---
title: "Validation of FastSurvival"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Validation of FastSurvival}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment  = "#>"
)
have_survival <- requireNamespace("survival", quietly = TRUE)
have_survRM2  <- requireNamespace("survRM2", quietly = TRUE)
have_survAH   <- requireNamespace("survAH", quietly = TRUE)
have_nph      <- requireNamespace("nph", quietly = TRUE)
have_nphRCT   <- requireNamespace("nphRCT", quietly = TRUE)
```

# Purpose

FastSurvival is built for speed, but a fast estimator is only useful if it
returns the same answer as the established implementation. This vignette
checks the numerical agreement between each FastSurvival function and a
reference, on a real clinical-trial dataset. We use the `gbsg` data from the
survival package, the German Breast Cancer Study Group cohort of 686 patients,
with recurrence-free survival time `rfstime` (in days), the event indicator
`status`, and the treatment indicator `hormon` (0 = no hormonal therapy,
1 = hormonal therapy). Throughout we take the no-hormone arm as the control
(`control = 0`) and report one-sided tests of treatment benefit (`side = 1`).

The references are: the survival package for the Kaplan-Meier estimate, the
log-rank test, the Cox hazard ratio, milestone survival, and the median
survival comparison; survRM2 for the restricted mean survival time; survAH for
the average hazard with survival weight; nph for the weighted log-rank test and
the max-combo test; and nphRCT for the robust modestly-weighted test. The
window mean survival time and the weighted Kaplan-Meier statistic, which have
no direct package counterpart on CRAN, are checked against a Kaplan-Meier
integral computed from `survfit()` and against the restricted mean survival
time identity, respectively. The same comparisons form the basis of the
automated test suite shipped with the package.

```{r load}
library(FastSurvival)
```

# Reference data

```{r data, eval = have_survival}
library(survival)

# German Breast Cancer Study Group cohort
str(gbsg[, c("rfstime", "status", "hormon")])
table(hormon = gbsg$hormon)
```

# Kaplan-Meier survival

`survfit_fast()` evaluates the Kaplan-Meier estimate at a single time point.
We compare against `summary(survfit(...))` at the same time (1000 days).

```{r km, eval = have_survival}
# survfit_fast assumes time-sorted input; the C++ core groups tied times in the
# survival::survfit risk-set convention, so the data only need ordering by time.
ord  <- order(gbsg$rfstime)
fast <- survfit_fast(gbsg$rfstime[ord], gbsg$status[ord],
                     t_eval = 1000, conf.type = "log-log")

fit  <- survfit(Surv(rfstime, status) ~ 1, data = gbsg)
ref  <- summary(fit, times = 1000)

data.frame(
  quantity = c("survival", "std.err"),
  fast     = c(unclass(fast)["surv"], unclass(fast)["std.err"]),
  survival = c(ref$surv, ref$std.err),
  row.names = NULL
)
```

# Log-rank test

`survdiff_fast()` with `side = 1` returns a signed Z-score. Its square is the
chi-square statistic from `survdiff()`.

```{r logrank, eval = have_survival}
fast_lr <- survdiff_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                         control = 0, side = 1)

ref_lr  <- survdiff(Surv(rfstime, status) ~ hormon, data = gbsg)

c(fast = as.numeric(fast_lr)^2, survival = ref_lr$chisq)
```

# Weighted log-rank test

`survdiff_fast()` also computes Fleming-Harrington G(rho, gamma) weighted
log-rank tests through the `weight = "fh"` argument. The
[nph](https://cran.r-project.org/package=nph) package provides
`logrank.test()` with the same family, so we compare the chi-square statistic
(the squared one-sided Z) across several weight choices.

```{r wlr, eval = have_survival && have_nph}
fh_grid <- data.frame(rho = c(0, 1, 0, 1), gamma = c(1, 0, 0, 1))

do.call(rbind, lapply(seq_len(nrow(fh_grid)), function(i) {
  r <- fh_grid$rho[i]
  g <- fh_grid$gamma[i]
  fast <- as.numeric(survdiff_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                                   control = 0, side = 1,
                                   weight = "fh", rho = r, gamma = g))
  nph_chisq <- nph::logrank.test(gbsg$rfstime, gbsg$status, gbsg$hormon,
                                 rho = r, gamma = g)$test$Chisq
  data.frame(rho = r, gamma = g, fast = fast^2, nph = nph_chisq)
}))
```

The two implementations agree to numerical precision across the weight family,
including the ordinary log-rank test recovered at `rho = 0, gamma = 0`.

# Cox hazard ratio

`coxph_fast()` returns the Pike-Halley Estimator, a closed-form approximation
to the Cox partial likelihood maximizer. We report the log hazard ratio from
both; the sign convention is the same and `side` does not affect the point
estimate.

