Spending time examples

Keaven M. Anderson

library(gsDesign)
library(gsDesign2)
library(tibble)
library(dplyr)
library(gt)

Overview

There are multiple scenarios where event-based spending for group sequential designs has limitations in terms of ensuring adequate follow-up and in ensuring adequate spending is preserved for the final analysis. Example contexts where this often arises is in trials where

• there may be a delayed treatment effect,
• control failure rates are different than expected, and
• multiple hypotheses are being tested.

In general, for such situations we have found that ensuring both adequate follow-up duration and an adequate number of events is important to fully evaluate the potential effectiveness of a new treatment. For testing of multiple hypotheses, carefully thinking through possible spending issues can be critical. In addition, for group sequential trials, preserving adequate \(\alpha\)-spending for a final evaluation of a hypothesis is important and difficult to do using traditional event-based spending.

In this document, we outline three examples to demonstrate these issues:

• For a delayed effect scenario we demonstrate:
  • the importance of both adequate events and adequate follow-up duration to ensure power in a fixed design, and
  • the importance of guaranteeing a reasonable amount of \(\alpha\)-spending for the final analysis in a group sequential design.
• For a trial examining an outcome in a biomarker positive and overall populations, we show the importance of considering how the design reacts to incorrect design assumptions on biomarker prevalence.

For the group sequential design options, we demonstrate that the concept of spending time is an effective way to adapt. Traditionally Lan and DeMets (1983), spending has been done according to targeting a specific number of events for an outcome at the end of the trial. However, for delayed treatment effect scenarios there is substantial literature (e.g., Lin et al. (2020), Roychoudhury et al. (2021)) documenting the importance of adequate follow-up duration in addition to requiring an adequate number of events under the traditional proportional hazards assumption.

While other approaches could be taken, we have found the spending time approach generalizes well for addressing a variety of scenarios. The fact that spending does not need to correspond to information fraction was perhaps first raised by Lan and DeMets (1989) where calendar-time spending was discussed. However, we note that Proschan, Lan, and Wittes (2006) has raised other scenarios where spending alternatives are considered. Two specific spending approaches are suggested here:

• Spending according to the minimum of planned and observed event counts. This is suggested for the delayed effect examples.
• Spending with a common spending time across multiple hypotheses; e.g., in the multiple population example, spending in the overall population at the same rate as in the biomarker positive subgroup regardless of event counts over time in the overall population. This is consistent with Follmann, Proschan, and Geller (1994) as applied when multiple experimental treatments are compared to a common control. Spending time in this case corresponds to the approach of Fleming, Harrington, and O’Brien (1984) where fixed incremental spending is set for a potentially variable number of interim analyses.

This document is fairly long in that it demonstrates a number of scenarios relevant to the spending time concept. The layout is intended to make it as easy as possibly to focus on the individual examples for those not interested in the full review. Code blocks can be unfolded for those interested in implementation. Rather than bog down the conceptual discussion with implementation details, we have tried to provide sufficient comments in the code to guide implementation for those who are interested in that.

Delayed effect scenario

We consider an example in a single stratum where there is a possibility of a delayed treatment effect. The next two sections will consider both a 1) fixed design with no interim analysis, and 2) a design with interim analysis. Following are the common assumptions:

• The control group time-to-event is exponentially distributed with a median of 12 months.
• 2.5% one-sided Type I error.
• 90% power.
• A constant enrollment rate with an expected enrollment duration of 12 months.
• A targeted trial duration of 30 months.
• A delayed effect for the experimental group compared to control, with a hazard ratio of 1 for the first 4 months and a hazard ratio of 0.6 thereafter.

The restrictions on constant control failure rate, only two hazard ratio time intervals and constant enrollment are not required, but simplify the example. The approach taken uses an average-hazard ratio approach for approximating treatment effect as in Mukhopadhyay et al. (2020) and the asymptotic group sequential theory of Tsiatis (1982).

# control median
m <- 12

# enrollment rate
enroll_rate <- define_enroll_rate(
  duration = 12, # expected enrollment duration of 12 months
  rate = 1 # here the rate is a ratio, which will be updated to achieve the desired sample size
)

# failure rate
fail_rate <- define_fail_rate(
  duration = c(4, 100), # hazard ratio of 1 for the first 4 months and a hazard ratio of 0.6 thereafter
  hr = c(1, .6),
  fail_rate = log(2) / m, # exponential distribution
  dropout_rate = .001
)

Fixed design, delayed effect

The sample size and events for this design are shown below. We see that the average hazard ratio (AHR) under the above assumptions is 0.7026, part way between the early HR of 1 and the later HR of 0.6 assumed for experimental versus control therapy.

# bounds for fixed design are just a fixed bound for nominal p = 0.025, 1-sided
z_025 <- qnorm(.975)

# fixed design, single stratum
# find sample size for 30 month trial under given
# enrollment and sample size assumptions
xx <- gs_design_ahr(enroll_rate,
  fail_rate,
  analysis_time = 30,
  upar = z_025,
  lpar = z_025
)

# get the summary table of the fixed design
summary(xx,
  analysis_vars = c("time", "n", "event", "ahr", "info_frac"),
  analysis_decimals = c(0, 0, 0, 4, 4)
) %>% as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 30 N: 510 Event: 340 AHR: 0.7026 Information fraction: 1
Efficacy	1.96	0.025	0.8084	0.9	0.025
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.

Power when assumptions design are wrong

Scenario 1: less experimental benefit

If we assume instead that the effect delay is 6 months instead of 4 and the control median is 10 months instead of 12, there is a substantial impact on power. Here, we have assumed only the targeted events is required to do the final analysis resulting in an expected final analysis time of 25 months instead of the planned 30 and an average hazard ratio of 0.78 at the expected time of analysis rather than the targeted average hazard ratio of 0.70 under the original assumptions.

