Summarize group sequential designs in gt tables

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

Overview

This vignette introduces publication quality table production for group sequential designs in the gsDesign2 package. It also demonstrates designs for an example scenario using multiple design approaches. We divide the document into 3 parts:

Design specification and derivation
Printing design summary tables
Details of output from design functions
Details on table output options

The reader can decide which of these sections is of interest to them.

The function used to generate bounds tables is gsDesign2::summary(). Users can use gsDesign2::as_gt() to format the above table using the gt package.

In this vignette, we introduce a general approach to bound summaries by examples using different design approaches for a time-to-event outcome:

the average hazard ratio (AHR) method extended from Mukhopadhyay et al. (2020) using gsDesign2::gs_design_ahr();
the weighted logrank (WLR) method of Yung and Liu (2020) using gsDesign2::gs_design_wlr();

Design specification and derivation

Design parameters

The design parameters we use across the different designs derived are:

# enrollment/failure rates
enroll_rate <- define_enroll_rate(
  duration = 12,
  rate = 30
)
fail_rate <- define_fail_rate(
  duration = c(4, 100),
  fail_rate = log(2) / 12,
  hr = c(1, .6),
  dropout_rate = .001
)

# Information fraction
info_frac <- (1:3) / 3
# Analysis times in months; first 2 will be ignored as info_frac will not be achieved
analysis_time <- c(.01, .02, 36)

# Experimental / Control randomization ratio
ratio <- 1

# 1-sided Type I error
alpha <- 0.025
# Type II error (1 - power)
beta <- 0.1

# Upper bound
upper <- gsDesign2::gs_spending_bound # alpha-spending bound
upar <- list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)

# Lower bound
lower <- gsDesign2::gs_spending_bound # beta-spending bound
lpar <- list(sf = gsDesign::sfHSD, total_spend = 0.1, param = 0, timing = NULL)

# Fleming-Harrington (FH) weight functions for weighted logrank (WLR)
wgt00 <- function(x, arm0, arm1) { # Equal weighting for logrank
  gsDesign2::wlr_weight_fh(x, arm0, arm1, rho = 0, gamma = 0)
}
wgt05 <- function(x, arm0, arm1) { # Early downweighting with FH(0,.5)
  gsDesign2::wlr_weight_fh(x, arm0, arm1, rho = 0, gamma = .5)
}

# Both of above tests for MaxCombo: logrank and FH(0,.5)
fh_test <- rbind(
  # Include logrank for all 3 analyses
  data.frame(rho = 0, gamma = 0, tau = -1, test = 1, Analysis = 1:3, analysis_time = c(12, 24, 36)),
  # Only include FH(0,.5) for analyses 2 and 3
  data.frame(rho = c(0, 0.5), gamma = 0.5, tau = -1, test = 2:3, Analysis = 3, analysis_time = 36)
)

Deriving designs

AHR design derivation

Using the design parameters above, the AHR design is derived as follows:

By using the design parameters above, one can generate an AHR model by gs_design_ahr as

x_design_ahr <- gs_design_ahr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  info_frac = info_frac,
  analysis_time = analysis_time,
  ratio = ratio,
  alpha = alpha,
  beta = beta,
  upper = upper,
  upar = upar,
  lower = lower,
  lpar = lpar
)

x_power_ahr <- gs_power_ahr(
  enroll_rate = x_design_ahr$enroll_rate,
  fail_rate = x_design_ahr$fail_rate,
  event = c(100, 200, 300),
  analysis_time = NULL,
  upper = upper,
  upar = upar,
  lower = lower,
  lpar = lpar
)

WLR design derivation

x_design_wlr <- gs_design_wlr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  weight = wgt05,
  info_frac = NULL,
  analysis_time = sort(unique(x_design_ahr$analysis$time)),
  ratio = ratio,
  alpha = alpha,
  beta = beta,
  upper = upper,
  upar = upar,
  lower = lower,
  lpar = lpar
) |> to_integer()

x_power_wlr <- gs_power_wlr(
  enroll_rate = x_design_wlr$enroll_rate,
  fail_rate = x_design_wlr$fail_rate,
  weight = wgt05,
  event = c(50, 100, 150),
  analysis_time = NULL,
  upper = upper,
  upar = upar,
  lower = lower,
  lpar = lpar
)

Default summary table production

Instead of outputting 4 detailed tables (a table of enrollment rates, a table of failure rates, a table of analysis summary, a table of bounds summary), users can get a com pensive summary table by calling summary(x), where x is the object returned either by gs_design_ahr or gs_design_wlr. The summary() function produces an overall summary table for bounds for publication in a protocol.

