Why the Baseline-Hazard Prior Matters
In a joint model, the baseline hazard sets the event rate before accounting for a subject’s longitudinal history and other predictors. Its prior therefore affects the time scale of predicted events, even when the prior for every regression parameter is unchanged. A prior that seems diffuse on a hazard’s numerical scale may still concentrate implied survival near very short or very long event times.
This tutorial focuses on priors for common parametric baseline hazards and for piecewise-constant hazards. The event-time distribution implied by a prior is often more informative than the prior’s parameter labels: the same family of priors can imply quite different survival curves under different parameterizations and units.
The hazard has units of inverse time. For example, a hazard of \(0.2\) per year is not the same numerical rate as \(0.2\) per month. Set the time unit before assigning a prior, and interpret prior values in that unit. Priors must also respect parameter constraints: rates, scales, and shapes are positive, while a Weibull shape determines whether the baseline hazard decreases, stays constant, or increases over time.
Exponential and Weibull Baseline Priors
For an exponential baseline, \(h_0(t;\lambda)=\lambda\), with \(\lambda>0\). Its cumulative baseline hazard is \(\lambda t\), so its baseline survival is \(\exp(-\lambda t)\). A Gamma prior is a common choice. Under the shape-rate convention, \(\lambda\sim\operatorname{Gamma}(a,b)\) has density proportional to \(\lambda^{a-1}\exp(-b\lambda)\), mean \(a/b\), and variance \(a/b^2\). If time is measured in years, \(b\) has units of years and the rate \(\lambda\) has units of inverse years.
A Gamma prior places all its probability on positive rates and can be calibrated through its mean and spread. But a rate prior should not be judged by its mean alone. For a fixed rate, survival is an exponential function of that rate; averaging survival over a rate distribution is generally different from evaluating survival at the mean rate.
For a Weibull baseline, use the scale-shape parameterization \(h_0(t;\rho,k)=\frac{k}{\rho}(t/\rho)^{k-1}\), where \(\rho>0\) is a scale in units of time and \(k>0\) is a dimensionless shape. The cumulative baseline hazard is \((t/\rho)^k\), and baseline survival is \(\exp\{-(t/\rho)^k\}\). The hazard decreases for \(k<1\), is constant for \(k=1\), and increases for \(k>1\). Independent lognormal priors on \(\rho\) and \(k\) are one flexible option: if \(\log(\rho)\sim N(m_\rho,s_\rho^2)\), then \(\rho\) is positive with median \(\exp(m_\rho)\), and similarly for \(k\).
Worked Example: A Gamma Prior for an Exponential Rate
Suppose event times are measured in years and the exponential baseline rate has prior \(\lambda\sim\operatorname{Gamma}(2,1)\), using shape-rate notation. The prior mean rate is \(2/1=2\) per year. For a given value of \(\lambda\), baseline survival at one year is \(S_0(1\mid\lambda)=\exp(-\lambda)\).
To average survival over the prior, use the Gamma Laplace transform. For \(\lambda\sim\operatorname{Gamma}(a,b)\), integrating its density gives \(E[\exp(-t\lambda)]=(b/(b+t))^a\). Thus, at one year, \(E[S_0(1\mid\lambda)]=(1/(1+1))^2=0.25\).
By contrast, survival evaluated at the prior mean rate is \(\exp(-2\times1)=\exp(-2)\approx0.1353\). These are distinct summaries: \(0.25\) is the prior average of conditional survival, while \(0.1353\) is survival at the mean rate. The calculation assumes the exponential baseline and the stated Gamma prior; it is not a claim that either quantity is the observed survival probability in a particular population.
Worked Example: Calibrating Weibull Priors
Suppose time is measured in years and choose independent priors \(\log(\rho)\sim N(\log 2,0.25^2)\) and \(\log(k)\sim N(0,0.2^2)\). These are proper distributions supported on positive scale and shape values. The prior median scale is \(2\) years, and the prior median shape is \(1\). Since \(\log(k)\) is centered at zero, the prior probability that \(k>1\) is \(0.5\); the prior therefore does not favor an increasing rather than decreasing baseline hazard by this criterion.
