covariates(fit_cov, group = "Good")
fitcov_medium <- set_covariates(fit_cov, group = "Medium")
fitcov_poor   <- set_covariates(fit_cov, group = "Poor")

Similar functionality is also available for fitted parametric models created using flexsurv::flexsurvreg() and flexsurv::flexsurvspline() wrapped into define_surv_fit()

library(survival)

km_1 <- survfit(
formula = Surv(futime, fustat) ~ 1,
data = ovarian
) |>
define_surv_fit()
km_cov <- survfit(
formula = Surv(rectime, censrec) ~ group,
data = bc
) |>
define_surv_fit()
plot(km_cov)
## No covariates provided, returning aggregate survival across all subjects.

km_good   <- set_covariates(km_cov, group = "Good")
km_medium <- set_covariates(km_cov, group = "Medium")
km_poor   <- set_covariates(km_cov, group = "Poor")

## Operations

Once defined, treatment effects of various types can be applied to any survival distribution:

• Hazard ratio: apply_hr().
• Odds ratio: apply_or().
• Acceleration factor: apply_af().
km_poor_ph <- apply_hr(km_poor, hr = 0.5)
km_medium_af <- apply_af(km_medium, af = 1.2)

In addition, distributions can be combined using a variety of operations:

• Join survival distributions together: join().
• Mix two (or more) survival distributions: mix().
• Combine two (or more) survival distributions as independent risks: add_hazards().
km_poor_join <- join(
km_poor,
fitcov_poor,
at = 365
)
models_all <- mix(
fitcov_good, fitcov_medium, fitcov_poor,
weights = c(0.25, 0.25, 0.5)
)
fit_w, fitcov_good
)

The transition or survival probabilities are computed with compute_surv(). Time (usually model_time or state_time) needs to be passed to the function as a time argument.

compute_surv(surv_dist_2, time = 1:5)
## [1] 8.780223e-06 2.271877e-05 3.500128e-05 4.649850e-05 5.747782e-05

All these operations can be chained with the |> pipe operator.

fit_cov |>
set_covariates(group = "Good") |>
apply_hr(hr = 2) |>
join(
fitcov_poor,
at = 3
) |>
mix(
fitcov_medium,
weights = c(0.25, 0.75)
) |>
fit_w
) |>
compute_surv(time = 1:5)
## [1] 0.0004011356 0.0004736851 0.0005069766 0.0005490092 0.0005692261

## Applications

### Transition matrix

For the example we define a simple model with only 1 strategy.

param <- define_parameters(
p1 = compute_surv(
surv_dist_1,
time = model_time # can also be state_time
),
p2 = km_1 |>
join(fit_w, at = 730) |>
compute_surv(
time = model_time,
cycle_length = 365  # time is in days in km_medium, in years in model_time
)
)

tm <- define_transition(
C, p1 - p2, p2,
0, C,       p2,
0, 0,       C
)
## No named state -> generating names.
plot(tm)

sA <-  define_state(
cost = 10, ut = 1
)
sB <-  define_state(
cost = 20, ut = .5
)
sC <-  define_state(
cost = 0, ut = 0
)

stratTM <- define_strategy(
transition = tm,
A = sA, B = sB, C = sC
)

resTM <- run_model(
parameters = param,
stratTM,
cycles = 15,
cost = cost, effect = ut
)
## No named model -> generating names.
## No covariates provided, returning aggregate survival across all subjects.
## No covariates provided, returning aggregate survival across all subjects.
plot(resTM)

### Partitioned survival model

A partitioned survival model can also be computed:

ps <- define_part_surv(
pfs = surv_dist_1,
os  = km_1 |>
join(fit_w, at = 730),
cycle_length = c(1, 365) # 1 for pfs, 365 for os
)
## No named state -> generating names.
stratPS <- define_strategy(
transition = ps,
A = sA, B = sB, C = sC
)

resPS <- run_model(
stratPS,
cycles = 15,
cost = cost, effect = ut
)
## No named model -> generating names.
## No covariates provided, returning aggregate survival across all subjects.
## No covariates provided, returning aggregate survival across all subjects.
plot(resPS)