Eliciting Priors for Cross-Species Parameters

From Biological Knowledge to a Prior

A cross-species prediction can be numerically precise because its parameters were fixed, yet biologically uncertain because those values were only approximate. Prior elicitation makes that uncertainty explicit before human data are available. The task is not to turn expert judgment into an unquestionable “true” distribution. It is to record what is currently plausible, on a stated scale, so that its consequences for a prediction can be examined.

For an allometric relationship, a common starting point is $CL \propto W^b$, where $CL$ is clearance, $W$ is body weight, and $b$ is the scaling exponent. A useful extension allows a residual species effect:

CL_h=CL_a\left(\frac{W_h}{W_a}\right)^b\exp(\delta).

Here $a$ and $h$ denote the animal and human, respectively, and $\delta$ is a log-scale species effect not already represented by the allometric term. Thus, $\exp(\delta)$ is a multiplicative adjustment. The model is positive for positive clearance and weight. It also assumes that this allometric form is an adequate working relationship for the prediction; a prior cannot compensate for a misspecified relationship.

Definition: A prior distribution assigns probabilities to parameter values before incorporating the data being analyzed. In translational prediction, it can encode prior knowledge about quantities such as $b$, a physiological parameter, or $\delta$. Its scale and interpretation must be stated, and its spread should reflect uncertainty rather than merely the range of observed values.

Elicitation usually starts with interpretable questions: What values seem typical? What range is still plausible? Is uncertainty symmetric in the parameter itself or in its logarithm? For positive quantities and multiplicative effects, log-scale distributions are often convenient. They avoid assigning probability to negative values and naturally express uncertainty as ratios.

Eliciting an Allometric Exponent

An exponent is not directly a fold change. If the body-weight ratio is $r=W_h/W_a$, the associated scaling factor is $r^b$. Consequently, an expert may find it easier to judge a plausible range for the scaling factor at a specified weight ratio, and then translate that range into a prior on $b$. This procedure requires the ratio $r$ to be stated: the same uncertainty in $b$ produces a wider range of scaling factors for a more distant species comparison.

Worked Example: Eliciting a Prior From a Plausible Scaling Range

Suppose experts consider an allometric scaling factor between 25 and 55 plausible for a human-to-animal body-weight ratio of $r=280$. For this illustration, interpret those bounds as the central 90% range of $r^b$, and use a normal prior for $b$. All values are hypothetical.

Since $r^b=\exp(b\log r)$, the endpoints on the log scale correspond to exponent values $\log(25)/\log(280)$ and $\log(55)/\log(280)$. A normal distribution has a central 90% range approximately 1.645 standard deviations on either side of its mean. The midpoint and half-width of the elicited log range therefore give:

\mu_b=\frac{\log(25)+\log(55)}{2\log(280)} =\frac{3.2189+4.0073}{2(5.6348)} \approx 0.6412,

\sigma_b=\frac{\log(55)-\log(25)}{2(1.645)\log(280)} =\frac{0.7885}{18.5385} \approx 0.0425.

The resulting elicited prior is $b\sim N(0.6412,0.0425^2)$, where the second argument is the variance. Its central 90% range is approximately $0.6412\pm1.645(0.0425)$, or 0.5713 to 0.7111. Substituting those endpoints into $280^b$ gives approximately 25 and 55, as intended.

This is a transparent translation of a particular judgment, not a universal prior for all drugs or endpoints. The numerical result depends on the assumed central range, the selected weight ratio, and the choice of a normal distribution on the exponent scale.

Positive Physiological Parameters and Lognormal Priors

Physiological quantities such as blood flow, organ volume, or enzyme abundance are positive. A lognormal prior is one option: if $\log Q\sim N(m,s^2)$, then $Q$ is lognormally distributed. Its median is $\exp(m)$, while its mean is $\exp(m+s^2/2)$. These are different whenever the prior has nonzero spread. A stated “typical value” should therefore be identified as a median, mean, or another summary.

Worked Example: Eliciting a Prior for Hepatic Blood Flow

Suppose a hypothetical adult-human hepatic blood-flow value $Q_H$ is assigned a median of 90 L/h, with log-scale standard deviation 0.20. The prior is:

\log Q_H\sim N(\log(90),0.20^2).

