Defining Overdose in a BLRM

What “Overdose” Means in a BLRM

In dose escalation, “overdose” is a model-based statement about the toxicity probability at a particular dose, judged against a limit chosen before examining the accumulating trial results. It is not simply a label for a dose that seems high, a dose outside the target interval, or a cohort in which many participants experienced a dose-limiting toxicity (DLT). Defining the event precisely matters because later safety decisions depend on distinguishing these ideas.

The target toxicity interval has a lower limit and an upper limit. A dose-specific DLT probability above the upper limit is an overdose under the definition used here. A probability below the lower limit is below target, but it is not an overdose. This distinction holds even though both probabilities lie outside the target interval.

Definition: Let $p(d;\theta)$ be the BLRM probability of a DLT at dose $d$, for model parameters $\theta$. Let $\phi_U$ be the prespecified upper toxicity limit of the target interval. The dose-specific overdose event is $$ \mathcal{O}(d)=\{p(d;\theta)>\phi_U\}. $$ Thus, a dose is an overdose at a particular value of $\theta$ only when its modeled DLT probability is strictly greater than $\phi_U$. A probability equal to $\phi_U$ is not greater than the limit under this definition.

The parameter vector $\theta$ is not known with certainty in a trial. Consequently, $\mathcal{O}(d)$ is an event about the unknown model parameters, not a directly observed fact. A posterior distribution over $\theta$ can quantify uncertainty about whether this event holds. The event itself must be defined first: it is the comparison of the dose-specific probability with $\phi_U$, not a comparison of an observed DLT count with a threshold.

Connecting the Event to the BLRM

For clarity, write the two-parameter BLRM in the usual dose-scaled form. At a positive dose $d$, with reference dose $d_{\mathrm{ref}}$, the model is

\operatorname{logit}\{p(d;\theta)\} = \log\left(\frac{p(d;\theta)}{1-p(d;\theta)}\right) = \beta_0+\beta_1\log\left(\frac{d}{d_{\mathrm{ref}}}\right), \qquad \theta=(\beta_0,\beta_1).

The probability is therefore

p(d;\theta) = \frac{\exp\left\{\beta_0+\beta_1\log(d/d_{\mathrm{ref}})\right\}} {1+\exp\left\{\beta_0+\beta_1\log(d/d_{\mathrm{ref}})\right\}}.

Here the outcome is whether a participant experiences a DLT within the trial’s defined observation window at the specified dose and regimen. The model treats that outcome as binary and describes its probability. The DLT definition and observation window must be applied consistently: changing which events count as DLTs, or how long participants are followed, changes the outcome whose probability is being modeled.

The overdose definition uses the probability scale, where the clinical upper limit $\phi_U$ is specified. The same comparison can be expressed on the log-odds scale, because the logit function is strictly increasing for probabilities between zero and one.

Why the Log-Odds Comparison Is Equivalent

The equivalence follows directly from the odds transformation. For $0<\phi_U<1$, start with the overdose condition and multiply by the positive denominator $1-p(d;\theta)$. Then take logarithms, which preserve the inequality because both sides are positive.

\begin{aligned} p(d;\theta)>\phi_U &\iff \frac{p(d;\theta)}{1-p(d;\theta)} > \frac{\phi_U}{1-\phi_U}\\ &\iff \operatorname{logit}\{p(d;\theta)\} > \operatorname{logit}(\phi_U). \end{aligned}

This is a useful check on interpretation: the threshold in the model’s linear predictor is the logit of the upper toxicity limit. It does not change the clinical definition; it is the same event expressed in different units. The calculation below uses the equivalence at a fixed dose. It does not, by itself, identify a dose at which the threshold is crossed.

Worked Example: Classifying Two Doses Against the Upper Limit

Suppose a hypothetical BLRM has $\beta_0=-2.4$, $\beta_1=1.8$, and $d_{\mathrm{ref}}=100$ mg. For this example, let the target interval be $[0.20,0.33]$, so $\phi_U=0.33$. Calculate the modeled DLT probabilities at 200 mg and 400 mg and classify each against the overdose definition.

At 200 mg, $\log(200/100)=\log(2)\approx0.6931$. The linear predictor and probability are

\begin{aligned} \eta(200) &=-2.4+1.8(0.6931)\\ &=-1.1524,\\ p(200;\theta) &=\frac{\exp(-1.1524)}{1+\exp(-1.1524)} \approx 0.240. \end{aligned}

Since $0.20\leq0.240\leq0.33$, this dose is in the target interval and is not an overdose. At 400 mg, $\log(400/100)=\log(4)\approx1.3863$, giving

\begin{aligned} \eta(400) &=-2.4+1.8(1.3863)\\ &=0.0953,\\ p(400;\theta) &=\frac{\exp(0.0953)}{1+\exp(0.0953)} \approx 0.524. \end{aligned}

Because $0.524>0.33$, the 400 mg dose satisfies the overdose event at these parameter values. The classification is based on the modeled probability and the stated upper limit, not on whether the dose is numerically larger than another dose.

