Lognormal and Log-Logistic Survival Models

Why Consider These Survival Distributions?

A parametric survival model specifies a full probability distribution for a positive event time, rather than leaving the baseline hazard unrestricted. The Weibull model can represent constant, increasing, or decreasing hazards, but its hazard is monotone unless it is constant. Lognormal and log-logistic models add useful alternatives: their hazards can rise and later decline, and their upper tails differ substantially. Those differences matter when the analysis concerns long-term survival or when risk may peak during follow-up.

Both distributions describe a positive event time \(T\), measured in a stated time unit. They are models for the event-time distribution; their use does not, by itself, account for censoring or establish that censoring is noninformative. For a simple unadjusted model, assume event times in the target population follow the stated distribution. If parameters are modeled conditional on covariates, the chosen distribution applies conditionally, with its parameters specified by that model.

Lognormal Event Times

Definition: An event time \(T\) has a lognormal distribution with location \(\mu\) and scale \(\sigma>0\) if \(\log T\sim N(\mu,\sigma^2)\). Thus \(T>0\), the median event time is \(\exp(\mu)\), and \(\sigma\) controls the spread of log event times. Write \(\Phi\) and \(\phi\) for the standard normal cumulative distribution function and density.

Let \(z(t)=(\log t-\mu)/\sigma\), for \(t>0\). Since \(P(T\le t)=P(\log T\le\log t)\), the cumulative distribution function is \(\Phi(z(t))\). Differentiating it gives the density; the factor \(1/(t\sigma)\) comes from differentiating \(z(t)\) with respect to \(t\).

F(t)=\Phi\!\left(\frac{\log t-\mu}{\sigma}\right),\qquad f(t)=\frac{1}{t\sigma}\phi\!\left(\frac{\log t-\mu}{\sigma}\right),\qquad S(t)=1-\Phi\!\left(\frac{\log t-\mu}{\sigma}\right).

The hazard is the instantaneous event rate at time \(t\), conditional on remaining event-free until \(t\). For a continuous event-time distribution, it equals density divided by survival probability:

h(t)=\frac{f(t)}{S(t)} =\frac{\phi(z(t))}{t\sigma\{1-\Phi(z(t))\}}.

The hazard has units of inverse time. Although \(\mu\) is a location on the log-time scale, \(\exp(\mu)\) is in the original time unit. The lognormal hazard approaches zero as \(t\) approaches zero and also as \(t\) grows without bound. For large \(t\), the normal-tail approximation \(1-\Phi(z)\approx\phi(z)/z\) gives \(h(t)\approx z(t)/(\sigma t)\), which tends to zero. The hazard therefore cannot be monotone increasing over the entire positive time axis; it rises and eventually falls, though the timing and prominence of the peak depend on the parameters.

Worked Example: Lognormal Survival and Hazard at 24 Months

Suppose a hypothetical event time in months is lognormal with \(\mu=\log(12)\) and \(\sigma=0.5\). Calculate survival at 24 months and the hazard at that time.

First standardize the log time: \(z(24)=\{\log(24)-\log(12)\}/0.5=\log(2)/0.5=1.3863\). Using standard normal values, \(1-\Phi(1.3863)\approx0.08283\) and \(\phi(1.3863)\approx0.15261\). Therefore, the probability of remaining event-free beyond 24 months under this model is approximately \(0.08283\), or 8.283%.

Substitute into the hazard formula, retaining the time unit of months: \(h(24)\approx0.15261/\{24(0.5)(0.08283)\}=0.15261/0.99396\approx0.1535\) per month. This is the model’s instantaneous event rate at month 24 among those still event-free then; it is not the probability of an event during the next month.

Log-Logistic Event Times

Definition: A positive event time \(T\) has a log-logistic distribution with scale \(\lambda>0\) and shape \(\kappa>0\) if \(\log T\) has a logistic distribution with location \(\log\lambda\) and scale \(1/\kappa\). Its cumulative distribution function is \(F(t)=1/\{1+(\lambda/t)^\kappa\}\), for \(t>0\). The median event time is \(\lambda\).

Subtracting this cumulative distribution function from one gives the survival function. Differentiating the cumulative distribution function gives the density, and dividing by survival gives the hazard. With \(x=t/\lambda\), these expressions are:

S(t)=\frac{1}{1+(t/\lambda)^\kappa},\qquad f(t)=\frac{\kappa}{\lambda} \frac{(t/\lambda)^{\kappa-1}}{\{1+(t/\lambda)^\kappa\}^2},\qquad h(t)=\frac{\kappa}{\lambda} \frac{(t/\lambda)^{\kappa-1}}{1+(t/\lambda)^\kappa}.

The scale \(\lambda\) has units of time, while \(\kappa\) is dimensionless. The shape parameter determines the hazard pattern. If \(\kappa\le1\), the hazard decreases with time (for \(\kappa=1\), it begins at \(1/\lambda\)). If \(\kappa>1\), it begins at zero, increases to a maximum, and then declines toward zero. To locate the maximum, differentiate the log hazard with respect to \(x\):

\frac{d}{dx}\log h =\frac{\kappa-1}{x}-\frac{\kappa x^{\kappa-1}}{1+x^\kappa} =\frac{\kappa-1-x^\kappa}{x(1+x^\kappa)}.

For \(\kappa>1\), the derivative is zero at \(x^\kappa=\kappa-1\), is positive before that point, and is negative after it. Thus the peak occurs at \(t=\lambda(\kappa-1)^{1/\kappa}\). For \(\kappa\le1\), the numerator is negative for every \(x>0\), so the hazard decreases throughout.

