# Plot survival curves from a Cox proportional hazard model with shared frailties

I am interested in deriving and plotting the survival functions from a Cox proportional hazards model with shared frailties. How can I do this?

This question has an answer here, but it is very high level and does not demonstrate a code implementation. I would like to confirm whether the implementation below makes sense. Following the advice here, I am asking this follow-up question as a new question, rather than as a comment on the original question.

In this example, I will use the eortc dataset from the coxme package.

Here's a summary of the steps I've taken:

1. Fit two Cox PH models: fit1 is a marginal model (which has no frailty terms), and fit2 is a shared frailty model
2. Use the muhaz function to get the baseline hazard function from fit1
3. Use the random effects from fit2 as multiples that act on the baseline hazard function, to give the hazard functions of specific groups
4. Derive group-specific cumulative hazard functions by integrating the hazard functions (using bayestestR::area_under_curve)
5. Derive group-specific survival curves from the group-specific cumulative hazard functions

library(tidyverse)
library(survival)
library(coxme)
library(ggfortify)
library(muhaz)
library(bayestestR)

data(eortc)

# fit models ------
fit1 <- coxph(Surv(y, uncens) ~ 1, data = eortc)  # marginal model
fit2 <- coxme(Surv(y, uncens) ~ (1 | center), data = eortc)  # shared frailty model

# estimate baseline hazard using muhaz: ----
muhaz_fit <- muhaz(eortc$$y, eortc$$uncens,
n.est.grid = 2000,
kern = "rectangle"
)
hazard_base_vector <- muhaz_fit$$haz.est times_vector <- muhaz_fit$$est.grid

# df with base hazard:
df_base_haz <-
tibble(hazard_base = hazard_base_vector,
time = times_vector) %>%
mutate(index = row_number())  # used to calculate cumulative hazard

# df with group-specific hazards: ----
# random effects act as multiples on the baseline hazard:
df_random_effect_haz <-
df_base_haz %>%
mutate(hazard_center1 = hazard_base * exp(ranef(fit2)$$center[1]), hazard_center2 = hazard_base * exp(ranef(fit2)$$center[2]))

# df with derived CHFs, hazards, and survival fns for the RE model: ----
# first define functions for getting chf values at time t
calc_chf <- function(index, hazard_colname, df = df_random_effect_haz){
df <- as.data.frame(df)
df2 <- df %>% dplyr::select(time, {{hazard_colname}}) %>% slice(1:index)

times_vector <- df2[,1]
haz_vector <- df2[,2]
return(area_under_curve(times_vector, haz_vector))
}

df_survival <-
df_random_effect_haz  %>%
mutate(cum_hazard_base = map_dbl(index, calc_chf, hazard_colname = "hazard_base"),
cum_hazard_center1 = map_dbl(index, calc_chf, hazard_colname = "hazard_center1"),
cum_hazard_center2 = map_dbl(index, calc_chf, hazard_colname = "hazard_center2"),

surv_base = exp(-cum_hazard_base),
surv_center1 = exp(-cum_hazard_center1),
surv_center2 = exp(-cum_hazard_center2)
)

# plot the derived survival curves: ----
df_survival %>%
pivot_longer(cols = c(surv_base, surv_center1, surv_center2),
names_to = "survival") %>%
ggplot(aes(x = time,
y = value,
group = survival,
col = survival)) +
geom_point() +
scale_x_continuous(limits = c(0, max(df_random_effect_haz\$time))) +
scale_y_continuous(limits = c(0, 1)) +
theme_minimal()



• What is the ultimate goal? What is an example of the events you are modeling? What made you choose a random effects approach? Jan 20 at 19:02
• I have data on the time-to-failure of several mechanical components. Each component can be used in different physical contexts. I am thinking of this as a multilevel structure, with the components as the higher level grouping, and the different contexts as the lower level grouping. However, each context on its own has a small fraction of all the data for the component, so the estimated survival curves are very noisy. The goal is to get smoother context-specific survival estimates by pooling information across contexts. Jan 20 at 19:30
• I was largely influenced by this paper: A tutorial on frailty models Jan 20 at 19:33
• this does sound like an ideal place for frailty models. Jan 20 at 23:31
• The vignettes by Therneau that come with the R survival package contain more information about that than I possess. Jan 22 at 12:39