At the pair of prior medians, \(\rho=2\) years and \(k=1\), baseline survival at two years is \(\exp\{-(2/2)^1\}=\exp(-1)\approx0.3679\). This is a useful reference calculation, not the prior-averaged survival at two years. Because both parameters are uncertain, that average would require integrating \(\exp\{-(2/\rho)^k\}\) over their joint prior. The scale’s time unit and the shape’s possible influence on hazard direction should both be considered when refining these priors.
Piecewise-Constant Baseline Hazards
A piecewise-constant baseline divides time into intervals and assigns one positive hazard to each. For cut points \(0=\tau_0<\tau_1<\cdots<\tau_J\), set \(h_0(t)=\lambda_j\) on \([\tau_{j-1},\tau_j)\). The baseline cumulative hazard is the sum of each rate multiplied by the amount of time spent in its interval. Independent Gamma priors are convenient and interpretable interval by interval, but they do not automatically make neighboring hazards similar.
The likelihood explains why Gamma priors are often used. Let \(y_i\) be the observed time (event time or censoring time), and \(\delta_i\) indicate an observed event. Conditional on the other model quantities, the event-time likelihood contribution is \(h_i(y_i)^{\delta_i}S_i(y_i)\). For interval \(j\), let \(d_j\) count observed events in that interval and define the exposure \(A_j=\sum_i\int_{\tau_{j-1}}^{\min(y_i,\tau_j)}\exp\{\eta_i(u)\}\,du\), where a subject contributes zero if follow-up ends before the interval begins. Conditional on the other parameters and random effects, the baseline-hazard terms in the likelihood are proportional to \(\lambda_j^{d_j}\exp(-\lambda_j A_j)\).
The calculation assumes the piecewise-constant hazard specification, the stated conditional likelihood, and a Gamma prior in the shape-rate convention. Censoring contributes exposure through the observed follow-up time, but a censored subject does not contribute an event count at their censoring time. The result is a conditional update for this parameter block; it does not mean that all dependence in the joint model disappears.
Worked Example: Prior Rates and Survival Across Two Intervals
Consider a hypothetical model with intervals \([0,0.5)\) and \([0.5,1)\), measured in years. Suppose the prior means for the two baseline rates are \(2\) and \(1\) per year, respectively. At those mean rates, the cumulative baseline hazard through year 1 is \(2(0.5)+1(0.5)=1.5\). Therefore baseline survival at one year, evaluated at the two prior mean rates, is \(\exp(-1.5)\approx0.2231\). The corresponding baseline event probability by one year is \(1-0.2231=0.7769\).
To see how uncertainty changes the prior-averaged survival, assign independent priors \(\lambda_1\sim\operatorname{Gamma}(2,1)\) and \(\lambda_2\sim\operatorname{Gamma}(2,2)\), both in shape-rate form. Their means are \(2/1=2\) and \(2/2=1\) per year. For each interval, the survival factor over its half-year width is \(\exp(-0.5\lambda_j)\). Applying the Gamma Laplace transform separately, the average survival is \((1/(1+0.5))^2(2/(2+0.5))^2=(2/3)^2(4/5)^2=64/225\approx0.2844\).
The prior-averaged survival, \(0.2844\), differs from survival at the mean rates, \(0.2231\). Both calculations are correct but answer different questions. These are baseline survival values; if \(\eta_i(t)\) is not zero, subject-specific survival also depends on its time-varying multiplier.
Calibration, Diagnostics, and Alternatives
A practical calibration begins with a clinically interpretable time horizon. Ask what range of baseline survival or event probability is plausible at that horizon, then examine which parameter values produce it. For an exponential baseline, a constant rate implies \(S_0(t)=\exp(-\lambda t)\). For a Weibull baseline, check both survival over time and whether the prior on \(k\) puts substantial weight on hazard patterns that are scientifically unreasonable. For a piecewise baseline, calculate cumulative hazard using each interval’s width and rate, and check whether cut points create implausible jumps.
