Choosing Among Parametric Survival Distributions

Why Distribution Choice Matters

A parametric survival model describes the event-time distribution with a specified family of curves. Choosing that family affects estimated survival probabilities, hazard shapes, extrapolations beyond observed follow-up, and, when covariates are included, estimated effects. A model with more flexibility can fit the observed data better, but may also spend parameters fitting random variation. Distribution choice is therefore a balance between fit, complexity, and scientific plausibility.

The aim is not to identify a family that is “true” from a single statistic. Instead, compare candidate models fitted to the same observations, examine whether their fitted curves reproduce important features of the data, and consider how sensitive the conclusions are to the choice. The models below are the exponential, Weibull, lognormal, and log-logistic families introduced in the earlier material on parametric survival models.

Likelihoods and Comparable Models

Let $T_i$ be subject $i$'s event time and $C_i$ the censoring time. For right-censored data, the observed time is $Y_i=\min(T_i,C_i)$, and $\delta_i=1$ indicates that the event was observed; $\delta_i=0$ indicates right censoring. Write $f(t)$, $S(t)$, and $h(t)=f(t)/S(t)$ for a candidate model's density, survival function, and hazard function.

Definition: Under independent, noninformative right censoring, a subject contributes $f(Y_i)^{\delta_i}S(Y_i)^{1-\delta_i}$ to the event-time likelihood. Thus, for $n$ independent subjects, the log-likelihood is $$ \ell(\theta)=\sum_{i=1}^{n}\left\{\delta_i\log f(Y_i;\theta)+(1-\delta_i)\log S(Y_i;\theta)\right\}. $$ A censored subject contributes a survival probability, not a density at the censoring time. This likelihood omits the censoring-time distribution because it is assumed not to depend on the event-time parameters being estimated.

The independence condition means that censoring does not carry additional information about a subject's event time, either unconditionally or after conditioning on covariates included in the model. The likelihood also assumes that subjects are independent and that the chosen event-time distribution is appropriate for the population and covariate structure being modeled. If censoring is informative, or the event process is otherwise misspecified, the resulting comparisons can be misleading.

For a positive event time $t$, common parameterizations are:

Exponential: rate $\lambda>0$, with $S(t)=\exp(-\lambda t)$ and $h(t)=\lambda$. Its hazard is constant.
Weibull: shape $\kappa>0$ and scale $a>0$, with $S(t)=\exp[-(t/a)^\kappa]$. The hazard increases if $\kappa>1$, decreases if $\kappa<1$, and is constant if $\kappa=1$. The exponential model is a special case.
Lognormal: $\log T\sim N(\mu,\sigma^2)$, with $\mu$ real and $\sigma>0$. Its density is $f(t)=\phi((\log t-\mu)/\sigma)/(t\sigma)$, where $\phi$ is the standard normal density.
Log-logistic: shape $\kappa>0$ and scale $a>0$, with $S(t)=1/[1+(t/a)^\kappa]$. Its hazard can rise and then fall when $\kappa>1$; it decreases when $\kappa\leq1$.

In this basic comparison, the exponential model has one distributional parameter, while each of the other three has two. A regression version may add covariate coefficients; the parameter count $k$ used in a criterion must include all estimated parameters in the fitted likelihood, not just the distributional ones.

Likelihood-Based Criteria

A maximized log-likelihood $\ell$ rewards fit: larger values indicate that the fitted model assigns greater likelihood to the observed event and censoring pattern. It cannot, by itself, fairly compare models of different flexibility, since adding parameters generally cannot reduce the maximized likelihood. Information criteria combine fit with a penalty for the number of estimated parameters.

Definition: For a model with $k$ estimated parameters fitted to $n$ independent subjects, the Akaike information criterion and Bayesian information criterion are $$ \mathrm{AIC}=-2\ell+2k,\qquad \mathrm{BIC}=-2\ell+k\log(n). $$ Among the candidates considered, the smaller value is preferred by that criterion. These are relative comparisons: neither value is a probability that a model is correct.

