Sensitivity of Optimal Dose to Prior Assumptions

Why Prior Sensitivity Matters

A Bayesian dose recommendation depends on both the observed dose-response data and the prior distribution assigned to unknown parameters. When data are informative, reasonable priors may lead to similar recommendations. When data are sparse, noisy, or weakly informative about the target region, prior assumptions can materially change posterior uncertainty and therefore the probability that a dose attains the target. A recommendation that is stable across plausible priors is more robust to prior choice than one that changes sharply.

This tutorial examines prior sensitivity within one specified dose-response model and one specified decision rule. It does not treat alternative structural models or average across them. The purpose is to ask: if the likelihood and clinical decision criterion are held fixed, how do plausible changes to the prior alter posterior target-attainment probabilities and the dose selected?

Model, Posterior, and Decision Rule

Consider a deliberately simple hypothetical setting in which the mean response at dose $d$ is $\beta d$, where $d>0$ is measured in mg and $\beta$ is an unknown slope measured in response units per mg. Suppose the data provide the likelihood summary $\hat{\beta}=3$ response units per mg with standard error $1$ response unit per mg. Equivalently, as a function of $\beta$, the likelihood is proportional to that of a normal observation with mean $\beta$, observed value $3$, and variance $1$. Assign the prior $\beta\sim N(\mu_0,\tau^2)$.

The normal-normal calculation assumes that the likelihood summary is adequately represented by a normal distribution with known standard error, the prior is normal on the same unconstrained scale as $\beta$, and the likelihood and prior are combined as specified. This example does not constrain $\beta$ to be positive. A positive-slope restriction would require a different, truncated prior and a correspondingly different posterior calculation.

Definition: Prior sensitivity is the assessment of how posterior summaries and decisions change when the prior distribution is varied across scientifically defensible alternatives while the likelihood, model, and decision rule are kept fixed. Here, the posterior mean $m$ and variance $v$ are $$ v=\left(\frac{1}{1}+\frac{1}{\tau^2}\right)^{-1}, \qquad m=v\left(\frac{3}{1}+\frac{\mu_0}{\tau^2}\right), \qquad \beta\mid\text{data}\sim N(m,v). $$

For a specified dose $d$, define the parameter-based target-attainment probability for target response $T$ as $P(d\beta\geq T\mid\text{data})$. This probability refers to uncertainty about the mean dose-response parameter; it is not the probability that an individual patient's observed response, including residual variability, exceeds $T$. With $T=6$ response units and $d>0$, attainment is equivalent to $\beta\geq 6/d$, so

P(d\beta\geq 6\mid\text{data}) = 1-\Phi\left(\frac{6/d-m}{\sqrt{v}}\right),

where $\Phi$ is the standard normal cumulative distribution function. Suppose the decision rule is to choose the smallest dose in the permitted range $0<d\leq5$ mg with attainment probability at least $q=0.90$. Because the posterior probability increases with $d$, solving the probability requirement gives the candidate continuous threshold

d_q=\frac{6}{m-z_{0.10}\sqrt{v}} =\frac{6}{m-1.2816\sqrt{v}},

provided the denominator is positive. If it is not positive, no finite positive dose can meet this probability requirement under this model and posterior. Even when the threshold exists, it is feasible only if it falls within the permitted dose range.

Worked Prior Comparisons

Worked Example: Changing the Prior Mean

Compare two priors with the same scale, $\tau^2=1$: Prior A is $\beta\sim N(2,1)$, and Prior B is $\beta\sim N(1,1)$. The data and decision rule are as specified above. Calculate each posterior and determine whether the 90% target-attainment requirement can be met within 0 to 5 mg.

For both priors, the posterior variance is $$ v=\left(1+\frac{1}{1}\right)^{-1}=0.5000. $$ For Prior A, the posterior mean is $m=(3+2)/2=2.500$ response units per mg. Its 10th posterior percentile is $$ 2.500-1.2816\sqrt{0.5000}=1.5938 $$ response units per mg. Thus the continuous threshold is $d_{0.90}=6/1.5938=3.765$ mg (using unrounded values). It is within the permitted range, so the minimum dose meeting the rule is approximately 3.765 mg.