```{r cox, eval = have_survival}
fast_hr <- coxph_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                      control = 0, side = 1)

ref_cox <- coxph(Surv(rfstime, status) ~ hormon, data = gbsg)

c(fast = unclass(fast_hr)["coef"], cox = unname(coef(ref_cox)))
```

Because the Pike-Halley Estimator is a closed-form approximation rather than
the exact partial-likelihood maximizer, the two log hazard ratios are not
expected to be identical. They differ slightly, here in the fourth decimal
place, which reflects the approximation and is not a sign of error.

# Restricted mean survival time

`rmst_fast()` integrates the Kaplan-Meier survival curve up to a horizon. We
compare the two-group RMST difference against `survRM2::rmst2()` at 1000 days.

```{r rmst, eval = have_survival && have_survRM2}
library(survRM2)

tau <- 1000

fast_rmst <- rmst_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                       control = 0, tau = tau, side = 1)

ref_rmst  <- rmst2(time = gbsg$rfstime, status = gbsg$status,
                   arm = gbsg$hormon, tau = tau)

c(fast = unclass(fast_rmst)["diff"],
  survRM2 = ref_rmst$unadjusted.result[1, 1])
```

# Window mean survival time

`wmst_fast()` integrates the Kaplan-Meier curve between a lower and an upper
window limit, generalizing the restricted mean survival time. CRAN has no
dedicated WMST package, so we validate the per-group windowed area directly
against a Kaplan-Meier step-function integral computed from `survfit()` over
the same window. A full comparison against the `survWMST` package, which is
distributed on GitHub, is provided in
`tools/compare_wmst_survwmst.R`.

```{r wmst, eval = have_survival}
tau1 <- 200
tau2 <- 1000

fast_wm <- wmst_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                     control = 0, side = 1, tau1 = tau1, tau2 = tau2)

# Integral of the Kaplan-Meier step function over [lo, hi] for one group.
km_window_area <- function(gi, lo, hi) {
  sf <- survfit(Surv(rfstime, status) ~ 1, data = gbsg[gbsg$hormon == gi, ])
  tk <- c(0, sf$time)
  sk <- c(1, sf$surv)
  brk  <- sort(unique(c(lo, hi, sf$time[sf$time > lo & sf$time < hi])))
  area <- 0
  for (b in seq_len(length(brk) - 1L)) {
    u  <- brk[b]
    su <- sk[max(which(tk <= u))]
    area <- area + su * (brk[b + 1L] - u)
  }
  area
}

data.frame(
  group    = c("control", "treatment"),
  fast     = unclass(fast_wm)[c("wmst.control", "wmst.treatment")],
  survfit  = c(km_window_area(0, tau1, tau2), km_window_area(1, tau1, tau2)),
  row.names = NULL
)
```

# Weighted Kaplan-Meier test

`wkm_fast()` computes the weighted Kaplan-Meier (Pepe-Fleming) test, the
weighted integral of the difference between the two Kaplan-Meier curves. With
the constant weight the weighted difference reduces exactly to the difference
in restricted mean survival time over the observed range, which gives a clean
internal check against `rmst_fast()` evaluated at the largest observed time.

```{r wkm, eval = have_survival}
tmax <- max(gbsg$rfstime)

fast_wk <- wkm_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                    control = 0, side = 1, weight = "constant")

ref_rm  <- rmst_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                     control = 0, tau = tmax, side = 1)

c(wkm.constant = unclass(fast_wk)["wdiff"], rmst = unclass(ref_rm)["diff"])
```

With the default Pepe-Fleming weight, `wkm_fast()` reproduces the weighted
Kaplan-Meier statistic of the `nphsim` package. Because `nphsim` is distributed
only on GitHub, that comparison is kept in
`tools/compare_wkm_nphsim.R` (which also bundles a survival-only
reproduction of the statistic) and in the test suite, rather than in this
vignette.

# Milestone survival

`milestone_fast()` compares Kaplan-Meier survival between two groups at a
milestone timepoint. The per-group survival probabilities match those from
`survfit()`.