For example, the default output of summary() for the AHR method is

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

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 11.7 N: 479.6 Event: 121.5 AHR: 0.85 Information fraction: 0.33
Futility	−0.9400	1.1856	0.8259	0.0338	0.1741
Efficacy	3.7100	0.5101	0.0001	0.0027	0.0001
Analysis: 2 Time: 20.3 N: 493.1 Event: 243 AHR: 0.74 Information fraction: 0.66
Futility	0.6300	0.9228	0.2656	0.0666	0.7368
Efficacy	2.5100	0.7246	0.0060	0.4135	0.0060
Analysis: 3 Time: 36 N: 493.1 Event: 364.6 AHR: 0.69 Information fraction: 1
Futility	1.9900	0.8118	0.0233	0.1006	0.9753
Efficacy	1.9900	0.8116	0.0231	0.9000	0.0245

Please note the summary() can also be applied to objected returned by gs_power_ahr(). For example,

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

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 10.4 N: 428.7 Event: 100 AHR: 0.87 Information fraction: 0.33
Futility	−1.1200	1.2537	0.8691	0.0341	0.1309
Efficacy	3.7100	0.4735	0.0001	0.0015	0.0001
Analysis: 2 Time: 16.7 N: 493.1 Event: 200 AHR: 0.78 Information fraction: 0.67
Futility	0.1000	0.9863	0.4616	0.0671	0.5450
Efficacy	2.5100	0.6984	0.0060	0.2262	0.0060
Analysis: 3 Time: 26.2 N: 493.1 Event: 300 AHR: 0.71 Information fraction: 1
Futility	1.3900	0.8503	0.0822	0.1006	0.9188
Efficacy	1.9900	0.7926	0.0231	0.8029	0.0249

And the default output of summary() for the WLR method is

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

Bound	Z	~wHR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 11.6 N: 360.3 Event: 91 AHR: 0.79 Information fraction: 0.14
Futility	−1.1900	1.2832	0.8828	0.0140	0.1172
Efficacy	3.7300	0.4580	0.0001	0.0034	0.0001
Analysis: 2 Time: 20.2 N: 372 Event: 183 AHR: 0.69 Information fraction: 0.46
Futility	0.5700	0.9196	0.2853	0.0462	0.7198
Efficacy	2.5200	0.6894	0.0059	0.4368	0.0060
Analysis: 3 Time: 36 N: 372 Event: 275 AHR: 0.64 Information fraction: 1
Futility	1.9900	0.7871	0.0236	0.1001	0.9756
Efficacy	2.0200	0.7836	0.0216	0.8957	0.0228

Note that summary() can also be applied to summarize an object returned by gs_power_wlr().

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

Bound	Z	~wHR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 8.3 N: 255.9 Event: 50 AHR: 0.86 Information fraction: 0.13
Futility	−1.8100	1.6700	0.9651	0.0134	0.0349
Efficacy	6.0600	0.1799	0.0000	0.0000	0.0000
Analysis: 2 Time: 12.3 N: 372 Event: 100 AHR: 0.78 Information fraction: 0.51
Futility	−0.5900	1.1248	0.7218	0.0512	0.2877
Efficacy	2.9400	0.5559	0.0017	0.0362	0.0017
Analysis: 3 Time: 16.6 N: 372 Event: 150 AHR: 0.72 Information fraction: 1
Futility	0.4100	0.9358	0.3422	0.1001	0.6745
Efficacy	1.9700	0.7250	0.0245	0.4561	0.0249

Detailed summary table formatting

Here we demonstrate options for formatting analysis rows, bound rows as well as other table parameters such as titles, labels and footnotes.

