Simulate and execute a single adaptive clinical trial design with a time-to-event endpoint

Usage

survival_adapt(
  hazard_treatment,
  hazard_control = NULL,
  cutpoints = 0,
  N_total,
  lambda = 0.3,
  lambda_time = 0,
  interim_look = NULL,
  end_of_study,
  prior = c(0.1, 0.1),
  block = 2,
  rand_ratio = c(1, 1),
  prop_loss = 0,
  alternative = "greater",
  h0 = 0,
  Fn = 0.05,
  Sn = 0.9,
  prob_ha = 0.95,
  N_impute = 10,
  N_mcmc = 10,
  method = "logrank",
  imputed_final = FALSE
)

Arguments

hazard_treatment: vector. Constant hazard rates under the treatment arm.
hazard_control: vector. Constant hazard rates under the control arm.
cutpoints: vector. Times at which the baseline hazard changes. Default is cutpoints = 0, which corresponds to a simple (non-piecewise) exponential model.
N_total: integer. Maximum sample size allowable
lambda: vector. Enrollment rates across simulated enrollment times. See enrollment for more details.
lambda_time: vector. Enrollment time(s) at which the enrollment rates change. Must be same length as lambda. See enrollment for more details.
interim_look: vector. Sample size for each interim look. Note: the maximum sample size should not be included. For two-arm designs, each interim look must be at least the (largest) block size (see block), ensuring both treatment arms are present at every interim analysis; a smaller look could enrol subjects from a single arm only, leaving the interim posterior undefined for the missing arm.
end_of_study: scalar. Length of the study; i.e. time at which endpoint will be evaluated.
prior: vector. The prior distributions for the piecewise hazard rate parameters are each $Gamma(a_0, b_0)$, where $a_0$ is the shape parameter and $b_0$ is the rate parameter (i.e., the inverse of the scale). This follows R's rgamma parameterization. The same prior is applied to all piecewise intervals and to both treatment arms. The default non-informative prior distribution used is Gamma(0.1, 0.1), which is specified by setting prior = c(0.1, 0.1).
block: scalar. Block size for generating the randomization schedule.
rand_ratio: vector. Randomization allocation for the ratio of control to treatment. Integer values mapping the size of the block. See randomization for more details.
prop_loss: scalar. Overall proportion of subjects lost to follow-up. Subjects are selected at random for LTFU regardless of treatment arm or event status. Each LTFU subject's observed time is drawn from a Uniform(0, t) distribution, where t is their potential event or censoring time. Since the LTFU time is always less than t, the event has not yet occurred at dropout and the subject is right-censored. Defaults to zero.
alternative: character. The string specifying the alternative hypothesis, must be one of "greater" (default), "less" or "two.sided". All three options are supported for method = "bayes", "logrank", and "cox". The chi-square test (method = "chisq") only supports "two.sided". For survival outcomes, "less" corresponds to the treatment group having a lower cumulative incidence (i.e., treatment is beneficial), and "greater" corresponds to the treatment group having a higher cumulative incidence.
h0: scalar. Null hypothesis value of $p_\textrm{treatment} - p_\textrm{control}$ when method = "bayes". Default is h0 = 0. The argument is ignored when method = "logrank" or = "cox"; in those cases the usual test of non-equal hazards is assumed.
Fn: vector of [0, 1] values. Each element is the probability threshold to stop at the $i$-th look early for futility. If there are no interim looks (i.e. interim_look = NULL), then Fn is not used in the simulations or analysis. The length of Fn should be the same as interim_look, else the values are recycled.
Sn: vector of [0, 1] values. Each element is the probability threshold to stop at the $i$-th look early for expected success. If there are no interim looks (i.e. interim_look = NULL), then Sn is not used in the simulations or analysis. The length of Sn should be the same as interim_look, else the values are recycled.
prob_ha: scalar [0, 1]. Probability threshold of alternative hypothesis.
N_impute: integer. Number of imputations for Monte Carlo simulation of missing data.
N_mcmc: integer. Number of samples to draw from the posterior distribution when using a Bayesian test (method = "bayes").
method: character. For an imputed data set (or the final data set after follow-up is complete), whether the analysis should be a log-rank (method = "logrank") test, Cox proportional hazards regression model Wald test (method = "cox"), a fully-Bayesian analysis (method = "bayes"), or a chi-square test (method = "chisq"). See Details section.
imputed_final: logical. Should the final analysis (after all subjects have been followed-up to the study end) be based on imputed outcomes for subjects who were LTFU (i.e. right-censored with time <end_of_study)? Default is TRUE. Setting to FALSE means that the final analysis would incorporate right-censoring.