```{r milestone, eval = have_survival}
tstar <- 1000

fast_ms <- milestone_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                          control = 0, tau = tstar, method = "loglog",
                          side = 1)

fit_g <- survfit(Surv(rfstime, status) ~ hormon, data = gbsg)
ref_g <- summary(fit_g, times = tstar)

data.frame(
  group    = c("control", "treatment"),
  fast     = fast_ms$surv[c("control", "treatment")],
  survival = ref_g$surv,
  row.names = NULL
)
```

# Median survival time

`medsurv_fast()` estimates the Kaplan-Meier median survival time and, for two
groups, their difference. The point estimate is the Kaplan-Meier median (the
first time at which the product-limit curve reaches 0.5), the same convention
as `survfit()`, so the per-group medians match those reported by `survfit()`
exactly.

```{r medsurv, eval = have_survival}
fast_med <- medsurv_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                         control = 0, side = 1, method = "nph")

fit_med <- survfit(Surv(rfstime, status) ~ hormon, data = gbsg)
med_ref <- summary(fit_med)$table[, "median"]

data.frame(
  group    = c("control", "treatment"),
  fast     = unclass(fast_med)[c("median.control", "median.treatment")],
  survival = c(med_ref[1], med_ref[2]),
  row.names = NULL
)
```

We use `survfit()` as the reference here, rather than `nph::nphparams()`,
because the two define the median on different survival curves: `survfit()`
and `medsurv_fast()` use the Kaplan-Meier (product-limit) median, whereas
`nph::nphparams()` reads the median off the Nelson-Aalen curve,
S(t) = exp(-H(t)), which lies above the Kaplan-Meier curve and so can cross 0.5
at a later time on tied data. The `method = "nph"` standard error reproduces
the `nph::nphparams()` standard error to numerical precision when the two
medians coincide; that comparison is run on tie-free scenarios in
`tools/compare_medsurv_nphparams.R` and in the package test suite.

# Average hazard with survival weight

`ahsw_fast()` computes the average hazard with survival weight of Uno and
Horiguchi. We compare the per-group average hazard against `survAH::ah2()`.

```{r ahsw, eval = have_survival && have_survAH}
library(survAH)

tau <- 1000

fast_ah <- ahsw_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                     control = 0, tau = tau, side = 1)

ref_ah  <- ah2(time = gbsg$rfstime, status = gbsg$status,
               arm = gbsg$hormon, tau = tau)

data.frame(
  quantity = c("AH (control)", "AH (treatment)"),
  fast     = unclass(fast_ah)[c("ah.ctrl", "ah.trt")],
  survAH   = c(ref_ah$ah["AH (arm0)", "Est."],
               ref_ah$ah["AH (arm1)", "Est."]),
  row.names = NULL
)
```

The average hazard is the ratio of the cumulative event probability to the
restricted mean survival time, so for the `gbsg` data, where follow-up is
measured in days, the values are on the order of 1e-04 per day in both groups.

# Average hazard ratio

`ahr_fast()` computes the Kalbfleisch-Prentice average hazard ratio between two
groups over a restricted interval. The reference implementation is `ahrKM()`
from the AHR package, but that package has been archived on CRAN, so here we
check the point estimates against a direct survival-based computation: each
group's Kaplan-Meier curve from `survfit()`, integrated to form the group
shares of the total hazard.

```{r ahr, eval = have_survival}
tau <- 1000

fast_ahr <- ahr_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                     control = 0, tau = tau, side = 1)

g  <- gbsg$hormon
ev <- c(gbsg$rfstime[g == 0 & gbsg$status == 1],
        gbsg$rfstime[g == 1 & gbsg$status == 1])
grid <- sort(unique(c(0, ev[ev <= tau], tau)))

km_on_grid <- function(gi) {
  sf <- survfit(Surv(rfstime, status) ~ 1, data = gbsg[g == gi, ])
  approxfun(sf$time, sf$surv, method = "constant",
            yleft = 1, rule = 2, f = 0)(grid)
}

S0  <- km_on_grid(0)
S1  <- km_on_grid(1)
m   <- length(grid)
dS0 <- S0 - c(1, S0[-m])
GL  <- S0[m] * S1[m]
ref_theta_ctrl <- -sum(S1 * dS0) / (1 - GL)