Customize the variables to be summarized for each analysis

In the above default table summary table generated by summary(x), the variables used to summarize each analysis includes analysis, time, n(sample size), event, AHR, and info_frac (information fraction). But users can customize these variables chosen using analysis_vars = ... and the corresponding decimals displayed using the argument analysis_decimals = .... For example

summary(
  x_design_ahr,
  analysis_vars = c("n", "event"),
  analysis_decimals = c(1, 1)
) %>%
  gt::gt() %>%
  gt::fmt_number(columns = c(3:6), decimals = 4)

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 N: 479.6 Event: 121.5
Futility	−0.9400	1.1856	0.8259	0.0338	0.1741
Efficacy	3.7100	0.5101	0.0001	0.0027	0.0001
Analysis: 2 N: 493.1 Event: 243
Futility	0.6300	0.9228	0.2656	0.0666	0.7368
Efficacy	2.5100	0.7246	0.0060	0.4135	0.0060
Analysis: 3 N: 493.1 Event: 364.6
Futility	1.9900	0.8118	0.0233	0.1006	0.9753
Efficacy	1.9900	0.8116	0.0231	0.9000	0.0245

Please note that there is no need to input "Analysis" into analysis_vars = ... as it will always appear.

Customize the bound names

Users can also customize the bound names. In the default output generated by summary(x), the bound name is c("Efficacy", "Futility"), which can be changed into c("A is better", "B is better") for a 2-sided design by using the argument bound_names = .... For example,

summary(
  x_design_ahr,
  bound_names = c("A is better", "B is better")
) %>%
  mutate_if(is.numeric, round, digits = 4) %>%
  gt::gt() %>%
  gt::fmt_number(columns = c(3:6), decimals = 4)
#> `mutate_if()` ignored the following grouping variables:
#> • Column `Analysis`

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 11.7 N: 479.6 Event: 121.5 AHR: 0.85 Information fraction: 0.33
B is better	−0.9400	1.1856	0.8259	0.0338	0.1741
A is better	3.7100	0.5101	0.0001	0.0027	0.0001
Analysis: 2 Time: 20.3 N: 493.1 Event: 243 AHR: 0.74 Information fraction: 0.66
B is better	0.6300	0.9228	0.2656	0.0666	0.7368
A is better	2.5100	0.7246	0.0060	0.4135	0.0060
Analysis: 3 Time: 36 N: 493.1 Event: 364.6 AHR: 0.69 Information fraction: 1
B is better	1.9900	0.8118	0.0233	0.1006	0.9753
A is better	1.9900	0.8116	0.0231	0.9000	0.0245

Customize into a gt table and add title/subtitle/footnotes/spanners

Users can also use as_gt() to get the the above R table into a gt table. Furthermore, they can edit the title/subtitle/spanner/footnotes of the gt table by using the arguments in summary.

summary(x_design_ahr) %>%
  as_gt(
    title = "Summary of the Crossing Probability",
    subtitle = "by Using gs_design_ahr",
    colname_spanner = "Cumulative boundary crossing probability",
    colname_spannersub = c("Alternate hypothesis", "Null hypothesis"),
    footnote = list(
      content = c(
        "approximate hazard ratio to cross bound.",
        "gs_design_ahr is a function in gsDesign2.",
        "AHR is average hazard ratio; info_frac is information fraction."
      ),
      location = c("~HR at bound", NA, NA),
      attr = c("colname", "subtitle", "analysis")
    )
  )

Bound	Z	Nominal p	~HR at bound²	Cumulative boundary crossing probability
Summary of the Crossing Probability
by Using gs_design_ahr¹
Bound	Z	Nominal p	~HR at bound²	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 11.7 N: 479.6 Event: 121.5 AHR: 0.85 Information fraction: 0.33³
Futility	-0.94	0.8259	1.1856	0.0338	0.1741
Efficacy	3.71	0.0001	0.5101	0.0027	0.0001
Analysis: 2 Time: 20.3 N: 493.1 Event: 243 AHR: 0.74 Information fraction: 0.66³
Futility	0.63	0.2656	0.9228	0.0666	0.7368
Efficacy	2.51	0.0060	0.7246	0.4135	0.0060
Analysis: 3 Time: 36 N: 493.1 Event: 364.6 AHR: 0.69 Information fraction: 1³
Futility	1.99	0.0233	0.8118	0.1006	0.9753
Efficacy	1.99	0.0231	0.8116	0.9000	⁴ 0.0245
¹ gs_design_ahr is a function in gsDesign2.
² approximate hazard ratio to cross bound.
³ AHR is average hazard ratio; info_frac is information fraction.
⁴ Cumulative alpha for final analysis (0.0245) 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 - 0.0005 = 0.0245) under the null hypothesis.