Worked Example: A Rising-Then-Falling Log-Logistic Hazard

Consider a hypothetical event time in months with \(\lambda=8\) months and \(\kappa=2\). Its median is 8 months. At that time, \(S(8)=1/\{1+(8/8)^2\}=1/2\). The hazard is \(h(8)=(2/8)(1)/(1+1)=0.125\) per month.

Because \(\kappa=2>1\), the hazard reaches its maximum at \(t=8(2-1)^{1/2}=8\) months. At 16 months, \(t/\lambda=2\), so \(h(16)=(2/8)(2)/(1+2^2)=0.5/5=0.1000\) per month. The model’s hazard is lower at 16 months than at its peak, even though the event-free group at that later time is a selected subset of the original population. The calculation describes the distribution’s hazard, not a causal effect of time.

Tail Behavior and Model Implications

The distributions differ in how quickly their survival probabilities decay at long times. For the log-logistic model, when \(t\) is large relative to \(\lambda\), \(S(t)=1/\{1+(t/\lambda)^\kappa\}\approx(\lambda/t)^\kappa\). This is a power-law tail, and its hazard behaves as \(h(t)\approx\kappa/t\). The lognormal survival tail instead decays faster than any inverse power of \(t\), but slower than an exponential tail. In particular, its hazard also approaches zero, at a rate approximately \(z(t)/(\sigma t)\).

Consequently, the log-logistic model can assign substantially more probability to extremely long event times than the lognormal model with a similar central event time. That difference can affect extrapolations beyond observed follow-up. Tail behavior should be evaluated in light of the scientific setting and data support rather than chosen solely because one model fits the center of the observed distribution.

Worked Example: Comparing Long-Term Survival

Compare two hypothetical distributions with the same median, 2 time units. Let the lognormal model have \(\mu=\log(2)\), \(\sigma=0.6\), and the log-logistic model have \(\lambda=2\), \(\kappa=2\). At \(t=10\), the lognormal standardized value is \(z=\{\log(10)-\log(2)\}/0.6=\log(5)/0.6\approx2.6824\). Thus \(S(10)=1-\Phi(2.6824)\approx0.00365\).

For the log-logistic model, \(S(10)=1/\{1+(10/2)^2\}=1/26\approx0.03846\). Both models have median 2, but at 10 time units the log-logistic survival probability is much larger. This numerical contrast illustrates the heavier power-law tail of the log-logistic distribution; it does not establish which model is appropriate for any particular data set.

Assumptions, Diagnostics, and Alternatives

These distributions assume a continuous, strictly positive event time and a particular parametric shape. The lognormal assumption is that log event times follow a normal distribution; the log-logistic assumption is that log event times follow a logistic distribution. In an adjusted analysis, this distributional assumption must be considered conditional on the covariates and the chosen parameter model. With right-censored data, inference also requires an appropriate assumption about the censoring mechanism; specifying an event-time distribution alone does not justify treating censored times as observed events.

Useful diagnostics compare the model-implied survival curve with a nonparametric estimate such as the Kaplan–Meier estimator over the time range where data are informative. Systematic disagreement, especially in the upper tail, can signal a poor distributional assumption. Hazard estimates can be noisy late in follow-up because few individuals remain at risk, so an apparent late rise or fall should not be interpreted without considering that uncertainty. Sensitivity to the chosen distribution is especially important when conclusions rely on extrapolated survival.

The Weibull model is a main parametric alternative: its hazard is monotone, except in the constant-hazard case. Lognormal and log-logistic models allow a hazard that rises and falls, with distinct tail behavior. The Cox proportional hazards model is a semiparametric alternative that does not impose one of these parametric baseline distributions, though it makes a proportional-hazards assumption for covariate effects. These approaches answer related but not identical modeling needs; no distribution is automatically preferable in every application.

A common mistake is to infer that equal medians imply similar long-term survival. The worked comparison shows why this is wrong: distributions can share a median and still assign very different probabilities to long event times. Another pitfall is to call a hazard of \(0.1535\) per month a 15.35% chance of an event in the next month. A hazard is an instantaneous rate conditional on survival to that time; it is not itself a finite-interval probability.

Key takeaway: Use the lognormal and log-logistic formulas with their parameterizations stated explicitly. Check both the hazard shape and the tail against the scientific question and the observed follow-up; a good fit near the median does not guarantee reliable tail behavior.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a lognormal event time, which parameter determines the median, and what is the median?
For a log-logistic model with \(\lambda=5\) days and \(\kappa=1\), calculate \(S(5)\) and \(h(5)\), including units.
For a log-logistic model with \(\lambda=4\) years and \(\kappa=3\), at what time does the hazard reach its maximum?
Which distribution has a power-law survival tail, and why can that matter when projecting beyond observed follow-up?

Solutions. (1) The median is \(\exp(\mu)\), because the median of \(\log T\) is \(\mu\). (2) \(S(5)=1/\{1+(5/5)^1\}=0.5\). Also, \(h(5)=(1/5)(1)/(1+1)=0.1000\) per day. (3) The peak time is \(4(3-1)^{1/3}=4(2^{1/3})\approx5.040\) years. (4) The log-logistic survival tail is approximately \((\lambda/t)^\kappa\), a power law. It can therefore yield more long-term survival than a lognormal model, making distributional choice consequential for extrapolation.