# update the median of control arm
am <- 10 # alternate control median (the original is 12)

# update the failure rate table
fail_rate$duration[1] <- 6 # the original is 4
fail_rate$fail_rate <- log(2) / am # the original is log(2)/12

# get the targeted number of events
target_events <- xx$analysis$event

# update the design and calculate the power under the targeted events
yy <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  # here we want to achieve the target events
  # and set analysis_time as NULL
  # so the analysis_time will be calculated according to the target events
  event = target_events,
  analysis_time = NULL,
  upar = z_025,
  lpar = z_025
)

yy %>%
  summary() %>%
  as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 25.2 N: 509.5 Event: 339.6 AHR: 0.78 Information fraction: 1
Futility	1.96	0.025	0.8071	0.364	0.975
Efficacy	1.96	0.025	0.8071	0.636	0.025
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.

Now we also require 30 months trial duration in addition to the targeted events. This improves the power from 63% above to 76% with an increase from 25 to 30 months duration and 340 to 377 expected events, an important gain. This is driven both by the average hazard ratio of 0.78 above compared to 0.76 below and by the increased expected number of events. It also ensures adequate follow-up to better describe longer-term differences in survival; this may be particularly important if early follow-up suggests a delayed effect or crossing survival curves. Thus, the adaptation of event-based design based to also require adequate follow-up can help ensure power for a large clinical trial investment where there is an clinically relevant underlying survival benefit.

yy <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  # here we want to achieve the targeted events,
  # but also keep the 30 month as the analysis_time
  event = target_events,
  analysis_time = 30,
  upar = z_025,
  lpar = z_025
)

# get the summary table of updated design
yy %>%
  summary() %>%
  as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 30 N: 509.5 Event: 376.2 AHR: 0.76 Information fraction: 1
Futility	1.96	0.025	0.816	0.2377	0.975
Efficacy	1.96	0.025	0.816	0.7623	0.025
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.

Scenario 2: low control event rates

Now we assume a longer than planned control median, 16 months to demonstrate the value of retaining the event count requirement. If we analyze after 30 months, the power of the trial is 87% with 288 events expected.

# alternate control median
am <- 16 # the original is 12

# update the failure rate
fail_rate$fail_rate <- log(2) / am
fail_rate$duration[1] <- 4

# calculate the power when trial duration is 30 month
yy <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  # here we set analysisTime as 30
  # and calculate the corresponding number of events
  event = NULL,
  analysis_time = 30,
  upper = gs_b, upar = z_025,
  lower = gs_b, lpar = z_025
)

yy %>%
  summary() %>%
  as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 30 N: 509.5 Event: 287.7 AHR: 0.69 Information fraction: 1
Futility	1.96	0.025	0.7916	0.127	0.975
Efficacy	1.96	0.025	0.7916	0.873	0.025
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.

If we also require adequate events, we restore power to 94.5, above the originally targeted level of 90%. The cost is that the expected trial duration becomes 38.5 months rather than 30; however, since the control median is now larger, the additional follow-up should be useful to characterize tail behavior. Note that for this scenario we are likely particularly interested in retaining power as the treatment effect is actually stronger than the original alternate hypothesis. Thus, for this example, the time cutoff alone would not have ensured sufficient follow-up to power the trial.

# calculate the power when trial duration is 30 month and the events is the targeted events
yy <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  # here we set trial duration as 30 month
  # and keep the events as the target events
  event = target_events,
  analysis_time = 30,
  upar = z_025,
  lpar = z_025
)

yy %>%
  summary() %>%
  as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 38.5 N: 509.5 Event: 339.6 AHR: 0.68 Information fraction: 1
Futility	1.96	0.025	0.8069	0.0536	0.975
Efficacy	1.96	0.025	0.8069	0.9464	0.025
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.

Conclusions for fixed design

In summary, we have demonstrated the value of requiring both adequate events and adequate follow-up duration over an approach where the analysis is done with only one of these requirements. Requiring both will retain both power and important treatment benefit characterization over time when there is potential for delayed onset of a positive beneficial treatment effect.

Group sequential design

Alternative spending strategies

We extend the above design to detect a delayed effect to a group sequential design with a single interim analysis after 80% of the final planned events have accrued. We will assume the final analysis will require both the targeted trial duration and events based on the fixed design based on the evaluations above. We assume the efficacy bound uses the Lan and DeMets (1983) spending function approximating an O’Brien-Fleming bound. No futility bound is planned, with the exception of a demonstration for one scenario. The interim analysis is far enough into the trial so that there is a substantial probability of stopping early under design assumptions.

Coding for the different strategies must be done carefully.

Spending approach 1: At the time of design, we specify only the spending function when specifying the use of information fraction for design.

# Spending for design with planned information fraction (IF)
upar_design_if <- list(
  # total_spend represents one-sided Type I error
  total_spend = 0.025,
  # Spending function and associated
  # parameter (NULL, in this case)
  sf = sfLDOF,
  param = NULL,
  # Do NOT specify spending time here as it will be set
  # by information fraction specified in call to gs_design_ahr()
  timing = NULL,
  # Do NOT specify maximum information here as it will be
  # set as the design maximum information
  max_info = NULL
)

Spending approach 2: If we wished to use 22 and 30 months as calendar analysis times and use calendar fraction for spending, we would need to specify spending time for the design.

# CF is for calendar fraction
upar_design_cf <- upar_design_if
# Now switch spending time to calendar fraction
upar_design_cf$timing <- c(22, 30) / 30

Spending approach 3: Next we show how to set up information-based spending for power calculation when timing of analysis is not based on information fraction; e.g., we will propose requiring not only achieving planned event counts, but also planned study duration before an analysis is performed. It is critical to set the maximum planned information to update the information fraction calculation in this case.