The above objective can also be realized by using functions in the R package gt for custom design of table layout. We note that as_gt() always produces a gt object and, thus, can be further customized with gt package formatting functions. In the future, we to support rich text format using a function as_rtf() in a fashion similar to as_gt().

Customize the variables to display

Users can select the variables to be displayed in the summary table by using the argument display_colunm = ....

summary(x_design_ahr) %>%
  as_gt(display_columns = c("Analysis", "Bound", "Z", "Probability"))

Bound	Z	Cumulative boundary crossing probability
Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 11.7 N: 479.6 Event: 121.5 AHR: 0.85 Information fraction: 0.33
Futility	-0.94	0.0338	0.1741
Efficacy	3.71	0.0027	0.0001
Analysis: 2 Time: 20.3 N: 493.1 Event: 243 AHR: 0.74 Information fraction: 0.66
Futility	0.63	0.0666	0.7368
Efficacy	2.51	0.4135	0.0060
Analysis: 3 Time: 36 N: 493.1 Event: 364.6 AHR: 0.69 Information fraction: 1
Futility	1.99	0.1006	0.9753
Efficacy	1.99	0.9000	¹ 0.0245
¹ Cumulative alpha for final analysis (0.0245) 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 - 0.0005 = 0.0245) under the null hypothesis.

Customize whether to show infinity bound or not

Users have options to either show the infinity bounds or not by taking advantage of display_inf_bound = ....

summary(x_design_ahr) %>%
  as_gt(display_inf_bound = FALSE)

Bound	Z	Nominal p¹	~HR at bound²	Cumulative boundary crossing probability
Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p¹	~HR at bound²	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 11.7 N: 479.6 Event: 121.5 AHR: 0.85 Information fraction: 0.33
Futility	-0.94	0.8259	1.1856	0.0338	0.1741
Efficacy	3.71	0.0001	0.5101	0.0027	0.0001
Analysis: 2 Time: 20.3 N: 493.1 Event: 243 AHR: 0.74 Information fraction: 0.66
Futility	0.63	0.2656	0.9228	0.0666	0.7368
Efficacy	2.51	0.0060	0.7246	0.4135	0.0060
Analysis: 3 Time: 36 N: 493.1 Event: 364.6 AHR: 0.69 Information fraction: 1
Futility	1.99	0.0233	0.8118	0.1006	0.9753
Efficacy	1.99	0.0231	0.8116	0.9000	³ 0.0245
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 (0.0245) 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 - 0.0005 = 0.0245) under the null hypothesis.

Details of output from design/power functions

There are four components in the objects returned by either gs_design_ahr()/gs_design_wlr() or gs_power_ahr()/gs_power_wlr(): 1. failure rates: a table summarizing failure rate and dropout rate. 1. enrollment rates: a table summarizing the enrollment rate. 1. bounds: a table summarize the bound of each analysis. 1. analysis: a table summarize the each analysis, with each one row for one analysis one hypothesis.

Failure rates

The failure rates of different gsDesign object can be obtained by using x$fail_rate, where x is the object returned either by gs_design_ahr or gs_design_wlr. For example, the failure rates of the AHR design derivation can be returned by calling

x_design_ahr$fail_rate %>%
  gt::gt() %>%
  gt::fmt_number(columns = 3:5, decimals = 4)

stratum	duration	fail_rate	dropout_rate	hr
All	4	0.0578	0.0010	1.0000
All	100	0.0578	0.0010	0.6000

Please note that both x_design_ahr and x_wlr returns the same failure rates, which is the same as that inputted as fail_rate. To verify, let’s take a look at the failure rate of the WLR design derivation, which are shown as below.

x_design_wlr$fail_rate %>%
  gt::gt() %>%
  gt::fmt_number(columns = 3:5, decimals = 4)

stratum	duration	fail_rate	dropout_rate	hr
All	4	0.0578	0.0010	1.0000
All	100	0.0578	0.0010	0.6000

