Priors for Association Parameters

Why the Association Prior Needs a Scale

In a shared-random-effects joint model, the association parameter connects a subject’s latent longitudinal trajectory to that subject’s event hazard. A prior on this parameter is not interpretable in isolation: its implications depend on how the longitudinal quantity is defined and measured. A coefficient per one biomarker unit can imply a very different range of hazard ratios from a coefficient per standard deviation, even when both describe the same model.

The goal is to specify a prior that expresses reasonable uncertainty about the association on a meaningful scale. This requires stating the association structure, identifying the contrast that the coefficient represents, and checking what the prior implies for relative hazards. These are prior implications, not evidence from observed data and not causal claims.

Definition: In a current-value association model, a common conditional hazard specification is $$ \lambda_i(t\mid b_i)=\lambda_0(t)\exp\{\gamma^\mathsf{T}w_i+\alpha m_i(t)\}, $$ where $\lambda_i(t\mid b_i)$ is subject $i$’s hazard at time $t$, conditional on their shared random effects $b_i$; $\lambda_0(t)$ is the baseline hazard; $w_i$ is a vector of event-model covariates; $m_i(t)$ is the latent longitudinal mean at time $t$; and $\alpha$ is the association parameter. For two subjects with the same $t$, $w_i$, and other event-model terms, whose latent values differ by $\Delta m$, the conditional hazard ratio is $\exp(\alpha\Delta m)$.

The parameter $\alpha$ is on the log-hazard scale per unit of $m_i(t)$. If the marker is measured in mg/dL, for example, $\alpha$ is per mg/dL; if the marker is measured in tens of mg/dL, it is per ten mg/dL. The same principle applies to other association structures. A slope association, for instance, uses a coefficient per unit change in the marker’s rate of change. The meaning and units of every association parameter follow from the longitudinal quantity multiplied by that parameter in the hazard model.

From a Coefficient Prior to Hazard-Ratio Beliefs

A normal prior is a common regularizing choice when positive and negative associations are both plausible. Write $\alpha\sim N(\mu_\alpha,\tau_\alpha^2)$, where $\mu_\alpha$ is the prior mean and $\tau_\alpha$ the prior standard deviation. For a fixed longitudinal contrast $\Delta m$, the log hazard ratio is $\alpha\Delta m$. Multiplying a normally distributed variable by a constant gives

\log(\mathrm{HR})=\alpha\Delta m \sim N(\Delta m\mu_\alpha,\;(\Delta m)^2\tau_\alpha^2).

Thus the prior mean and spread on the log hazard-ratio scale depend on the contrast being considered. If $\mu_\alpha=0$, a central 95% prior interval for this log hazard ratio is approximately $\pm1.96|\Delta m|\tau_\alpha$; exponentiating its endpoints gives the corresponding central 95% prior interval for the hazard ratio. This is a description of prior mass, not a confidence or credible interval based on data.

Worked Example: Choosing a Prior for a Ten-Unit Contrast

Suppose the longitudinal marker is recorded in mg/dL and the model uses its current value. Researchers consider a difference of 10 mg/dL to be a useful contrast. They assign $\alpha\sim N(0,0.02^2)$ per mg/dL. Find the prior distribution of the log hazard ratio for a 10 mg/dL difference and its central 95% prior interval.

For this contrast, $\Delta m=10$. Therefore, $$ \log(\mathrm{HR})=10\alpha,\qquad \log(\mathrm{HR})\sim N(0,\;10^2(0.02)^2)=N(0,0.2^2). $$ The prior standard deviation of the log hazard ratio is $0.2$. The central 95% interval on that scale is $$ 0\pm1.96(0.2)=\pm0.392. $$ Exponentiating gives $(\exp(-0.392),\exp(0.392))\approx(0.676,1.480)$. Under this prior, the central 95% prior interval for the conditional hazard ratio comparing subjects whose latent values differ by 10 mg/dL is approximately 0.676 to 1.480, holding the other terms fixed. Since the prior is symmetric about zero, $P(\alpha>0)=0.5$; the prior does not favor either direction of association.

A useful way to assess a proposed prior is therefore to choose a clinically or scientifically interpretable marker contrast and calculate its hazard-ratio implications. Calling a prior “weak” or “diffuse” without stating the marker scale and contrast does not establish that it is reasonable.