# We now need to change max_info from spending as specified for design
upar_actual_info_frac <- upar_design_if
# Note that we still have timing = NULL, unchanged from information-based design
upar_actual_info_frac <- NULL
# Replace NULL maximum information with planned maximum null hypothesis
# information from design
# This max will be updated for each planned design later
upar_actual_info_frac$max_info <- 100

Spending approach 4: The final case will be to replace information fraction for a design to a specific spending time which will be plugged into the spending function to compute incremental \(\alpha\)-spending for each analysis. For our case, we will use planned information fraction from the design, which is 0.8 at the interim analysis and 1 for the final analysis. This will be used regardless of what scenario we are using to compute power, but recall that information fraction is still used for computing correlations in the asymptotic distribution approximation for design tests.

# Copy original upper planned spending
upar_planned_info_frac <- upar_design_if
# Interim and final spending time will always be the same, regardless of
# expected events or calendar timing of analysis
upar_planned_info_frac$timing <- c(0.8, 1)
# We will reset planned maximum information later

Planned design

We extend the design studied above to a group sequential design with a single interim analysis after 80% of the final planned events have accrued. We will assume the final analysis will require both the targeted trial duration and events based on the fixed design evaluations made above. We assume the efficacy bound uses the Lan-DeMets spending function approximating an O’Brien-Fleming bound. No futility bound is planned. The interim analysis is far enough into the trial that there is a substantial probability of stopping early under design assumptions.

# Control median
m <- 12

# Planned information fraction at interim(s) and final
planned_info_frac <- c(.8, 1)

# No futility bound
lpar <- rep(-Inf, 2)

# enrollment rate
enroll_rate <- define_enroll_rate(
  duration = 12,
  rate = 1
)

# failure rate
fail_rate <- define_fail_rate(
  duration = c(4, 100),
  hr = c(1, .6),
  fail_rate = log(2) / m,
  dropout_rate = .001
)

# get the group sequential design model
xx <- gs_design_ahr(
  enroll_rate,
  fail_rate,
  # final analysis time set to targeted study duration;
  # analysis times before are 'small' to ensure use of information fraction for timing
  analysis_time = c(1, 30),
  # timing here matches what went into planned_info_frac above
  info_frac = planned_info_frac,
  # upper bound : spending approach 1
  upper = gs_spending_bound,
  upar = upar_design_if,
  # lower bound: no futility bound
  lower = gs_b,
  lpar = lpar
)

# get the summary table
xx %>%
  summary() %>%
  as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22.1 N: 517 Event: 275.7 AHR: 0.73 Information fraction: 0.8
Efficacy	2.25	0.0122	0.7625	0.6363	0.0122
Analysis: 2 Time: 30 N: 517 Event: 344.6 AHR: 0.7 Information fraction: 1
Efficacy	2.02	0.0214	0.8040	0.9000	0.0250
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.

Two alternate approaches

We consider two alternate approaches to demonstrate the spending time concept that may be helpful in practice. However, skipping the following two subsections can be done if these are not of interest. The first demonstrates calendar spending as in Lan and DeMets (1989). The second is a basically the method of Fleming, Harrington, and O’Brien (1984) where a fixed incremental spend is used for a potentially variable number of interim analyses, with the final bound computed based on the unspent one-sided Type I error assigned to a hypothesis.

Calendar spending

We use the same sample size as above, but change efficacy bound spending to calendar-based. The reason this spending is different than information-based spending is mainly due to the fact that the expected information is not linear in time. In this case, the calendar fraction at interim is less than the information fraction, but exactly the opposite would be true earlier in the trial. We just note that if calendar-based spending is chosen, it may be worth comparing the design bounds with bounds using the same spending function, but with information-based spending to see if there are important differences to the trial team or possibly to the scientific or regulatory community. We note also that there is risk there will not be enough events to achieve targeted power at the final analysis under a calendar-based spending strategy. We will not examine calendar-based spending further in this document.

yy <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = xx$fail_rate,
  # Planned time will drive timing since information accrues faster
  event = 1:2,
  # Interim time rounded
  analysis_time = c(22, 30),
  # upper bound: use calendar fraction
  upper = gs_spending_bound,
  upar = upar_design_cf,
  # lower bound: no futility bound
  lower = gs_b,
  lpar = lpar
)

yy %>%
  summary() %>%
  as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22 N: 517 Event: 274.2 AHR: 0.73 Information fraction: 0.79
Efficacy	2.37	0.0089	0.7486	0.5845	0.0089
Analysis: 2 Time: 30 N: 517 Event: 344.6 AHR: 0.7 Information fraction: 1
Efficacy	1.99	0.0231	0.8053	0.9059	0.0250
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.

Fixed incremental spend with a variable number of analyses

As noted, this method was proposed by Fleming, Harrington, and O’Brien (1984). The general strategy demonstrated is to do an interim analyses every 6 months until a both a final targeted follow-up time and cumulative number of events is achieved. Once efficacy analyses start, a fixed incremental spend of 0.001 is used at each interim. When the criteria for final analysis is met, the remaining \(\alpha\) is spent. Cumulative spending at months 18 and 24 will be 0.001 and 0.002, respectively, with the full cumulative \(\alpha\)-spending of 0.025 at the final analysis. This is done by setting the spending time at 18 and 24 months to 1/25, 2/25 and 1; i.e., 1/25 incremental \(\alpha\)-spending is incorporated at each interim analysis and any remaining \(\alpha\) is spent at the final analysis. This enables a strategy such as analyzing every 6 months until both a minimum targeted follow-up and minimum number of events are observed, at which time the final analysis is performed. We will skip efficacy analyses at the first two interim analyses at months 6 and 12.

For futility, we simply use a nominal 1-sided p-value of 0.05 favoring control at each interim. We note that this only raises a flag if the futility bound is crossed and a Data Monitoring Committee (DMC) can choose to continue the trial even if a futility bound is crossed. However, the bound may be more effective in providing a DMC guidance not to stop for futility prematurely. For comparison with the above designs, we will leave the enrollment rates, failure rates, dropout rates and final analysis time as before.

