Right-Censoring and Age-Dependent Unaffected Status

Why Age at Examination Matters

For a late-onset disease, an unaffected person has not necessarily escaped the disease for life. A relative examined at age 25 who has no diagnosis has so far remained disease-free; the same observation at age 75 carries different information about the chance of eventual disease. Treating both people as lifetime noncases discards the information in their ages and can distort a segregation likelihood.

The key is to represent disease onset as a time-to-event variable. An examination or end of follow-up supplies an age through which onset has not been observed. If the person has not developed disease by that age, the onset time is right-censored: it is known to exceed the observed age, but its later value is unknown.

Definition: Let \(T_i\) be the age at disease onset for relative \(i\), and let \(g_i\) denote that relative’s genotype. The age-specific penetrance is \(F_{g_i}(a)=P(T_i\leq a\mid g_i)\), the probability of onset by age \(a\) conditional on genotype. The disease-free survival function is \(S_{g_i}(a)=P(T_i>a\mid g_i)=1-F_{g_i}(a)\). If onset time has a continuous density, write that density as \(f_{g_i}(a)\).

Thus, an unaffected observation at age \(a\) contributes \(S_{g_i}(a)\), not the probability of never developing disease. In the usual continuous-time formulation, onset exactly at a specified age has probability zero; \(f_{g_i}(a)\) is a density, with units of inverse age, rather than a probability at that exact age. A probability for onset in a short interval of width \(da\) near age \(a\) is approximately \(f_{g_i}(a)\,da\).

Observed Onset and Right-Censored Onset

Let \(C_i\) be the age at which observation ends, whether because of study end or loss to follow-up. The recorded age is \(Y_i=\min(T_i,C_i)\), and the event indicator is \(\delta_i=1\) if onset is observed by the censoring age and \(\delta_i=0\) otherwise. For an observed onset, the disease-time contribution is the onset density at the recorded age. For a person still disease-free at the end of observation, it is the probability of remaining disease-free through that age.

L_i(g_i)=\{f_{g_i}(Y_i)\}^{\delta_i}\{S_{g_i}(Y_i)\}^{1-\delta_i}.

This expression is the disease-time likelihood conditional on the censoring schedule, or the disease component of the likelihood under suitable independent-censoring conditions. It does not mean that an event and a non-event are equally informative across all ages: both terms depend on age. An observed onset at 50 contributes \(f_{g_i}(50)\); a person censored disease-free at 50 contributes \(S_{g_i}(50)\).

The assumptions behind that contribution matter. For the conditional disease-time product across relatives used below, disease-onset times must be conditionally independent across relatives given the complete genotype configuration. This is an assumption about dependence between relatives; it is not implied merely by assumptions about onset and censoring within each person. If censoring is random, the censoring processes must also be handled appropriately. In particular, a joint product of person-level observed-data contributions requires the relevant onset-censoring pairs to factor across relatives conditional on the genotype configuration, or else a joint censoring model is needed.

Assumption: In the basic segregation likelihood below, condition on the censoring ages and assume disease-onset times are conditionally independent across relatives given the complete genotype configuration \(\mathbf g\). Then the disease-time likelihood for that configuration factors into the individual \(f\) and \(S\) terms. If censoring ages are random rather than conditioned on, their mechanism must be noninformative for the disease model and handled consistently; dependence among relatives’ censoring processes cannot be ignored when modeling their joint observed data.

How the Censoring Assumption Enters

If onset time \(T_i\) and censoring time \(C_i\) are conditionally independent within a person given \(g_i\), and both have continuous distributions, the joint observed-data density for that person is

p(y_i,\delta_i\mid g_i) = \{f_{g_i}(y_i)S_{C_i\mid g_i}(y_i)\}^{\delta_i} \{f_{C_i\mid g_i}(y_i)S_{g_i}(y_i)\}^{1-\delta_i}.

Here \(S_{C_i\mid g_i}\) and \(f_{C_i\mid g_i}\) describe the censoring time. For an observed disease event, the person must also have remained under observation until that age; for a censored observation, censoring occurs at that age while onset has not yet occurred. If the censoring mechanism is fixed and conditioned on, the censoring factors are not part of the disease-time likelihood. If it is random but independent of genotype and has parameters separate from the disease model, its factors can often be treated as nuisance terms. However, if censoring depends on genotype, those factors may vary across genotype configurations and cannot generally be discarded inside a sum over unknown genotypes.

