Decision Table for the TITE-BOIN Design
Source:R/tite_boin_decision_table.R
tite_boin_decision_table.RdTabulate the dose assignment rule of the time-to-event Bayesian optimal interval (TITE-BOIN) design, which allows new patients to be treated while the DLT assessment of earlier patients is still pending. For every attainable combination of patients treated, DLTs observed and patients pending at the current dose, the table gives the decision or, when the decision depends on how long the pending patients have been followed, the values of the follow-up statistic at which it changes.
Usage
tite_boin_decision_table(
target,
max_n,
method = c("imputation", "ess"),
p_saf = NULL,
p_tox = NULL,
cutoff_eli = 0.95,
max_pending_ratio = NULL,
min_completed = NULL,
min_follow_up = 0
)Arguments
- target
Numeric scalar. Target DLT probability, for example 0.30.
- max_n
Integer scalar. Largest number of patients treated at a single dose.
- method
Character scalar. How the pending patients enter the estimate of the DLT probability:
"imputation"(the default) for the single mean imputation of Yuan et al. (2018), or"ess"for the effective sample size of Lin and Yuan (2020). See the details.- p_saf
Numeric scalar. Highest DLT probability deemed subtherapeutic. Defaults to
0.6 * target.- p_tox
Numeric scalar. Lowest DLT probability deemed overly toxic. Defaults to
1.4 * target.- cutoff_eli
Numeric scalar. Posterior probability cutoff for dose elimination. Defaults to 0.95.
- max_pending_ratio
Numeric scalar greater than 0 and at most 1. Accrual is suspended when the proportion of the patients at the current dose whose assessment is pending exceeds this value, unless the dose is de-escalated whatever their outcomes. Defaults to 0.5 for
"imputation", following Yuan et al. (2018), and to 1, which never suspends on this ground, for"ess".- min_completed
Integer scalar. While some patients are pending, escalation requires at least this many patients at the current dose to have completed the assessment, and accrual is suspended instead when fewer have. Defaults to 0 for
"imputation"and to 2 for"ess", following Lin and Yuan (2020).- min_follow_up
Numeric scalar between 0 and 1. While some patients are pending, escalation also requires the shortest follow-up among the pending patients at the current dose, as a fraction of the assessment window (MF), to be at least this value, and accrual is suspended instead when it is not. Defaults to 0, which never suspends on this ground. Chen et al. (2025) recommend 0.25, their rule 2 with B = 25. The rule leaves the boundaries on the follow-up statistic unchanged, so it is recorded as an attribute rather than in the rows of the table.
Value
An object of class tite_boin_decision_table, which is a data frame
with one row per attainable state of the current dose, ordered by
n, n_tox and n_pending, and with the columns
- n
Number of patients treated at the current dose, from 1 to
max_n.- n_tox
Number of DLTs observed, which can only come from patients who completed the assessment.
- n_pending
Number of patients whose assessment is pending.
- decision
Decision code.
"E"escalates,"S"stays,"D"de-escalates,"DE"de-escalates and eliminates the dose together with all higher doses, and"SUS"suspends accrual until more data are available. These do not depend on the follow-up of the pending patients. The codes"E/S","S/D"and"E/S/D"mark states whose decision depends on the follow-up statistic throughesc_boundanddeesc_bound; in them"SUS"takes the place of"E"when escalation is blocked bymin_completed.- esc_bound
Escalate, or suspend when escalation is blocked, when the follow-up statistic is at least this value.
NAwhen no such boundary applies.- deesc_bound
De-escalate when the follow-up statistic is at most this value.
NAwhen no such boundary applies.
The design parameters are stored as attributes: method,
statistic ("STFT" or "ESS"), target,
p_saf, p_tox, lambda_e, lambda_d,
cutoff_eli, max_pending_ratio, min_completed and
min_follow_up. When min_follow_up is positive, an
escalation (code "E", or "E" as the first part of a code
with a slash) in a state with pending patients takes place only when MF is
at least min_follow_up, and accrual is suspended otherwise. The
object has print and plot methods.