Centering and Standardizing the Marker

Centering changes the reference value, while scaling changes the units. Suppose $m_i^*(t)=(m_i(t)-c)/s$, where $c$ is a fixed centering value and $s>0$ is a fixed scale, such as a reference standard deviation. Then $m_i(t)=c+s m_i^*(t)$, and the association contribution becomes $$ \alpha m_i(t)=\alpha c+(\alpha s)m_i^*(t). $$

The constant term $\alpha c$ can be absorbed into the baseline hazard level when the model allows that level to vary freely. The coefficient on the rescaled marker is $\alpha^*=s\alpha$. To preserve the same model and prior beliefs after rescaling, transform the prior as well: if $\alpha\sim N(\mu_\alpha,\tau_\alpha^2)$, then $\alpha^*\sim N(s\mu_\alpha,s^2\tau_\alpha^2)$. Assigning the same numerical prior after changing units generally changes the implied hazard-ratio beliefs.

Worked Example: Translating a Prior to a Standard-Deviation Scale

Suppose a marker has a reference standard deviation of 12 mg/dL. The model uses the marker in mg/dL, with $\alpha\sim N(0,0.015^2)$ per mg/dL. Translate this prior to the coefficient for a one-standard-deviation increase, and calculate the central 95% prior interval for the hazard ratio for that increase.

For a one-standard-deviation increase, the log hazard ratio is $12\alpha$. Its prior standard deviation is $12(0.015)=0.18$, so the equivalent coefficient on the standard-deviation scale has prior $N(0,0.18^2)$. The central 95% log hazard-ratio interval is $$ \pm1.96(0.18)=\pm0.3528. $$ Thus the hazard-ratio interval is $(\exp(-0.3528),\exp(0.3528))\approx(0.703,1.423)$. The original prior also implies that for a one mg/dL increase, the central 95% interval is $(\exp(-1.96(0.015)),\exp(1.96(0.015)))\approx(0.971,1.030)$. These are consistent: a one-standard-deviation increase is 12 times as large as a one-unit increase, so its log-hazard contrast has 12 times the prior standard deviation.

If the marker is standardized and the model is refit, the equivalent prior standard deviation is 0.18, not 0.015. Using 0.015 after standardizing would encode a substantially tighter prior on the one-standard-deviation association.

More Than One Association Parameter

Some models link the hazard to several features of the latent trajectory. For example, a value-and-slope association can include $\alpha_v m_i(t)+\alpha_s m_i'(t)$, where $m_i'(t)$ is the derivative of the latent mean with respect to time. For a contrast between two trajectories at a common time, the log hazard ratio is $$ L=\alpha_v\Delta m+\alpha_s\Delta m'. $$

If the association coefficients have a joint normal prior with covariance matrix $\Sigma$, then $L$ is normal with mean $(\Delta m,\Delta m')$ multiplied by the prior mean vector and variance $$ \operatorname{Var}(L)= (\Delta m)^2\operatorname{Var}(\alpha_v)+ (\Delta m')^2\operatorname{Var}(\alpha_s)+ 2\Delta m\Delta m'\operatorname{Cov}(\alpha_v,\alpha_s). $$ The covariance term matters whenever the coefficients are not independent. The units also differ: $\alpha_v$ is per marker unit, while $\alpha_s$ is per marker unit per time unit. A time unit must therefore be specified when interpreting a slope contrast.

Worked Example: A Prior for a Value-and-Slope Contrast

Consider two hypothetical trajectories at the same time. One has a latent marker value 1.2 units higher and a latent slope 0.4 units per year lower than the other, so $\Delta m=1.2$ units and $\Delta m'=-0.4$ units/year. Suppose the independent priors are $\alpha_v\sim N(0,0.25^2)$ per unit and $\alpha_s\sim N(0,0.15^2)$ per unit/year. Find the prior variance of the log hazard ratio and its central 95% hazard-ratio interval.

The contrast is $L=1.2\alpha_v-0.4\alpha_s$. Independence makes the covariance term zero, giving $$ \begin{aligned} \operatorname{Var}(L) &=1.2^2(0.25^2)+(-0.4)^2(0.15^2)\\ &=1.44(0.0625)+0.16(0.0225)\\ &=0.0900+0.0036=0.0936. \end{aligned} $$ The log hazard ratio has prior mean zero and standard deviation $\sqrt{0.0936}\approx0.3059$. Its central 95% interval is approximately $\pm1.96(0.3059)=\pm0.5996$. Exponentiating gives $(\exp(-0.5996),\exp(0.5996))\approx(0.549,1.821)$. Thus, under these priors, the central 95% prior interval for the hazard ratio comparing the specified trajectories is approximately 0.549 to 1.821.