For Prior B, $m=(3+1)/2=2.000$ response units per mg, with the same variance $v=0.5000$. The 10th posterior percentile is $2.000-1.2816\sqrt{0.5000}=1.0938$ response units per mg. Therefore $d_{0.90}=6/1.0938=5.486$ mg using unrounded values. This exceeds the maximum permitted dose of 5 mg, so the target is not attainable at the required 90% probability within the allowed range. The threshold is a mathematical extrapolation of the decision rule, not a permitted recommendation.

The lower prior mean shifts the posterior downward and changes the decision from a feasible dose to an infeasible target. This example isolates prior location: the likelihood and prior variance are unchanged.

Worked Example: Changing the Prior Scale

Now hold the prior mean fixed at 2 response units per mg and compare a relatively concentrated prior, $\beta\sim N(2,0.25)$, with a more diffuse prior, $\beta\sim N(2,4)$. Calculate the posterior and the 90% threshold in each case.

For the concentrated prior, prior precision is $1/0.25=4$, so $$ v=(1+4)^{-1}=0.2000, \qquad m=0.2000(3+4\times2)=2.200. $$ The lower 10th posterior percentile is $2.200-1.2816\sqrt{0.2000}=1.6269$ response units per mg. The threshold is $6/1.6269=3.688$ mg. It lies within the permitted range.

For the diffuse prior, prior precision is $1/4=0.25$, giving $$ v=(1+0.25)^{-1}=0.8000, \qquad m=0.8000(3+0.25\times2)=2.800. $$ The lower 10th posterior percentile is $2.800-1.2816\sqrt{0.8000}=1.6537$ response units per mg. The threshold is $6/1.6537=3.628$ mg, also within the permitted range. The posterior variance is larger under the diffuse prior, and the resulting required dose is higher. Here the likelihood is informative enough that even the diffuse prior yields a finite, feasible threshold.

The posterior mean is not necessarily between the two prior means and the likelihood estimate by a simple unweighted average; it is a precision-weighted average. In the concentrated-prior case, the data still shift the mean from 2 to 2.2 because the likelihood is more precise than that prior, but the prior scale affects how much posterior uncertainty remains.

Worked Example: Attainment at a Fixed Dose

At a dose of 3 mg, compare the probability of attaining 6 response units under the four posteriors from the two examples. The required slope is $6/3=2$ response units per mg.

Under Prior A from the first example, $m=2.500$ and $\sqrt v=0.7071$. Thus $z=(2-2.500)/0.7071=-0.7071$, and $$ P(3\beta\geq6\mid\text{data})=1-\Phi(-0.7071)\approx0.760. $$ Under Prior B, the standardized value is $(2-2.000)/0.7071=0$, giving probability $0.500$.

Under the concentrated prior in the second example, $z=(2-2.200)/\sqrt{0.2000}=-0.4472$, so the probability is approximately $0.673$. Under the diffuse prior, $z=(2-2.800)/\sqrt{0.8000}=-0.8944$, giving approximately $0.814$. Thus the attainment probability at the same 3 mg dose ranges from 0.500 to 0.814 across these prior choices. These values quantify posterior uncertainty about the model's mean response, not individual response variability.

Why the Dose Threshold Formula Holds

The threshold calculation follows directly from the posterior quantile. Under the normal posterior, the requirement $P(d\beta\geq6\mid\text{data})\geq0.90$ is equivalent to

P\left(\beta\geq\frac{6}{d}\mid\text{data}\right)\geq0.90 \quad\Longleftrightarrow\quad \frac{6}{d}\leq m-z_{0.10}\sqrt{v}.

The right-hand side is the posterior 10th percentile of $\beta$. If that percentile is positive, rearranging the inequality gives $d\geq6/(m-z_{0.10}\sqrt v)$, and the smallest dose meeting the rule is the equality threshold. If the percentile is zero or negative, then $6/d>0$ for every finite positive $d$, so the posterior probability of $\beta\geq6/d$ cannot reach 0.90. If the percentile is positive but the resulting threshold exceeds 5 mg, the target may be attainable at some larger dose under the model but is infeasible in the specified range.

This justification also explains why checking only the posterior mean is inadequate. A dose based on $6/m$ would ignore posterior uncertainty and generally fail to enforce the stated probability requirement. The quantile, not just the mean, determines this risk-based threshold.

