This vignette walks through the basic functions in npsurvSS. By the end, users should be able to:

- create an object that captures the distributional assumptions for a treatment arm
- visualize the distributional assumptions
- calculate power and sample size for a two-arm study
- approximate an event-driven study by a time-driven study
- simulate a two-arm study

The cornerstone of npsurvSS lies in objects of class “arm”. These objects are lists that capture for a treatment arm assumptions regarding its sample size, accrual, survival, censoring, and duration of follow-up. Once created, they serve as inputs for other functions, including functions for power/sample size calculation and trial simulation.

The following code creates two arms, an `active`

arm and a
`control`

arm. Both arms will accrue 120 patients uniformly
over 6 months and follow them for an additional 12 months. Patients will
be subjected to loss of follow-up at an exponential rate of 0.00578.
`active`

and `control`

patients will experience
event at exponential rates of 0.0462 and 0.0578, respectively. The
hazard ratio between the two arms is therefore approximately 0.8.

```
library(npsurvSS)
active <- create_arm(size=120,
accr_time=6,
surv_scale=0.0462,
loss_scale=0.00578,
follow_time=12)
control <- create_arm(size=120,
accr_time=6,
surv_scale=0.0578,
loss_scale=0.00578,
follow_time=12)
```

In practice, investigators seldom consider exponential distributions
on the hazard rate scale. Instead, they consider the median survival or
survival probability at some milestone *t*. We have defined
additional functions to facilitate this practice. `per2haz`

is a simple code that can convert an exponential survival percentile to
the hazard rate and vice versa.

```
active <- create_arm(size=120,
accr_time=6,
surv_scale=per2haz(15), # corresponds to 15 month median
loss_scale=per2haz(120), # corresponds to 120 month median
follow_time=12)
control <- create_arm(size=120,
accr_time=6,
surv_scale=per2haz(12), # corresponds to 12 month median
loss_scale=per2haz(120),
follow_time=12)
per2haz(15) # convert median survival to hazard rate
#> [1] 0.04620981
per2haz(0.0462) # convert hazard rate to median survival
#> [1] 15.00319
```

Alternatively, `create_arm_lachin`

allows investigators to
specify exponential survival and censoring distributions by providing
median survivals or milestone survivals.

```
active <- create_arm_lachin(size=120,
accr_time=6,
surv_median=15,
loss_milestone=c(120, 0.5), # corresponds to 120 month median
follow_time=12)
control <- create_arm_lachin(size=120,
accr_time=6,
surv_milestone=c(12, 0.5), # corresponds to 12 month median
loss_median=120,
follow_time=12)
class(active)
#> [1] "list" "lachin" "arm"
```

Note that objects created by `create_arm_lachin`

belong to
class “lachin” and “arm”. “lachin” is a subclass of “arm”. It is named
after the class of distributions considered by Lachin (1986), which
covers uniform/truncated-exponential accrual, exponential survival, and
exponential censoring. Check out the R Documentation
`?create_arm`

for examples of “arm” objects with more
sophisticated assumptions, such as piecewise-uniform accrual,
piecewise-exponential/Weibull survival, and Weibull censoring. While
objects created by `create_arm_lachin`

are always of class
“lachin” and “arm”, objects created by `create_arm`

are
always of class “arm”, but not necessarily of class “lachin”.

Having created an “arm” object, visualizing its assumptions is easy. For example, the following code plots the accrual cumulative distribution function (CDF):

```
x <- seq(0, 6, 0.1)
plot(x, paccr(q=x, arm=control),
xlab="Time from first patient in (month)",
ylab="Accrual CDF",
type="l")
```

Likewise, the survival function:

```
x <- seq(0, 18, 0.1)
plot(x, psurv(q=x, arm=control, lower.tail=F),
xlab="Time from study entry (month)",
ylab="Survival function",
type="l")
```

Just as `pbinom`

in the R package stats is accompanied by
functions for the density, quantile, and random generation,
`paccr`

is similarly accompanied by `daccr`

,
`qaccr`

, and `raccr`

. Distribution functions
`psurv`

and `ploss`

are further accompanied by
`hsurv`

and `hloss`

for the hazard.

Given an `active`

arm and a `control`

arm,
calculating power and sample size is also easy. The following code
calculates power under the default setting of an unweighted log-rank
test with one-sided alpha 0.025:

To calculate power for other tests:

```
# unweighted log-rank
power_two_arm(control, active, test=list(test="weighted logrank"))
#> [1] 0.2366524
# Gehan-Breslow weighted log-rank
power_two_arm(control, active, test=list(test="weighted logrank", weight="n"))
#> [1] 0.2210357
# difference in 12 month survival
power_two_arm(control, active, test=list(test="survival difference", milestone=12))
#> [1] 0.2050328
# ratio of 12 month RMST
power_two_arm(control, active, test=list(test="rmst ratio", milestone=12))
#> [1] 0.1823979
```

