Time-to-Event CRM (TITE-CRM): A Worked Example

because the working assumption is that higher doses are associated with greater toxicity probability.

The skeleton is not necessarily treated as the true toxicity curve. Instead, it provides the initial shape of the dose-toxicity relationship that is subsequently updated using observed data.

The Target Toxicity Level

The investigator specifies a target probability of DLT, denoted by:

For example, suppose the target toxicity probability is 25%:

The goal is to identify the dose whose estimated DLT probability is closest to 25%.

This dose is often referred to as the estimated maximum tolerated dose, or MTD, although the precise terminology and decision rule should be defined in the protocol.

The CRM Dose-Toxicity Model

A common one-parameter CRM model uses the power model:

where:

• \(p_j\) is the skeleton probability at dose \(j\)
• \(\theta\) is an unknown model parameter
• \(p_j(\theta)\) is the modeled probability of DLT at dose \(j\)

The exponential transformation ensures that the model remains monotone in the dose index when the skeleton is monotone.

When:

the model reproduces the skeleton:

If \(\theta>0\), the probabilities are shifted downward because \(\exp(\theta)>1\) and \(0

If \(\theta<0\), the probabilities are shifted upward.

The Bayesian Prior

CRM uses a prior distribution for the unknown parameter.

For this tutorial, suppose:

The posterior distribution is then obtained by combining this prior with the likelihood contributed by the observed patient data.

In practice, prior specification and skeleton calibration should be performed carefully because the prior can influence early dose assignments, particularly when little information has accumulated.

What Changes in TITE-CRM?

The fundamental CRM model remains the same.

The key modification is the treatment of incomplete follow-up.

For patient \(i\), define:

• \(Y_i=1\) if a DLT has occurred
• \(Y_i=0\) if no DLT has yet occurred
• \(u_i\) = amount of follow-up currently observed
• \(\tau\) = planned DLT observation window

For the basic TITE-CRM weighting scheme:

The no-DLT weight is typically bounded at 1:

A patient who experiences a DLT receives weight 1 because the event has already been observed.

A patient with no DLT receives a weight reflecting the fraction of the planned observation period that has elapsed.

Examples of TITE Weights

The weight is therefore a measure of the amount of available follow-up under this simple weighting approach.

How the Weighted Likelihood Works

Suppose patient \(i\) is treated at dose \(j\), and let the modeled DLT probability at that dose be:

For a conventional binary outcome, the likelihood contribution would be:

TITE-CRM raises this contribution to the patient's follow-up weight:

Equivalently:

The total weighted likelihood is:

The posterior is proportional to the prior multiplied by this likelihood:

Why a DLT Gets Weight 1

Suppose a patient develops a DLT on day 10 of a 28-day observation window.

The DLT is already known.

There is no need to discount the event simply because the patient did not complete the remaining 18 days.

The relevant outcome for the dose-toxicity model is already observed:

The remaining follow-up may matter for other safety endpoints, but it does not make the already-observed DLT more or less of a DLT.

Why a DLT-Free Patient Gets a Partial Weight

Now suppose a patient has completed only 7 days without a DLT.

There are still 21 days during which the patient could experience a DLT.

Under the simple linear weighting approach:

The patient therefore contributes a fractional likelihood contribution.

This allows the trial to use the accumulating information without pretending that the patient's outcome is already fully known.

Staggered Patient Entry

The major operational advantage of TITE-CRM is that patients do not necessarily have to wait for previous patients to complete the entire DLT observation window.

For example:

This can substantially reduce trial duration when toxicity outcomes are late relative to the desired enrollment rate.

A Complete Worked Example

Consider an oncology Phase I trial evaluating five dose levels.

The primary endpoint is the occurrence of a dose-limiting toxicity during a 28-day observation period.

The target toxicity probability is:

The investigators specify the following skeleton:

The working model is:

and the prior is:

Trial Configuration

Patient-Level Data at an Interim Look

Suppose eight patients have entered the study.

The important feature is that several patients are still being followed.

