knitr::opts_chunk$set( collapse = TRUE, comment = "#>", dev = "ragg_png", dpi = 96, fig.retina = 1, fig.width = 7.2916667, fig.asp = 0.618, fig.align = "center", out.width = "80%" ) options(width = 58)
library(gsDesign) library(tidyr) library(knitr) library(tibble)
This article/vignette provides a summary of functions in the gsDesign package supporting design and evaluation of trial designs for time-to-event outcomes. We do not focus on detailed output options, but what numbers summarizing the design are based on. If you are not looking for this level of detail and just want to see how to design a fixed or group sequential design for a time-to-event endpoint, see the vignette Basic time-to-event group sequential design using gsSurv.
The following functions support use of the very straightforward @Schoenfeld1981 approximation for 2-arm trials:
nEvents()
: number of events to achieve power or power given number of events with no interim analysis.zn2hr()
: approximate the observed hazard ratio (HR) required to achieve a targeted Z-value for a given number of events.hrn2z()
: approximate Z-value corresponding to a specified HR and event count.hrz2n()
: approximate event count corresponding to a specified HR and Z-value.The above functions do not directly support sample size calculations. This is done with the @LachinFoulkes method. Functions include:
nSurv()
: More flexible enrollment scenarios; single analysis.gsSurv()
: Group sequential design extension of nSurv()
.nSurvival()
: Sample size restricted to single enrollment rate, single analysis; this has been effectively replaced and generalized by nSurv()
and gsSurv()
.Output for survival design information is supported in various formats:
gsBoundSummary()
: Tabular summary of a design in a data frame.plot.gsDesign()
: Various plot summaries of a design.gsHR()
: Approximate HR required to cross a bound.We will assume a hazard ratio $\nu < 1$ represents a benefit of experimental treatment over control. We let $\delta = \log\nu$ denote the so-called natural parameter for this case. Asymptotically the distribution of the Cox model estimate $\hat{\delta}$ under the proportional hazards assumption is (@Schoenfeld1981) $$\hat\delta\sim \text{Normal}(\delta=\log\nu, (1+r)^2/nr).$$ The inverse of the variance is the statistical information: $$\mathcal I = nr/(1 + r)^2.$$ Using a Cox model to estimate $\delta$, the Wald test for $\text{H}_0: \delta=0$ can be approximated with the asymptotic variance from above as:
$$Z_W\approx \frac{\sqrt {nr}}{1+r}\hat\delta=\frac{\ln(\hat\nu)\sqrt{nr}}{1+r}.$$
Also, we know that the Wald test $Z_W$ and a standard normal version of the logrank $Z$ are both asymptotically efficient and therefore asymptotically equivalent, at least under a local hypothesis framework. We denote the standardized effect size as
$$\theta = \delta\sqrt r / (1+r)= \log(\nu)\sqrt r / (1+r).$$ Letting $\hat\theta = -\sqrt r/(1+r)\hat\delta$ and $n$ representing the number of events observed, we have $$\hat \theta \sim \text{Normal}(\theta, 1/ n).$$ Thus, the standardized Z version of the logrank is approximately distributed as
$$Z\sim\text{Normal}(\sqrt n\theta,1).$$ Treatment effect favoring experimental treatment compared to control in this notation corresponds to a hazard ratio $\nu < 1$, as well as negative values of the standardized effect $\theta$, natural parameter $\delta$ and standardized Z-test.
