Reproducing PROC SEQDESIGN survival designs in gsDesign

Starting point: SAS PROC SEQDESIGN survival example

Consider the example described in the SAS Documentation: Computing Sample Size for Survival Data with Uniform Accrual. The first PROC SEQDESIGN call in that example does not specify ACCTIME; SAS therefore derives a range of possible accrual times. The comparison below uses the subsequent SAS call with ACCTIME=18, which fixes the maximum sample size at 15 * 18 = 270 subjects and asks SAS to solve the follow-up time. Note that the SAS survival sample size model is specified as uses the Schoenfeld formula for the targeted event counts. The computed sample size and event counts at each analysis are continuous values, not rounded to integers.

proc seqdesign; /* Group sequential design procedure */
    ErrorSpend: design /* Label for this design block */
                nstages=4 /* Total analyses including final */
                method=errfuncobf /* Lan-DeMets O'Brien-Fleming spending */
                alt=twosided /* Two-sided alternative hypothesis */
                stop=reject /* Early stop only for efficacy */
                alpha=0.05 /* Total two-sided Type I error */
                beta=0.10 /* Type II error (90% power) */
                ;
    samplesize model(ceiladjdesign=include)= /* Survival model */
        twosamplesurvival( /* Two-sample survival endpoint */
        nullhazard = 0.03466 /* Control hazard under H0 */
        hazard     = 0.01733 /* Experimental hazard under H1 */
        accrual    = uniform /* Uniform enrollment */
        accrate    = 15 /* Subjects enrolled per time unit */
        acctime    = 18 /* Accrual duration in time units */
    );
run;

SAS assumptions summary:

• 4-stage group sequential design
• Lan-DeMets O’Brien-Fleming error spending function
• Two-sided symmetric test, total alpha = 0.05
• 90% power (beta = 0.10)
• Equally spaced information fractions: 0.25, 0.50, 0.75, 1.00
• Schoenfeld formula for required number of events
• Fixed accrual rate and duration, so the maximum sample size is \(N = \text{ACCRATE} \times \text{ACCTIME} = 15 \times 18 = 270\)
• Study duration \(T\) solved to achieve the required events
• Hazard ratio \(HR = 0.01733 / 0.03466 = 0.5\)

SAS does produce a sample size in this example. In the “Sample Size Summary” table for the ACCTIME=18 case, it reports Max Sample Size = 270, Expected Sample Size (Null Ref) = 269.9206, Expected Sample Size (Alt Ref) = 263.1141, Follow-up Time = 7.133226, and Total Time = 25.13323. The maximum sample size is not calculated from the event formula; it is implied directly by the fixed accrual rate and accrual duration: 15 subjects/time unit * 18 time units = 270 subjects.

SAS reports two sets of analysis quantities. We focus first on the fractional-time design, where the analysis times are not rounded, and then return to the CEILING=TIME adjusted design later.

sas_fractional <- data.frame(
  Analysis = 1:4,
  Events = c(22.26962, 44.53924, 66.80886, 89.07847),
  Calendar_time = c(11.2631, 16.2875, 20.4926, 25.13323),
  N = c(168.95, 244.31, 270.00, 270.00),
  Upper_Z = c(4.33263, 2.96333, 2.35902, 2.01409)
)

sas_ceiling_time <- data.frame(
  Analysis = 1:4,
  Events = c(25.11225, 48.22068, 69.38712, 92.92776),
  Calendar_time = c(12, 17, 21, 26),
  N = c(180, 255, 270, 270),
  Upper_Z = c(4.15591, 2.90189, 2.36973, 2.01362)
)

To reproduce the SAS fractional-time output exactly while specifying analysis timing in calendar units, we use gsDesign::gsSurvCalendar(calendarTime = sas_fractional$Calendar_time). We keep one-sided alpha = 0.025, use method = "Schoenfeld", and keep accrual fixed at 15 subjects per time unit for 18 time units by setting R = 18 with minfup = max(calendarTime) - R.

des <- gsSurvCalendar(
  test.type = 2,
  alpha     = 0.025,
  beta      = 0.10,
  sfu       = sfLDOF, # Lan-DeMets O'Brien--Fleming
  calendarTime = sas_fractional$Calendar_time,
  spending  = "information",
  lambdaC   = 0.03466,
  hr        = 0.5,
  eta       = 0, # No dropout
  gamma     = 15, # Uniform accrual rate of 15/month
  R         = 18, # Accrual duration (months)
  minfup    = max(sas_fractional$Calendar_time) - 18,
  ratio     = 1, # 1:1 randomization
  method    = "Schoenfeld"
)