Likewise, independence of \(T_i\) and \(C_i\) within each person does not establish independence of the observed data across relatives. Shared follow-up rules, family-level loss to follow-up, or other dependence may require a joint censoring model. The simple product used for disease times is justified by the stated conditional independence of onset times across relatives, with censoring conditioned on or factored as required.

From Individual Contributions to a Pedigree Likelihood

In a pedigree, some genotypes may be unobserved. Let \(\mathbf g\) denote a complete genotype configuration for all relatives, and let \(P(\mathbf g)\) be its probability under the pedigree transmission model and any specified founder-genotype distribution. Given the conditional-independence assumption above, the disease-time likelihood conditional on that configuration is the product of the relatives’ age-specific contributions. The likelihood for the observed disease histories is obtained by summing over possible configurations, weighting each configuration by its probability.

L_{\mathrm{disease}} = \sum_{\mathbf g}P(\mathbf g) \prod_i \{f_{g_i}(Y_i)\}^{\delta_i} \{S_{g_i}(Y_i)\}^{1-\delta_i}.

The product is not automatic just because the observations belong to different people. It follows here from conditional independence of disease-onset times across relatives given the complete \(\mathbf g\). If onset times remain dependent after conditioning on \(\mathbf g\), for example because of an unmodeled shared familial factor, the product must be replaced by the appropriate joint conditional likelihood. If random censoring is included in the observed-data likelihood, its cross-relative factorization must also be justified, or the joint censoring distribution must be modeled.

Worked Example: An Unaffected Relative at Two Examination Ages

Suppose, purely hypothetically, that a genotype-specific penetrance model gives \(F_g(30)=0.08\) and \(F_g(70)=0.55\). Calculate the disease-time likelihood contribution for a relative known to be unaffected at age 30, and compare it with the contribution if the relative is known to be unaffected at age 70.

For an unaffected observation at age \(a\), the contribution is \(S_g(a)=1-F_g(a)\). At age 30:

$$ S_g(30)=1-0.08=0.92. $$

At age 70:

$$ S_g(70)=1-0.55=0.45. $$

The age-30 observation is compatible with this genotype with disease-time likelihood contribution \(0.92\); the age-70 observation contributes \(0.45\). Neither is a probability that the genotype is correct: each is a likelihood contribution conditional on that genotype. If the model also says \(F_g(\infty)=0.80\), then treating the age-30 relative as a lifetime noncase would substitute \(1-0.80=0.20\) for the correct \(0.92\). That substitution incorrectly asserts knowledge about disease status after age 30.

Worked Example: Event Density Versus Censoring Survival

Assume, for illustration, an exponential onset-time model conditional on genotype, with constant onset hazard \(\lambda_g\). Then \(S_g(t)=e^{-\lambda_g t}\) and \(f_g(t)=\lambda_g e^{-\lambda_g t}\). For genotype \(g_1\), let \(\lambda_{g_1}=0.020\) per year. Compare a disease onset observed at age 40 with a relative censored disease-free at age 40.

For observed onset, \(\delta=1\), so the contribution is the density:

f_{g_1}(40) = 0.020e^{-(0.020)(40)} = 0.020e^{-0.8} \approx 0.008987\ \text{per year}.

For censoring while disease-free at age 40, \(\delta=0\), so the contribution is the survival probability:

S_{g_1}(40) = e^{-(0.020)(40)} = e^{-0.8} \approx 0.4493.

These are different kinds of quantities: the first is a density with units of inverse years, while the second is a dimensionless probability. The calculations use the specified exponential model, which assumes a constant onset hazard within this genotype. They show why one cannot replace an observed event density with a survival probability, or treat a censored observation as if onset had never occurred.

Marginalizing Age-Dependent Evidence in a Pedigree

The effect of censoring is especially important when different genotype configurations imply different age-specific survival probabilities. The next calculation uses only censored observations to make the weighting transparent. Suppose the two listed configurations exhaust the possible configurations in this hypothetical example, and their probabilities already incorporate the pedigree transmission information.

Worked Example: Two Censored Relatives and Two Genotype Configurations

Suppose two relatives are disease-free at ages 30 and 60. Under configuration A, their respective survival probabilities are \(0.90\) and \(0.50\), and \(P(A)=0.40\). Under configuration B, the probabilities are \(0.98\) and \(0.80\), and \(P(B)=0.60\). Assume disease-onset times are conditionally independent across these two relatives given either complete configuration, and condition on their examination ages.