Patients 3, 4, and 7 have not completed their full 28-day DLT windows.

Nevertheless, they contribute information through their partial follow-up weights.

Step 1: Calculate the Follow-Up Weights

For Patient 3:

For Patient 4:

For Patient 7:

Patient 6 experienced a DLT, so:

The complete vector of weights is therefore:

Step 2: Write the Dose-toxicity Model

The five skeleton probabilities are:

For any value of \(\theta\), the modeled probabilities are:

Step 3: Construct the Weighted Likelihood

Patient 1 was treated at Dose 1 and had no DLT with complete follow-up. His contribution is:

Patient 3 was treated at Dose 2 and had no DLT after 21 days. His contribution is:

Patient 4 was treated at Dose 3 and had no DLT after 14 days. His contribution is:

Patient 6 experienced a DLT at Dose 4. His contribution is:

Patient 7 was treated at Dose 4 and had no DLT after 7 days. His contribution is:

Multiplying all eight contributions gives the complete weighted likelihood.

The posterior distribution is proportional to:

Step 4: Estimate the Posterior Parameter

The posterior distribution does not generally have a convenient closed-form expression under this model.

Numerical integration can therefore be used to calculate the posterior mean:

For the illustrative dataset above, numerical integration gives approximately:

This is the posterior mean used in the subsequent dose-toxicity calculation.

Step 5: Convert the Posterior Parameter into DLT Probabilities

Using:

the fitted DLT probabilities are approximately:

The target toxicity probability is:

Dose 4 has an estimated DLT probability of approximately 26.2%, which is very close to the target.

Step 6: Select the Dose Closest to the Target

The standard CRM decision rule is:

For the example:

Therefore:

Under this illustrative model and dataset, Dose 4 is the dose whose posterior estimated DLT probability is closest to the 25% target.

What If Patient 7 Had Completed 28 Days?

This provides a useful way to understand the weighting mechanism.

At seven days:

After completing the full observation window:

The patient's contribution would therefore become:

rather than:

The statistical information contributed by the patient increases as follow-up accumulates.

What If Patient 7 Develops a DLT?

Suppose Patient 7 develops a DLT on day 12.

The patient is no longer treated as a pending DLT-free observation.

Instead:

The contribution becomes:

This is an important property of time-to-event designs: the exact timing of the event determines when the event becomes known, while a DLT-free patient accumulates information progressively over time.

Conventional CRM vs. TITE-CRM

The Core Difference in One Equation

The conventional CRM uses a complete binary outcome:

TITE-CRM supplements this with:

and therefore uses:

within the weighted likelihood.

The conceptual transition is:

Why TITE-CRM Can Shorten a Trial

Suppose the DLT window is 28 days.

A traditional sequential design that waits for every patient to complete the window could require approximately 28 days between dose decisions.

TITE-CRM can allow patients to enter while previous patients are still being followed.

The actual trial duration then depends on:

• Patient accrual rate
• DLT observation-window length
• Timing of observed DLTs
• Frequency of interim model updates
• Operational safety rules
• Maximum sample size

Therefore, TITE-CRM is particularly attractive when the toxicity observation window is long relative to the desired accrual interval.

But TITE-CRM Does Not Eliminate Uncertainty

A patient with seven days of DLT-free follow-up is not equivalent to a patient with 28 days of DLT-free follow-up.

The weighting scheme attempts to reflect this difference.

Consequently, the model is making decisions with incomplete information.

Time-to-Event Information Is Not Necessarily Linear

The simple weight:

assumes that information accumulates linearly with follow-up.

That assumption is convenient but need not be clinically realistic.

If DLTs are much more likely early in the observation window, seven days of follow-up may provide more information than one quarter of the total information.

Conversely, if DLT risk is concentrated near the end of the window, linear weighting may overstate the information obtained early.

This motivates alternative time-to-event weighting approaches.

Generalized Follow-Up Weights

Instead of a linear weight, one can define:

where \(g(\cdot)\) reflects an assumed pattern of information accumulation.