Details
Suppose that n patients have been treated at the current dose, that r of them
have completed the DLT assessment with n_tox DLTs, and that
c = n - r are pending. The standardized total follow-up time (STFT) is the
sum of the follow-up times of the pending patients divided by the length of
the assessment window, so that STFT lies between 0 and c. With no pending
patient the decision is that of the BOIN design, taken from
boin_boundary.
method = "imputation" replaces each pending outcome by its expected
value under a uniform time to DLT. The estimate of the DLT probability is
(n_tox + q * (c - STFT)) / n, where q = p / (1 - p) and
p = (n_tox + a) / (r + 1) is the posterior mean under a
Beta(a, 1 - a) prior with a = target / 2 (Yuan et al., 2018,
Supplementary Appendix A). The estimate decreases as STFT grows, so the rule
becomes a pair of boundaries on STFT. As in the BOIN design, escalation is
possible only while n_tox / n is below the target and de-escalation
only once it reaches the target.
method = "ess" uses the approximated likelihood of Lin and Yuan
(2020), under which the estimate is n_tox / ESS with the effective
sample size ESS = r + STFT, which lies between r and n. The dose is escalated
when ESS is at least n_tox / lambda_e and de-escalated when it is at
most n_tox / lambda_d. The boundaries in the table refer to ESS.
The rules are applied in this order: elimination; de-escalation that holds
whatever the outcomes of the pending patients, that is when
n_tox / n is at least lambda_d; suspension for too many
pending patients; and finally the boundaries on the follow-up statistic, in
which an escalation blocked by min_completed becomes a suspension.
An escalation that remains is then subject to min_follow_up.
Elimination requires Pr(p > target | data) > cutoff_eli computed
with all n treated patients, the pending ones counted as without DLT, which
is how both articles define it. It is not evaluated before three patients
have been treated.
Neither the length of the assessment window nor the weights given to the follow-up times change the table, because both enter only through STFT. A weighted STFT, such as that of a piecewise uniform time to DLT, is compared with the same boundaries.
Agreement with the published tables
With the defaults, method = "imputation" reproduces every entry of
Table 1 (target 0.2) of Yuan et al. (2018) and of Table S1 (target 0.3) of
its supplementary appendix. These tables allow de-escalation when
n_tox / n equals the target (Table 1, 15 patients with 3 DLTs), which
equation (5) of the appendix, read literally, would not. The implementation
follows the tables, which agree with the verbal description of the rule in
the appendix.
With max_pending_ratio = 0.49 and min_follow_up = 0.25,
method = "imputation" reproduces every entry of Table 1 (target 0.3,
up to six patients) of Chen et al. (2025) and of Table A1 (target 0.25, up
to nine patients) of the supplementary materials of Chen et al. (2026).
Their rule 1 suspends accrual when fewer than 51 percent of the patients
have completed the assessment, which for whole numbers of patients is the
same as more than 49 percent pending, unless the observed DLT rate already
exceeds lambda_d; this exception is the de-escalation that holds
whatever the outcomes of the pending patients.
For method = "ess" no table for the BOIN design has been published;
the tables of Lin and Yuan (2020) are for the keyboard and mTPI designs.
References
Yuan, Y., Lin, R., Li, D., Nie, L. and Warren, K. E. (2018). Time-to-Event Bayesian Optimal Interval Design to Accelerate Phase I Trials. Clinical Cancer Research, 24(20), 4921-4930.
Lin, R. and Yuan, Y. (2020). Time-to-Event Model-Assisted Designs for Dose-Finding Trials with Delayed Toxicity. Biostatistics, 21(4), 807-824.
Chen, K., Chen, T.-Y., Zhang, Y., Lin, R. and Yuan, Y. (2025). Practical Considerations for Using the TITE-BOIN Design to Handle Late-Onset Toxicity or Fast Accrual in Phase I Trials. Clinical Cancer Research, 31(13), 2573-2580.
