library(eha) options(digits = 4, fig.width=8)
Cox regression is not very suitable in the analysis of huge data sets with a lot of events (e.g., deaths). For instance, consider analyzing the mortality of the Swedish population aged 60--110 during the years 1968-2019, where we can count to more than four million deaths.
The obvious way to handle that situation is by tabulation and applying a piecewise constant hazard function, because it is a well-known fact that any continuous function can arbitrary well be approximated by a step function, simply by taking small enough steps.
The data sets swepop
and swedeaths
in eha
contain age and sex
specific yearly information on population size and number of deaths,
respectively. They both cover the full Swedish population the years 1968--2019.
The first few rows of each:
head(swepop) head(swedeaths)
The funny rownames and the column id
are created by the function reshape
,
which was used to transform the original tables, given in wide format, to
long format. In the original data, downloaded from
Statistics Sweden, the population size refers to the last
day, December 31, of the given year, but here it refers to an average of that
value and the corresponding one the previous year. In that way we get an estimate
of the number of person years, which allows us to consider number of
occurrences and exposure time in each cell of the data. This information will
allow us to fit proportional hazards survival models. So we start by joining
the two data sets and remove irrelevant stuff:
dat <- swepop[, c("age", "sex", "year", "pop")] dat$deaths <- swedeaths$deaths rownames(dat) <- 1:NROW(dat) # Simplify rownames. head(dat) tail(dat)
We note that the age column ends with age == 100
, which in fact means
age >= 100
. There are in total r sum(dat$deaths)
observed deaths during the years
1968--2019, or r as.integer(round(sum(dat$deaths) / (2019 - 1967)))
deaths per
year on average. There are 101 age groups, two sexes, and 52 years, in all 10504
cells (rows in the data frame).
Assuming a piecewise constant hazards model on the 101 age groups, we can fit a
proportional hazards model by Poisson regression, utilizing the fact that two
likelihood functions in fact are identical. In R, we use glm
.
fit.glm <- glm(deaths ~ offset(log(pop)) + I(year - 2000) + sex + factor(age), data = dat, family = poisson) summary(fit.glm)$coefficients[2:3, ]
The 101 coefficients corresponding to the intercept and the age factor can be
used to estimate the hazard function: The intercept, r fit.glm$coefficients[1]
,
is the log of the hazard in the age interval 0-1, and the rest are differences to
that value, so we can reconstruct the baseline hazard by
lhaz <- coefficients(fit.glm)[-(2:3)] n <- length(lhaz) lhaz[-1] <- lhaz[-1] + lhaz[1] haz <- exp(lhaz)
and plot the result, see Figure \@ref(fig:glmbasefig).
oldpar <- par(las = 1, lwd = 1.5, mfrow = c(1, 2)) plot(0:(n-1), haz, type = "s", main = "log(hazards)", xlab = "Age", ylab = "", log = "y") plot(0:(n-1), haz, type = "s", main = "hazards", xlab = "Age", ylab = "Deaths / Year")
par(oldpar) # Restore the default
While it straightforward to use glm and Poisson regression to fit the model, it
takes some efforts to get it right. That is the reason for the creation of
the function tpchreg
("Tabular Piecewise Constant Hazards REGression"), and
with it, the "Poisson analysis" is performed by
fit <- tpchreg(oe(deaths, pop) ~ I(year - 2000) + sex, time = age, last = 101, data = dat)
Note:
The function oe
("occurrence/exposure") takes two arguments, the first
is the number of events (deaths in our example), and the second is exposure time,
or person years.
The argument time
is the defining time intervals variable. It can be either character, like
c("0-1", "1-2", ..., "100-101") or numeric (as here). If numeric, the value refers
to the start of the corresponding interval, and the next start is the end of the
previous interval. This leaves the last interval's endpoint undefined, and if not given
by the last
argument (see below), it is chosen so that the length of the last interval is
one.
The argument last
closes the last interval, if is not already closed, see
above. The exact value of last is only important for plotting and for the calculation
of the restricted mean survival time, (RMST) see the summary result below.
summary(fit)
The restricted mean survival time is defined as the integral of the survivor function over the given time interval. Note that if the lower limit of the interval is larger than zero, it gives the conditional restricted mean survival time, given survival to the lower endpoint.
Graphs of the hazards and the log(hazards) functions are shown in Figure \@ref(fig:tpplot).
oldpar <- par(mfrow = c(1, 2), las = 1, lwd = 1.5) plot(fit, fn = "haz", log = "y", main = "log(hazards)", xlab = "Age") plot(fit, fn = "haz", log = "", main = "hazards", xlab = "Age", ylab = "Deaths / Year")
Same results as with glm
and Poisson regression, but a lot simpler.
Sometimes you have a large data file in classical, individual form, suitable for
Cox regression with coxreg
, but the mere size makes it impractical, or even
impossible. Then help is close by tabulating and assuming a piecewise constant
hazard function, returning to the method in the previous section, that is, using
tpchreg
.
The helper function is toTpch
, and we illustrate its use on the oldmort
data
frame:
head(oldmort[, c("enter", "exit", "event", "sex", "civ", "birthdate")]) oldmort$birthyear <- floor(oldmort$birthdate) - 1800 om <- toTpch(Surv(enter, exit, event) ~ sex + civ + birthyear, cuts = seq(60, 100, by = 2), data = oldmort) head(om)
Note two things:
The creation of a new variable, birthyear
. The original birthdate
is
given with precision days and contains r length(unique(oldmort$birthdate))
unique values, which will contribute to creating a very large table, so the
transformation gives birth year with r length(unique(oldmort$birthyear))
unique values. Further, the new variable is given a
reference value of 1800 by subtraction, necessary so that the baseline does
not coincide with the birth of Christ. Will foremost affect plotting of the
estimated survivor function. However, regression parameter estimates are
unaffected, as long as no interaction effect including birthyear
is present.
The length of the time pieces is set to two years, in order to avoid empty intervals in the upper age range. This choice has only a marginal effect on the final results of the analyses. You can try it out yourself. Note that it is not necessary to use equidistant cut points, it is chosen here only for convenience.
Now we can run tpchreg
as before
fit3 <- tpchreg(oe(event, exposure) ~ sex + civ + birthyear, time = age, data = om) summary(fit3)
And the hazards graphs are shown in Figure \@ref(fig:plotom).
oldpar <- par(mfrow = c(1, 2), las = 1, lwd = 1.5) plot(fit3, fn = "haz", log = "y", main = "log(hazards)", xlab = "Age", ylab = "log(Deaths / Year)", col = "blue") plot(fit3, fn = "haz", log = "", main = "hazards", xlab = "Age", ylab = "Deaths / Year", col = "blue")
The plots of the survivor and cumulative hazards functions are "smoother", see Figure \@ref(fig:plotsurcum).
oldpar <- par(mfrow = c(1, 2), las = 1, lwd = 1.5) plot(fit3, fn = "cum", log = "y", main = "Cum. hazards", xlab = "Age", col = "blue") plot(fit3, fn = "sur", log = "", main = "Survivor function", xlab = "Age", col = "blue") par(oldpar)
A comparison with Cox regression on the original data.
fit4 <- coxreg(Surv(enter, exit, event) ~ sex + civ + I(birthdate - 1800), data = oldmort) summary(fit4)
And the graphs, see Figure \@ref(fig:coxgraphs).
oldpar <- par(mfrow = c(1, 2), lwd = 1.5, las = 1) plot(fit4, main = "Cumulative hazards", xlab = "Age", col = "blue") plot(fit4, main = "Survivor function", xlab = "Age", fn = "surv", col = "blue")
par(oldpar)
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