These checks are prior-design diagnostics, not evidence that the fitted model is correct. A prior may be mathematically proper but place much of its probability on event times far outside the relevant follow-up window. Conversely, an apparently broad prior on parameters can imply a concentrated event-time distribution. Inspecting several time horizons and relevant subject-level predictor values helps reveal this mismatch.
| Baseline specification | Typical prior choice | Strength | Potential limitation |
|---|---|---|---|
| Exponential rate | Gamma prior on \(\lambda\) | Simple constant hazard and transparent rate units | Cannot represent a changing baseline hazard |
| Weibull scale and shape | Lognormal priors on \(\rho\) and \(k\) | Allows a monotone increasing or decreasing hazard | Only a restricted range of baseline shapes is available |
| Piecewise-constant rates | Gamma prior for each \(\lambda_j\) | Flexible over selected time intervals; conditional Gamma form | Independent priors can yield abrupt, noisy changes between intervals |
Alternatives should match the intended structure. A parametric Weibull prior may be preferable when a smooth monotone hazard is credible; a piecewise specification allows more local flexibility but depends on cut points and regularization. Independent Gamma priors are not the only option for piecewise rates: priors that relate neighboring log-rates can discourage erratic jumps. Such dependence changes the simple independent interval-wise update derived above. No prior family can compensate for a baseline form that is incompatible with the scientific setting.
Common Mistakes and Limitations
- Confusing a rate with a scale. In the Gamma shape-rate convention, \(b\) is a rate parameter for the Gamma density, while \(\lambda\) is the hazard rate. Check which parameter appears in the exponential term and state units.
- Using survival at the mean as the prior average. In general, \(E[\exp(-\lambda t)]\ne\exp\{-E(\lambda)t\}\). Integrate over the prior when the question concerns prior-averaged survival.
- Ignoring time units. A prior calibrated for hazards per year cannot be carried unchanged to times recorded in months.
- Assuming independent piecewise priors imply a smooth hazard. Independence gives no mechanism for neighboring interval rates to resemble one another.
- Interpreting baseline survival as a subject-specific prediction. A nonzero \(\eta_i(t)\) changes the cumulative hazard and therefore the survival curve.
The baseline prior’s effect is conditional on the rest of the joint model. It does not alone determine observed event times, since the longitudinal process, random effects, association structure, and censoring pattern also matter. Its role is nevertheless fundamental: it controls the hazard shape and time scale available to the model before the event data update that information.
Check Your Understanding
Answer each question, then compare with the solutions below.
- An exponential baseline rate has prior \(\operatorname{Gamma}(3,2)\) in shape-rate form. What are its prior mean rate and units if time is recorded in months?
- For this prior, calculate prior-averaged baseline survival at one month and compare it with survival evaluated at the prior mean rate.
- A piecewise interval has prior \(\operatorname{Gamma}(1.5,3)\), with conditional event count \(d_j=2\) and weighted exposure \(A_j=4\). What is the conditional posterior distribution?
- For a Weibull baseline, what does \(k<1\) imply about the hazard over time? What role does \(\rho\) play?
Solutions. (1) The mean is \(a/b=3/2=1.5\) per month. (2) The prior average is \((2/(2+1))^3=8/27\approx0.2963\). Survival at the mean is \(\exp(-1.5)\approx0.2231\) at one month, so the two summaries differ. (3) The posterior is \(\operatorname{Gamma}(1.5+2,3+4)=\operatorname{Gamma}(3.5,7)\), still in shape-rate form. (4) If \(k<1\), the Weibull baseline hazard decreases over time; \(\rho\) is the positive scale in time units and sets the event-time scale through cumulative hazard \((t/\rho)^k\).