Produces a data frame that is sorted by stratum and time. Included in this is only the times at which one or more event occurs. The output dataset contains stratum, TTE (time-to-event), at risk count, and count of events at the specified TTE sorted by stratum and TTE.

## Value

A data frame grouped by `stratum`

and sorted within stratum by `tte`

.
Remain rows with at least one event in the population, at least one subject
is at risk in both treatment group and control group.
Other variables in this represent the following within each stratum at
each time at which one or more events are observed:

`event_total`

: Total number of events`event_trt`

: Total number of events at treatment group`n_risk_total`

: Number of subjects at risk`n_risk_trt`

: Number of subjects at risk in treatment group`s`

: Left-continuous Kaplan-Meier survival estimate`o_minus_e`

: In treatment group, observed number of events minus expected number of events. The expected number of events is estimated by assuming no treatment effect with hypergeometric distribution with parameters total number of events, total number of events at treatment group and number of events at a time. (Same assumption of log-rank test under the null hypothesis)`var_o_minus_e`

: Variance of`o_minus_e`

under the same assumption.

## Details

The function only considered two group situation.

The tie is handled by the Breslow's Method.

The output produced by `counting_process()`

produces a
counting process dataset grouped by stratum and sorted within stratum
by increasing times where events occur. The object is assigned the class
"counting_process". It also has the attributes "n_ctrl" and "n_exp",
which are the totals of the control and experimental treatments,
respectively, from the input time-to-event data.

## Examples

```
# Example 1
x <- data.frame(
stratum = c(rep(1, 10), rep(2, 6)),
treatment = rep(c(1, 1, 0, 0), 4),
tte = 1:16,
event = rep(c(0, 1), 8)
)
counting_process(x, arm = 1)
#> stratum event_total event_trt tte n_risk_total n_risk_trt s
#> 1 1 1 1 2 9 5 1.0000000
#> 2 1 1 0 4 7 4 0.8888889
#> 3 1 1 1 6 5 3 0.7619048
#> 4 1 1 0 8 3 2 0.6095238
#> 5 2 1 0 12 5 2 1.0000000
#> 6 2 1 1 14 3 1 0.8000000
#> o_minus_e var_o_minus_e
#> 1 0.4444444 0.2469136
#> 2 -0.5714286 0.2448980
#> 3 0.4000000 0.2400000
#> 4 -0.6666667 0.2222222
#> 5 -0.4000000 0.2400000
#> 6 0.6666667 0.2222222
# Example 2
x <- sim_pw_surv(n = 400)
y <- cut_data_by_event(x, 150) |> counting_process(arm = "experimental")
# Weighted logrank test (Z-value and 1-sided p-value)
z <- sum(y$o_minus_e) / sqrt(sum(y$var_o_minus_e))
c(z, pnorm(z))
#> [1] -3.808578e+00 6.988404e-05
```