We see in the following table summarizing efficacy bounds and power that there is little impact on the total power by having futility analyses as specified. While the cumulative \(\alpha\)-spending is 0.001 and 0.002 at the efficacy interim analyses, we see that the nominal p-value bound at the second interim is 0.0015, more then the 0.001 incremental \(\alpha\)-spend. We also note that with these nominal p-values for testing, the approximate hazard ratio required to cross the bounds would presumably help justify consideration of completing the trial based on a definitive interim efficacy finding. Also, with the small interim spend, the final nominal p-value is not reduced much from the overall \(\alpha=0.025\) Type I error set for the group sequential design.

We also examine the futility bound. The nominal p-value of 0.05 at each analysis is the one-sided p-value in favor of control over experimental treatment. We can see that the probability of stopping early under the alternate hypothesis (\(\beta\)-spending) is not substantial even given the early delayed effect. Also, the substantial approximate observed hazard ratios to cross a futility bound seem reasonable given the timing and number of events observed; the exception to this is the small number of events at the first interim, but a larger number could be observed by this time if there were early excess risk. It may be useful to plan additional analyses if a futility bound is crossed to support stopping or not. For example, looking in subgroups or evaluating smoothed hazard rates over time for each treatment group may be useful. A clinical trial study team should have a complete discussion of futility bound considerations at the time of design.

# Cumulative spending at IA3 and IA4 will be 0.001 and 0.002, respectively.
# Power spending function sfPower with param = 1 is linear in timing
# which makes setting the above cumulative spending targets simple by
# setting timing variable the the cumulative proportion of spending at each analysis.
# There will be no efficacy testing at IA1 or IA2.
# Thus, incremental spend, which will be unused, is set very small for these analyses.
upar_fho <- list(
  total_spend = 0.025,
  sf = sfPower,
  param = 1,
  timing = c((1:2) / 250, (1:2) / 25, 1)
)

fho <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = xx$fail_rate,
  event = NULL,
  analysis_time = seq(6, 30, 6),
  upper = gs_spending_bound,
  upar = upar_fho,
  # No efficacy testing at IA1 or IA2
  # Thus, the small alpha the spending function would have
  # allocated will not be used
  test_upper = c(FALSE, FALSE, TRUE, TRUE, TRUE),
  lower = gs_b,
  lpar = c(rep(qnorm(.05), 4), -Inf)
)

fho %>%
  summary() %>%
  as_gt()

Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p1	~HR at bound2	Cumulative boundary crossing probability
				Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 6 N: 258.5 Event: 39.2 AHR: 0.96 Information fraction: 0.12
Futility	-1.64	0.9500	1.6929	0.0388	0.0500
Analysis: 2 Time: 12 N: 517 Event: 133.7 AHR: 0.84 Information fraction: 0.39
Futility	-1.64	0.9500	1.3323	0.0414	0.0865
Analysis: 3 Time: 18 N: 517 Event: 227.3 AHR: 0.76 Information fraction: 0.66
Futility	-1.64	0.9500	1.2468	0.0414	0.1106
Efficacy	3.09	0.0010	0.6607	0.1428	0.0010
Analysis: 4 Time: 24 N: 517 Event: 294.4 AHR: 0.72 Information fraction: 0.85
Futility	-1.64	0.9500	1.2138	0.0414	0.1259
Efficacy	2.97	0.0015	0.7047	0.4169	0.0020
Analysis: 5 Time: 30 N: 517 Event: 344.6 AHR: 0.7 Information fraction: 1
Efficacy	1.96	0.0249	0.8080	0.8726	3 0.0249
1 One-sided p-value for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.
2 Approximate hazard ratio to cross bound.
3 Cumulative alpha for final analysis (0.0249) is less than the full alpha (0.025) when the futility bound is non-binding. The smaller value subtracts the probability of crossing a futility bound before crossing an efficacy bound at a later analysis (0.025 - 0.0001 = 0.0249) under the null hypothesis.

Scenario with less treatment effect

As before, we compute power under the assumption of changing the median control group time-to-event to 10 months rather than the assumed 12 and the delay in effect onset is 6 months rather than 4. We otherwise do not change enrollment, dropout or hazard ratio assumptions. In both of the following examples, we require both the targeted number of events and targeted trial duration from the group sequential design before doing the interim and final analyses. The first example, which uses interim spending based on the event count observed over the originally planned final event count has the information fraction 323 / 355 = 0.91. This gives event-based spending of 0.0191, substantially above the targeted information fraction of 284 / 355 = 0.8 with targeted interim spending of 0.0122. This reduces the power overall from 76% to 73% and lowers the nominal p-value bound at the final analysis from 0.0218 to 0.0165; see the following two tables. Noting that the average hazard ratio is 0.8 at the interim and 0.76 at the final analysis emphasizes the value of preserving \(\alpha\)-spending until the final analysis. Thus, in this example it is valuable to limit spending at the interim analysis to the minimum of planned spending as opposed to using event-based spending.

