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Group sequential design with calendar-based timing of analyses

Source: R/gsSurvCalendar.R
gsSurvCalendar.Rd

This is like gsSurv(), but the timing of analyses is specified in calendar time units. Information fraction is computed from the input rates and the calendar times. Spending can be based on information fraction as in Lan and DeMets (1983) or calendar time units as in Lan and DeMets (1989).

Usage

gsSurvCalendar(
  test.type = 4,
  alpha = 0.025,
  sided = 1,
  beta = 0.1,
  astar = 0,
  sfu = gsDesign::sfHSD,
  sfupar = -4,
  sfl = gsDesign::sfHSD,
  sflpar = -2,
  sfharm = gsDesign::sfHSD,
  sfharmparam = -2,
  calendarTime = c(12, 24, 36),
  spending = c("information", "calendar"),
  lambdaC = log(2)/6,
  hr = 0.6,
  hr0 = 1,
  eta = 0,
  etaE = NULL,
  gamma = 1,
  R = 12,
  S = NULL,
  minfup = 18,
  ratio = 1,
  r = 18,
  tol = .Machine$double.eps^0.25,
  testUpper = TRUE,
  testLower = TRUE,
  testHarm = TRUE,
  method = c("LachinFoulkes", "Schoenfeld", "Freedman", "BernsteinLagakos")
)

Arguments

test.type

1=one-sided
2=two-sided symmetric
3=two-sided, asymmetric, beta-spending with binding lower bound
4=two-sided, asymmetric, beta-spending with non-binding lower bound
5=two-sided, asymmetric, lower bound spending under the null hypothesis with binding lower bound
6=two-sided, asymmetric, lower bound spending under the null hypothesis with non-binding lower bound
7=two-sided, asymmetric, with binding futility and binding harm bounds
8=two-sided, asymmetric, with non-binding futility and non-binding harm bounds.
See details, examples and manual.

alpha

Type I error rate. Default is 0.025 since 1-sided testing is default.

sided

1 for 1-sided testing, 2 for 2-sided testing.

beta

Type II error rate. Default is 0.10 (90% power); NULL if power is to be computed based on other input values.

astar

Total spending for the lower (test.type 5 or 6) or harm (test.type 7 or 8) bound under the null hypothesis. Default is 0. For test.type 5 or 6, astar specifies the total probability of crossing a lower bound at all analyses combined. For test.type 7 or 8, astar specifies the total probability of crossing the harm bound at all analyses combined under the null hypothesis. If astar = 0, it will be changed to \(1 - \)alpha.

sfu

A spending function or a character string indicating a boundary type (that is, “WT” for Wang-Tsiatis bounds, “OF” for O'Brien-Fleming bounds and “Pocock” for Pocock bounds). For one-sided and symmetric two-sided testing is used to completely specify spending (test.type=1, 2), sfu. The default value is sfHSD which is a Hwang-Shih-DeCani spending function. See details, vignette("SpendingFunctionOverview"), manual and examples.

sfupar

Real value, default is \(-4\) which is an O'Brien-Fleming-like conservative bound when used with the default Hwang-Shih-DeCani spending function. This is a real-vector for many spending functions. The parameter sfupar specifies any parameters needed for the spending function specified by sfu; this is not needed for spending functions (sfLDOF, sfLDPocock) or bound types (“OF”, “Pocock”) that do not require parameters. Note that sfupar can be specified as a positive scalar for sfLDOF for a generalized O'Brien-Fleming spending function.

sfl

Specifies the spending function for lower boundary crossing probabilities when asymmetric, two-sided testing is performed (test.type = 3, 4, 5, or 6). Unlike the upper bound, only spending functions are used to specify the lower bound. The default value is sfHSD which is a Hwang-Shih-DeCani spending function. The parameter sfl is ignored for one-sided testing (test.type=1) or symmetric 2-sided testing (test.type=2). See details, spending functions, manual and examples.