Value

A data frame containing some input parameters (arguments) as well as statistics from the analysis, including:

N_treatment:: integer. The number of patients enrolled in the treatment arm for each simulation.
N_control:: integer. The number of patients enrolled in the control arm for each simulation.
est_final:: scalar. The treatment effect that was estimated at the final analysis. Final analysis occurs when either the maximum sample size is reached and follow-up complete, or the interim analysis triggered an early stopping of enrollment/accrual and follow-up for those subjects is complete.
post_prob_ha:: scalar. The corresponding posterior probability from the final analysis. If imputed_final is true, this is calculated as the posterior probability of efficacy (or equivalent, depending on how alternative: and h0 were specified) for each imputed final analysis dataset, and then averaged over the N_impute imputations. If method = "logrank", post_prob_ha is calculated in the same fashion, but value represents $1 - P$, where $P$ denotes the frequentist $P$-value.
stop_futility:: integer. A logical indicator of whether the trial was stopped early for futility.
stop_expected_success:: integer. A logical indicator of whether the trial was stopped early for expected success.

Details

Implements the Goldilocks design method described in Broglio et al. (2014). At each interim analysis, two probabilities are computed:

The posterior predictive probability of eventual success. This is calculated as the proportion of imputed datasets at the current sample size that would go on to be success at the specified threshold. At each interim analysis it is compared to the corresponding element of Sn, and if it exceeds the threshold, accrual/enrollment is suspended and the outstanding follow-up allowed to complete before conducting the pre-specified final analysis.
The posterior predictive probability of final success. This is calculated as the proportion of imputed datasets at the maximum threshold that would go on to be successful. Similar to above, it is compared to the corresponding element of Fn, and if it is less than the threshold, accrual/enrollment is suspended and the trial terminated. Typically this would be a binding decision. If it is not a binding decision, then one should also explore the simulations with Fn = 0.

Hence, at each interim analysis look, 3 decisions are allowed:

Stop for expected success
Stop for futility
Continue to enroll new subjects, or if at maximum sample size, proceed to final analysis.

At each interim (and final) analysis methods as:

Log-rank test (method = "logrank"). Each (imputed) dataset with both treatment and control arms can be compared using a standard log-rank test. The output is a P-value, and there is no treatment effect reported. The function returns $1 - P$, which is reported in post_prob_ha. Whilst not a posterior probability, it can be contrasted in the same manner. For example, if the success threshold is $P < 0.05$, then one requires post_prob_ha $> 0.95$. The reason for this is to enable simple switching between Bayesian and frequentist paradigms for analysis. When alternative = "less" or "greater", a one-sided P-value is computed from the log-rank z-statistic.
Cox proportional hazards regression Wald test (method = "cox"). Similar to the log-rank test, a P-value is calculated and $1 - P$ is reported in post_prob_ha. When alternative = "two.sided", the standard two-sided Wald P-value is used. When alternative = "less" or "greater", a one-sided P-value is derived from the Wald z-statistic. The treatment effect (log hazard ratio) is also reported.
Bayesian absolute difference (method = "bayes"). Each imputed dataset is used to update the conjugate Gamma prior (defined by prior), yielding a posterior distribution for the piecewise exponential rate parameters. In turn, the posterior distribution of the cumulative incidence function ($1 - S(t)$, where $S(t)$ is the survival function) evaluated at time end_of_study is calculated. If a single arm study, then this summarizes the treatment effect, else, if a two-armed study, the independent posteriors are used to estimate the posterior distribution of the difference. A posterior probability is calculated according to the specification of the test type (alternative) and the value of the null hypothesis (h0).
Chi-square test (method = "chisq"). Each (imputed) dataset with both treatment and control arms can be compared using a standard chi-square test on the final event status, which discards the event time information. The output is a P-value, and there is no treatment effect reported. The function returns $1 - P$, which is reported in post_prob_ha. Whilst not a posterior probability, it can be contrasted in the same manner. For example, if the success threshold is $P < 0.05$, then one requires post_prob_ha $> 0.95$. The reason for this is to enable simple switching between Bayesian and frequentist paradigms for analysis. Because the chi-square test cannot handle right-censored observations, subjects lost to follow-up are excluded from the final analysis when imputed_final = FALSE. When imputed_final = TRUE, LTFU subjects are imputed before the test is applied, so all subjects are included.
Imputed final analysis (imputed_final). The overall final analysis conducted after accrual is suspended and follow-up is complete can be analyzed on imputed datasets (default) or on the non-imputed dataset. Since the imputations/predictions used during the interim analyses assume all subjects are imputed (since loss to follow-up is not yet known), it would seem most appropriate to conduct the trial in the same manner, especially if loss to follow-up rates are appreciable. Note, this only applies to subjects who are right-censored due to loss to follow-up, which we assume is a non-informative process. This can be used with any method.

When method = "bayes" and imputation is involved (either at interim analyses or via imputed_final = TRUE), a two-stage posterior procedure is used. First, the posterior distribution of the piecewise hazard rates is estimated from the observed data and used to draw imputed event times for censored subjects. Second, a new posterior is estimated from the combined observed and imputed data, and this posterior is used for inference. This is consistent with the predictive probability framework described in Broglio et al. (2014), but users should be aware that the imputation model's posterior influences the analysis posterior. For frequentist methods ("logrank", "cox", "chisq"), the second stage uses a standard test rather than a posterior, so this feedback loop does not arise.

At each interim look, follow-up times are masked (censored) to reflect the calendar time of the analysis. Subjects enrolled at the exact interim boundary have zero follow-up time, which is incompatible with survSplit. These times are clamped to .Machine$double.eps (approximately $2.2 \times 10^{-16}$) so that they contribute negligible but non-zero exposure. This affects at most one subject per interim look.

References

Broglio KR, Connor JT, Berry SM. Not too big, not too small: a Goldilocks approach to sample size selection. Journal of Biopharmaceutical Statistics, 2014; 24(3): 685–705.

Examples

# RCT with exponential hazard (no piecewise breaks)
# Note: the number of imputations is small to enable this example to run
#       quickly on CRAN tests. In practice, much larger values are needed.
survival_adapt(
 hazard_treatment = -log(0.85) / 36,
 hazard_control = -log(0.7) / 36,
 cutpoints = 0,
 N_total = 600,
 lambda = 20,
 lambda_time = 0,
 interim_look = 400,
 end_of_study = 36,
 prior = c(0.1, 0.1),
 block = 2,
 rand_ratio = c(1, 1),
 prop_loss = 0.30,
 alternative = "less",
 h0 = 0,
 Fn = 0.05,
 Sn = 0.9,
 prob_ha = 0.975,
 N_impute = 10,
 N_mcmc = 10,
 method = "bayes")
#>   prob_threshold margin alternative N_treatment N_control N_enrolled N_max
#> 1          0.975      0        less         300       300        600   600
#>   post_prob_ha  est_final ppp_success stop_futility stop_expected_success
#> 1            1 -0.1309505           0             0                     0