The central 90% prior interval is obtained by exponentiating the endpoints on the log scale:

\left[90\exp\{-1.645(0.20)\},\;90\exp\{1.645(0.20)\}\right] \approx [64.8,\;125.1]\ {\rm L/h}.

The median is 90 L/h by construction. The mean is $90\exp(0.20^2/2)=90\exp(0.02)\approx91.82$ L/h. To show how the prior can affect a prediction, suppose, only for this calculation, that a clearance component equals $0.35Q_H$, with the fraction 0.35 treated as fixed. The resulting median is $0.35(90)=31.5$ L/h, and its central 90% interval is $0.35[64.8,125.1]$, or approximately 22.7 to 43.8 L/h. This calculation isolates uncertainty in $Q_H$; it does not claim that the fraction is known without uncertainty in an actual application.

If the elicited information is instead a plausible range, determine whether it is naturally symmetric on the original or log scale. For example, equal multiplicative departures above and below a median are symmetric after taking logarithms, not in the original units. The distributional choice should follow the scientific interpretation, not just mathematical convenience.

Representing Residual Species Effects

After known scaling and physiological mechanisms have been accounted for, there may still be systematic cross-species differences. A log-scale effect $\delta$ represents such a difference multiplicatively. A prior centered at zero, such as $\delta\sim N(0,\tau^2)$, says that no direction of residual effect is favored before the relevant evidence is considered. It does not assert that species are identical: the spread $\tau$ determines how much departure is allowed. If biological knowledge supports a directional difference, the prior mean can instead be nonzero.

Worked Example: Translating a Multiplicative Species Range

Suppose experts judge that, after allometric scaling, a residual species multiplier between $1/1.39$ and $1.39$ covers about 90% of plausible values. This range is symmetric on the log scale. If $\delta$ is normal with mean zero, the upper 95th percentile of $\exp(\delta)$ is $\exp(1.645\tau)$. Thus:

\tau=\frac{\log(1.39)}{1.645}\approx 0.2002, \qquad \delta\sim N(0,0.2002^2).

For a hypothetical animal clearance of 0.50 L/h and weight ratio $r=280$, let the exponent be fixed at 0.67 for this example. The prediction before applying the residual species effect is:

0.50(280)^{0.67} =0.50\exp\{0.67(5.6348)\} \approx 0.50(43.61) =21.81\ {\rm L/h}.

The prior median species multiplier is $\exp(0)=1$, so the median prediction remains 21.81 L/h. Its central 90% interval is obtained by multiplying the unrounded prediction by the elicited multiplier limits: $21.8055/1.39\approx15.69$ L/h and $21.8055(1.39)\approx30.31$ L/h. These are prior-based prediction limits under the stated model and fixed inputs, not a confidence interval from sampled human observations.

How Prior Uncertainty Enters a Prediction

The log form of the scaling model makes the contribution of the exponent and species effect explicit. Let $r=W_h/W_a$. If $CL_a$ is treated as known, then:

\log CL_h=\log CL_a+b\log r+\delta.

If $b$ and $\delta$ are independent normal variables, this sum is normal. Its mean and variance follow from the mean and variance rules for independent random variables. Exponentiating gives a lognormal human prediction. In particular, the exponent's standard deviation is multiplied by $|\log r|$ on the prediction's log scale: uncertainty in $b$ therefore matters more when the species are farther apart in body weight.

Result: If $\log CL_a$ is fixed, $b\sim N(\mu_b,\sigma_b^2)$, $\delta\sim N(\mu_\delta,\tau^2)$, and $b$ and $\delta$ are independent, then $$ \log CL_h\sim N\left(\log CL_a+\mu_b\log r+\mu_\delta,\;(\log r)^2\sigma_b^2+\tau^2\right). $$ The prediction median is the exponential of the log-scale mean. A central 90% prior interval is the exponential of that mean plus or minus 1.645 times the log-scale standard deviation.

For a short justification, write the random part as $b\log r+\delta$. Multiplication of $b$ by the constant $\log r$ multiplies its variance by $(\log r)^2$. Independence makes the covariance zero, so the variance of the sum is the sum of the two variances. If $b$ and $\delta$ are correlated, a covariance term must be included; omitting it can either understate or overstate prediction uncertainty.