AIC's penalty is motivated by an estimate of the optimism in a model's in-sample log-likelihood as an estimate of predictive fit: under regularity conditions, that optimism is approximately $k$ on the log-likelihood scale, or $2k$ on the $-2\ell$ scale. BIC applies a penalty that grows with sample size and has a large-sample model-selection interpretation under its own assumptions. They answer related but not identical questions, so they need not select the same candidate.

Use the same subjects, event definition, censoring information, and likelihood convention for every candidate. AIC or BIC values from models fitted to different outcome records or different likelihood data are not a meaningful head-to-head comparison. Smaller is better, but a very small difference is weak evidence of a practical distinction; the criterion does not establish that the preferred model fits adequately.

Worked Example: Comparing Four Fits by AIC and BIC

Suppose four distributions are fitted to the same hypothetical right-censored data from $n=120$ subjects. Assume independent censoring and that each fit converged to its maximum. The reported maximized log-likelihoods include all estimated parameters. The exponential fit has $k=1$; each other fit has $k=2$.

Model	$\ell$	$k$	AIC	BIC
Exponential	-302	1	606	608.8
Weibull	-286	2	576	581.6
Lognormal	-283	2	570	575.6
Log-logistic	-284	2	572	577.6

For the lognormal model, $\mathrm{AIC}=-2(-283)+2(2)=566+4=570$. Its BIC is $-2(-283)+2\log(120)=566+9.575=575.6$. For the exponential model, $\mathrm{AIC}=604+2=606$, and $\mathrm{BIC}=604+\log(120)=608.8$. Applying the same calculations to the other rows gives the table.

Both criteria rank the lognormal model first among these candidates, with the log-logistic model next. This is not proof that event times are lognormal: it is a relative preference given the candidate set, fitted likelihoods, and criterion. The exponential fit is substantially worse by these criteria, but the graphical checks still matter.

Graphical Checks and Model Shape

A useful first check is to overlay each fitted survival curve on a Kaplan–Meier estimate. Look for systematic departures across the time range of interest: for example, a model that consistently places survival above the empirical estimate in the middle of follow-up and below it later may be missing a change in curve shape. The Kaplan–Meier estimate is itself uncertain, especially in the tail where few subjects remain at risk. Use confidence bands or other uncertainty cues, and do not interpret every small visual gap as evidence of failure.

Distribution-specific plots can reveal patterns that a curve overlay makes less obvious. For the Weibull model, a plot of $\log[-\log S(t)]$ against $\log t$ should be approximately linear, with slope $\kappa$. For a lognormal model, plotting the standard normal quantile of the event distribution against $\log t$ should be approximately linear. For a log-logistic model, $\log\{F(t)/[1-F(t)]\}$ against $\log t$ should be approximately linear, where $F(t)=1-S(t)$. Censoring complicates these plots: they should be constructed using appropriate survival-data estimates, not by treating censored times as observed events.

Worked Example: Reading a Survival-Curve Overlay

Consider hypothetical data on 80 subjects, with independent right censoring. At times 1, 2, 4, and 8 months, suppose the Kaplan–Meier estimates are respectively $0.88, 0.68, 0.37,$ and $0.14$. Fitted survival probabilities at those same times are:

Time (months)	Kaplan–Meier	Exponential	Weibull	Lognormal	Log-logistic
1	0.88	0.81	0.90	0.91	0.89
2	0.68	0.66	0.67	0.66	0.69
4	0.37	0.43	0.34	0.35	0.36
8	0.14	0.19	0.13	0.12	0.16

The exponential curve is below the estimated survival at month 1 and above it at months 4 and 8, suggesting that a constant hazard may not capture the observed curve's shape. The other three curves are closer at these selected times. Their differences are small in this display, so it does not establish a winner; the full curves, uncertainty, and likelihood criteria should also be considered. Moreover, the tail comparison at month 8 may be unstable if few subjects remain under observation.

Balancing Flexibility and Fit

The exponential model is restrictive: one rate determines its entire curve. The Weibull relaxes that restriction by estimating a shape parameter but still constrains hazard shape to be monotone or constant. Lognormal and log-logistic distributions can accommodate nonmonotone hazards, though their tail behavior differs. Flexibility is not automatically beneficial: it is useful only if the data contain enough information to estimate the extra shape, and if the resulting shape is credible for the application.