# Alternate control median
am <- 10

# Update the failure rate
fail_rate$fail_rate <- log(2) / am
fail_rate$duration[1] <- 6

# Set planned maximum information from planned design
max_info0 <- max(xx$analysis$info)
upar_actual_info_frac <- upar_design_if
upar_actual_info_frac$max_info <- max_info0

# compute power if actual information fraction relative to original
# planned total is used
yy <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  # Planned time will drive timing since information accrues faster
  event = 1:2,
  analysis_time = xx$analysis$time,
  upper = gs_spending_bound,
  upar = upar_actual_info_frac,
  lower = gs_b,
  lpar = lpar
)

yy %>%
  summary() %>%
  filter(Bound == "Efficacy") %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22.1 N: 517 Event: 315 AHR: 0.8 Information fraction: 0.82
Efficacy	2.0500	0.7922	0.0201	0.4832	0.0201
Analysis: 2 Time: 30 N: 517 Event: 381.8 AHR: 0.76 Information fraction: 1
Efficacy	2.1700	0.7997	0.0150	0.7091	0.0250

Just as important, the general design principle of making interim analysis criteria more stringent that final is ensured for this alternate scenario. There are multiple trials where delayed effects have been observed where this difference in the final nominal p-value bound would have made a difference to ensure a statistically significant finding.

yz <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  event = xx$analysis$Events,
  analysis_time = xx$analysis$time,
  upper = gs_spending_bound,
  upar = upar_planned_info_frac,
  lpar = lpar
)
#> Warning: Unknown or uninitialised column: `Events`.

yz %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22.1 N: 517 Event: 315 AHR: 0.8 Information fraction: 0.82
Efficacy	2.2500	0.7745	0.0122	0.4054	0.0122
Analysis: 2 Time: 30 N: 517 Event: 381.8 AHR: 0.76 Information fraction: 1
Efficacy	2.0200	0.8126	0.0219	0.7549	0.0250

Scenario with longer control median

Now we return to the example where the control median is longer than expected to confirm that spending according to the planned level alone without considering the actual number of events will also result in a power reduction. While the power gain is not great (94.2% vs 95.0%) the interim and final p-value bounds are more aligned with the intent of emphasizing the final analysis where a smaller average hazard ratio is expected (0.680 vs 0.723 at the interim). First, we show the result using planned spending.

# Alternate control median
am <- 16

# Update the failure rate
fail_rate$fail_rate <- log(2) / am
# Return to 4 month delay with HR=1 before HR = 0.6
fail_rate$duration[1] <- 4

# Start with spending based on planned information
# which is greater than actual information
yy <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  event = c(1, max(xx$analysis$event)),
  analysis_time = xx$analysis$time,
  upper = gs_spending_bound,
  upar = upar_planned_info_frac,
  lower = gs_b,
  lpar = lpar
)

yy %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22.1 N: 517 Event: 226.3 AHR: 0.72 Information fraction: 0.65
Efficacy	2.2500	0.7385	0.0122	0.5774	0.0122
Analysis: 2 Time: 38.5 N: 517 Event: 344.6 AHR: 0.68 Information fraction: 1
Efficacy	2.0700	0.7982	0.0191	0.9464	0.0250

Since the number of events was less than expected, if we had used the actual number of events the interim bound would be more stringent than above and we obtain slightly greater power.

yz <- gs_power_ahr(
  enroll_rate = xx$enroll_rate,
  fail_rate = fail_rate,
  event = c(1, max(xx$analysis$event)),
  analysis_time = xx$analysis$time,
  upper = gs_spending_bound,
  upar = upar_actual_info_frac,
  lower = gs_b,
  lpar = lpar
)

yz %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22.1 N: 517 Event: 226.3 AHR: 0.72 Information fraction: 0.65
Efficacy	2.5100	0.7135	0.0061	0.4770	0.0061
Analysis: 2 Time: 38.5 N: 517 Event: 344.6 AHR: 0.68 Information fraction: 1
Efficacy	1.9900	0.8051	0.0230	0.9567	0.0250

Summary for spending time motivation assuming delayed benefit

In summary, using the minimum of planned and actual spending to adapt the design based on event-based spending adapts the interim bound to be more stringent than the final bound under different scenarios and ensures better power than event-based interim analysis and spending.

Testing multiple hypotheses

Assumptions

We consider a simple case where we use the method of Maurer and Bretz (2013) to test both in the overall population and in a biomarker subgroup for the same endpoint. We assume an exponential failure rate with a median of 12 for the control group regardless of population. The hazard ratio in the biomarker positive subgroup will be assumed to be 0.6, and in the negative population 0.8. We assume the biomarker positive group represents half of the population, meaning that enrollment rates will be assumed to be the same in negative and positive patients. The only difference between failure rates in the two strata is the hazard ratio. For this case, we assume proportional hazards within negative (HR = 0.8) and positive (HR = 0.6) patients.

# we assume an exponential failure rate with a median of 12
# for the control group regardless of population.
m <- 12

# the enrollment rate of both subgroup and population is the same
enroll_rate <- define_enroll_rate(
  stratum = c("Positive", "Negative"),
  duration = 12,
  rate = 20
)

# the hazard ratio in the biomarker positive subgroup will be assumed to be 0.6,
# and in the negative population 0.8.
fail_rate <- define_fail_rate(
  stratum = c("Positive", "Negative"),
  hr = c(0.6, 0.8),
  duration = 100,
  fail_rate = log(2) / m,
  dropout_rate = 0.001
)

For illustrative purposes, we are choosing a strategy based on the possible feeling of much less certainty at study start as to whether there is any underlying benefit in the biomarker negative population. We wish to ensure power for the biomarker positive group, but allow a good chance of a positive overall population finding if there is a lesser benefit in the biomarker negative population. If an alternative trial strategy is planned, an alternate approach to the following should be considered. In any case, we design first for the biomarker positive population with one-sided Type I error controlled at \(\alpha = 0.0125\):

Planned design for biomarker positive population

# Since execution will be event-based for biomarker population,
# there will be no need to change spending plan for different scenarios.