A simple linear approach is:

More sophisticated approaches can incorporate assumptions about the hazard or timing distribution of DLTs.

Safety Restrictions Are Essential

The original CRM framework attracted criticism because an unconstrained model could recommend aggressive dose escalation when little toxicity information was available.

Modern implementations commonly incorporate safety modifications.

Examples include:

• No dose skipping during escalation
• Maximum permitted escalation per decision
• Minimum number of patients before escalation
• Requirements for sufficient follow-up at a dose
• Posterior overdose-control rules
• Trial stopping rules
• Maximum number of patients treated at a dose

These modifications are not interchangeable with the underlying TITE-CRM model. They are additional design components intended to control the practical risks of adaptive dose assignment.

No Dose Skipping

A common safety restriction is that the trial may not skip an untested dose during escalation.

For example, if the current dose is Dose 2, the next patient may be permitted to receive Dose 3 but not Dose 4.

Whether dose skipping is allowed is a design choice and must be explicitly specified.

Overdose Control

Another approach is to require that a dose have sufficiently low posterior probability of being excessively toxic before it can be selected.

Suppose the target is:

and investigators define an overdose threshold:

A possible safety criterion could require:

where \(c\) is a prespecified probability cutoff.

This is conceptually different from simply choosing the dose with toxicity probability closest to the target.

Handling Pending Patients

One of the most important practical questions in TITE-CRM is how pending patients are represented.

Suppose the trial currently has:

The model incorporates all four patients simultaneously.

Patient A provides a complete DLT event.

Patients B and C provide partial information.

Patient D provides a complete DLT-free observation.

This is what makes TITE-CRM different from simply analyzing completed patients.

Repeated Model Updating

TITE-CRM is called a continual reassessment method because the model is re-estimated as new information becomes available.

For example:

Stopping Rules

A TITE-CRM protocol should include explicit stopping rules.

Examples include:

• Maximum total sample size reached
• Maximum number of patients treated at a dose reached
• Excessive posterior probability that all doses exceed the target
• Insufficient evidence that any dose is acceptably safe
• Predefined safety signal
• Operational stopping criteria

The exact stopping criteria are design-specific.

Final MTD Selection

At the end of the trial, the estimated MTD is generally selected using the same prespecified model and decision rule used during dose escalation.

For example:

where \(\hat{\theta}\) is the final posterior summary specified by the protocol, such as the posterior mean.

The final selection rule should be prespecified rather than chosen after examining the final toxicity curve.

TITE-CRM Is a Bayesian Adaptive Design

There are three important components:

The Bayesian updating mechanism is therefore:

Interpreting the Posterior Dose-Toxicity Curve

The skeleton provides the initial curve.

The patient data shift the curve according to the evidence accumulated during the trial.

A DLT generally pushes the fitted toxicity curve upward.

DLT-free follow-up generally pushes the curve downward, with the magnitude of the contribution depending on the available follow-up.

This produces a continually evolving estimate of the toxicity probability at each dose.

What Happens After a DLT?

Suppose a DLT occurs at Dose 4.

The DLT contributes a complete event:

The posterior toxicity curve will generally shift upward.

The next dose recommendation may therefore remain at Dose 4, move downward, or—depending on the complete accumulated data and safety restrictions—remain near the same dose.

The important point is that the decision uses the entire accumulated dataset, not simply the most recent patient.

What Happens After a DLT-Free Patient?

Suppose a patient at Dose 4 remains DLT-free for 14 days.

The weight is:

The contribution is:

This provides evidence that the toxicity probability at Dose 4 may not be as high as initially assumed.

However, because the patient is still within the DLT window, the evidence is only partial.

Why the Model Borrows Information Across Doses

A major difference between CRM and simple rule-based designs is that the model connects the dose levels.

Suppose there are no observed DLTs at Dose 3.

That information does not affect only Dose 3.

Because the dose-toxicity model is shared across doses, the posterior estimate of \(\theta\) changes, which changes the estimated toxicity probabilities at all dose levels:

This is one reason CRM can be more statistically efficient than independent dose-by-dose rules.