nEvents()
Based on the above, the power for the logrank test when $n$ events have been observed is approximated by
$$P[Z\le z]=\Phi(z -\sqrt n\theta)=\Phi(z- \sqrt{nr}/(1+r)\log\nu).$$ Thus, assuming $n=100$ events and $\delta = \log\nu=-\log(.7)$, and $r=1$ (equal randomization) we approximate power for the logrank test when $\alpha=0.025$ as
n <- 100 hr <- .7 delta <- log(hr) alpha <- .025 r <- 1 pnorm(qnorm(alpha) - sqrt(n * r) / (1 + r) * delta)
We can compute this with gsDesign::nEvents()
as:
nEvents(n = n, alpha = alpha, hr = hr, r = r)
We solve for the number of events $n$ to see how many events are required to obtain a desired power
$$1-\beta=P(Z\ge \Phi^{-1}(1-\alpha))$$ with
$$n = \left(\frac{\Phi^{-1} (1-\alpha)+\Phi^{-1}(1-\beta)}{\theta}\right)^2 =\frac{(1+r)^2}{r(\log\nu)^2}\left({\Phi^{-1} (1-\alpha)+\Phi^{-1}(1-\beta)}\right)^2.$$ Thus, the approximate number of events required to power for HR=0.7 with $\alpha=0.025$ one-sided and power $1-\beta=0.9$ is
beta <- 0.1 (1 + r)^2 / r / log(hr)^2 * ((qnorm(1 - alpha) + qnorm(1 - beta)))^2
which, rounding up, matches (with tabular output):
nEvents(hr = hr, alpha = alpha, beta = beta, r = 1, tbl = TRUE) %>% kable()
The notation delta
in the above table changes the sign for the standardized treatment effect $\theta$ in the above:
theta <- delta * sqrt(r) / (1 + r) theta
The se
in the table is the estimated standard error for the log hazard ratio $\delta=\log\hat\nu$
(1 + r) / sqrt(331 * r)
We can create a group sequential design for the above problem either with $\theta$ or with the fixed design sample size. The parameter delta
in gsDesign()
corresponds to standardized effect size with sign changed $-\theta$ in notation used above and by @JTBook, while the natural parameter, $\log(\text{HR})$ is in the parameter delta1
passed to gsDesign()
.
The name of the effect size is specified in deltaname
and the parameter logdelta = TRUE
indicates that delta
input needs to be exponentiated to obtain HR in the output below.
This example code can be useful in practice.
We begin by passing the number of events for a fixed design in the parameter n.fix
(continuous, not rounded) to adapt to a group sequential design.
By rounding to integer event counts with the toInteger()
function we increase the power slightly over the targeted 90%.
Schoenfeld <- gsDesign( k = 2, n.fix = nEvents(hr = hr, alpha = alpha, beta = beta, r = 1), delta1 = log(hr) ) %>% toInteger() Schoenfeld %>% gsBoundSummary(deltaname = "HR", logdelta = TRUE, Nname = "Events") %>% kable(row.names = FALSE)
Exactly the same result can be obtained with the following, passing the standardized effect size theta
from above to the parameter delta
in gsDesign()
.
Schoenfeld <- gsDesign(k = 2, delta = -theta, delta1 = log(hr)) %>% toInteger()
We noted above that the asymptotic variance for $\hat\theta$ is $1/n$ which corresponds to statistical information $\mathcal I=n$ for the parameter $\theta$. Thus, the value
Schoenfeld$n.I
corresponds both to the number of events and the statistical information for the standardized effect size $\theta$ required to power the trial at the desired level. Note that if you plug in the natural parameter $\delta= -\log\nu > 0$, then $n.I$ returns the statistical information for the log hazard ratio.
gsDesign(k = 2, delta = -log(hr))$n.I
The reader may wish to look above to derive the exact relationship between events and statistical information for $\delta$.
Another application of the @Schoenfeld1981 method is to approximate boundary characteristics of a design.
We will consider zn2hr()
, gsHR()
and gsBoundSummary()
to approximate the treatment effect required to cross design bounds.
zn2hr()
is complemented by the functions hrn2z()
and hrz2n()
.
We begin with the basic approximation used across all of these functions in this section and follow with a sub-section with example code to reproduce some of what is in the table above.