# upper bound: spending based on information fraction
upar_design_spend <- list(
  sf = gsDesign::sfLDOF, # spending function
  total_spend = 0.0125, # total alpha spend is now 0.0125
  timing = NULL, # to select maximum planned information for information fraction
  param = NULL
)

# lower bound: no futility bound
lpar <- rep(-Inf, 2) # Z = -infinity for lower bound

# we will base the combined hypothesis design to ensure power in the biomarker subgroup
positive <- gs_design_ahr(
  # enroll/failure rates
  enroll_rate = enroll_rate %>% filter(stratum == "Positive"),
  fail_rate = fail_rate %>% filter(stratum == "Positive"),
  # Following drives information fraction for interim
  info_frac = c(.8, 1),
  # Total study duration driven by final analysis_time value, i.e., 30
  # Enter small increasing values before that
  # so information fraction in planned_info_frac drives timing of interims
  analysis_time = c(1, 30),
  # upper bound
  upper = gs_spending_bound,
  upar = upar_design_spend,
  # lower lower
  lower = gs_b,
  lpar = lpar
)

positive %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22.6 N: 292.9 Event: 151.6 AHR: 0.6 Information fraction: 0.79
Efficacy	2.5600	0.6597	0.0052	0.7176	0.0052
Analysis: 2 Time: 30 N: 292.9 Event: 189.5 AHR: 0.6 Information fraction: 1
Efficacy	2.2900	0.7167	0.0109	0.9000	0.0125

Planned design for overall population

We adjust the overall study enrollment rate to match the design requirement for the biomarker positive population.

# Get enrollment rate inflation factor compared to originally input rate
inflation_factor <- positive$enroll_rate$rate[1] / enroll_rate$rate[1]

# Using this inflation factor, set planned enrollment rates
planned_enroll_rate <- enroll_rate %>% mutate(rate = rate * inflation_factor)
planned_enroll_rate %>% gt()

stratum	duration	rate
Positive	12	24.40523
Negative	12	24.40523

# Store overall enrollment rates for future use
overall_enroll_rate <- planned_enroll_rate %>%
  summarize(
    stratum = "All",
    duration = first(duration),
    rate = sum(rate)
  )

overall_enroll_rate %>% gt()

stratum	duration	rate
All	12	48.81047

Now we can examine the power for the overall population based on hazard ratio assumptions in biomarker negative and biomarker positive subgroups and the just calculated enrollment assumption. We use the analysis times from the biomarker positive population design. We see that the interim information fraction for the overall population is slightly greater than the biomarker positive population above. To compensate for this and to enable flexibility below as biomarker positive prevalence changes, we use the same spending time as the biomarker positive subgroup regardless of the true fraction of final planned events at each analysis. Thus, the interim nominal p-value bound is the same for both the biomarker positive and overall populations. While this does not make much difference here, we see that we have a very natural way to adapt the design if the observed biomarker positive prevalence is different than what was assumed for the design.

# Set total spend for overall population, O'Brien-Fleming spending function, and
# same spending time as biomarker subgroup
upar_overall_planned_info_frac <- list(
  sf = gsDesign::sfLDOF, # O'Brien-Fleming spending function
  param = NULL,
  total_spend = 0.0125, # alpha
  timing = c(.8, 1), # same spending time as biomarker subgroup
  max_info = NULL # we will use actual final information as planned initially
)

overall_planned_bounds <- gs_power_ahr(
  # enroll/failure rates
  enroll_rate = planned_enroll_rate,
  fail_rate = fail_rate,
  # analysis time: the planned analysis time for biomarker positive population
  analysis_time = positive$analysis$time,
  # events will be determined by expected events at planned analysis times
  event = NULL,
  # upper bound: planned spending times are specified the same as before
  upper = gs_spending_bound,
  upar = upar_overall_planned_info_frac,
  # lower bound: no futility
  lower = gs_b,
  lpar = lpar
)

overall_planned_bounds %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 22.6 N: 585.7 Event: 317.1 AHR: 0.7 Information fraction: 0.8
Efficacy	2.5600	0.7484	0.0052	0.7407	0.0052
Analysis: 2 Time: 30 N: 585.7 Event: 394.1 AHR: 0.7 Information fraction: 1
Efficacy	2.2900	0.7929	0.0110	0.9100	0.0125

Alternate scenarios overview

We divide our further evaluations into three subsections:

1. one with a higher prevalence of biomarker positive patients than expected;
2. one with a lower biomarker prevalence;
3. differing event rate and hazard ratio assumptions.

For each case, we will assume the total enrollment rate of 48.8 per month as planned above. We also assume that we enroll until the targeted biomarker positive subgroup enrollment of 293 from above is achieved, regardless of the overall enrollment.

The specify interim analysis timing to require both 80% of the planned final analysis events in the biomarker positive population and at least 10 months of minimum follow-up; thus, for the biomarker population we will never vary events or spending here. The same spending time will be used for the overall population, but we will compare with event-based spending. The above choices are arbitrary. While we think they are reasonable, the design planner should think carefully about other variations to suit their clinical trial team needs.

## Setting spending alternatives

# Using information (event)-based spending time relative to overall population plan
# Set total spend for overall population, O'Brien-Fleming spending function.
# For design information-spending, we set timing =  NULL and max_info to plan from above
upar_overall_planned_info_frac <- list(
  sf = gsDesign::sfLDOF, # O'Brien-Fleming spending function
  total_spend = 0.0125, # alpha
  max_info = max(overall_planned_bounds$info0), # we will use planned final information for
  # overall population from design to
  # compute information fraction relative to plan
  param = NULL,
  timing = planned_info_frac
)
#> Warning in max(overall_planned_bounds$info0): no non-missing arguments to max;
#> returning -Inf

# Using planned information fraction will demonstrate problems below.
# Set total spend for overall population, O'Brien-Fleming spending function, and
# same spending time as biomarker subgroup
upar_overall_actual_info_frac <- list(
  sf = gsDesign::sfLDOF, # O'Brien-Fleming spending function
  total_spend = 0.0125, # alpha
  max_info = max(overall_planned_bounds$info0), # we will use planned final information
  # for overall population from design
  param = NULL,
  timing = NULL
)
#> Warning in max(overall_planned_bounds$info0): no non-missing arguments to max;
#> returning -Inf

Biomarker subgroup prevalence higher than planned

Biomarker subgroup power

We suppose the biomarker prevalence is 60%, higher then the 50% prevalence the design anticipated. The enrollment rates by positive versus negative patients and expected enrollment duration are now:

# update the enrollment rate due to 60% prevalence
positive_60_enroll_rate <- rbind(
  overall_enroll_rate %>% mutate(stratum = "Positive", rate = 0.6 * rate),
  overall_enroll_rate %>% mutate(stratum = "Negative", rate = 0.4 * rate)
)

# update the enrollment duration
positive_60_enroll_rate$duration <- max(positive$analysis$n) /
  overall_enroll_rate$rate /
  0.6

# display the updated enrollment rate table
positive_60_enroll_rate %>%
  gt() %>%
  fmt_number(columns = "rate", decimals = 1)

stratum	duration	rate
Positive	10	29.3
Negative	10	19.5

Now we can compute the power for the biomarker positive group with the targeted events. Since we have a simple proportional hazards model, they only thing that is changing here from the original design is that this takes slightly less time.

positive_60_power <- gs_power_ahr(
  # enrollment/failure rate
  enroll_rate = positive_60_enroll_rate %>% filter(stratum == "Positive"),
  fail_rate = fail_rate %>% filter(stratum == "Positive"),
  # number of events
  event = positive$analysis$event,
  # analysis time will be calcuated to achieve the targeted events
  analysis_time = NULL,
  # upper bound
  upper = gs_spending_bound,
  upar = upar_design_spend,
  # lower bound
  lower = gs_b,
  lpar = lpar
)

positive_60_power %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 21.5 N: 292.9 Event: 151.6 AHR: 0.6 Information fraction: 0.79
Efficacy	2.5600	0.6557	0.0052	0.7176	0.0052
Analysis: 2 Time: 28.9 N: 292.9 Event: 189.5 AHR: 0.6 Information fraction: 1
Efficacy	2.2900	0.7143	0.0109	0.9001	0.0125

Overall population power

Now we use the same spending as above for the overall population, resulting in full \(\alpha\)-spending at the end of the trial even though the originally targeted events are not expected to be achieved. We note that the information fraction computed here is based on the originally planned events for the overall population. Given this and the larger proportion of patients that are biomarker positive, the average hazard ratio is stronger than originally planned and the power for the overall population is still over 90%.

gs_power_ahr(
  # set the enrollment/failure rate
  enroll_rate = positive_60_enroll_rate,
  fail_rate = fail_rate,
  # set evnets and analysis time
  event = NULL,
  analysis_time = positive_60_power$analysis$time,
  # set upper bound: use planned spending in spite of lower overall information
  upper = gs_spending_bound,
  upar = upar_overall_planned_info_frac,
  # set lower bound: no futility
  lower = gs_b,
  lpar = rep(-Inf, 2)
) %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 21.5 N: 488.1 Event: 262 AHR: 0.68 Information fraction: 0.8
Efficacy	2.5600	0.7266	0.0052	0.7214	0.0052
Analysis: 2 Time: 28.9 N: 488.1 Event: 325.9 AHR: 0.68 Information fraction: 1
Efficacy	2.2900	0.7745	0.0110	0.8994	0.0125

If we had used information-based (i.e., event-based) spending, we would not have reached full spending at final analysis and thus would have lower power.

gs_power_ahr(
  # set the enrollment/failure rate
  enroll_rate = positive_60_enroll_rate,
  fail_rate = fail_rate,
  # set evnets and analysis time
  event = NULL,
  analysis_time = positive_60_power$analysis$time,
  # upper bound: use actual spending which uses less than complete alpha
  upper = gs_spending_bound,
  upar = upar_overall_actual_info_frac,
  # lower bound: no futility
  lower = gs_b,
  lpar = lpar
) %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 21.5 N: 488.1 Event: 262 AHR: 0.68 Information fraction: 0.8
NA	NA	NA	NA	NA	NA
Analysis: 2 Time: 28.9 N: 488.1 Event: 325.9 AHR: 0.68 Information fraction: 1
NA	NA	NA	NA	NA	NA

Biomarker subgroup prevalence lower than planned

We suppose the biomarker prevalence is 40%, lower than the 50% prevalence the design anticipated. The enrollment rates by positive versus negative patients and expected enrollment duration will now be:

# set the enrollment rate under 40% prevalence
positive_40_enroll_rate <- rbind(
  overall_enroll_rate %>% mutate(stratum = "Positive", rate = 0.4 * rate),
  overall_enroll_rate %>% mutate(stratum = "Negative", rate = 0.6 * rate)
)

# update the duration of enrollment table
positive_40_enroll_rate$duration <- max(positive$analysis$n) /
  positive_40_enroll_rate$rate[1]

# display the enrollment table
positive_40_enroll_rate %>%
  gt() %>%
  fmt_number(columns = "rate", decimals = 1)

stratum	duration	rate
Positive	15	19.5
Negative	15	29.3

Biomarker positive subgroup power

Now we can compute the power for the biomarker positive group with the targeted events.

upar_actual_info_frac$total_spend <- 0.0125
upar_actual_info_frac$max_info <- max(positive$analysis$info)

positive_40_power <- gs_power_ahr(
  # set enrollment/failure rate
  enroll_rate = positive_40_enroll_rate %>% filter(stratum == "Positive"),
  fail_rate = fail_rate %>% filter(stratum == "Positive"),
  # set events/analysis time
  event = positive$analysis$event,
  analysis_time = NULL,
  # set upper bound
  upper = gs_spending_bound,
  upar = upar_actual_info_frac,
  # set lower bound
  lower = gs_b,
  lpar = rep(-Inf, 2)
)

positive_40_power %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 24.3 N: 292.9 Event: 151.6 AHR: 0.6 Information fraction: 0.79
Efficacy	2.5300	0.6591	0.0057	0.7276	0.0057
Analysis: 2 Time: 31.7 N: 292.9 Event: 189.5 AHR: 0.6 Information fraction: 1
Efficacy	2.3000	0.7133	0.0107	0.8985	0.0125