Chen, K., Zhao, Y., Takeda, K. and Yuan, Y. (2026). BE-BOIN: A Dose Optimization Design Accommodating Backfill and Late-Onset Toxicity. Therapeutic Innovation and Regulatory Science. doi:10.1007/s43441-026-00994-0
Examples
# Table 1 of Yuan et al. (2018)
decisions <- tite_boin_decision_table(target = 0.2, max_n = 15)
print(decisions, cohort_size = 3)
#> TITE-BOIN decision table with target 0.2
#> Method: single mean imputation (Yuan et al., 2018); boundaries refer to STFT
#>
#> Patients DLTs Pending Escalate Stay De-escalate
#> 3 0 <= 1 Y
#> 3 0 >= 2 Suspend Suspend Suspend
#> 3 1 <= 2 Y
#> 3 >= 2 <= 1 Y & Elim
#> 6 0 <= 3 Y
#> 6 0 >= 4 Suspend Suspend Suspend
#> 6 1 <= 3 Y
#> 6 1 >= 4 Suspend Suspend Suspend
#> 6 2 <= 4 Y
#> 6 >= 3 <= 3 Y & Elim
#> 9 0 <= 4 Y
#> 9 0 >= 5 Suspend Suspend Suspend
#> 9 1 <= 2 Y
#> 9 1 3 >= 0.77 < 0.77
#> 9 1 4 >= 2.15 < 2.15
#> 9 1 >= 5 Suspend Suspend Suspend
#> 9 2 0 Y
#> 9 2 1 > 0.52 <= 0.52
#> 9 2 2 > 1.59 <= 1.59
#> 9 2 3 > 2.66 <= 2.66
#> 9 2 4 > 3.73 <= 3.73
#> 9 2 >= 5 Suspend Suspend Suspend
#> 9 3 <= 6 Y
#> 9 >= 4 <= 5 Y & Elim
#> 12 0 <= 6 Y
#> 12 0 >= 7 Suspend Suspend Suspend
#> 12 1 <= 5 Y
#> 12 1 6 >= 1.24 < 1.24
#> 12 1 >= 7 Suspend Suspend Suspend
#> 12 2 <= 6 Y
#> 12 2 >= 7 Suspend Suspend Suspend
#> 12 3, 4 <= 9 Y
#> 12 >= 5 <= 7 Y & Elim
#> 15 0 <= 7 Y
#> 15 0 >= 8 Suspend Suspend Suspend
#> 15 1 <= 7 Y
#> 15 1 >= 8 Suspend Suspend Suspend
#> 15 2 <= 2 Y
#> 15 2 3 >= 1.14 < 1.14
#> 15 2 4 >= 2.31 < 2.31
#> 15 2 5 >= 3.48 < 3.48
#> 15 2 6 >= 4.65 < 4.65
#> 15 2 7 >= 5.82 < 5.82
#> 15 2 >= 8 Suspend Suspend Suspend
#> 15 3 <= 2 Y
#> 15 3 3 > 1.16 <= 1.16
#> 15 3 4 > 2.34 <= 2.34
#> 15 3 5 > 3.53 <= 3.53
#> 15 3 6 > 4.72 <= 4.72
#> 15 3 7 > 5.90 <= 5.90
#> 15 3 >= 8 Suspend Suspend Suspend
#> 15 4, 5 <= 11 Y
#> 15 >= 6 <= 9 Y & Elim
#>
#> Patients = treated at the current dose, DLTs = DLTs observed, Pending = assessment not yet completed
#> STFT = total follow-up time of the pending patients divided by the length of the assessment window
#> Accrual is suspended when more than 50% of the patients are pending
#> Y & Elim = de-escalate and eliminate this dose and all higher doses
# Nine patients, one DLT and four pending: escalate once STFT reaches 2.15
decisions[decisions$n == 9 & decisions$n_tox == 1 & decisions$n_pending == 4, ]
#> TITE-BOIN decision table with target 0.2
#> Method: single mean imputation (Yuan et al., 2018); boundaries refer to STFT
#>
#> Patients DLTs Pending Escalate Stay De-escalate
#> 9 1 4 >= 2.15 < 2.15
#>
#> Patients = treated at the current dose, DLTs = DLTs observed, Pending = assessment not yet completed
#> STFT = total follow-up time of the pending patients divided by the length of the assessment window
#> Accrual is suspended when more than 50% of the patients are pending
#> Y & Elim = de-escalate and eliminate this dose and all higher doses
# Table 1 of Chen et al. (2025): at least 51 percent of the patients with a
# completed assessment, and escalation only once every pending patient has
# been followed for a quarter of the window
decisions_chen <- tite_boin_decision_table(target = 0.3, max_n = 6,
max_pending_ratio = 0.49,
min_follow_up = 0.25)