We return to the following equation from above:
$$Z\approx Z_W\approx \frac{\sqrt {nr}}{1+r}\hat\delta=\frac{\ln(\hat\nu)\sqrt{nr}}{1+r}.$$ By fixing $Z=z, n$ we can solve for $\hat\nu$ from the above:
$$\hat{\nu} = \exp(z(1+r)/\sqrt{rn}).$$ By fixing $\hat\nu$ and $z$, we can solve for the corresponding number of events required: $$ n = (z(1+r)/\log(\hat{\nu}))^2/r.$$
We continue with the Schoenfeld
example event counts:
Schoenfeld$n.I
We reproduce the approximate hazard ratios required to cross efficacy bounds using the Schoenfeld approximations above:
gsHR( z = Schoenfeld$upper$bound, # Z-values at bound i = 1:2, # Analysis number x = Schoenfeld, # Group sequential design from above ratio = r # Experimental/control randomization ratio )
For the following examples, we assume $r=1$.
r <- 1
1) Assuming a Cox model estimate $\hat\nu$ and a corresponding event count, approximately what Z-value (p-value) does this correspond to? We use the first equation above:
hr <- .73 # Observed hr events <- 125 # Events in analysis z <- log(hr) * sqrt(events * r) / (1 + r) c(z, pnorm(z)) # Z- and p-value
We replicate the Z-value with
hrn2z(hr = hr, n = events, ratio = r)
2) Assuming an efficacy bound Z-value and event count, approximately what hazard ratio must be observed to cross the bound? We use the second equation above:
z <- qnorm(.025) events <- 120 exp(z * (1 + r) / sqrt(r * events))
We can reproduce this with zn2hr()
by switching the sign of z
above; note that the default is ratio = 1
for all of these functions and often is not specified:
zn2hr(z = -z, n = events, ratio = r)
3) Finally, if we want an observed hazard ratio $\hat\nu = .8$ to represent a positive result, how many events would be need to observe to achieve a 1-sided p-value of 0.025? assuming 2:1 randomization? We use the third equation above:
r <- 2 hr <- .8 z <- qnorm(.025) events <- (z * (1 + r) / log(hr))^2 / r events
This is replicated with
hrz2n(hr = hr, z = z, ratio = r)
For the purpose of sample size and power for group sequential design, the @LachinFoulkes is recommended based on substantial evaluation not documented further here.
We try to make clear here what some of the strengths and weaknesses of both the @LachinFoulkes method as well as its implementation in the gsDesign::nSurv()
(fixed design) and gsDesign::gsSurv()
(group sequential) functions.
For historical and testing purposes, we also discuss use of the less flexible gsDesign::nSurvival()
function that was independently programmed and can be used for some limited validations of gsDesign::nSurv()
.
Some detail in specification comes With the flexibility allowed by the @LachinFoulkes method. The model assumes
Other than the proportional hazards assumption, this allows a great deal of flexibility in trial design assumptions.
While @LachinFoulkes adjusts the piecewise constant enrollment rates proportionately to derive a sample size, gsDesign::nSurv()
also enables the approach of @KimTsiatis which fixes enrollment rates and extends the final enrollment rate duration to power the trial; the minimum follow-up period is still assumed with this approach.
We do not enable the drop-in option proposed in @LachinFoulkes.
The two practical differences the @LachinFoulkes method has from the @Schoenfeld1981 method are:
1) By assuming enrollment, failure and dropout rates the method delivers sample size $N$ as well as events required. 2) The variance for the log hazard ratio $\hat\delta$ is computed differently and both a null ($\sigma^2_0$) and alternate hypothesis ($\sigma^2_1$) variance are incorporated through the formula $$N = \left(\frac{\Phi^{-1}(1-\alpha)\sigma_0 + \Phi^{-1}(1-\beta)\sigma_1}{\delta}\right).$$ The null hypothesis is derived by averaging the alternate hypothesis rates, weighting according to the proportion randomized in each group.