Enrollment

The enrollment rate of a gs design derivation can be collected by using x$fail_rate, where x is the object returned either by gs_design_ahr or gs_design_wlr. For example, the enrollment rates of the AHR/WLR design derivation is

x_design_ahr$enroll_rate %>%
  gt::gt() %>%
  gt::fmt_number(columns = 3, decimals = 4)

stratum	duration	rate
All	12	41.0885

x_design_wlr$enroll_rate %>%
  gt::gt() %>%
  gt::fmt_number(columns = 3, decimals = 4)

stratum	duration	rate
All	12	31.0000

It can be seen that, although the design derivation is different, the enrollment rate table share the same table structure, same enrollment period durations for each rate. Yet, the enrollment rates differ between designs only by a multiplicative constant.

Analysis

The analysis summary table has the structure of one row per analysis per hypothesis. And columns can vary with different defaults for each design option. This type of tables are useful for understanding commonalities in how designs are summarized for different models. To get analysis summary table, users can call x$analysis, where x is the object returned either by gs_design_ahr or gs_design_wlr. For example, the analysis summary of the AHR/WLR design derivation is

x_design_ahr$analysis %>%
  gt::gt() %>%
  gt::fmt_number(columns = 2:8, decimals = 4)

analysis	time	n	event	ahr	theta	info	info0	info_frac
1	11.6713	479.5569	121.5178	0.8487	0.1641	29.8909	30.3795	0.3322753
2	20.2531	493.0625	243.0357	0.7427	0.2974	59.4218	60.7589	0.6605497
3	36.0000	493.0625	364.5535	0.6917	0.3686	89.9581	91.1384	1.0000000

x_design_wlr$analysis %>%
  gt::gt() %>%
  gt::fmt_number(columns = 2:8, decimals = 4)

analysis	time	n	event	ahr	theta	info	info0	info_frac	info_frac0
1	11.6211	360.2556	91.0000	0.7926	0.5447	3.4558	3.4759	0.1384101	0.1317923
2	20.2095	372.0000	182.9999	0.6898	0.6949	11.5089	11.7455	0.4609515	0.4453419
3	35.9888	372.0000	275.0000	0.6443	0.6819	24.9676	26.3742	1.0000000	1.0000000

Bounds

The analysis summary table has the structure of One row per analysis per bound per hypothesis. Columns can vary with different defaults for each design option. To get a bonus summary table, users can call x$analysis, where x is the object returned either by gs_design_ahr or gs_design_wlr. For example, the bounds summary of the AHR/WLR design derivation is

x_design_ahr$bound %>%
  gt::gt() %>%
  gt::fmt_number(columns = c(3, 5:7), decimals = 4)

analysis	bound	probability	probability0	z	~hr at bound	nominal p
1	upper	0.0027	0.0001035057	3.7103	0.5101	0.0001
1	lower	0.0338	0.1740609319	−0.9382	1.1856	0.8259
2	upper	0.4135	0.0060482351	2.5114	0.7246	0.0060
2	lower	0.0666	0.7368187135	0.6262	0.9228	0.2656
3	upper	0.9000	0.0245385477	1.9931	0.8116	0.0231
3	lower	0.1006	0.9753350071	1.9905	0.8118	0.0233

x_design_wlr$bounds %>%
  gt::gt() %>%
  gt::fmt_number(columns = c(3, 5:7), decimals = 4)

analysis	bound	probability	probability0	z	~hr at bound	nominal p
1	upper	0.0034	9.762582e-05	3.7251	0.4580	0.0001
1	lower	0.0140	1.171678e-01	−1.1893	1.2832	0.8828
2	upper	0.4368	5.998566e-03	2.5158	0.6894	0.0059
2	lower	0.0462	7.198316e-01	0.5672	0.9196	0.2853
3	upper	0.8957	2.283770e-02	2.0223	0.7836	0.0216
3	lower	0.1001	9.756012e-01	1.9851	0.7871	0.0236

References

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.

Yung, Godwin, and Yi Liu. 2020. “Sample Size and Power for the Weighted Log-Rank Test and Kaplan-Meier Based Tests with Allowance for Nonproportional Hazards.” Biometrics 76 (3): 939–50.

Yujie Zhao