Posterior Distributions of Target-Attaining Doses

From Parameter Uncertainty to Target-Dose Uncertainty

A fitted dose-response model describes response as a function of dose and unknown parameters. When those parameters have a posterior distribution, the dose required to reach a specified response is uncertain too. The target-dose distribution expresses that uncertainty on the dose scale, which is often more directly useful than separate intervals for model parameters.

The essential operation is to evaluate the target-dose calculation for each posterior draw of the parameters. This is not generally the same as calculating a dose from posterior mean parameter values: target dose is often a nonlinear function, and different parameter draws can imply very different doses. Some draws may not support a finite dose at the target at all. Such draws must be identified rather than silently omitted.

Definition: Let \(y\) denote the observed data, \(\theta\) the dose-response parameters, and \(m(d;\theta)\) the model-predicted response at dose \(d\). For a specified target response \(r\), a target-attaining dose \(d^*(\theta)\) is a dose solving \(m(d;\theta)=r\), subject to the stated dose domain and any rule for choosing among roots. A posterior distribution of target-attaining doses is the distribution induced by the posterior \(p(\theta\mid y)\) through this dose calculation, on draws for which the calculation is defined.

The definition requires care. A finite unique solution does not exist for every possible model and parameter draw. In the main summaries below, \(A\) denotes the set of parameter values for which the specified target has a unique finite dose in the allowed dose domain. Summaries conditional on \(A\) describe dose uncertainty among attainable draws; the posterior probability \(P(A\mid y)\) describes how often the target is attainable under the posterior. Report both when \(P(A\mid y)\) is less than one.

A Saturating Dose-Response Model and Its Target Dose

A useful example is the increasing Emax model

m(d;\theta)=E_0+\frac{E_{\max}d}{ED_{50}+d}, \qquad d\geq 0,

where \(E_0\) is the response at zero dose, \(E_{\max}>0\) is the maximum increase above \(E_0\), and \(ED_{50}>0\) is the dose producing half that increase. Suppose the target is \(r\) and \(q=r-E_0\) is positive. Solving for dose gives

q=\frac{E_{\max}d}{ED_{50}+d} \quad\Longrightarrow\quad d^*(\theta)=ED_{50}\frac{q}{E_{\max}-q}.

This is a finite positive dose precisely when \(0<q<E_{\max}\). If \(q=E_{\max}\), the target is the asymptote and no finite dose reaches it. If \(q>E_{\max}\), the model's maximum response is below the target. Thus, for this model and target, \(A\) consists of draws satisfying \(E_{\max}>q\), when \(q>0\). These conditions depend on the parameter draw; a target attainable at a posterior mean parameter value need not be attainable on every posterior draw.

Worked Example: Transforming One Posterior Draw

Consider a hypothetical model with \(E_0=10\) response units, \(E_{\max}=80\) response units, and \(ED_{50}=20\) mg on one posterior draw. The target is \(r=50\) response units. Here \(q=50-10=40\) response units, so \(0<40<80\) and a unique finite dose exists.

Substituting the draw into the target-dose formula gives

d^*=20\text{ mg}\times\frac{40\text{ response units}}{80-40\text{ response units}} =20\text{ mg}.

The result is 20 mg. Checking in the original model, \(m(20)=10+80(20)/(20+20)=50\) response units, exactly the target. The check also confirms that the dose is expressed in mg and the predicted response is in response units.

Induced Posterior Distribution and Summaries

If a unique finite target dose exists for almost every posterior draw, its cumulative distribution function is the probability that the transformed dose is at most \(x\):

F_{D^*}(x\mid y) = P\{d^*(\theta)\leq x\mid y\} = \int I\{d^*(\theta)\leq x\}\,p(\theta\mid y)\,d\theta.

This expression assumes that \(d^*(\theta)\) is defined for almost every draw. If some draws have no unique finite target dose, the expression cannot be used as written: the dose and the indicator are undefined on those draws. A conditional distribution among attainable draws is instead

P\{d^*(\theta)\leq x\mid A,y\} = \frac{\int_A I\{d^*(\theta)\leq x\}\,p(\theta\mid y)\,d\theta} {P(A\mid y)}, \qquad P(A\mid y)>0.

A posterior median is the 0.5 quantile of the stated dose distribution. A central 95% credible interval is bounded by its 0.025 and 0.975 quantiles. It is a posterior interval for the target dose under the model and prior, conditional on the observed data; it is not a statement that a fixed parameter has a 95% chance of lying in the realized interval. A tail probability such as \(P\{d^*(\theta)\leq d_0\mid A,y\}\) answers the posterior question of how much attainable-dose mass is at or below a specified dose \(d_0\).