sflpar

Real value, default is \(-2\), which, with the default Hwang-Shih-DeCani spending function, specifies a less conservative spending rate than the default for the upper bound.

sfharm

A spending function for the harm bound, used with test.type = 7 or test.type = 8. Default is sfHSD. See spendingFunction for details.

sfharmparam

Real value, default is \(-2\). Parameter for the harm bound spending function sfharm.

calendarTime

Vector of increasing positive numbers with calendar times of analyses. Time 0 is start of randomization.

spending

Select between calendar-based spending and information-based spending.

lambdaC

Scalar, vector or matrix of event hazard rates for the control group; rows represent time periods while columns represent strata; a vector implies a single stratum. Note that rates corresponding the final time period are extended indefinitely.

hr

Hazard ratio (experimental/control) under the alternate hypothesis (scalar, > 0, must differ from hr0). Both hr < hr0 (experimental is beneficial when lower hazard is better) and hr > hr0 (e.g., time-to-response or safety designs) are supported.

hr0

Hazard ratio (experimental/control) under the null hypothesis (scalar, > 0, must differ from hr).

eta

Scalar, vector or matrix of dropout hazard rates for the control group; rows represent time periods while columns represent strata; if entered as a scalar, rate is constant across strata and time periods; if entered as a vector, rates are constant across strata.

etaE

Matrix dropout hazard rates for the experimental group specified in like form as eta; if NULL, this is set equal to eta.

gamma

A scalar, vector or matrix of rates of entry by time period (rows) and strata (columns); if entered as a scalar, rate is constant across strata and time periods; if entered as a vector, rates are constant across strata.

R

A scalar or vector of durations of time periods for recruitment rates specified in rows of gamma. Length is the same as number of rows in gamma. Note that when variable enrollment duration is specified (input T = NULL), the final enrollment period is extended as long as needed.

S

A scalar or vector of durations of piecewise constant event rates specified in rows of lambda, eta and etaE; this is NULL if there is a single event rate per stratum (exponential failure) or length of the number of rows in lambda minus 1, otherwise. The final time period is extended indefinitely for each stratum.

minfup

A non-negative scalar less than the maximum value in calendarTime. Enrollment will be cut off at the difference between the maximum value in calendarTime and minfup.

ratio

Randomization ratio of experimental treatment divided by control; normally a scalar, but may be a vector with length equal to number of strata.

r

Integer value (>= 1 and <= 80) controlling the number of numerical integration grid points. Default is 18, as recommended by Jennison and Turnbull (2000). Grid points are spread out in the tails for accurate probability calculations. Larger values provide more grid points and greater accuracy but slow down computation. Jennison and Turnbull (p. 350) note an accuracy of \(10^{-6}\) with r = 16. This parameter is normally not changed by users.

tol

Tolerance for error passed to the gsDesign function.

testUpper

Indicator of which analyses should include an upper (efficacy) bound. A single value of TRUE (default) indicates all analyses have an efficacy bound. Otherwise, a logical vector of length k indicating which analyses will have an efficacy bound. Overridden to all TRUE for test.type 1 and 2. Must be TRUE at the final analysis to achieve targeted power. At each analysis, at least one of testUpper, testLower, or testHarm must be TRUE. Where testUpper is FALSE, the upper bound is set to +20 (effectively Inf) and displayed as NA in output.

testLower

Indicator of which analyses should include a lower (futility) bound. A single value of TRUE (default) indicates all analyses have a lower bound; FALSE indicates none. Otherwise, a logical vector of length k. Ignored for test.type 1 (one-sided, no lower bound). Overridden to all TRUE for test.type 2 (symmetric). For test.type 3–8, at least one analysis must be TRUE. Where testLower is FALSE, the lower bound is set to -20 (effectively -Inf) and displayed as NA in output.

testHarm

Indicator of which analyses should include a harm bound. A single value of TRUE (default) indicates all analyses have a harm bound; FALSE indicates none. Otherwise, a logical vector of length k. Only used for test.type 7 or 8; at least one analysis must be TRUE for those types. Where testHarm is FALSE, the harm bound is set to -20 (effectively -Inf) and displayed as NA in output.

method

One of "LachinFoulkes" (default), "Schoenfeld", "Freedman", or "BernsteinLagakos". Note: "Schoenfeld" and "Freedman" methods only support superiority testing (hr0 = 1). "Freedman" does not support stratified populations.

