`nSurvival()`

is used to calculate the sample size for a clinical trial
with a time-to-event endpoint. The Lachin and Foulkes (1986) method is used.
`nEvents`

uses the Schoenfeld (1981) approximation to provide sample
size and power in terms of the underlying hazard ratio and the number of
events observed in a survival analysis. The functions `hrz2n()`

,
`hrn2z()`

and `zn2hr()`

also use the Schoenfeld approximation to
provide simple translations between hazard ratios, z-values and the number
of events in an analysis; input variables can be given as vectors.

`nSurvival()`

produces an object of class "nSurvival" with the number
of subjects and events for a set of pre-specified trial parameters, such as
accrual duration and follow-up period. The calculation is based on Lachin
and Foulkes (1986) method and can be used for risk ratio or risk difference.
The function also consider non-uniform (exponential) entry as well as
uniform entry.

If the logical `approx`

is `TRUE`

, the variance under alternative
hypothesis is used to replace the variance under null hypothesis. For
non-uniform entry, a non-zero value of `gamma`

for exponential entry
must be supplied. For positive `gamma`

, the entry distribution is
convex, whereas for negative `gamma`

, the entry distribution is
concave.

`nEvents()`

uses the Schoenfeld (1981) method to approximate the number
of events `n`

(given `beta`

) or the power (given `n`

).
Arguments may be vectors or scalars, but any vectors must have the same
length.

The functions `hrz2n`

, `hrn2z`

and `zn2hr`

also all apply the
Schoenfeld approximation for proportional hazards modeling. This
approximation is based on the asymptotic normal distribtuion of the logrank
statistic as well as related statistics are asymptotically normal. Let
\(\lambda\) denote the underlying hazard ratio (`lambda1/lambda2`

in
terms of the arguments to `nSurvival`

). Further, let \(n\) denote the
number of events observed when computing the statistic of interest and
\(r\) the ratio of the sample size in an experimental group relative to a
control. The estimated natural logarithm of the hazard ratio from a
proportional hazards ratio is approximately normal with a mean of
\(log{\lambda}\) and variance \((1+r)^2/nr\). Let \(z\) denote a
logrank statistic (or a Wald statistic or score statistic from a
proportional hazards regression model). The same asymptotic theory implies
\(z\) is asymptotically equivalent to a normalized estimate of the hazard
ratio \(\lambda\) and thus \(z\) is asymptotically normal with variance
1 and mean $$\frac{log{\lambda}r}{(1+r)^2}.$$ Plugging the estimated
hazard ratio into the above equation allows approximating any one of the
following based on the other two: the estimate hazard ratio, the number of
events and the z-statistic. That is, $$\hat{\lambda}=
\exp(z(1+r)/\sqrt{rn})$$ $$z=\log(\hat{\lambda})\sqrt{nr}/(1+r)$$ $$n=
(z(1+r)/\log(\hat{\lambda}))^2/r.$$

`hrz2n()`

translates an observed interim hazard ratio and interim
z-value into the number of events required for the Z-value and hazard ratio
to correspond to each other. `hrn2z()`

translates a hazard ratio and
number of events into an approximate corresponding Z-value. `zn2hr()`

translates a Z-value and number of events into an approximate corresponding
hazard ratio. Each of these functions has a default assumption of an
underlying hazard ratio of 1 which can be changed using the argument
`hr0`

. `hrn2z()`

and `zn2hr()`

also have an argument
`hr1`

which is only used to compute the sign of the computed Z-value in
the case of `hrn2z()`

and whether or not a z-value > 0 corresponds to a
hazard ratio > or < the null hazard ratio `hr0`

.

```
# S3 method for nSurvival
print(x, ...)
```nSurvival(
lambda1 = 1/12,
lambda2 = 1/24,
Ts = 24,
Tr = 12,
eta = 0,
ratio = 1,
alpha = 0.025,
beta = 0.1,
sided = 1,
approx = FALSE,
type = c("rr", "rd"),
entry = c("unif", "expo"),
gamma = NA
)

nEvents(
hr = 0.6,
alpha = 0.025,
beta = 0.1,
ratio = 1,
sided = 1,
hr0 = 1,
n = 0,
tbl = FALSE
)

zn2hr(z, n, ratio = 1, hr0 = 1, hr1 = 0.7)

hrn2z(hr, n, ratio = 1, hr0 = 1, hr1 = 0.7)

hrz2n(hr, z, ratio = 1, hr0 = 1)

x

An object of class "nSurvival" returned by `nSurvival()`

(optional: used for output; "months" or "years" would be the 'usual'
choices).

...

Allows additional arguments for `print.nSurvival()`

.

lambda1, lambda2

event hazard rate for placebo and treatment group respectively.

Ts

maximum study duration.