# des |> gsBoundSummary(tdigits = 2, ddigits = 2) |> gt::gt()

Side-by-side comparison table

# Get results for comparison from gsDesign object
events_a2 <- round(des$n.I, 2) # Event counts
z_a2 <- round(des$upper$bound, 4) # Z-boundaries
N2a <- sum(des$eNC[4, ]) + sum(des$eNE[4, ]) # Total sample size

knitr::kable(
  data.frame(
    Analysis = 1:4,
    Events_SAS = round(sas_fractional$Events, 2),
    Events_gsDesign = events_a2,
    Z_SAS = round(sas_fractional$Upper_Z, 4),
    Z_gsDesign = z_a2
  ),
  caption = "Side-by-side comparison of events and Z-boundaries at each look."
)

The rest of this vignette identifies the key assumptions in each system and explains why alternative parameter translations can produce different results. Those who would like to know more details about the design of the time-to-event functionality in gsDesign should consult vignette("SurvivalOverview").

1. Event formula

• SAS: Schoenfeld (1981). Uses only the null-hypothesis variance.
• gsDesign: Lachin and Foulkes (1986) by default. Uses both null and alternative hypothesis variances. L-F is slightly more conservative, giving ~1-2% more events than Schoenfeld for the same parameters. Use method = "Schoenfeld" to match the SAS event formula.

2. Alpha handling in gsDesign() and gsSurv()

gsDesign() stores and spends the one-sided Type I error. Thus, for a symmetric two-sided design (test.type = 2), alpha = 0.025 means 0.025 in each tail, or 0.05 total two-sided Type I error.

This convention is deliberate: most group sequential computations in gsDesign() are organized around the upper-bound crossing probability. A symmetric two-sided design is represented by mirroring that upper-bound spending in the lower tail. Thus the alpha argument is still the per-tail crossing probability, even though the design has both an upper and lower efficacy boundary.

gsSurv() follows the same convention internally. If a user prefers to enter the total two-sided alpha, alpha = 0.05, sided = 2 is equivalent to alpha = 0.025, sided = 1 for this example because gsSurv() passes alpha / sided to gsDesign().

Here we use test.type = 2 and alpha = 0.025 to make the symmetric two-sided gsDesign() object match the SAS two-sided total alpha of 0.05.

3. Accrual duration and follow-up time

With ACCTIME=18, SAS fixes the accrual duration and total maximum sample size and solves for the additional follow-up time. To match this in gsSurv(), set both T = NULL and minfup = NULL. This tells gsSurv() to keep the input accrual rate and accrual duration fixed, then solve the follow-up duration needed for the final group sequential event requirement. If analysis times are specified directly in calendar units, the corresponding gsSurvCalendar() setup is calendarTime = ... with minfup = max(calendarTime) - accrual_duration.

This is a common source of apparent disagreement. For a fixed-duration survival design, T is the total study duration and minfup is the minimum follow-up after enrollment closes. With scalar accrual, specifying both T and minfup makes the implied accrual duration T - minfup; specifying only one of them can therefore change which quantity gsSurv() solves for. The SAS comparison fixes accrual at 18 time units and lets the total time be derived.

4. Fractional time vs. ceiling time

The original SAS fractional-time design has equal information fractions and final events 89.07847. The SAS ceiling-time adjusted design rounds analysis times to 12, 17, 21, and 26. These rounded calendar times imply unequal information fractions and final events 92.92776. Both are valid SAS outputs; they should not be mixed when comparing against gsDesign.

The distinction matters because SAS’s fractional-time table answers “what analysis times give the target information fractions?”, while the ceiling-time table answers “what happens after those analysis times are rounded up to whole time units?” Once calendar times are rounded up, subjects are followed longer, more events are expected, and the O’Brien-Fleming spending times are no longer exactly 0.25, 0.50, 0.75, and 1.00.

