# Examples of using survtab

#### Joonas Miettinen

#### 2016-04-27

# Overview

This vignette aims to clarify the usage of the `survtab_ag` and `survtab` functions included in this package. `survtab_ag` estimates various survival functions and cumulative incidence functions (CIFs) non-parametrically using aggregated data, and `survtab` is a wrapper for `survtab_ag`, to which `Lexis` data is supplied.

Two methods (`surv.method`) are currently supported: The `"lifetable"` (actuarial) method only makes use of counts when estimating any of the supported survival time functions. The default method (`"hazard"`}) estimates appropriate hazards and transforms them into survival function or CIF estimates.

For relative survival estimation we need also to enumerate the expected hazard levels for the subjects in the data. This is done by merging expected hazards to individuals’ subintervals (which divide their survival time lines to a number of small intervals). For Pohar-Perme-weighted analyses one must additionally compute various weighted figures at the level of split subject data.

If one has subject-level data, the simplest way of computing survival function estimates with `popEpi` is by defining a `Lexis` object and using `survtab`, which will do the rest. For pre-aggregated data one may use the `survtab_ag` function instead. One can also use the `lexpand` function to split, merge population hazards, and aggregate in a single function call and then use `survtab_ag` if that is convenient.

# Using `survtab`

It is straightforward to estimate various survival time functions with `survtab` once a `Lexis` object has been defined (see `?Lexis` in package `Epi` for details):

`r library(popEpi) library(Epi) library(survival)`

    data(sire)
    ## NOTE: recommended to use factor status variable
    x <- Lexis(entry = list(FUT = 0, AGE = dg_age, CAL = get.yrs(dg_date)),
               exit = list(CAL = get.yrs(ex_date)),
               data = sire[sire$dg_date < sire$ex_date, ],
               exit.status = factor(status, levels = 0:2,
                                    labels = c("alive", "canD", "othD")),
               merge = TRUE)

    ## NOTE: entry.status has been set to "alive" for all.

    ## pretend some are male
    set.seed(1L)
    x$sex <- rbinom(nrow(x), 1, 0.5)

    ## observed survival - explicit method
    st <- survtab(Surv(time = FUT, event = lex.Xst) ~ sex, data = x,
                  surv.type = "surv.obs",
                  breaks = list(FUT = seq(0, 5, 1/12)))

    ## observed survival - easy method (assumes lex.Xst in x is the status variable)
    st <- survtab(FUT ~ sex, data = x,
                  surv.type = "surv.obs",
                  breaks = list(FUT = seq(0, 5, 1/12)))

    ## printing gives the used settings and
    ## estimates at the middle and end of the estimated
    ## curves; more information available using summary()
    st

    ##
    ## Call:
    ##  survtab(formula = FUT ~ sex, data = x, breaks = list(FUT = seq(0, 5, 1/12)), surv.type = "surv.obs")
    ##
    ## Type arguments:
    ##  surv.type: surv.obs --- surv.method: hazard
    ##
    ## Confidence interval arguments:
    ##  level: 95 % --- transformation: log-log
    ##
    ## Totals:
    ##  person-time:23993 --- events: 3636
    ##
    ## Stratified by: 'sex'
    ##    sex Tstop surv.obs.lo surv.obs surv.obs.hi SE.surv.obs
    ## 1:   0   2.5      0.6174   0.6328      0.6478    0.007751
    ## 2:   0   5.0      0.4962   0.5126      0.5288    0.008321
    ## 3:   1   2.5      0.6235   0.6389      0.6539    0.007748
    ## 4:   1   5.0      0.5006   0.5171      0.5334    0.008370

Plotting by strata (men = blue, women = red):

    plot(st, col = c("blue", "red"))

Note that the correct usage of the `formula` argument in `survtab` specifies the time scale in the `Lexis` object over which survival is computed (here `"FUT"` for follow-up time). This is used to identify the appropriate time scale in the data. When only supplying the survival time scale as the right-hand-side of the formula, the column `lex.Xst` in the supplied `Lexis` object is assumed to be the (correctly formatted!) status variable. When using `Surv()` to be explicit, we effectively (and exceptionally) pass the starting times to the `time` argument in `Surv()`, and `time2` is ignored entirely. The function will fail if `time` does not match exactly with a time scale in data.