Power for multiple tests can be calculated simulateously:

```
power_two_arm(control, active, test=list(list(test="weighted logrank"),
list(test="weighted logrank", weight="n"),
list(test="survival difference", milestone=12),
list(test="rmst ratio", milestone=12)
))
#> test power
#> 1 1 0.2366524
#> 2 2 0.2210357
#> 3 3 0.2050328
#> 4 4 0.1823979
```

To calculate sample size required to achieve 80% power:

```
size_two_arm(control, active,
test=list(list(test="weighted logrank"),
list(test="weighted logrank", weight="n"),
list(test="survival difference", milestone=12),
list(test="rmst ratio", milestone=12)
))
#> test n0 n1 n d0 d1 d
#> 1 1 609.7478 609.7478 1219.496 339.3015 292.4822 631.7837
#> 2 2 663.7018 663.7018 1327.404 369.3249 318.3627 687.6876
#> 3 3 729.6092 729.6092 1459.218 405.9998 349.9770 755.9768
#> 4 4 848.3117 848.3117 1696.623 472.0532 406.9159 878.9692
```

Note that `size_two_arm`

returns the required sample size
*n* and expected number of events *d* (per arm and total).
When calculating the required sample size per arm, it considers as input
the specified ratio between the two arms (e.g. 120:120) while ignoring
their individual values (e.g. 120 and 120). Thus, the following two
“arm” objects result in the same sample size calculation for the
unweighted log-rank test:

```
control_new <- control
active_new <- active
control_new$size <- 1
active_new$size <- 1
size_two_arm(control_new, active_new)
#> n0 n1 n d0 d1 d
#> 609.7478 609.7478 1219.4956 339.3015 292.4822 631.7837
```

Sample size for a trial with 2:1 randomization in favor of the active arm can be calculated like so:

By containing the keys `follow_time`

and
`total_time`

, “arm” objects intrinsically apply to
time-driven trials that end when a fixed period of time has elapsed
after the last patient in. However, they can also be used to approximate
event-driven trials, trials in which the study ends when a desired
number of events has been observed. Specifically, a trial requiring
*d* events can be approximated by a trial of length *t*,
where the expected number of events at *t* is equal to
*d*. The functions `exp_events`

and
`exp_duration`

can be useful for this purpose:

```
exp_events(control, active) # expected number of events
#> [1] 124.3367
tau <- exp_duration(control, active, d=150) # study duration for expected number of events to equal d
tau
#> [1] 23.75
```

Therefore, under the given assumptions, a trial requiring 150 events
can be approximated by a 23.75-month long trial. When updating the trial
duration in an “arm” object, it is important to update both the
`follow_time`

and `total_time`

to ensure their
consistency:

Finally, time-driven and event-driven trials can be simulated using the following code:

```
trial1 <- simulate_trial(control, active, duration=18)
head(trial1, 5)
#> arm time.accr time.obs time.total censor reason
#> 122 1 1.0114577 0.2993216 1.310779 1 event
#> 110 0 0.0382217 1.5187492 1.556971 1 event
#> 109 0 1.5549119 0.3500164 1.904928 1 event
#> 12 0 0.3402699 1.5805129 1.920783 1 event
#> 39 0 0.1547186 1.9131597 2.067878 1 event
table(trial1$arm, trial1$reason)
#>
#> administration event followup
#> 0 45 71 4
#> 1 48 69 3
max(trial1$time.total)
#> [1] 18
trial2 <- simulate_trial(control, active, events=150)
head(trial2, 5)
#> arm time.accr time.obs time.total censor reason
#> 81 0 0.1689503 0.6329578 0.8019081 1 event
#> 222 1 0.3934911 0.6720223 1.0655134 1 event
#> 88 0 1.2510261 0.1418356 1.3928617 1 event
#> 149 1 0.6253446 0.9906663 1.6160108 1 event
#> 198 1 0.5424634 1.3130129 1.8554763 0 followup
sum(trial2$censor)
#> [1] 150
```

If both `duration`

and `events`

are provided,
the study will end whenever one of the criteria is met:

```
trial3 <- simulate_trial(control, active, duration=18, events=150)
max(trial3$time.total)
#> [1] 18
sum(trial3$censor)
#> [1] 105
```

Should it be desired, investigators may also simulate complete data (accrual, survival, censoring) for each individual treatment arm. Note that no cutoff (by number of events or time) is applied. Hence, no patients are administratively censored:

```
control_sim <- simulate_arm(control)
head(control_sim, 5)
#> arm time.accr time.obs time.total censor reason time.surv time.loss
#> 1 1 1.396755 0.7678287 2.164583 1 event 0.7678287 385.47039
#> 2 1 1.912468 16.9183206 18.830789 1 event 16.9183206 41.05433
#> 3 1 2.019832 11.7896546 13.809486 1 event 11.7896546 75.81059
#> 4 1 0.100030 4.6107775 4.710807 1 event 4.6107775 642.99622
#> 5 1 3.885474 17.4757370 21.361211 1 event 17.4757370 27.76870
table(control_sim$arm, control_sim$reason)
#>
#> event followup
#> 1 113 7
```