print(decisions_chen, cohort_size = 3)
#> TITE-BOIN decision table with target 0.3
#> Method: single mean imputation (Yuan et al., 2018); boundaries refer to STFT
#>
#> Patients DLTs Pending Escalate Stay De-escalate
#> 3 0 0 Y
#> 3 0 1 MF >= 0.25
#> 3 0 >= 2 Suspend Suspend Suspend
#> 3 1 0 Y
#> 3 1 1 > 0.88 <= 0.88
#> 3 1 >= 2 Suspend Suspend Suspend
#> 3 2 <= 1 Y
#> 3 3 0 Y & Elim
#> 6 0 0 Y
#> 6 0 1-2 MF >= 0.25
#> 6 0 >= 3 Suspend Suspend Suspend
#> 6 1 0 Y
#> 6 1 1 MF >= 0.25
#> 6 1 2 >= 0.60 & MF >= 0.25 < 0.60
#> 6 1 >= 3 Suspend Suspend Suspend
#> 6 2 0 Y
#> 6 2 1 > 0.73 <= 0.73
#> 6 2 2 > 1.80 <= 1.80
#> 6 2 >= 3 Suspend Suspend Suspend
#> 6 3 <= 3 Y
#> 6 >= 4 <= 2 Y & Elim
#>
#> Patients = treated at the current dose, DLTs = DLTs observed, Pending = assessment not yet completed
#> STFT = total follow-up time of the pending patients divided by the length of the assessment window
#> Accrual is suspended when more than 49% of the patients are pending
#> MF = shortest follow-up time of the pending patients divided by the length of the assessment window
#> While patients are pending, escalation requires MF >= 0.25; otherwise accrual is suspended
#> Y & Elim = de-escalate and eliminate this dose and all higher doses
# The effective sample size of Lin and Yuan (2020)
decisions_ess <- tite_boin_decision_table(target = 0.3, max_n = 12, method = "ess")
print(decisions_ess, cohort_size = 3)
#> TITE-BOIN decision table with target 0.3
#> Method: effective sample size (Lin and Yuan, 2020); boundaries refer to ESS
#>
#> Patients DLTs Pending Escalate Stay De-escalate
#> 3 0 <= 1 Y
#> 3 0 >= 2 Suspend Suspend Suspend
#> 3 1 0 Y
#> 3 1 >= 1 > 2.79 <= 2.79
#> 3 2 <= 1 Y
#> 3 3 0 Y & Elim
#> 6 0 <= 4 Y
#> 6 0 >= 5 Suspend Suspend Suspend
#> 6 1 <= 1 Y
#> 6 1 2-3 >= 4.23 < 4.23
#> 6 1 4 >= 4.23 > 2.79, < 4.23 <= 2.79
#> 6 1 >= 5 Suspend if >= 4.23 > 2.79, < 4.23 <= 2.79
#> 6 2 0 Y
#> 6 2 >= 1 > 5.58 <= 5.58
#> 6 3 <= 3 Y
#> 6 >= 4 <= 2 Y & Elim
#> 9 0 <= 7 Y
#> 9 0 >= 8 Suspend Suspend Suspend
#> 9 1 <= 4 Y
#> 9 1 5-6 >= 4.23 < 4.23
#> 9 1 7 >= 4.23 > 2.79, < 4.23 <= 2.79
#> 9 1 >= 8 Suspend if >= 4.23 > 2.79, < 4.23 <= 2.79
#> 9 2 0 Y
#> 9 2 1-3 >= 8.46 < 8.46
#> 9 2 >= 4 >= 8.46 > 5.58, < 8.46 <= 5.58
#> 9 3 0 Y
#> 9 3 >= 1 > 8.37 <= 8.37
#> 9 4 <= 5 Y
#> 9 >= 5 <= 4 Y & Elim
#> 12 0 <= 10 Y
#> 12 0 >= 11 Suspend Suspend Suspend
#> 12 1 <= 7 Y
#> 12 1 8-9 >= 4.23 < 4.23
#> 12 1 10 >= 4.23 > 2.79, < 4.23 <= 2.79
#> 12 1 >= 11 Suspend if >= 4.23 > 2.79, < 4.23 <= 2.79
#> 12 2 <= 3 Y
#> 12 2 4-6 >= 8.46 < 8.46
#> 12 2 >= 7 >= 8.46 > 5.58, < 8.46 <= 5.58
#> 12 3 <= 3 Y
#> 12 3 >= 4 > 8.37 <= 8.37
#> 12 4 0 Y
#> 12 4 >= 1 > 11.16 <= 11.16
#> 12 5, 6 <= 7 Y
#> 12 >= 7 <= 5 Y & Elim
#>
#> Patients = treated at the current dose, DLTs = DLTs observed, Pending = assessment not yet completed
#> STFT = total follow-up time of the pending patients divided by the length of the assessment window
#> ESS = number of patients with a completed assessment plus STFT
#> While patients are pending, escalation requires at least 2 patients with a completed assessment
#> Y & Elim = de-escalate and eliminate this dose and all higher doses