Overdose Is Not the Same as Being Outside the Target Interval

Suppose the target interval is $[\phi_L,\phi_U]$. There are three distinct comparisons: below target when $p(d;\theta)<\phi_L$; in target when $\phi_L\leq p(d;\theta)\leq\phi_U$; and overdose when $p(d;\theta)>\phi_U$. Under this convention, equality at either endpoint belongs to the interval. A probability below the lower endpoint is outside the interval, but it does not meet the overdose definition.

This distinction also separates the overdose event from the maximum tolerated dose (MTD) and from a dose recommendation. Those concepts answer different questions. The overdose event asks whether the modeled DLT probability at a specified dose exceeds a toxicity limit. An MTD is a dose defined by a toxicity criterion, while a dose recommendation is a decision that may also account for uncertainty, available doses, and trial rules. They should not be treated as interchangeable labels.

Worked Example: Below Target Is Not Overdose

Consider a hypothetical model with $\beta_0=-1.8$, $\beta_1=1.2$, and $d_{\mathrm{ref}}=50$ mg. Let the target interval be $[0.20,0.30]$. Calculate the modeled probabilities at 50 mg, 100 mg, and 200 mg.

At 50 mg, the log-dose term is zero:

p(50;\theta) = \frac{\exp(-1.8)}{1+\exp(-1.8)} \approx 0.142.

This probability is below the lower target limit of $0.20$. It is below target, not an overdose, because $0.142$ is not greater than the upper limit $0.30$.

At 100 mg, $\log(100/50)=\log(2)\approx0.6931$, so

\begin{aligned} \eta(100)&=-1.8+1.2(0.6931)=-0.9683,\\ p(100;\theta)&=\frac{\exp(-0.9683)}{1+\exp(-0.9683)} \approx0.275. \end{aligned}

Since $0.20\leq0.275\leq0.30$, 100 mg is in the target interval. At 200 mg, $\log(200/50)=\log(4)\approx1.3863$, which gives

\begin{aligned} \eta(200)&=-1.8+1.2(1.3863)=-0.1364,\\ p(200;\theta)&=\frac{\exp(-0.1364)}{1+\exp(-0.1364)} \approx0.466. \end{aligned}

Because $0.466>0.30$, 200 mg is an overdose at these parameter values. The three classifications are therefore below target, in target, and overdose, respectively.

Assumptions and What the Classification Does Not Say

The definition is simple, but its interpretation depends on a coherent model and a clearly specified clinical threshold. In applying it, check the following assumptions and choices:

DLT outcome: the binary outcome, DLT criteria, and observation window are specified in advance and applied consistently across participants.
Dose and regimen: $d$ refers to the dose and schedule represented by the model. A probability for one regimen should not silently be interpreted as the probability for a different schedule or combination.
Threshold: $\phi_U$ is the protocol’s prespecified upper toxicity limit, expressed as a probability on the same DLT definition and time scale. It is not selected after seeing which value would make a preferred dose admissible.
Model interpretation: the BLRM probability is conditional on its parameters and assumptions. If the model uses a positive dose slope, it encodes increasing toxicity probability with increasing dose; this monotonicity is a modeling assumption, not a guarantee that every observed cohort rate will increase.
Parameter uncertainty: the unknown $\theta$ means that a single estimated probability does not fully represent uncertainty about the overdose event. A fitted curve or point estimate can be informative, but it is not the event’s posterior probability.

A posterior distribution over the parameters can assign uncertainty to whether $p(d;\theta)>\phi_U$. Keep the two objects distinct: $\mathcal{O}(d)$ is the event for a given unknown parameter value, while a posterior probability summarizes how much posterior mass lies in that event. Neither object is the same as the observed fraction of participants with DLTs.

Worked Example: An Observed Rate Is Not the Overdose Definition

Suppose a hypothetical cohort of 10 participants treated at a dose has 6 DLTs, giving an observed cohort proportion of $6/10=0.60$. For illustration, suppose a particular BLRM parameter value gives a modeled DLT probability of $0.28$ at that dose, and the prespecified upper limit is $\phi_U=0.30$. Does the observed proportion make the dose an overdose under the definition?

No. At the stated parameter value, the event is determined by comparing the model probability with the upper limit:

0.28>0.30\quad\text{is false}, \qquad\text{so}\qquad \mathcal{O}(d)\text{ does not hold at this parameter value.}

The observed proportion is $0.60$, but it is a sample statistic based on 10 outcomes, not the model’s underlying probability. The discrepancy deserves examination: the data may shift the posterior, the model may fit poorly, or random variation may produce a cohort proportion far from the underlying probability. This example does not establish that the dose is safe or appropriate. It shows only why an observed cohort rate cannot be substituted for the defined overdose event.