The translation used below is summarized as follows:

gsSurv() with aligned parameters

Let’s start our work with gsDesign by defining parameters to match the SAS fractional-time design:

k <- 4
alpha_sas <- 0.05 # Two-sided total alpha (SAS convention)
alpha_gsdesign <- alpha_sas / 2 # gsDesign uses one-sided alpha
beta <- 0.10 # 1 - power = 0.10 -> 90% power
lambdaC <- 0.03466 # Control hazard rate
lambdaE <- 0.01733 # Experimental hazard rate
HR <- lambdaE / lambdaC # = 0.5
timing <- c(0.25, 0.50, 0.75, 1.00) # Equally spaced information fractions
accrate <- 15 # Uniform accrual rate (subjects per time unit)
accrual_duration <- 18 # Accrual duration (time units)
sas_total_time <- 25.13323
sas_followup_time <- sas_total_time - accrual_duration
N <- accrate * accrual_duration

Instead of starting with a call to gsDesign::gsDesign(), we begin with gsDesign::gsSurv(). The gsSurv() function combines nSurv() with gsDesign() (group sequential boundaries) in one call. It is the standard gsDesign function for designing time-to-event trials.

The call below uses the same two-sided symmetric structure as SAS:

• test.type = 2 for symmetric two-sided boundaries;
• alpha = 0.025, the one-sided alpha corresponding to SAS’s total two-sided alpha of 0.05;
• method = "Schoenfeld" for the SAS event formula;
• T = NULL and minfup = NULL so the accrual rate and accrual duration remain fixed while gsSurv() solves the follow-up duration.

des_2 <- gsSurv(
  k = 4,
  test.type = 2, # Symmetric two-sided design
  alpha = alpha_gsdesign, # One-sided alpha; SAS total alpha is 2 * this
  beta = 0.10,
  sfu = sfLDOF,
  timing = c(.25, .50, .75, 1),
  lambdaC = 0.03466,
  hr = 0.5,
  eta = 0, # Assume no dropout
  gamma = accrate,
  R = accrual_duration,
  T = NULL,
  minfup = NULL,
  ratio = 1,
  method = "Schoenfeld"
)
des_2
#> Group sequential design (method=Schoenfeld; k=4 analyses; Two-sided symmetric)
#> HR=0.500 vs HR0=1.000 | alpha=0.025 (sided=2) | power=90.0%
#> N=270.0 subjects | D=89.1 events | T=25.1 study duration | accrual=18.0 Accrual duration | minfup=7.1 minimum follow-up | ratio=1 randomization ratio (experimental/control)
#> 
#> Spending functions:
#>   Bounds derived using a  Lan-DeMets O'Brien-Fleming approximation spending function (no parameters).
#> 
#> Analysis summary:
#> Method: Schoenfeld 
#>    Analysis              Value Efficacy Futility
#>   IA 1: 25%                  Z   4.3326  -4.3326
#>      N: 170        p (1-sided)   0.0000   0.0000
#>  Events: 23       ~HR at bound   0.1594   6.2728
#>   Month: 11   P(Cross) if HR=1   0.0000   0.0000
#>             P(Cross) if HR=0.5   0.0035   0.0000
#>   IA 2: 50%                  Z   2.9631  -2.9631
#>      N: 246        p (1-sided)   0.0015   0.0015
#>  Events: 45       ~HR at bound   0.4115   2.4302
#>   Month: 16   P(Cross) if HR=1   0.0015   0.0015
#>             P(Cross) if HR=0.5   0.2579   0.0000
#>   IA 3: 75%                  Z   2.3590  -2.3590
#>      N: 272        p (1-sided)   0.0092   0.0092
#>  Events: 67       ~HR at bound   0.5615   1.7811
#>   Month: 20   P(Cross) if HR=1   0.0096   0.0096
#>             P(Cross) if HR=0.5   0.6853   0.0000
#>       Final                  Z   2.0141  -2.0141
#>      N: 272        p (1-sided)   0.0220   0.0220
#>  Events: 90       ~HR at bound   0.6526   1.5323
#>   Month: 25   P(Cross) if HR=1   0.0250   0.0250
#>             P(Cross) if HR=0.5   0.9000   0.0000
#> 
#> Key inputs (names preserved):
#>                                desc    item value input
#>                     Accrual rate(s)   gamma    15    15
#>            Accrual rate duration(s)       R    18    18
#>              Control hazard rate(s) lambdaC 0.035 0.035
#>             Control dropout rate(s)     eta     0     0
#>        Experimental dropout rate(s)    etaE     0  etaE
#>  Event and dropout rate duration(s)       S  NULL     S

Observations:

Using gsSurv() with parameters that align with SAS settings, the fractional-time output agrees with SAS. The final follow-up duration is 7.13321, giving total study duration 25.13321.

gs_fractional <- data.frame(
  Analysis = 1:k,
  Events_SAS = sas_fractional$Events,
  Events_gsDesign = des_2$n.I,
  Time_SAS = sas_fractional$Calendar_time,
  Time_gsDesign = des_2$T,
  N_SAS = sas_fractional$N,
  N_gsDesign = rowSums(des_2$eNC) + rowSums(des_2$eNE),
  Z_SAS = sas_fractional$Upper_Z,
  Z_gsDesign = des_2$upper$bound
)

knitr::kable(
  round(gs_fractional, 5),
  caption = "Fractional-time SAS output compared with gsSurv()."
)

The final event count is 89.07847 in both systems. The print() and gsBoundSummary() methods display rounded integer sample sizes and events, but the object retains the fractional values shown above.

