In keaven/gsDesign2: Group Sequential Design with Non-Constant Effect

library(gsDesign2)
library(gt)
library(dplyr)
library(tibble)
library(ggplot2)

Overview

We set up futility bounds under a non-proportional hazards assumption. We consider methods presented by @kornfreidlin2018 for setting such bounds and then consider an alternate futility bound based on $\beta-$spending under a delayed or crossing treatment effect to simplify implementation. Finally, we show how to update this $\beta-$spending bound based on blinded interim data. We will consider an example to reproduce a line of @kornfreidlin2018 Table 1 with the alternative futility bounds considered.

Initial design set-up for fixed analysis

@kornfreidlin2018 considered delayed effect scenarios and proposed a futility bound that is a modification of an earlier method proposed by @wieand. We begin with the enrollment and failure rate assumptions which @kornfreidlin2018 based on an example by @ttchen2013.

# Enrollment assumed to be 680 patients over 12 months with no ramp-up
enrollRates <- tibble(Stratum = "All", duration = 12, rate = 680 / 12)
# Failure rates
## Control exponential with median of 12 mos
## Delayed effect with HR = 1 for 3 months and HR = .693 thereafter
## Censoring rate is 0
failRates <- tibble(Stratum = "All", duration = c(3, 100), 
                   failRate = -log(.5) / 12, hr = c(1, .693), dropoutRate = 0)
## Study duration was 34.8 in Korn & Freidlin Table 1
## We change to 34.86 here to obtain 512 expected events more precisely
studyDuration <- 34.86

We now derive a fixed sample size based on these assumptions. Ideally, we would allow a targeted event count and variable follow-up in fixed_design() so that the study duration will be computed automatically.

fixedevents <- fixed_design(x = "AHR", alpha = 0.025, power = NULL, 
                      enrollRates = enrollRates,
                      failRates = failRates,
                      studyDuration = studyDuration)
fixedevents %>% summary() %>% 
  select(-Bound) %>%
  as_gt(footnote="Power based on 512 events") %>%
  fmt_number(columns = 3:4, decimals = 2) %>% 
  fmt_number(columns = 5:6, decimals = 3)

Modified Wieand futility bound

The @wieand rule recommends stopping after 50% of planned events accrue if the observed HR > 1. kornfreidlin2018 modified this by adding a second interim analysis after 75% of planned events and stop if the observed HR > 1 This is implemented here by requiring a trend in favor of control with a direction $Z$-bound at 0 resulting in the Nominal p bound being 0.5 for interim analyses in the table below. A fixed bound is specified with the 'gs_b()function forupperandlowerand its correspoinding parametersuparfor the upper (efficacy) bound andlparfor the lower (futility) bound. The final efficacy bound is for a 1-sided nominal p-value of 0.025; the futility bound lowers this to 0.0247 as noted in the lower-right-hand corner of the table below. In the last row under *Alternate hypothesis* below we see the power is 88.44%. @kornfreidlin2018 computed 88.4% power for this design with 100,000 simulations which estimate the standard error for the power calculation to ber paste(100 * round(sqrt(.884 * (1 - .884)/100000),4),'%',sep='')`.

wieand <- gs_power_ahr(enrollRates = enrollRates, failRates = failRates,
                       upper = gs_b, upar = c(rep(Inf, 2), qnorm(.975)),
                       lower = gs_b, lpar = c(0, 0, -Inf),
                       events = 512 * c(.5, .75, 1))
wieand %>% summary() %>% 
  as_gt(title="Group sequential design with futility only at interim analyses",
        subtitle="Wieand futility rule stops if HR > 1")

Beta-spending futility bound with AHR

Need to summarize here.

betaspending <- gs_power_ahr(enrollRates = enrollRates, failRates = failRates,
                       upper = gs_b, upar = c(rep(Inf, 2), qnorm(.975)),
                       lower = gs_spending_bound, 
                       lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025,
                                   param = NULL, timing = NULL),
                       events = 512 * c(.5, .75, 1),
                       test_lower = c(TRUE, TRUE, FALSE))
betaspending %>% 
  summary() %>% as_gt(title="Group sequential design with futility only",
                      subtitle="Beta-spending futility bound")

Classical beta-spending futility bound

A classical $\beta-$spending bound would assume a constant treatment effect over time using the proportional hazards assumption. We use the average hazard ratio at the fixed design analysis for this purpose.

Korn and Freidlin futility bound

The @kornfreidlin2018 futility bound is set when at least 50% of the expected events have occurred and at least two thirds of the observed events have occurred later than 3 months from randomization. The expected timing for this is demonstrated below.

Accumulation of events by time interval

We consider the accumulation of events over time that occur during the no-effect interval for the first 3 months after randomization and events after this time interval. This is done for the overall trial without dividing out by treatment group using the gsDesign2::AHR() function. We consider monthly accumulation of events through the 34.86 months planned trial duration. We note in the summary of early expected events below that all events during the first 3 months on-study are expected prior to the first interim analysis.

event_accumulation <- 
AHR(enrollRates = enrollRates,
    failRates = failRates,
    totalDuration = c(1:34, 34.86),
    ratio = 1,
    simple = FALSE)
head(event_accumulation, n = 7) %>% gt()

We can look at the proportion of events after the first 3 months as follows:

event_accumulation %>% 
  group_by(Time) %>%
  summarize(`Total events` = sum(Events), "Proportion early" = first(Events) /  `Total events`) %>%
  ggplot(aes(x=Time, y=`Proportion early`)) + geom_line()

For the @kornfreidlin2018 bound the targeted timing is when both 50% of events have occurred and at least 2/3 are more than 3 months after enrollment with 3 months being the delayed effect period. We see above that about 1/3 of events are still within 3 months of enrollment at month 20.