Estimating density from interval [start, stop] data in R

Rather than calculate for finer and finer time intervals you can just keep a cumulative count of the number of patients at a particular age

setDT(ccdeath)
x <- rbind(
  ccdeath[, .(age = age, num_patients = 1)],
  ccdeath[, .(age = age_last, num_patients = -1)]
)[, .(num_patients = sum(num_patients)), keyby = age]

cccdeath <- x[x[, .(age = unique(age))], on = 'age']
cccdeath[, num_patients := cumsum(num_patients)]
ggplot(cccdeath, aes(x = age, y = num_patients)) + geom_step()

The sawtooth pattern is because every patient is assumed to start at an integer age. Had some thoughts about how you'd smooth this and came up with this idea - assign equal probabilites to a set of evenly spaced ages between the given age and age+1. You get something like this,

smooth_param <- 12
x <- rbindlist(lapply(
  (1:smooth_param-0.5)/smooth_param,
  function(s) {
    rbind(
      ccdeath[, .(age = age+s, num_patients = 1/smooth_param)],
      ccdeath[, .(age = age_last+s, num_patients = -1/smooth_param)]
    )
  }
))[, .(num_patients = sum(num_patients)), keyby = age]

cccdeath <- x[x[, .(age = sort(unique(age)))], on = 'age']
cccdeath[, num_patients := cumsum(num_patients)]
ggplot(cccdeath, aes(x = age, y = num_patients)) + geom_step()

I would proceed along these lines:

If you are interested in knowing the age distribution after t days in the study, the age will simply be the age at enrollment plus t days. The exceptions that you need to handle those who have died or have been right-censored. In your example, you seem to have kept people's age "frozen" at the time they left the study. Personally I think the age distribution of survivors who have not been censored is more useful in a survival analysis, but I will stick to your set-up for this example.

The two possibilities for each patient at time t then are to have age at enrollment plus t if t is less than follow-up time. Otherwise the age will be the age at enrollment plus the follow-up time.

If you wrap this in a function, you can see how the age distribution changes throughout the study. For completeness we will always plot a faint density of age at enrollment, and a line indicating the current mean age:

age_distribution <- function(df, t = 0)
{
  df %>% 
    mutate(age_at_t = age + ifelse(time > t, t, time) / 365.25) %>%
    ggplot() +
    geom_density(aes(age), linetype = 2, colour = "gray50") +
    geom_density(aes(age_at_t)) +
    geom_vline(aes(xintercept = mean(age_at_t)), color = "red", linetype = 2) +
    labs(x = paste("Age at day", t, "of study"),
         y = "Density",
         title = paste("Age distribution after", t, "days in study"))
}

So, for example:

age_distribution(ccdeath, 0)

And after 1 year:

age_distribution(ccdeath, 365)

And after 5 years:

age_distribution(ccdeath, 5 * 365.25)

For completeness, the equivalent function with censored / dead patients removed would be like this:

age_distribution <- function(df, t = 0)
{
  df %>% 
    filter(time > t) %>%
    mutate(age_at_t = age + t / 365.25) %>%
    ggplot() +
    geom_density(data = df, aes(age), linetype = 2, colour = "gray50") +
    geom_density(aes(age_at_t)) +
    geom_vline(aes(xintercept = mean(age_at_t)), color = "red", linetype = 2) +
    labs(x = paste("Age at day", t, "of study"),
         y = "Density",
         title = paste("Age distribution after", t, "days in study"))
}

So we can see how the distribution's shape changes after 5 years of follow-up:

age_distribution(ccdeath, 5 * 365.25)

This shows more clearly that there is a disproportionate loss of older people from the initial cohort.