References

Lan KKG and DeMets DL (1983), Discrete Sequential Boundaries for Clinical Trials. Biometrika, 70, 659-663.

Lan KKG and DeMets DL (1989), Group Sequential Procedures: Calendar vs. Information Time. Statistics in Medicine, 8, 1191-1198.

Schoenfeld D (1981), The Asymptotic Properties of Nonparametric Tests for Comparing Survival Distributions. Biometrika, 68, 316-319.

Freedman LS (1982), Tables of the Number of Patients Required in Clinical Trials Using the Logrank Test. Statistics in Medicine, 1, 121-129.

See also

gsSurv, gsDesign, gsBoundSummary

Examples

# First example: while timing is calendar-based, spending is event-based
x <- gsSurvCalendar() |> toInteger()
gsBoundSummary(x)
#>     Analysis              Value Efficacy Futility
#>    IA 1: 29%                  Z   3.0856  -0.4349
#>       N: 130        p (1-sided)   0.0010   0.6682
#>   Events: 50       ~HR at bound   0.4178   1.1309
#>    Month: 12   P(Cross) if HR=1   0.0010   0.3318
#>              P(Cross) if HR=0.6   0.1018   0.0122
#>    IA 2: 79%                  Z   2.3279   1.3991
#>       N: 196        p (1-sided)   0.0100   0.0809
#>  Events: 137       ~HR at bound   0.6718   0.7874
#>    Month: 24   P(Cross) if HR=1   0.0106   0.9213
#>              P(Cross) if HR=0.6   0.7505   0.0606
#>        Final                  Z   2.0154   2.0154
#>       N: 196        p (1-sided)   0.0219   0.0219
#>  Events: 173       ~HR at bound   0.7361   0.7361
#>    Month: 36   P(Cross) if HR=1   0.0228   0.9772
#>              P(Cross) if HR=0.6   0.9001   0.0999

# Second example: both timing and spending are calendar-based
# This results in less spending at interims and leaves more for final analysis
y <- gsSurvCalendar(spending = "calendar") |> toInteger()
gsBoundSummary(y)
#>     Analysis              Value Efficacy Futility
#>    IA 1: 29%                  Z   3.0107  -0.3784
#>       N: 126        p (1-sided)   0.0013   0.6474
#>   Events: 49       ~HR at bound   0.4231   1.1142
#>    Month: 12   P(Cross) if HR=1   0.0013   0.3526
#>              P(Cross) if HR=0.6   0.1123   0.0148
#>    IA 2: 79%                  Z   2.5581   1.1380
#>       N: 190        p (1-sided)   0.0053   0.1276
#>  Events: 133       ~HR at bound   0.6417   0.8209
#>    Month: 24   P(Cross) if HR=1   0.0062   0.8785
#>              P(Cross) if HR=0.6   0.6593   0.0437
#>        Final                  Z   1.9854   1.9854
#>       N: 190        p (1-sided)   0.0235   0.0235
#>  Events: 168       ~HR at bound   0.7361   0.7361
#>    Month: 36   P(Cross) if HR=1   0.0237   0.9763
#>              P(Cross) if HR=0.6   0.9006   0.0994

# Note that calendar timing for spending relates to planned timing for y
# rather than timing in y after toInteger() conversion

# Values plugged into spending function for calendar time
y$usTime
#> [1] 0.3333333 0.6666667 1.0000000
# Actual calendar fraction from design after toInteger() conversion
y$T / max(y$T)
#> [1] 0.3317243 0.6637288 1.0000000