If the SAS follow-up time is already known, the equivalent fixed-follow-up translation is T = NULL with minfup set to the SAS follow-up time. In that case, gsSurv() holds the follow-up duration fixed and solves the accrual duration needed for the final group sequential event target:

des_fixed_followup <- gsSurv(
  k = 4,
  test.type = 2,
  alpha = alpha_gsdesign,
  beta = 0.10,
  sfu = sfLDOF,
  timing = c(.25, .50, .75, 1),
  lambdaC = 0.03466,
  hr = 0.5,
  eta = 0,
  gamma = accrate,
  R = accrual_duration,
  T = NULL,
  minfup = sas_followup_time,
  ratio = 1,
  method = "Schoenfeld"
)

fixed_followup_check <- data.frame(
  Quantity = c(
    "Final study duration",
    "Accrual duration",
    "Final events",
    "Final N"
  ),
  Solved_followup = c(
    des_2$T[k],
    sum(des_2$R),
    des_2$n.I[k],
    sum(des_2$eNC[k, ] + des_2$eNE[k, ])
  ),
  Specified_followup = c(
    des_fixed_followup$T[k],
    sum(des_fixed_followup$R),
    des_fixed_followup$n.I[k],
    sum(des_fixed_followup$eNC[k, ] + des_fixed_followup$eNE[k, ])
  )
)
fixed_followup_check[-1] <- lapply(fixed_followup_check[-1], round, digits = 5)

knitr::kable(
  fixed_followup_check,
  caption = "Both fixed-accrual translations produce the same final design."
)

Matching the SAS ceiling-time adjusted design

The SAS example also reports a CEILING=TIME adjusted design. This is not the same design as the equal-information fractional-time output above. The analysis times are rounded up to 12, 17, 21, and 26, which increases the expected events and changes the information fractions. For calendar-scheduled analyses in general, this would naturally be specified with gsSurvCalendar(calendarTime = c(12, 17, 21, 26), spending = "information"). To match the SAS CEILING=TIME table exactly, we retain fixed accrual and use the implied event counts at those rounded calendar times.

We can reproduce the ceiling-time event counts directly from the fixed accrual rate, fixed accrual duration, and exponential event rates. With equal randomization, the control and experimental accrual rates are each 7.5 per time unit.

event_count_at_time <- function(time) {
  control_events <- eEvents(
    lambda = lambdaC,
    gamma = accrate / (1 + 1),
    R = accrual_duration,
    T = time
  )$d
  experimental_events <- eEvents(
    lambda = lambdaE,
    gamma = accrate / (1 + 1),
    R = accrual_duration,
    T = time
  )$d
  sum(control_events) + sum(experimental_events)
}

ceiling_times <- ceiling(sas_fractional$Calendar_time)
ceiling_events <- vapply(ceiling_times, event_count_at_time, numeric(1))

des_ceiling <- gsDesign(
  k = k,
  test.type = 2,
  alpha = alpha_gsdesign,
  beta = beta,
  sfu = sfLDOF,
  n.I = ceiling_events,
  timing = ceiling_events / max(ceiling_events)
)

gs_ceiling <- data.frame(
  Analysis = 1:k,
  Events_SAS = sas_ceiling_time$Events,
  Events_gsDesign = ceiling_events,
  Time_SAS = sas_ceiling_time$Calendar_time,
  Time_gsDesign = ceiling_times,
  N_SAS = sas_ceiling_time$N,
  N_gsDesign = pmin(accrate * ceiling_times, N),
  Z_SAS = sas_ceiling_time$Upper_Z,
  Z_gsDesign = des_ceiling$upper$bound
)

knitr::kable(
  round(gs_ceiling, 5),
  caption = "Ceiling-time SAS output compared with gsDesign()."
)

The remaining small differences in the printed Z-values are numerical rounding differences. The important conceptual distinction is that the SAS fractional-time and ceiling-time tables are different designs: one uses equal information fractions, and the other uses rounded calendar analysis times.