Choosing Among Prior Families

The prior family should reflect both the scientific information available and the role of regularization. Common choices include:

Prior choice	Typical use	Important implication
Normal centered at zero	Both association directions are plausible; large effects should be regularized.	Its scale should be checked on a stated marker contrast.
Student t centered at zero	Both signs are plausible, but heavier tails than a normal are desired.	More prior probability lies far from zero; the degrees of freedom and scale affect tail behavior.
Half-normal or another positive-support prior	A nonnegative association is required by substantive knowledge or model design.	The sign restriction is a substantive assumption, not a neutral default.

A prior can instead be specified on a hazard ratio for a chosen contrast. For a positive hazard ratio $h=\exp(\alpha\Delta m)$, a prior on $h$ is a prior on the transformed coefficient. If the prior is specified through a density, the change of variables includes the Jacobian: $p_h(h)=p_\alpha(\log(h)/\Delta m)/(h|\Delta m|)$, for $\Delta m\ne0$. This approach can make elicitation intuitive, but the chosen contrast must be explicit; a prior on the hazard ratio for one marker unit is not the same as one for a standard-deviation increase.

Assumptions, Diagnostics, and Common Pitfalls

The calculations above presume that the stated conditional hazard model is the model being considered, that the contrast is defined at a common time with other hazard terms held fixed, and that the prior is placed on the coefficient in the specified parameterization. A hazard ratio from this model is conditional on the shared effects and covariates; it is not automatically a marginal population hazard ratio. The calculations describe prior distributions, not posterior conclusions.

Before using a prior, check that the marker’s units, centering constant, scaling factor, and time unit for any slope are documented. Then calculate prior hazard-ratio implications for one or more relevant contrasts. If those implications are implausibly extreme or implausibly concentrated near one, reconsider the prior scale or the chosen parameterization. This is a check of the prior’s meaning, not a substitute for evaluating the fitted model.

Using the same numerical scale after standardization. This changes the prior unless its standard deviation is transformed with the marker scale.
Interpreting $\alpha$ as a hazard ratio. The coefficient is on the log-hazard scale; a contrast $\Delta m$ gives the hazard ratio $\exp(\alpha\Delta m)$.
Ignoring the contrast size. A small per-unit coefficient can still imply a wide hazard-ratio range for a large marker difference.
Restricting the sign without justification. A positive-only prior rules out negative associations by assumption; it does not merely express uncertainty about their size.
Confusing conditional association with causation. A shared association parameter describes a model-based relationship, not the effect of intervening on the biomarker.

A normal prior is often convenient and transparent, while a heavier-tailed prior allows more probability for large coefficients. Neither is universally preferable. Their comparison should focus on the implied hazard ratios for relevant contrasts, while respecting the model’s chosen association structure and the available substantive knowledge.

Key takeaway: Specify the association parameter together with its units and longitudinal contrast. Then translate its prior into hazard-ratio implications. When the marker is recentered or rescaled, transform the prior so that the intended prior beliefs are preserved.

Check Your Understanding

Answer each question, then compare with the solutions.

If $\alpha\sim N(0,0.1^2)$ and the contrast is 2 marker units, what is the prior standard deviation of the log hazard ratio?
A marker is rescaled by dividing its centered value by 5. If the original coefficient has prior standard deviation 0.04, what is the equivalent prior standard deviation for the rescaled coefficient?
Why is a positive-only prior on an association parameter not merely a more precise version of a prior centered at zero?
For independent $\alpha_v$ and $\alpha_s$, why does the variance of a value-and-slope log hazard contrast contain no covariance term?

Solutions. (1) The log hazard ratio is $2\alpha$, so its prior standard deviation is $2(0.1)=0.2$. (2) The rescaled marker is the original centered marker divided by 5, so its coefficient is $5\alpha$; its prior standard deviation is $5(0.04)=0.20$. (3) A positive-only prior assigns zero probability to negative association, whereas a prior centered at zero allows both signs; the former imposes a directional assumption. (4) Independence implies $\operatorname{Cov}(\alpha_v,\alpha_s)=0$, making the cross-product covariance term zero in the variance formula.