TITE-CRM vs. 3+3

The important distinction is not that one design is universally superior. Rather, the designs solve the dose-finding problem using fundamentally different statistical mechanisms.

TITE-CRM vs. TITE-BOIN

TITE-CRM and TITE-BOIN both address the same operational problem: DLT outcomes may not be immediately available.

Their underlying statistical approaches differ.

Both approaches can be useful, but they should not be treated as equivalent algorithms.

R Implementation: Define the Design

The following R code establishes the example design.

skeleton <- c(
  0.05,
  0.10,
  0.20,
  0.30,
  0.45
)

target <- 0.25

tau <- 28

sigma <- 1.34

max_n <- 12

Create the Patient-Level Dataset

data <- data.frame(
  patient = 1:8,

  dose = c(
    1, 2, 2, 3,
    3, 4, 4, 3
  ),

  dlt = c(
    0, 0, 0, 0,
    0, 1, 0, 0
  ),

  followup = c(
    28, 28, 21, 14,
    28, 10, 7, 28
  )
)

data

Calculate TITE Weights

data$weight <- ifelse(
  data$dlt == 1,
  1,
  pmin(data$followup / tau, 1)
)

data

The resulting weights are:

data$weight

# 1.00
# 1.00
# 0.75
# 0.50
# 1.00
# 1.00
# 0.25
# 1.00

Define the CRM Model

tox_prob <- function(theta, dose) {

  skeleton[dose]^exp(theta)

}

For example, the Dose 4 probability is:

tox_prob(
  theta = 0.106,
  dose = 4
)

# approximately 0.262

Define the Weighted Log-Likelihood

log_likelihood <- function(theta, data) {

  p <- tox_prob(
    theta,
    data$dose
  )

  sum(
    data$weight *
      (
        data$dlt * log(p) +
        (1 - data$dlt) * log(1 - p)
      )
  )
}

The weighting enters directly into the likelihood.

A fully observed DLT-free patient contributes weight 1. A patient with only 25% of the planned follow-up contributes weight 0.25.

Define the Normal Prior

log_prior <- function(theta) {

  dnorm(
    theta,
    mean = 0,
    sd = sigma,
    log = TRUE
  )

}

Construct the Log Posterior

log_posterior <- function(theta, data) {

  log_likelihood(theta, data) +
    log_prior(theta)

}

The posterior is proportional to:

Numerically Integrate the Posterior

One simple approach is to evaluate the posterior on a sufficiently fine grid of possible \(\theta\) values.

theta_grid <- seq(
  -3,
  1.5,
  length.out = 18001
)

log_post <- sapply(
  theta_grid,
  log_posterior,
  data = data
)

post <- exp(
  log_post -
    max(log_post)
)

post <- post /
  sum(post)

posterior_mean <- sum(
  theta_grid * post
)

posterior_mean

For the illustrative dataset:

posterior_mean

# approximately 0.106

Calculate Posterior DLT Probabilities

posterior_probs <- tox_prob(
  posterior_mean,
  1:length(skeleton)
)

posterior_probs

The resulting probabilities are approximately:

# Dose 1   0.0358
# Dose 2   0.0773
# Dose 3   0.1671
# Dose 4   0.2622
# Dose 5   0.4116

Select the Dose Closest to the Target

next_dose <- which.min(
  abs(
    posterior_probs -
      target
  )
)

next_dose

# 4

Therefore, the model recommends Dose 4 in this illustrative example.

Implement a No-Skipping Restriction

Suppose the current dose is Dose 2. Even if the unrestricted CRM recommends Dose 4, a no-skipping rule can restrict the next dose to Dose 3.

current_dose <- 2

allowed_doses <- c(
  current_dose - 1,
  current_dose,
  current_dose + 1
)

allowed_doses <- allowed_doses[
  allowed_doses >= 1 &
  allowed_doses <= length(skeleton)
]

restricted_dose <-
  allowed_doses[
    which.min(
      abs(
        posterior_probs[
          allowed_doses
        ] - target
      )
    )
  ]

restricted_dose

This separates two concepts:

• Statistical recommendation: what the model would choose.
• Operational assignment: what the prespecified safety rules permit.