With posterior draws \(\theta^{(1)},\ldots,\theta^{(M)}\), calculate \(d^{*(m)}=d^*(\theta^{(m)})\) wherever the target dose exists. If all draws are attainable, empirical quantiles and event proportions approximate the full posterior summaries. If only some are attainable, calculate dose quantiles and dose-tail proportions using the valid draws, and separately report the fraction of draws attaining the target as an estimate of \(P(A\mid y)\). The empirical proportion is a simulation estimate, not an exact posterior probability.

Worked Example: Summarizing a Posterior Dose Distribution

Suppose eight hypothetical posterior draws share \(E_0=10\) response units and \(ED_{50}=20\) mg, with target \(r=50\) response units. Thus \(q=40\) response units. The table gives the draw-specific \(E_{\max}\) and the transformed target dose. All draws satisfy \(E_{\max}>40\), so all eight doses are defined.

Draw	\(E_{\max}\) (response units)	Target dose (mg)
1	106.667	12
2	97.143	14
3	90.000	16
4	84.444	18
5	80.000	20
6	73.333	24
7	68.571	28
8	62.222	36

For example, draw 6 gives \(20\times40/(73.333-40)=24\) mg. The ordered target doses are 12, 14, 16, 18, 20, 24, 28, and 36 mg. Using the usual interpolated empirical quantile convention, the median for eight draws is the midpoint of the two central values: \((18+20)/2=19\) mg.

For a central 50% interval, the empirical first and third quartiles use positions \(1+(8-1)(0.25)=2.75\) and \(1+(8-1)(0.75)=6.25\) in the ordered list. Therefore \(Q_{0.25}=14+0.75(16-14)=15.5\) mg and \(Q_{0.75}=24+0.25(28-24)=25\) mg. The estimated central 50% credible interval is 15.5 to 25 mg. The estimated posterior probability that the target dose is at most 20 mg is \(5/8=0.625\). These are deliberately small illustrative draws; real analyses require enough well-behaved posterior draws to estimate the summaries adequately.

Partial Attainability Must Be Reported

When a target is unattainable on some draws, two distinct uncertainties matter: whether the model draw permits a finite target dose, and the size of the dose among draws that do. A conditional median or interval answers only the second question. Discarding unattainable draws without reporting their frequency can make the target appear more reliably achievable than the posterior supports.

Worked Example: Reporting Attainability and Conditional Dose

Use the same hypothetical Emax model, \(E_0=10\), target \(r=50\), and hence \(q=40\) response units. Consider six hypothetical posterior draws, all with \(ED_{50}=20\) mg and with \(E_{\max}\) values 60, 80, 100, 45, 35, and 40 response units.

The first four draws have \(E_{\max}>40\), so their target doses are \(20(40)/(E_{\max}-40)\): 40, 20, 13.333, and 160 mg. The last two draws have \(E_{\max}=35\) and \(E_{\max}=40\). The target is above the asymptote in the first of these and equal to the asymptote in the second, so neither has a finite target dose.

The estimated posterior probability of attainability is \(4/6=0.667\). Conditional on attainment, the ordered doses are 13.333, 20, 40, and 160 mg. The conditional median is \((20+40)/2=30\) mg. The estimated conditional probability of a dose at most 30 mg is \(2/4=0.500\), and the estimated conditional probability of a dose above 50 mg is \(1/4=0.250\). A responsible summary reports all of these with their conditioning clear: for example, “the target was attainable on 4 of 6 draws; among attainable draws, the median dose was 30 mg.” The six-draw estimate of attainability is highly imprecise, but the calculation illustrates the distinction.

Key distinction: \(P(A\mid y)\) is the posterior probability that a unique finite target dose exists. A conditional dose quantile or tail probability describes \(d^*(\theta)\) given \(A\). Neither quantity substitutes for the other.

Nonlinearity, Diagnostics, and Alternatives

The transformation is nonlinear, so transforming a representative parameter value need not produce a representative target dose. In general, the dose calculated at posterior mean parameters is not the posterior mean dose, and it need not equal the posterior median dose. Parameter values must also be transformed as joint draws: pairing values independently from separate marginal summaries can destroy posterior dependence and produce a distribution that the fitted model's posterior does not support.

Worked Example: A Hill Emax Curve and a Dose-Tail Probability

For a hypothetical increasing Hill Emax model, let

m(d;\theta)=E_0+\frac{E_{\max}d^\gamma}{ED_{50}^\gamma+d^\gamma}, \qquad d\geq0,

with \(E_0=5\) response units, \(ED_{50}=12\) mg, \(\gamma=2\), and target \(r=35\) response units. Then \(q=30\) response units. Solving \(q=E_{\max}d^\gamma/(ED_{50}^\gamma+d^\gamma)\) gives \(d^*=ED_{50}\{q/(E_{\max}-q)\}^{1/\gamma}\) when \(0<q<E_{\max}\). For a draw with \(E_{\max}=50\), the target dose is

d^*=12\text{ mg}\left(\frac{30}{50-30}\right)^{1/2} =12\sqrt{1.5}\text{ mg} \approx14.697\text{ mg}.