Conducting and Interpreting a Sensitivity Analysis

A useful analysis varies prior assumptions that have a scientific rationale, rather than choosing alternatives solely to produce a preferred recommendation. Prior means can represent plausible typical slopes; prior scales represent the degree of prior uncertainty. For each prior, keep the likelihood, target, dose range, and attainment requirement fixed, then compare posterior summaries and decisions side by side.

Specify the decision calculation.
State the dose-response parameter, target, dose range, and required posterior attainment probability. Clarify whether attainment concerns a mean response or a future individual outcome.

Choose defensible prior alternatives.
Vary location, scale, or both over values that could reasonably represent prior knowledge. Record the rationale and units for each alternative.

Recompute posterior and decision quantities.
For every prior, calculate the posterior, target-attainment probabilities at relevant doses, and the dose threshold under the same rule.

Report stability and feasibility.
Show which conclusions remain similar, which change, and whether any threshold is outside the allowed range or does not exist.

Prior sensitivity is distinct from changing the data model. Here, the likelihood summary is fixed and only the prior changes. It is also distinct from changing the decision rule: raising the required probability from 0.90 to 0.95 would change the selected dose even if the posterior stayed exactly the same. Results should make these sources of change explicit.

Diagnostics, Alternatives, and Limitations

For a conjugate calculation, check the precision-weighted posterior formulas and units. The posterior variance must be positive and, in this example, smaller than either component variance because both sources of information contribute positive precision. Check that the lower posterior quantile used for the 90% rule is $m-1.2816\sqrt v$, not $m+1.2816\sqrt v$. Then verify the threshold by substituting it back into the attainment probability and confirming that the result is approximately 0.90. At a dose below the threshold, the probability should be lower; above it, the probability should be higher.

A prior predictive check asks whether responses implied by a proposed prior are scientifically plausible before considering the data. A posterior predictive check assesses whether the fitted model can reproduce relevant features of observed data. These checks address different concerns from prior sensitivity: sensitivity tells how conclusions respond to prior choices, but does not establish that the dose-response model itself is adequate. A linear response through the origin may be inappropriate for saturating dose-response behavior, a nonzero baseline, or a dose range with substantial extrapolation. In such settings, the model and its assumptions must be reconsidered, not merely the prior.

A common mistake is to call the most favorable prior the “best” prior because it gives the lowest recommended dose. That reasoning selects assumptions by the desired answer rather than by prior scientific knowledge. The correction is to define plausible priors independently of the decision outcome and report the range of resulting decisions. Another mistake is to interpret $P(d\beta\geq6\mid\text{data})=0.90$ as a 90% chance that a treated individual will respond above 6. It is a posterior probability about the parameter-based mean response in this model; individual outcome variability would need to be included for a predictive probability.

Sensitivity analysis is not a proof that the prior is correct, and a narrow range of results across a few priors does not guarantee robustness to all plausible assumptions. Conversely, a large change is informative: it can signal that the data do not dominate prior information for the decision-relevant part of the parameter distribution. Report the prior alternatives, posterior quantities, units, and decision rule so a reader can see what drives the recommendation.

Key takeaway: A Bayesian dose recommendation is a function of the likelihood, the prior, and the decision rule. To assess prior sensitivity, vary scientifically plausible prior assumptions while holding the other two fixed, then compare posterior target probabilities and feasible dose thresholds.

Check Your Understanding

Answer each question, then compare with the solutions below.

With likelihood summary $\hat{\beta}=3$, standard error $1$, and prior $\beta\sim N(2,1)$, what are the posterior mean and variance?
At a fixed dose of 2 mg, what slope value is required to attain a target of 6 response units?
Why can a positive continuous threshold still fail to produce an allowed recommendation?
What does a 90% posterior target-attainment probability mean in the model used here, and what does it not mean?

Solutions. (1) The posterior variance is $v=(1+1)^{-1}=0.5000$, and the posterior mean is $m=0.5000(3+2)=2.500$ response units per mg. (2) Attainment requires $d\beta\geq6$, so at $d=2$ mg the required slope is $\beta\geq6/2=3$ response units per mg. (3) The threshold may exceed the permitted maximum dose; for example, a threshold above 5 mg is not feasible when the allowed range ends at 5 mg. (4) It is the posterior probability that the model's mean response at that dose meets or exceeds the target, given the data and prior. It is not automatically the probability that an individual patient's observed response meets the target, because that would require accounting for residual outcome variability.