Overall population power

We see that by adapting the overall sample size and spending according to the biomarker subgroup, we retain 90% power. In spite of the lower overall effect size, the larger adapted sample size ensures power retention.

gs_power_ahr(
  enroll_rate = positive_40_enroll_rate,
  fail_rate = fail_rate,
  event = 1:2,
  analysis_time = positive_40_power$analysis$time,
  upper = gs_spending_bound,
  upar = upar_overall_planned_info_frac,
  lower = gs_b,
  lpar = rep(-Inf, 2)
) %>%
  summary() %>%
  gt() %>%
  fmt_number(columns = 3:6, decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 24.3 N: 732.2 Event: 399.8 AHR: 0.72 Information fraction: 0.8
Efficacy	2.5600	0.7727	0.0052	0.7751	0.0052
Analysis: 2 Time: 31.7 N: 732.2 Event: 496.3 AHR: 0.72 Information fraction: 1
Efficacy	2.2900	0.8133	0.0110	0.9279	0.0125

Summary of findings

We suggested two overall findings when planning and executing a trial with a potentially delayed treatment effect:

• Require both a targeted event count minimum follow-up before completing analysis of a trial helps ensure both powering the trial appropriately and having a better description of the tail behavior that may be essential if long-term results are key to establishing a potentially positive risk-benefit.
• Do not over-spend Type I error at interim analyses by using event-based spending. This helps to ensure the least stringent bounds are at the final analysis when the most complete risk-benefit assessment can be made. We gave two options to this:
  • Use a fixed, small incremental \(\alpha\)-spend at each interim such as proposed by Fleming, Harrington, and O’Brien (1984) with a variable number of interim analyses to ensure adequate follow-up.
  • Use the minimum of planned and actual spending at interim analyses.

When implementing the Fleming, Harrington, and O’Brien (1984) approach, we also suggested a simple approach to futility that may be quite useful practically in a scenario with a potentially delayed onset of treatment effect. This basically looks for evidence of a favorable control group effect relative to experimental by setting a nominal p-value cutoff at a 1-sided 0.05 level for early interim futility analyses. Where crossing survival curves or inferior survival curves may exist, this may be a useful way to ensure continuing a trial is ethical; this approach is perhaps most useful when the experimental treatment is replacing components of the control treatment or in a case where add-on treatment may be toxic or potentially have other detrimental effects.

In addition to the delayed effect example, we considered an example testing in both a biomarker positive subgroup and the overall population. Using a common spending time for all hypotheses with a common interim analysis strategy as advocated by Follmann, Proschan, and Geller (1994) can be helpful to implement spending so that all hypotheses have adequate \(\alpha\) to spend at the final analysis and also to ensure full utilization of \(\alpha\)-spending. We suggested again using the minimum of planned and actual spending at interim analysis. Spending can be based on a key hypothesis (e.g., the biomarker positive population) or the minimum spending time among all hypotheses being tested. Taking advantage of know correlations to ensure full \(\alpha\) utilization in multiple hypothesis testing is also more simply implemented with this strategy Anderson et al. (2022).

In summary, we have illustrated both the motivation and the illustration of the spending time approach through examples we have commonly encountered. Approaches suggested included an implementation of Fleming, Harrington, and O’Brien (1984) with a fixed incremental \(\alpha\)-spend at each interim analysis as well as the use of the minimum of planned and actual spending at interim analyses.

References

Anderson, Keaven M, Zifang Guo, Jing Zhao, and Linda Z Sun. 2022. “A Unified Framework for Weighted Parametric Group Sequential Design.” Biometrical Journal 64 (7): 1219–39.

Fleming, Thomas R, David P Harrington, and Peter C O’Brien. 1984. “Designs for Group Sequential Tests.” Controlled Clinical Trials 5 (4): 348–61.

Follmann, Dean A, Michael A Proschan, and Nancy L Geller. 1994. “Monitoring Pairwise Comparisons in Multi-Armed Clinical Trials.” Biometrics 50 (2): 325–36.

Lan, K. K. Gordon, and David L DeMets. 1983. “Discrete Sequential Boundaries for Clinical Trials.” Biometrika 70 (3): 659–63.

———. 1989. “Group Sequential Procedures: Calendar Versus Information Time.” Statistics in Medicine 8 (10): 1191–98.

Lin, Ray S, Ji Lin, Satrajit Roychoudhury, Keaven M Anderson, Tianle Hu, Bo Huang, Larry F Leon, et al. 2020. “Alternative Analysis Methods for Time to Event Endpoints Under Nonproportional Hazards: A Comparative Analysis.” Statistics in Biopharmaceutical Research 12 (2): 187–98.

Maurer, Willi, and Frank Bretz. 2013. “Multiple Testing in Group Sequential Trials Using Graphical Approaches.” Statistics in Biopharmaceutical Research 5 (4): 311–20.

Mukhopadhyay, Pralay, Wenmei Huang, Paul Metcalfe, Fredrik Öhrn, Mary Jenner, and Andrew Stone. 2020. “Statistical and Practical Considerations in Designing of Immuno-Oncology Trials.” Journal of Biopharmaceutical Statistics 30 (6): 1130–46.

Proschan, Michael A., K. K. Gordon Lan, and Janet Turk Wittes. 2006. Statistical Monitoring of Clinical Trials: A Unified Approach. New York, NY: Springer.

Roychoudhury, Satrajit, Keaven M Anderson, Jiabu Ye, and Pralay Mukhopadhyay. 2021. “Robust Design and Analysis of Clinical Trials with Nonproportional Hazards: A Straw Man Guidance from a Cross-Pharma Working Group.” Statistics in Biopharmaceutical Research, 1–15.

Tsiatis, Anastasios A. 1982. “Repeated Significance Testing for a General Class of Statistics Used in Censored Survival Analysis.” Journal of the American Statistical Association 77 (380): 855–61.