Priors for Longitudinal Fixed Effects

Why Fixed-Effect Priors Need Careful Calibration

A prior for a longitudinal fixed effect is a statement about plausible population-average biomarker patterns before the observed data are used. Its meaning depends on the units and coding of the predictors. A normal prior that is moderately spread for a change measured in milligrams per liter per year could be extremely diffuse, or extremely restrictive, if the outcome or time scale is different.

In a joint model, longitudinal fixed effects help describe the average biomarker trajectory, while the event-time component and shared latent effects contribute information to the joint posterior. This tutorial focuses on specifying and interpreting priors for the longitudinal fixed-effect vector. It does not specify priors for the random-effects covariance matrix, baseline hazard, or association parameters.

For a concrete model, let \(Y_{ij}\) be the biomarker measurement for subject \(i\) at time \(t_{ij}\), and let \(\mathbf{x}_{ij}\) be a vector of fixed-effect predictors. One common specification is

Here, \(\boldsymbol{\beta}\) is the vector of longitudinal fixed effects, \(\mathbf{b}_i\) contains subject-specific random effects, and \(\varepsilon_{ij}\) is measurement error. For illustration, suppose the longitudinal measurements are conditionally independent given the random effects and covariates, with \(\varepsilon_{ij}\sim N(0,\sigma^2)\). The prior discussed here is on \(\boldsymbol{\beta}\), not on \(\mathbf{b}_i\), \(\sigma^2\), or other parameters. The joint posterior combines the prior with the joint data likelihood; consequently, information from the event-time part can influence the posterior for \(\boldsymbol{\beta}\) through the shared model structure.

Normal Priors and Their Interpretation

A common starting point is a multivariate normal prior,

The vector \(\mathbf{m}_{\beta}\) gives prior means, and \(\boldsymbol{\Sigma}_{\beta}\) gives prior variances and covariances. With independent coefficient priors, each coefficient has a univariate normal distribution and \(\boldsymbol{\Sigma}_{\beta}\) is diagonal. For example, \(\beta_t\sim N(0,3^2)\) assigns a normal prior with mean zero and standard deviation 3 to a time slope. The second argument of \(N\) is the variance, so the variance here is 9, not 3.

A weakly informative prior is not a synonym for “no prior information.” It is a proper distribution that regularizes implausibly large values while allowing a broad range of effects that remain plausible on the scientific scale. Its location and spread should be chosen by considering the coefficient’s units and meaning. A zero-centered prior is suitable when positive and negative effects of similar magnitude are plausible; substantive prior knowledge may instead justify a nonzero mean or an asymmetric distribution.

For a normal prior with mean \(m\) and standard deviation \(s\), about 95% of its probability lies between \(m-1.96s\) and \(m+1.96s\). This provides a practical calibration: first state a plausible range for the coefficient, then choose a mean and standard deviation that express that range. It is a probability statement about the prior distribution, not a confidence interval calculated from data.

Scaling and Changing Units

A coefficient’s numerical value changes when its predictor or outcome is rescaled. A prior must change with it if the intended scientific beliefs are to remain the same. Consider a standardized outcome \(Y^*=(Y-c_Y)/s_Y\) and a time variable \(t^*=t/s_t\), where \(s_Y\) is a positive outcome scale and \(s_t\) is a positive time scale. If the standardized model contains an intercept \(\alpha\) and slope \(\gamma\),

then substitution gives

Therefore, on the original scale the intercept is \(c_Y+s_Y\alpha\), and the slope per original time unit is \((s_Y/s_t)\gamma\). If \(\gamma\sim N(m,s^2)\), multiplying by the constant \(s_Y/s_t\) gives an original-scale slope distributed as \(N((s_Y/s_t)m, (s_Y/s_t)^2s^2)\). The units transform along with the coefficient.

The same issue arises when predictors are centered. If \(x^*=(x-c)/s\), then a model \(\alpha+\gamma x^*\) is equivalent to a model with an intercept \(\beta_0\) and raw-scale slope \(\beta_x\):

Even if \(\alpha\) and \(\gamma\) are assigned independent normal priors, the induced priors on \(\beta_0\) and \(\beta_x\) will generally be dependent. Thus changing the origin or scale of a predictor can change the prior on the raw-scale parameters, particularly when new independent priors are assigned after the change.

Intercepts, Coding, and Prior Dependence

An intercept is the expected biomarker value when every predictor equals zero, under the fixed-effects part of the model. That value may be scientifically meaningful, but it may also refer to an implausible reference subject or time. Centering time at baseline or age at a representative value can make the intercept easier to interpret. The prior for the intercept should then be calibrated for the expected biomarker level at that reference point.

For a binary covariate coded 0 for control and 1 for treatment, its coefficient is the treatment-minus-control difference in the conditional mean at the other predictor values held fixed. If time interacts with treatment, the treatment coefficient is the difference at the time value coded as zero; it is not necessarily the difference at every follow-up time. Prior interpretation must follow the model’s actual coding.

Choosing Among Prior Specifications

Independent normal priors are convenient and interpretable when each coefficient can be calibrated separately. A multivariate normal prior can represent prior dependence among fixed effects when that dependence is substantively justified. But independence is not invariant to reparameterization: independent priors on an intercept and slope at one time origin generally imply correlated priors on the intercept and slope at another origin. Choose a parameterization that makes the coefficients meaningful, and state clearly whether the prior is intended to apply in that parameterization or to preserve a specified distribution under transformation.

A proper normal prior has support over the entire real line. That is appropriate when positive and negative coefficient values are possible, but it can assign probability to scientifically impossible trajectories or mean values. If the parameter itself has a genuine constraint, the model and prior should respect that constraint rather than relying on a broad normal prior to do so. A weakly informative prior should regularize extreme values without silently imposing a strong belief near zero.

Diagnostics and Common Pitfalls

Before fitting the model, check the prior’s implications in the units that will be reported. For a slope, translate its scale into expected biomarker change over one year and over the full follow-up period. For an intercept, identify the covariate values at which it applies. For an interaction, state the reference values at which its coefficient is interpreted. These checks can reveal a prior that looks numerically moderate but implies implausibly large biomarker trajectories.

• Ignoring units. A standard deviation of 1 for a slope means something different per month than per year. Convert it before interpreting or comparing priors.
• Using an unexplained “large” scale. A prior is not weakly informative merely because its standard deviation is large relative to 1. Compare it with plausible changes on the biomarker’s scale.
• Forgetting the intercept reference. With centered time or age, the intercept refers to the chosen reference value, not necessarily time zero or age zero.
• Re-centering but reusing the same independent priors. This may change the implied prior for trajectories. Transform the prior distribution if preserving the original beliefs is intended.
• Calling a prior-data conflict a software error. If plausible observed trajectories lie in regions to which the prior assigns little probability, revisit the scientific calibration and model specification. Do not widen priors automatically without considering why the conflict arose.

A frequent mistake is to say, “The prior is weak because its standard deviation is 10.” That conclusion has no meaning without the coefficient’s units and interpretation. A standard deviation of 10 mg/L/year may be broad for one biomarker and restrictive for another. The correction is to state the implied range and trajectory changes in context, then decide whether those implications are genuinely weak relative to the scientific information available.