When using `Surv()`, one must also pass the status variable, which can be something other than the `lex.Xst` variable created by `Lexis()`, though usually ``lex.Xst` is what you want to use (especially if the data has already been split using e.g. `splitLexis` or `splitMulti`, which is allowed). It is recommended to use a factor status variable to pass to `Surv()`, though a numeric variable will work in simple cases (0 = alive, 1 = dead; also `FALSE` = alive, `TRUE` = dead). Using `Surv()` also allows easy passing of transformations of `lex.Xst`, e.g. `Surv(FUT, lex.Xst %in% 1:2)`.

The argument `breaks` must be a named list of breaks by which to split the `Lexis` data (see `?splitMulti`). It is mandatory to assign breaks at least to the survival time scale (`"FUT"` in our example) so that `survtab` knows what intervals to use to estimate the requested survival time function(s). The breaks also determine the window used: It is therefore easy to compute so called period estimates by defining the roof and floor along the calendar time scale, e.g.

`breaks = list(FUT = seq(0, 5, 1/12), CAL = c(2000, 2005))`

would cause `survtab` to compute period estimates for 2000-2004 (breaks given here as fractional years, so 2005 is effectively 2004.99999…).

## Relative/net survival

Relative/net survival estimation requires knowledge of the expected hazard levels for the individuals in the data. In `survtab` this is accomplished by passing a long-formt `data.frame` of population hazards via the `pophaz` argument. E.g. the `popmort` dataset included in `popEpi` (Finnish overall mortality rates for men and women).

    data(popmort)
    pm <- data.frame(popmort)
    names(pm) <- c("sex", "CAL", "AGE", "haz")
    head(pm)

    ##   sex  CAL AGE         haz
    ## 1   0 1951   0 0.036363176
    ## 2   0 1951   1 0.003616547
    ## 3   0 1951   2 0.002172384
    ## 4   0 1951   3 0.001581249
    ## 5   0 1951   4 0.001180690
    ## 6   0 1951   5 0.001070595

The `data.frame` should contain a variable named `"haz"` indicating the population hazard at the level of one subject-year. Any other variables are considered to be variables, by which to merge population hazards to the (split) subject-level data within `survtab`. These merging variables may correspond to the time scales in the used `Lexis` object. This allows for e.g. merging in different population hazards for the same subject as they get older.

The following causes `survtab` to estimate EdererII relative survival:

    st.e2 <- survtab(Surv(time = FUT, event = lex.Xst) ~ sex, data = x,
                     surv.type = "surv.rel", relsurv.method = "e2",
                     breaks = list(FUT = seq(0, 5, 1/12)),
                     pophaz = pm)

    plot(st.e2, y = "r.e2", col = c("blue", "red"))

Note that the curves diverge due to merging in the “wrong” population hazards for some individuals which we randomized earlier to be male though all the individuals in data are actually female. Pohar-Perme-weighted estimates can be computed by

    st.pp <- survtab(Surv(time = FUT, event = lex.Xst) ~ sex, data = x,
                     surv.type = "surv.rel", relsurv.method = "pp",
                     breaks = list(FUT = seq(0, 5, 1/12)),
                     pophaz = pm)

Compare with EdererII estimates:

    plot(st.e2, y = "r.e2", col = c("blue", "red"), lty = 1)
    lines(st.pp, y = "r.pp", col = c("blue", "red"), lty = 2)

## Adjusting estimates

`survtab` also allows for adjusting the survival curves by categorical variables — typically by age groups. The following demonstrates how:

    ## an age group variable
    x$agegr <- cut(x$dg_age, c(0, 60, 70, 80, Inf), right = FALSE)

    ## using "internal weights" - see ?ICSS for international weights standards
    w <- table(x$agegr)
    w

    ##
    ##   [0,60)  [60,70)  [70,80) [80,Inf)
    ##     1781     1889     2428     2129

    w <- list(agegr = as.numeric(w))

    st.as <- survtab(Surv(time = FUT, event = lex.Xst) ~ sex + adjust(agegr),
                     data = x, weights = w,
                     surv.type = "surv.rel", relsurv.method = "e2",
                     breaks = list(FUT = seq(0, 5, 1/12)),
                     pophaz = pm)

    plot(st.as, y = "r.e2.as", col = c("blue", "red"))

We now have age-adjusted EdererII relative/net survival estimates. The `weights` argument allows for either a list of weigths (with one or multiple variables to adjust by) or a `data.frame` of weights. Examples:

    list(sex = c(0.4, 0.6), agegr = c(0.2, 0.2, 0.4, 0.2))

    ## $sex
    ## [1] 0.4 0.6
    ##
    ## $agegr
    ## [1] 0.2 0.2 0.4 0.2

    wdf <- merge(0:1, 1:4)
    names(wdf) <- c("sex", "agegr")
    wdf$weights <- c(0.1, 0.1, 0.1, 0.1, 0.2, 0.2, 0.1, 0.1)
    wdf

    ##   sex agegr weights
    ## 1   0     1     0.1
    ## 2   1     1     0.1
    ## 3   0     2     0.1
    ## 4   1     2     0.1
    ## 5   0     3     0.2
    ## 6   1     3     0.2
    ## 7   0     4     0.1
    ## 8   1     4     0.1

The weights do not have to sum to one when supplied as they are internally forced to do so within each stratum. In the `data.frame` of weights, the column of actual weights to use must be named “weights”. When there are more than one variable to adjust by, and a list of weights has been supplied, the variable-specific weights are first multiplied together (cumulatively) and then scaled to sum to one.

This adjusting can be done to any survival time function that `survtab` (and `survtab_ag`) estimates. One can also supply adjusting variables via the `adjust` argument if convenient:

    st.as <- survtab(Surv(time = FUT, event = lex.Xst) ~ sex,
                     adjust = "agegr",
                     data = x, weights = w,
                     surv.type = "surv.rel", relsurv.method = "e2",
                     breaks = list(FUT = seq(0, 5, 1/12)),
                     pophaz = pm)

Where `adjust` could also be `adjust = agegr`, `adjust = list(agegr)` or

`adjust = list(agegr = cut(dg_age, c(0, 60, 70, 80, Inf), right = FALSE))`

for exactly the same results. When adjusting by multiple variables, one must supply a vector of variable names in data or a list of multiple elements (as in the base function `aggregate`).

## Other survival time functions

One can also estimate cause-specific survival functions, cumulative incidence functions (CIFs, a.k.a. crude risk a.k.a. absolute risk functions), and CIFs based on the excess numbers of events. Cause-specific survival is close to net survival as they are philosophically highly similar concepts:

    st.ca <- survtab(Surv(time = FUT, event = lex.Xst) ~ 1,
                     data = x,
                     surv.type = "surv.cause",
                     breaks = list(FUT = seq(0, 5, 1/12)))

    st.pp <- survtab(Surv(time = FUT, event = lex.Xst) ~ 1, data = x,
                     surv.type = "surv.rel", relsurv.method = "pp",
                     breaks = list(FUT = seq(0, 5, 1/12)),
                     pophaz = pm)

    plot(st.ca, y = "surv.obs.canD", col = "blue")
    lines(st.pp, y = "r.pp", col = "red")

Absolute risk:

    st.cif <- survtab(Surv(time = FUT, event = lex.Xst) ~ 1,
                      data = x,
                      surv.type = "cif.obs",
                      breaks = list(FUT = seq(0, 5, 1/12)))
    plot(st.cif, y = "CIF_canD", conf.int = FALSE)
    lines(st.cif, y = "CIF_othD", conf.int = FALSE, col = "red")