For four hypothetical posterior draws with \(E_{\max}=50,60,45,\) and \(90\) response units, respectively, the doses are approximately 14.697, 12.000, 16.971, and 8.485 mg. Each draw is attainable because \(E_{\max}>30\). Two of these four doses are at most 13 mg, so the empirical estimate of \(P(d^*\leq13\text{ mg}\mid y)\) is \(2/4=0.500\). This is a probability about the model-implied target dose under posterior uncertainty, not a guarantee that a patient receiving 13 mg will attain the response.

Diagnostics should examine both the parameter draws and the transformed doses. Check that posterior computation has mixed adequately and that summaries are stable across chains or longer runs. Tabulate the fraction of draws with no root, multiple roots, or roots outside the allowed dose range. Inspect extreme target doses: a few very large values can make a mean misleading, and may signal draws with \(E_{\max}\) just above the target increment. For a monotone model, verify numerically that substituting each computed dose reproduces the target within a stated tolerance. For a non-monotone model, specify which root is scientifically relevant before summarizing; “the” target dose is not well-defined if several roots are possible.

Posterior target-dose summaries differ from the earlier material on delta-method uncertainty, profile likelihood, and bootstrap distributions of effective doses. Those approaches construct uncertainty summaries using local approximations, likelihood-based inversion, or resampling, respectively. Posterior transformation instead uses the posterior distribution of parameters and naturally yields posterior probabilities and credible intervals. Their numerical answers may differ because their uncertainty frameworks and assumptions differ; none removes the need to define the target, dose domain, and root rule.

The method inherits the fitted model's assumptions, prior, and data limitations. A narrow posterior dose interval cannot compensate for an inappropriate response model or a target outside the informative dose range. Nor does a target-dose distribution by itself incorporate safety constraints, patient-to-patient variability, or uncertainty about the target definition. Interpret its summaries as conditional on the specified model, target, and dose domain.

Common Mistakes and Practical Reporting

Leaving unattainable draws undefined but treating them as ordinary dose values. Define the attainable set and report \(P(A\mid y)\), or explicitly state a chosen extended-value convention. For conditional summaries, use only attainable draws and state the conditioning.
Reporting a conditional median as if all posterior draws attained the target. Pair it with the estimated posterior probability of attainability.
Calculating from marginal parameter means or independently shuffled parameter draws. Transform each intact joint posterior draw; nonlinear transformations and posterior dependence matter.
Ignoring the dose domain or choosing a root after seeing the result. State the allowed doses and root-selection rule before interpreting the distribution.
Calling a credible interval a confidence interval or a patient-level prediction interval. It summarizes posterior uncertainty in the model-implied target dose, not individual response variability.

Reporting checklist: State the response target and its units, model and dose domain, root or attainment rule, posterior probability of attainability, and the conditioning for every dose quantile or tail probability. Include the dose units and describe the posterior computation used.

Check Your Understanding

Answer each question, then compare with the solutions below.

For the Emax model, \(E_0=8\), \(ED_{50}=10\) mg, \(E_{\max}=50\) response units, and target \(r=28\) response units, find the target dose and check attainability.
For that same target increment, what happens on a draw with \(E_{\max}=20\) response units? What if \(E_{\max}=20\) exactly?
In 100 posterior draws, 80 have a unique finite target dose. Among those 80, 52 doses are at most 25 mg. Give both relevant empirical probability estimates and state their meanings.
Why is transforming posterior mean parameters generally not a substitute for transforming each joint posterior draw?

Solutions. (1) The target increment is \(q=28-8=20\) response units, which is positive and less than \(E_{\max}=50\), so a finite dose exists. It is \(10\times20/(50-20)=6.667\) mg. Substitution gives \(8+50(6.667)/(10+6.667)\approx28\) response units. (2) If \(E_{\max}=20\), the target equals the asymptotic response increment, so no finite dose reaches it. If \(E_{\max}=20\) exactly, it is still an asymptote, not a finite root. (3) The estimated attainability probability is \(80/100=0.80\). Conditional on attainability, the estimated probability of a dose at most 25 mg is \(52/80=0.65\). The first is about existence of a finite dose; the second is about dose size among attainable draws. (4) The target-dose function is nonlinear, so its value at parameter means need not equal a posterior dose mean or median. Independent marginal draws can additionally destroy dependence among parameters; transform the parameters together for each posterior draw.