A semiparametric Cox model and the nonparametric Kaplan–Meier estimator are important alternatives. The Kaplan–Meier estimator describes survival without choosing one of these parametric families, while the Cox model leaves the baseline hazard unspecified and models covariate effects through proportional hazards. Parametric models can offer smoother estimates and direct extrapolation, but their extrapolation depends strongly on the distributional assumption. AIC and BIC compare likelihood-based candidates; they do not make these approaches interchangeable or remove the need to state the modeling goal.

Worked Example: When AIC and BIC Disagree

In a second hypothetical data set, $n=40$ independent subjects have right-censored event times under independent censoring. Compare an exponential model ($k=1$) with a Weibull model ($k=2$); the latter includes the former when $\kappa=1$. Suppose their maximized log-likelihoods are $-91.0$ and $-89.4$, respectively.

The exponential AIC is $-2(-91.0)+2(1)=182+2=184.0$. The Weibull AIC is $-2(-89.4)+2(2)=178.8+4=182.8$, so AIC favors the Weibull by $1.2$ points. For BIC, the exponential value is $182+\log(40)=185.69$, while the Weibull value is $178.8+2\log(40)=186.18$. BIC slightly favors the exponential, by about $0.49$ points.

The Weibull improves the maximized log-likelihood by $1.6$, but its extra parameter is penalized differently by the two criteria. The disagreement is a signal to examine the estimated shape, uncertainty, graphical fit, and intended use rather than to treat either ranking as definitive. Both BIC values are close, so the BIC preference is especially weak.

Diagnostics, Limitations, and a Common Pitfall

Before trusting a ranking, check that the likelihood maximization converged, parameter estimates are finite and plausible, and all models used the same observations and censoring indicators. Inspect fitted curves over the time interval that matters, not only where events are dense. A model may fit the early data well but extrapolate implausibly; sparse late follow-up cannot reliably determine tail behavior. Also consider whether covariates, changing hazards, or dependence between subjects call for a richer model than any of these four distributions.

Common mistake: “The model with the largest likelihood must be best, so I will choose it.” This ignores the fact that a more flexible model can improve likelihood simply by using extra parameters. Compare models with a complexity-aware criterion such as AIC or BIC, and then assess fit with graphical and subject-matter checks. The reverse mistake is to treat the smallest AIC as proof of adequate fit: a best-ranked model can still fit poorly if all candidates are misspecified.

A likelihood-ratio test is appropriate only for nested models under its regularity conditions; it is not a general substitute for AIC or BIC when comparing nonnested distributions. The exponential model is nested within the Weibull, so a test can be considered for that particular comparison, but it answers a different question from predictive or information-criterion ranking. Finally, results may be sensitive to which candidate distributions are considered, how covariates enter, and the amount of censoring. Report the comparison transparently rather than presenting a single selected family as inevitable.

Key takeaway: Compare parametric survival distributions using a common censored-data likelihood, penalize flexibility with AIC or BIC, and check fitted curves against the observed survival pattern. A criterion ranks candidates; it does not certify model adequacy.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a subject censored at $Y_i=5$, what is the subject's event-time likelihood contribution?
Two models fitted to the same data have log-likelihoods $-50$ and $-48$, with $k=1$ and $k=2$, respectively. Calculate their AIC values.
What hazard shape does a Weibull model imply when its shape parameter is below 1?
Why is a close overlay with the Kaplan–Meier estimate at late times not necessarily strong evidence of good tail fit?

Solutions. (1) Since $\delta_i=0$, the contribution is $S(5)$, the probability of surviving beyond the censoring time under the model. (2) The first AIC is $-2(-50)+2(1)=102$; the second is $-2(-48)+2(2)=100$. AIC favors the second model by 2 points, but this does not establish adequate fit. (3) Its hazard decreases over time. (4) Few subjects may remain under observation in the tail, so the Kaplan–Meier estimate is uncertain there; visual agreement should be interpreted alongside the amount of follow-up and uncertainty.)