A Simple TITE-CRM Algorithm in R

tite_crm <- function(
  data,
  skeleton,
  target = 0.25,
  tau = 28,
  sigma = 1.34
) {

  data$weight <- ifelse(
    data$dlt == 1,
    1,
    pmin(data$followup / tau, 1)
  )

  tox_prob <- function(
    theta,
    dose
  ) {
    skeleton[dose]^exp(theta)
  }

  log_likelihood <- function(
    theta
  ) {

    p <- tox_prob(
      theta,
      data$dose
    )

    sum(
      data$weight *
        (
          data$dlt * log(p) +
          (1 - data$dlt) *
          log(1 - p)
        )
    )
  }

  theta_grid <- seq(
    -3,
    1.5,
    length.out = 18001
  )

  log_post <- sapply(
    theta_grid,
    function(theta) {
      log_likelihood(theta) +
        dnorm(
          theta,
          0,
          sigma,
          log = TRUE
        )
    }
  )

  post <- exp(
    log_post -
      max(log_post)
  )

  post <- post /
    sum(post)

  theta_hat <-
    sum(
      theta_grid * post
    )

  fitted_probs <-
    tox_prob(
      theta_hat,
      1:length(skeleton)
    )

  recommended_dose <-
    which.min(
      abs(
        fitted_probs -
          target
      )
    )

  list(
    posterior_mean = theta_hat,
    toxicity_probabilities =
      fitted_probs,
    recommended_dose =
      recommended_dose
  )
}

Run the Example

result <- tite_crm(
  data = data,
  skeleton = skeleton,
  target = 0.25
)

result

The important outputs are:

result$posterior_mean

result$toxicity_probabilities

result$recommended_dose

Interpreting the R Output

The posterior mean summarizes the updated dose-toxicity curve.

The toxicity probability vector gives the estimated DLT probability for each dose.

The recommended dose identifies the dose closest to the target toxicity probability.

For the worked example:

and:

so Dose 4 is closest to:

Calculating the Information Contribution of Each Patient

A useful diagnostic is to display each patient's weight.

Why Follow-Up Time Is Different From Calendar Time

TITE-CRM is driven by patient-level follow-up, not simply by the number of days since the trial started.

For example, two patients could enter the study on the same calendar day but have different observation histories because of treatment delays or different event timing.

The statistical weight should therefore be calculated from the prespecified follow-up definition used by the protocol.

What Counts as Follow-Up?

The protocol should define what constitutes valid observation time.

Potential considerations include:

• Date of first treatment
• Date of DLT onset
• Date of treatment discontinuation
• Date of withdrawal
• Date of loss to follow-up
• Date of data cutoff

The exact handling of treatment discontinuation or loss to follow-up is a protocol-level decision.

Interim Dose Assignment Timing

TITE-CRM does not imply that the model must be updated continuously after every minute of follow-up.

The protocol can specify decision times based on:

• Each new patient
• Each cohort
• Scheduled interim reviews
• Availability of new DLT information

The operational schedule should be selected together with the accrual plan and safety monitoring process.

Accrual Rate Matters

The benefit of TITE-CRM depends heavily on how quickly patients enter the trial.

Suppose the DLT window is 28 days.

If a new patient enters every 30 days, there is little need for a time-to-event design because the previous patient will generally have completed follow-up.

If a new patient enters every three days, however, many patients may be simultaneously pending.

TITE-CRM becomes much more relevant in the second situation.

Accrual and Follow-Up Timeline

At calendar day 21, every patient contributes some information.

A conventional approach that required complete follow-up might still have only one fully evaluable patient.

The Price of Staggered Enrollment

The advantage of faster accrual is accompanied by greater dependence on the statistical model.