Tr

accrual (recruitment) duration.

eta

equal dropout hazard rate for both groups.

ratio

randomization ratio between placebo and treatment group. Default is balanced design, i.e., randomization ratio is 1.

alpha

type I error rate. Default is 0.025 since 1-sided testing is default.

beta

type II error rate. Default is 0.10 (90% power). Not needed for
`nEvents()`

if n is provided.

sided

one or two-sided test? Default is one-sided test.

approx

logical. If `TRUE`

, the approximation sample size formula
for risk difference is used.

type

type of sample size calculation: risk ratio (“rr”) or risk difference (“rd”).

entry

patient entry type: uniform entry (`"unif"`

) or
exponential entry (`"expo"`

).

gamma

rate parameter for exponential entry. `NA`

if entry type
is `"unif"`

(uniform). A non-zero value is supplied if entry type is
`"expo"`

(exponential).

hr

Hazard ratio. For `nEvents`

, this is the hazard ratio under
the alternative hypothesis (>0).

hr0

Hazard ratio under the null hypothesis (>0, for `nEvents`

,
`!= hr`

).

n

Number of events. For `nEvents`

may be input to compute power
rather than sample size.

tbl

Indicator of whether or not scalar (vector) or tabular output is
desired for `nEvents()`

.

z

A z-statistic.

hr1

Hazard ratio under the alternate hypothesis for ```
hrn2z,
zn2hr
```

(>0, `!= hr0`

)

`nSurvival`

produces a list with the following component
returned:

As input.

As input.

Sample size required (computed).

Number of events required (computed).

As input.

As input.

As input.

As input.

As input.

As input.

As input.

As input.

As input.

As input.

nEvents produces a scalar or vector of sample sizes (or powers) when tbl=FALSE or, when tbl=TRUE a data frame of values with the following columns:

As input.

If `n[1]=0`

on input
(default), output contains the number of events need to obtain the input
Type I and II error. If `n[1]>0`

on input, the input value is
returned.

As input.

If `n[1]=0`

on input
(default), `beta`

is output as input. Otherwise, this is the computed
Type II error based on the input `n`

.

One minus the
output `beta`

. When `tbl=FALSE, n[1]>0`

, this is the value or
vector of values returned.

Standardized effect size represented by input difference between null and alternative hypothesis hazard ratios.

Ratio of experimental to control sample size where 'experimental' is the same as the group with hazard represented in the numerator of the hazard ratio.

Estimated standard error for the observed log(hazard ratio) with the given sample size.

hrz2n outputs a number of events required to approximately have the input hazard ratio, z-statistic and sample size correspond. hrn2z outputs an approximate z-statistic corresponding to an input hazard ratio and number of events. zn2hr outputs an approximate hazard ratio corresponding to an input z-statistic and number of events.

Lachin JM and Foulkes MA (1986), Evaluation of Sample Size and
Power for Analyses of Survival with Allowance for Nonuniform Patient Entry,
Losses to Follow-Up, Noncompliance, and Stratification. *Biometrics*,
42, 507-519.

Schoenfeld D (1981), The Asymptotic Properties of Nonparametric Tests for
Comparing Survival Distributions. *Biometrika*, 68, 316-319.

# NOT RUN { library(ggplot2) # consider a trial with # 2 year maximum follow-up # 6 month uniform enrollment # Treatment/placebo hazards = 0.1/0.2 per 1 person-year # drop out hazard 0.1 per 1 person-year # alpha = 0.025 (1-sided) # power = 0.9 (default beta=.1) ss <- nSurvival( lambda1 = .2, lambda2 = .1, eta = .1, Ts = 2, Tr = .5, sided = 1, alpha = .025 ) # group sequential translation with default bounds # note that delta1 is log hazard ratio; used later in gsBoundSummary summary x <- gsDesign( k = 5, test.type = 2, n.fix = ss$nEvents, nFixSurv = ss$n, delta1 = log(ss$lambda2 / ss$lambda1) ) # boundary plot plot(x) # effect size plot plot(x, plottype = "hr") # total sample size x$nSurv # number of events at analyses x$n.I # print the design x # overall design summary cat(summary(x)) # tabular summary of bounds gsBoundSummary(x, deltaname = "HR", Nname = "Events", logdelta = TRUE) # approximate number of events required using Schoenfeld's method # for 2 different hazard ratios nEvents(hr = c(.5, .6), tbl = TRUE) # vector output nEvents(hr = c(.5, .6)) # approximate power using Schoenfeld's method # given 2 sample sizes and hr=.6 nEvents(hr = .6, n = c(50, 100), tbl = TRUE) # vector output nEvents(hr = .6, n = c(50, 100)) # approximate hazard ratio corresponding to 100 events and z-statistic of 2 zn2hr(n = 100, z = 2) # same when hr0 is 1.1 zn2hr(n = 100, z = 2, hr0 = 1.1) # same when hr0 is .9 and hr1 is greater than hr0 zn2hr(n = 100, z = 2, hr0 = .9, hr1 = 1) # approximate number of events corresponding to z-statistic of 2 and # estimated hazard ratio of .5 (or 2) hrz2n(hr = .5, z = 2) hrz2n(hr = 2, z = 2) # approximate z statistic corresponding to 75 events # and estimated hazard ratio of .6 (or 1/.6) # assuming 2-to-1 randomization of experimental to control hrn2z(hr = .6, n = 75, ratio = 2) hrn2z(hr = 1 / .6, n = 75, ratio = 2) # }