We will use the same hazard ratio 0.7 as for the @Schoenfeld1981 sample size calculations above. We assume further that the trial will enroll for a constant rate for 12 months, have a control group median of 8 months (exponential failure rate $\lambda = \log(2)/8$), a dropout rate of 0.001 per month, and 16 months of minimum follow-up. As before, we assume a randomization ratio $r=1$, one-sided Type I error $\alpha=0.025$, 90% power which is equivalent to Type II error $\beta=0.1$.
r <- 1 # Experimental/control randomization ratio alpha <- 0.025 # 1-sided Type I error beta <- 0.1 # Type II error (1 - power) hr <- 0.7 # Hazard ratio (experimental / control) controlMedian <- 8 dropoutRate <- 0.001 # Exponential dropout rate per time unit enrollDuration <- 12 minfup <- 16 # Minimum follow-up Nlf <- nSurv( lambdaC = log(2) / controlMedian, hr = hr, eta = dropoutRate, T = enrollDuration + minfup, # Trial duration minfup = minfup, ratio = r, alpha = alpha, beta = beta ) cat(paste("Sample size: ", ceiling(Nlf$n), "Events: ", ceiling(Nlf$d), "\n"))
Recall that the @Schoenfeld1981 method recommended
r ceiling(nEvents(hr=hr, alpha=alpha, beta=beta, ratio=r))
events. The two methods tend to yield very similar event count recommendations, but not the same. Other methods will also differ slightly; see @LachinFoulkes. Sample size recommendations can vary more between methods.
We can get the same result with the nSurvival()
routine since only a single enrollment, failure and dropout rate is proposed for this example.
lambda1 <- log(2) / controlMedian nSurvival( lambda1 = lambda1, lambda2 = lambda1 * hr, Ts = enrollDuration + minfup, Tr = enrollDuration, eta = dropoutRate, ratio = r, alpha = alpha, beta = beta )
Now we produce a group sequential design with a default asymmetric design with a futility bound based on $\beta$-spending. We round interim event counts and round up the final event count to ensure the targeted power.
k <- 2 # Total number of analyses lfgs <- gsSurv( k = 2, lambdaC = log(2) / controlMedian, hr = hr, eta = dropoutRate, T = enrollDuration + minfup, # Trial duration minfup = minfup, ratio = r, alpha = alpha, beta = beta ) %>% toInteger() lfgs %>% gsBoundSummary() %>% kable(row.names = FALSE)
Although we did not use the @Schoenfeld1981 for sample size, it is still used for the approximate HR at bound calculation above:
events <- lfgs$n.I z <- lfgs$upper$bound zn2hr(z = z, n = events) # Schoenfeld approximation to HR
There are various plots available. The approximate hazard ratios required to cross bounds again use the @Schoenfeld1981 approximation. For a ggplot2 version of this plot, use the default base = FALSE
.
plot(lfgs, pl = "hr", dgt = 2, base = TRUE)
The variance calculations for the Lachin and Foulkes method are mostly determined by expected event accrual under the null and alternate hypotheses. The null hypothesis characterized above is seemingly designed so that event accrual will be similar under each hypothesis. You can see the expected events accrued at each analysis under the alternate hypothesis with:
tibble::tibble( Analysis = 1:2, `Control events` = lfgs$eDC, `Experimental events` = lfgs$eDE ) %>% kable()
It is worth noting that if events accrue at the same rate in both the null and alternate hypothesis, then the expected duration of time to achieve the targeted events would be shortened. Keep in mind that there can be many reasons events will accrue at a different rate than in the design plan.
The expected event accrual of events over time for a design can be computed as follows:
Month <- seq(0.025, enrollDuration + minfup, .025) plot( c(0, Month), c(0, sapply(Month, function(x) { nEventsIA(tIA = x, x = lfgs) })), type = "l", xlab = "Month", ylab = "Expected events", main = "Expected event accrual over time" )
On the other hand, if you want to know the expected time to accrue 25% of the final events and what the expected enrollment accrual is at that time, you compute using:
b <- tEventsIA(x = lfgs, timing = 0.25) cat(paste( " Time: ", round(b$T, 1), "\n Expected enrollment:", round(b$eNC + b$eNE, 1), "\n Expected control events:", round(b$eDC, 1), "\n Expected experimental events:", round(b$eDE, 1), "\n" ))
For expected accrual of events without a design returned by gsDesign::gsSurv()
, see the help file for gsDesign::eEvents()
.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.