At any interim decision, some patients may have very little follow-up.

Consequently, the design should be calibrated under realistic combinations of:

• True dose-toxicity curves
• DLT probabilities
• DLT event-time distributions
• Patient accrual rates
• Dropout rates
• Starting dose
• Cohort size
• Safety restrictions

Simulation Is Essential

Unlike a simple fixed-sample calculation, a TITE-CRM design has many interacting components.

A simulation study can estimate:

• Probability of selecting the true target dose
• Probability of overdosing patients
• Average number of patients treated at each dose
• Average trial duration
• Probability of early stopping
• Number of dose escalations
• Number of dose de-escalations
• Probability of selecting a dose above the target
• Probability of selecting a dose below the target

Example Simulation Scenario

Suppose the true DLT probabilities are:

The target is 25%, so Dose 4 is the true target dose.

A simulation can repeatedly generate trials under this toxicity curve and record how frequently the design selects Dose 4.

Generating a DLT Outcome

For a simulated patient treated at Dose \(j\), generate:

If \(Y_i=1\), generate a DLT time from the assumed event-time distribution.

If \(Y_i=0\), the patient remains DLT-free through the current follow-up time.

The simulation then advances calendar time according to the planned accrual process.

Operating Characteristics

A simulation report might include:

Common TITE-CRM Design Choices

Several choices must be made before the study begins.

• Target toxicity probability
• Number and spacing of dose levels
• Skeleton probabilities
• Working dose-toxicity model
• Prior distribution
• Follow-up weighting function
• Starting dose
• Cohort size
• Accrual schedule
• Maximum sample size
• No-skipping restrictions
• Overdose-control rules
• Minimum follow-up requirements
• Stopping criteria
• Final MTD selection rule

Choosing the Starting Dose

The starting dose should not simply be the dose that happens to be closest to the target according to the skeleton.

The initial dose should reflect the total clinical context, including preclinical evidence, prior human experience, pharmacology, and safety considerations.

The statistical model should not be treated as a substitute for clinical judgment about a safe starting dose.

Choosing the Skeleton

The skeleton should represent plausible prior toxicity probabilities across the dose range.

For example:

implies increasing toxicity with increasing dose.

The target of 25% lies between the skeleton probabilities for Dose 3 and Dose 4.

This is a sensible structure for an illustrative example because the dose range contains levels on both sides of the target.

Why the Dose Range Matters

Suppose the skeleton were:

If the true target is 25%, every dose is far below the target.

The trial would have difficulty identifying a dose near the target because the available dose range is poorly positioned.

Conversely, if every skeleton probability were far above the target, the dose range could be excessively toxic.

Model Misspecification

The CRM model is a working model.

The true dose-toxicity relationship may not exactly follow:

This is one reason simulation studies should consider multiple possible true toxicity curves rather than only the curve assumed by the working model.

For example, simulations might consider scenarios in which:

• Toxicity increases slowly at low doses and rapidly at high doses.
• Toxicity increases rapidly at low doses and then plateaus.
• The skeleton is approximately correct.
• The skeleton is substantially misspecified.

The Long-Memory Property

CRM methods can exhibit a phenomenon sometimes described as long memory: early observations can continue to influence the posterior model throughout the trial.

This is a consequence of the model-based approach.

A few early DLTs can substantially affect the posterior curve, particularly when the sample size is small.

This is not necessarily a flaw, but it highlights the importance of:

• Prior calibration
• Starting-dose selection
• Safety restrictions
• Simulation under alternative scenarios

Pending Patients and Safety Decisions

One of the most important operational questions is whether the trial should allow escalation when many patients are still incompletely observed.

A statistical model can incorporate partial information, but the protocol may still impose additional restrictions.

For example, a design might require:

overdose_limit <- 0.35

dose4_probs <-
  tox_prob(
    theta_grid,
    dose = 4
  )

posterior_overdose_prob <-
  sum(
    post[
      dose4_probs >
        overdose_limit
    ]
  )

posterior_overdose_prob