Diagnostics, Alternatives, and Common Pitfalls

Before using an overdose classification, check the event’s inputs rather than relying on a label in a table. Confirm that the probability and upper limit refer to the same DLT definition, follow-up window, and regimen. Verify that the probability is between zero and one, that the intended dose has been used in the dose scaling, and that the upper limit is the protocol value. Check the direction of the comparison: overdose requires $p(d;\theta)>\phi_U$, not $p(d;\theta)<\phi_U$. If the log-odds form is used, confirm that the comparison is with $\operatorname{logit}(\phi_U)$.

A dose may appear to have an unexpected classification when the model’s dose scaling is entered incorrectly, a threshold is expressed on the wrong scale, or a different DLT window is used. A mismatch between the observed data and modeled curve is another warning to investigate model fit and uncertainty; it is not a reason to redefine overdose after seeing the data. With a monotone positive-slope BLRM, the modeled probability increases with dose for fixed parameters, but this does not mean observed DLT proportions must rise across successive cohorts.

Quantity or rule	Question it answers	Why it is not interchangeable with overdose
Target interval $[\phi_L,\phi_U]$	Is the modeled probability below, within, or above the desired range?	Only being above $\phi_U$ is overdose; being below $\phi_L$ is below target.
Observed DLT proportion	What fraction of an observed cohort had DLTs?	It is a sample statistic, not the underlying dose-specific probability.
MTD	Which dose meets a specified maximum-tolerated-dose criterion?	Its definition depends on the criterion and is not, by itself, a full statement about uncertainty or recommendation.
Dose recommendation	Which dose should the trial consider under its decision rules?	It is a decision that may use overdose uncertainty and other constraints, not the overdose event alone.

Common mistake: “The modeled probability is below the lower target limit, so this dose is an overdose because it is outside the target interval.” The correction is to compare with the upper limit specifically. A probability below $\phi_L$ is below target; overdose means strictly greater than $\phi_U$. Another common mistake is to call a dose an overdose solely because one cohort had a high observed DLT proportion. The correction is to distinguish that observed proportion from the BLRM probability and to account for uncertainty rather than treating a small cohort’s rate as the underlying truth.

Key takeaway: In a BLRM, overdose at dose $d$ means the model’s dose-specific DLT probability exceeds the prespecified upper toxicity limit: $p(d;\theta)>\phi_U$. It is distinct from being below the target interval, from an observed cohort DLT rate, and from a final dose recommendation.

Check Your Understanding

Answer each question using the strict upper-limit definition, then compare your reasoning with the solutions.

The target interval is $[0.18,0.28]$, and a dose has modeled probability $0.15$. Is it an overdose, in target, or below target?
With the same interval, a dose has modeled probability $0.28$. Is it an overdose under the definition in this tutorial?
A cohort has 4 DLTs among 6 participants, while the modeled probability at a stated parameter value is $0.25$ and $\phi_U=0.30$. Does the observed cohort proportion alone establish the overdose event?
Why is comparing $\operatorname{logit}\{p(d;\theta)\}$ with $\operatorname{logit}(\phi_U)$ equivalent to comparing the probabilities directly?

Solutions. (1) $0.15<0.18$, so the dose is below target, not an overdose. (2) No. The probability equals the upper limit, and the definition requires a value strictly greater than $0.28$. (3) No. The observed proportion is $4/6\approx0.667$, but it is not the modeled probability; at the stated parameter value, $0.25$ is not greater than $0.30$. The observed data may affect uncertainty, but the cohort proportion alone does not define the event. (4) The logit function is strictly increasing on probabilities between zero and one, so it preserves the direction of the inequality: $p(d;\theta)>\phi_U$ if and only if $\operatorname{logit}\{p(d;\theta)\}>\operatorname{logit}(\phi_U)$.

Quantity or rule	Question it answers	Why it is not interchangeable with overdose
Target interval \([\phi_L,\phi_U]\)	Is the modeled probability below, within, or above the desired range?	Only being above \(\phi_U\) is overdose; being below \(\phi_L\) is below target.
Observed DLT proportion	What fraction of an observed cohort had DLTs?	It is a sample statistic, not the underlying dose-specific probability.
MTD	Which dose meets a specified maximum-tolerated-dose criterion?	Its definition depends on the criterion and is not, by itself, a full statement about uncertainty or recommendation.
Dose recommendation	Which dose should the trial consider under its decision rules?	It is a decision that may use overdose uncertainty and other constraints, not the overdose event alone.

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