The result also shows why a prior can affect more than the width of a prediction. Changing the prior mean of $b$ changes the log prediction by $\log r$ times that mean change. Changing the prior spread alters the prediction interval, with an effect amplified by the distance between species. A nonzero mean for $\delta$ shifts the prediction multiplicatively.

Checking Prior Choices and Comparing Alternatives

A useful check is to calculate or simulate the implications of the priors before relying on a human prediction. Ask whether the implied exponents, physiological values, species multipliers, and final clearance ranges are scientifically plausible. If a modest change in a prior produces a large shift in the prediction, report that dependence and investigate which judgment drives it. This is a check on prior implications; it does not establish that the translation model itself is correct.

Several reasonable approaches may be compared:

Approach	Useful when	Main caution
Informative elicited prior	Relevant mechanistic or expert knowledge supports a specific center and spread.	It can look more objective than the judgments and assumptions used to construct it.
Weakly informative prior	Some regularization is appropriate, but a narrow quantitative judgment is not justified.	“Weak” depends on parameterization; a broad prior can still imply extreme predictions.
Fixed parameter value	A sensitivity calculation needs a clear reference scenario.	It represents no uncertainty for that parameter and can make predictions falsely narrow.

A practical analysis can retain an evidence-based primary prior while repeating the prediction under defensible alternatives. This is prior sensitivity analysis: it asks how results change under alternative prior specifications. It is not a procedure for selecting whichever prior gives a preferred answer. Clearly label which assumptions changed and keep the biological model and other inputs fixed when comparing their effect.

Common Pitfalls and Limitations

Confusing a prior range with observed variability. A prior describes uncertainty about a parameter before using the data in question; the spread of individual animal measurements may instead include biological variability and measurement error.
Using a symmetric range on the wrong scale. For a multiplicative effect, symmetric limits such as 0.72 and 1.39 are approximately reciprocal. Treating them as symmetric additive differences around 1 changes the implied prior.
Centering every species effect at zero by default. Zero is a defensible center only when no directional residual difference is supported after the modeled mechanisms are considered.
Counting the same uncertainty twice. A species difference already represented by a physiological covariate should not automatically be added again as an unrestricted residual effect without a scientific reason.
Reporting only a prior mean. A point estimate hides how much the prediction depends on uncertainty. State the distribution, parameterization, and relevant spread.

Elicitation remains limited by the information available and by the assumed distributional form. Experts may disagree, evidence may be sparse, and judgments about an unobserved human parameter can be weakly informed. A precisely calculated interval does not remove those limitations. The value of explicit priors is that assumptions become inspectable, comparable, and revisable when new evidence is considered.

Check Your Understanding

Work each problem from the stated assumptions. The solutions follow the questions.

For $r=100$, experts give a central 90% range of 16 to 36 for $r^b$. Under a normal prior for $b$, find its mean and standard deviation.
If $\log Q\sim N(\log 50,0.10^2)$, what are the median and approximate central 90% interval for $Q$?
A residual species multiplier has a central 90% range of 0.80 to 1.25 and is modeled as $\exp(\delta)$. Is a zero-centered normal prior on $\delta$ consistent with these limits approximately? Find its standard deviation.
For fixed animal clearance and independent normal $b$ and $\delta$, what happens to the contribution of exponent uncertainty to log-prediction variance when $|\log r|$ doubles?

Solutions. (1) The exponent endpoints are $\log(16)/\log(100)$ and $\log(36)/\log(100)$. Thus $\mu_b=[\log(16)+\log(36)]/[2\log(100)]=\log(24)/\log(100)\approx0.6901$, and $\sigma_b=[\log(36)-\log(16)]/[2(1.645)\log(100)]\approx0.0535$. (2) The median is 50 in the units of $Q$. The interval is $50\exp\{\pm1.645(0.10)\}$, approximately 42.4 to 58.9 in those same units. (3) The limits are approximately reciprocal because $1/1.25=0.80$. Therefore $\tau=\log(1.25)/1.645\approx0.1356$, so $\delta\sim N(0,0.1356^2)$ is consistent with the stated central range. (4) The exponent contribution is $(\log r)^2\sigma_b^2$. Doubling $|\log r|$ multiplies this variance contribution by four.