data.frame(
  quantity  = c("theta (control)", "theta (treatment)", "AHR"),
  fast      = c(fast_ahr$theta[[1]], fast_ahr$theta[[2]], fast_ahr$ahr),
  reference = c(ref_theta_ctrl, 1 - ref_theta_ctrl,
                (1 - ref_theta_ctrl) / ref_theta_ctrl),
  row.names = NULL
)
```

The point estimates match the survival-based reference to numerical precision.
The average hazard ratio was additionally cross-checked against the archived
AHR package, the reference implementation used by Dormuth et al. (2024): the
two group shares, the average hazard ratio, the variances, and both test
statistics agree to within the order of 1e-15. That external comparison is
reproducible with the `tools/compare_ahr_ahrKM.R` script shipped in the package
sources.

# Max-combo test

`maxcombo_fast()` computes the max-combo test, the most extreme of a set of
Fleming-Harrington weighted log-rank statistics. The `nph` package provides
`logrank.maxtest()`, whose default weight set is FH(0, 0), FH(0, 1), and
FH(1, 0). We request the same three weights from `maxcombo_fast()` and compare
the component Z-scores. The individual Z-scores are kept in the `z` attribute,
signed so that a negative value indicates benefit for the treatment group under
the package convention. Because `nph` orients the contrast in the opposite
direction, the signs are mirrored, so we compare absolute values.

```{r maxcombo, eval = have_survival && have_nph}
rho   <- c(0, 0, 1)
gamma <- c(0, 1, 0)

fast_mc <- maxcombo_fast(gbsg$rfstime, gbsg$status, gbsg$hormon,
                         control = 0, side = 1, rho = rho, gamma = gamma)

nph_mc <- nph::logrank.maxtest(gbsg$rfstime, gbsg$status, gbsg$hormon)

data.frame(
  weight = c("FH(0,0)", "FH(0,1)", "FH(1,0)"),
  fast   = abs(attr(fast_mc, "z")),
  nph    = abs(nph_mc$tests$z),
  row.names = NULL
)
```

The component statistics agree in absolute value. The final max-combo statistic
is the largest of these, which for the one-sided test is the most extreme
component Z. The p-values from the two packages are close but not identical,
since they use different numerical methods for the joint distribution (a
multivariate normal integral here, a Bonferroni-style combination in `nph`).

# Robust modestly-weighted log-rank test

`rmw_fast()` computes the robust modestly-weighted (rMW) test of Magirr and
Öhrn, the maximum of the standard log-rank statistic and a single
modestly-weighted log-rank statistic. Each component is a weighted log-rank
test, so we check the two component Z-scores against
[nphRCT](https://cran.r-project.org/package=nphRCT), the implementation used by
the method's authors. The modestly-weighted component uses the
survival-threshold parameterization, so `s_star = 1` recovers the standard
log-rank test and `s_star = 0.5` gives the modestly-weighted component. As with
the max-combo test the package orients a negative Z toward treatment benefit
while `nphRCT` uses the opposite sign, so we compare absolute values.

```{r rmw, eval = have_survival && have_nphRCT}
gbsg_df <- data.frame(
  time  = gbsg$rfstime,
  event = gbsg$status,
  arm   = factor(ifelse(gbsg$hormon == 0, "control", "experimental"),
                 levels = c("control", "experimental"))
)

fit_rmw <- rmw_fast(gbsg_df$time, gbsg_df$event, gbsg_df$arm,
                    control = "control", side = 1, s_star = 0.5)

z_lr_nph <- nphRCT::wlrt(Surv(time, event) ~ arm, data = gbsg_df,
                         method = "mw", s_star = 1)$z
z_mw_nph <- nphRCT::wlrt(Surv(time, event) ~ arm, data = gbsg_df,
                         method = "mw", s_star = 0.5)$z

data.frame(
  component = c("log-rank (s_star = 1)", "modestly-weighted (s_star = 0.5)"),
  fast      = abs(attr(fit_rmw, "z")),
  nphRCT    = abs(c(z_lr_nph, z_mw_nph)),
  row.names = NULL
)
```

The component statistics agree in absolute value. The null correlation of the
two components, reported in the `corr` attribute, matches the covariance
computed by the authors' `find_cor` routine to numerical precision. The
combined statistic and one-sided p-value then follow from the bivariate normal
distribution of the two components, evaluated with `mvtnorm::pmvnorm`. On the
POPLAR overall-survival data analyzed in the original article, `rmw_fast()`
reproduces the published one-sided p-values of 0.0028 for the log-rank test,
0.0009 for the modestly-weighted test, and 0.0012 for the rMW test, with a
component correlation of 0.97.

# Summary

Across all functions the FastSurvival results reproduce the reference values on
the `gbsg` data. The point estimates and test statistics agree to numerical
precision for the Kaplan-Meier, log-rank, weighted log-rank, RMST, milestone,
median, and average-hazard quantities; the window mean survival time matches a
Kaplan-Meier integral and the weighted Kaplan-Meier statistic matches the RMST
identity; the average hazard ratio matches a survival-based reference; and the
closed-form Cox hazard ratio agrees with the partial-likelihood maximizer to
the order expected for the Pike-Halley approximation. This agreement is verified
continuously by the package test suite.

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