The “relative CIF” attempts to be close to the true CIF without using knowledge about the types of events, e.g. causes of death:

    st.cir <- survtab(Surv(time = FUT, event = lex.Xst) ~ 1,
                      data = x,
                      surv.type = "cif.rel",
                      breaks = list(FUT = seq(0, 5, 1/12)),
                      pophaz = pm)
    plot(st.cif, y = "CIF_canD", conf.int = FALSE, col = "blue")
    lines(st.cir, y = "CIF.rel", conf.int = FALSE, col = "red")

# Using `survtab_ag`

Arguments concerning the types and methods of estimating of survival time functions work the same in `survtab_ag` as in `survtab` (the latter uses the former). However, with aggregated data one must explicitly supply the various count and person-time variables. Also, usage of the `formula` argument is different.

For demonstration purposes we form an aggregated data set using `lexpand`; see `?lexpand` for more information on that function.

    sire$sex <- rbinom(nrow(sire), size = 1, prob = 0.5)
    ag <- lexpand(sire, birth = "bi_date", entry = "dg_date", exit = "ex_date",
                  status = "status", breaks = list(fot = seq(0, 5, 1/12)),
                  aggre = list(sex, fot))

    ## dropped 16 rows where entry == exit

    head(ag)

    ##    sex        fot     pyrs at.risk from0to0 from0to1 from0to2
    ## 1:   0 0.00000000 336.2780    4112       18      126       11
    ## 2:   0 0.08333333 323.8440    3957       28       89       14
    ## 3:   0 0.16666667 313.6453    3826       23       95       13
    ## 4:   0 0.25000000 304.2891    3695       19       63       10
    ## 5:   0 0.33333333 296.7140    3603       20       62        9
    ## 6:   0 0.41666667 289.5723    3512       17       51       11

Now simply do:

    st <- survtab_ag(fot ~ sex, data = ag, surv.type = "surv.obs",
                     surv.method = "hazard",
                     d = c("from0to1", "from0to2"), pyrs = "pyrs")

Or:

    st <- survtab_ag(fot ~ sex, data = ag, surv.type = "surv.obs",
                     surv.method = "lifetable",
                     d = c("from0to1", "from0to2"), n = "at.risk",
                     n.cens = "from0to0")

Note that e.g. argument `d` could also have been supplied as

`list(from0to1, from0to2)`

or

`list(canD = from0to1, othD = from0to2)`

for identical results. The last is convenient for e.g. `surv.cause` computations:

    st.ca <- survtab_ag(fot ~ sex, data = ag, surv.type = "surv.cause",
                        surv.method = "hazard",
                        d = list(canD = from0to1, othD = from0to2), pyrs = "pyrs")
    plot(st.ca, y = "surv.obs.canD", col = c("blue", "red"))

One has to supply the most variables when computing Pohar-Perme estimates (though it is probably rare to have third-source aggregated data with Pohar-Perme weighted figures, it is implemented here to be used as a workhorse for `survtab`). For this we must aggregate again to get the Pohar-Perme weighted counts and subject-times:

    ag <- lexpand(sire, birth = "bi_date", entry = "dg_date", exit = "ex_date",
                  status = "status", breaks = list(fot = seq(0, 5, 1/12)),
                  pophaz = popmort, pp = TRUE,
                  aggre = list(sex, fot))

    ## dropped 16 rows where entry == exit

    ## NOTE: 83 rows in split data had values of 'age' higher than max of pophaz's 'agegroup'; the hazard values at 'agegroup' == 100 were used for these

    st.pp <- survtab_ag(fot ~ sex, data = ag, surv.type = "surv.rel",
                        surv.method = "hazard", relsurv.method = "pp",
                        d = list(from0to1 + from0to2), pyrs = "pyrs",
                        d.pp = list(from0to1.pp + from0to2.pp),
                        d.pp.2 = list(from0to1.pp.2 + from0to2.pp.2),
                        pyrs.pp = "ptime.pp", d.exp.pp = "d.exp.pp")
    plot(st.pp, y = "r.pp", col = c("blue", "red"))

Here it is best to supply only one column to each argument since Pohar-Perme estimates will not be computed for several types of events at the same time.