Under configuration A, the conditional disease-time likelihood is the product of the two survival contributions:

$$ L_A=0.90(0.50)=0.45. $$

Under configuration B:

$$ L_B=0.98(0.80)=0.784. $$

Marginalizing over the two configurations gives:

L_{\mathrm{disease}} = P(A)L_A+P(B)L_B = 0.40(0.45)+0.60(0.784) = 0.18+0.4704 = 0.6504.

The sum, rather than a product of separately averaged individual contributions, preserves the fact that both relatives’ contributions are evaluated under the same complete genotype configuration. This calculation also depends on conditional independence across relatives within each configuration; if that assumption fails, \(L_A\) and \(L_B\) must instead be the appropriate joint disease-time likelihoods.

A Short Derivation and a Practical Workflow

The right-censored contribution follows directly from what is observed. If a person is censored at age \(c\), the data say \(T_i>c\). Therefore the probability of the observed disease-free history, conditional on genotype, is \(P(T_i>c\mid g_i)=S_{g_i}(c)\). If onset is observed at \(t\), the data locate the continuous onset time in a small interval \([t,t+dt)\), whose probability is approximately \(f_{g_i}(t)\,dt\). Dividing by the interval width gives the density contribution \(f_{g_i}(t)\). These are the event and censoring terms in the likelihood.

For a pedigree, the conditional likelihood is a product of these terms only when onset times are conditionally independent across relatives given the complete genotype configuration. The law of total probability then averages that conditional likelihood over configurations:

P(\text{observed disease history}) = \sum_{\mathbf g} P(\mathbf g)\, P(\text{observed disease history}\mid\mathbf g).

In applications, a reliable sequence is:

Record each person’s observation.
For every relative, specify the onset age if disease occurred, or the last known disease-free age if observation ended first.

Choose the matching contribution.
Use \(f_{g_i}(Y_i)\) for observed onset and \(S_{g_i}(Y_i)\) for right-censoring.

State the dependence assumptions.
Before multiplying person-level disease contributions, state conditional independence of onset times across relatives given the complete genotype configuration. Specify how censoring is conditioned on or modeled.

Average over unknown configurations.
Weight each configuration-specific likelihood by its pedigree probability and sum across configurations.

Common Pitfalls and Interpretation

The most common error is to code every unaffected relative as a lifetime noncase. An unaffected person examined at age 25 contributes \(S_g(25)\), not \(1-F_g(\infty)\). The distinction can be consequential for late-onset disease because many genetically susceptible people may remain unaffected at younger ages.

Ignoring examination age. Two relatives with the same current disease status can provide different evidence if one was examined much later. Use each person’s observed age in the survival contribution.
Calling censoring absence of disease for life. Right-censoring means onset has not been observed through a stated age; it does not establish that onset will never occur.
Multiplying without a dependence assumption. The product over relatives requires conditional independence of disease-onset times given the complete genotype configuration. Within-person independence of onset and censoring does not establish this across-person condition.
Dropping censoring terms automatically. Conditioning on a fixed censoring schedule is different from modeling random censoring. If censoring depends on genotype or is dependent across relatives, a simple disease-only product may not be the full observed-data likelihood.
Treating a density as a probability. An event contribution \(f_g(t)\) has units of inverse age. A censored contribution \(S_g(c)\) is a probability and has no units.

Right-censoring also requires substantive care. For example, if death prevents a later disease diagnosis, treating death as ordinary independent censoring may be inappropriate when death and disease onset are related. The censoring assumptions should reflect how follow-up ended and what processes could influence observation.

Key takeaway: An unaffected relative contributes the probability of remaining disease-free through the age at examination. A relative with observed onset contributes an onset density at the observed age. In pedigree likelihoods, multiply across relatives only under conditional independence of disease-onset times given the complete genotype configuration, then marginalize over configurations; handle censoring assumptions explicitly.

Check Your Understanding

Answer each question by identifying the relevant observation and likelihood contribution.

A relative is disease-free at age 42. Which function of age and genotype gives the disease-time likelihood contribution?
In a continuous onset-time model, explain why an observed onset contributes a density rather than the probability of onset at exactly that age.
State the across-relative assumption needed to multiply disease-time contributions conditional on a complete genotype configuration.
Why does conditional independence of onset and censoring within each person not, by itself, justify a product over relatives?
When might censoring terms need to be retained or modeled rather than omitted from a disease likelihood?