Missing Covariates In Absolute-Risk Prediction

Why Missing Predictors Change the Prediction Target

A competing-risk prediction is defined for a specified covariate profile and time horizon. For cause $k$, write $F_k(t\mid x)$ for the probability of experiencing cause $k$ by time $t$, before any competing event, for a person with covariates $x$. If a predictor is missing, the profile is not fully specified. There is therefore no single risk obtained merely by inserting a blank value into the prediction model.

The appropriate target depends on the question. For an individual with observed covariates $x_{\mathrm{obs}}$ and missing covariates $Z$, one useful target is the risk averaged over the plausible values of $Z$ among people with those observed characteristics. For a cohort, the target may be the average individual risk across everyone in that cohort, including people with missing predictors. These targets differ from the risk among complete cases unless additional assumptions justify treating the complete cases as representative.

Definition: If $Z$ is a missing predictor and $X_{\mathrm{obs}}$ denotes observed predictors, the risk for cause $k$ at horizon $t$, averaged over missing values, is $$ \widetilde F_k(t\mid x_{\mathrm{obs}}) = \int F_k(t\mid x_{\mathrm{obs}},z)\, dP(z\mid x_{\mathrm{obs}},\text{available prediction information}). $$ For a discrete $Z$, the integral is a sum over its possible values. The distribution used for averaging must be appropriate to the prediction target and the information available when the prediction is made.

This is a conditional expectation: it averages the cause-specific cumulative incidence over uncertainty about the missing predictor, while retaining the competing-event structure inside each prediction $F_k$. It is not the same as predicting for a hypothetical person whose covariates equal their mean values.

Missingness Assumptions and Prediction Data

Missingness assumptions describe how the chance that a value is absent relates to the data. They do not, by themselves, guarantee that a prediction model or imputation procedure is correct.

Missing completely at random (MCAR): the probability a predictor is missing does not depend on observed or unobserved data.
Missing at random (MAR): conditional on observed information, the probability of missingness does not depend on the missing value itself. The observed information used for this condition must include relevant variables and missingness patterns.
Missing not at random (MNAR): even after conditioning on observed information, missingness still depends on the unseen value, or on other unobserved information.

Under MCAR, complete cases can represent the original cohort with respect to the missingness process, although sampling variability and model misspecification remain. Under MAR, complete cases are generally a selected subset; an appropriate imputation or weighting strategy may use the observed predictors related to missingness. Under MNAR, observed data alone usually cannot identify the missing-value distribution without additional assumptions, sensitivity analyses, or external information.

Prediction-time information deserves particular care. In model development, an imputation model may use observed follow-up information, including event type and event time, to preserve relationships needed when fitting the prediction model. For a new person being predicted at baseline, future event information is unavailable and must not be used to impute that person's baseline predictors. The development and deployment procedures must therefore be distinguished.

Deriving the Risk-Averaging Rule

The averaging rule follows directly from conditioning on the missing predictor. Let $Y_k(t)$ equal 1 if cause $k$ occurs by $t$ before any competing event, and 0 otherwise. For an individual with observed profile $x_{\mathrm{obs}}$, the conditional probability of this outcome is the risk averaged over the possible values of $Z$:

\begin{aligned} P\{Y_k(t)=1\mid x_{\mathrm{obs}}\} &=\sum_z P\{Y_k(t)=1\mid x_{\mathrm{obs}},Z=z\} P(Z=z\mid x_{\mathrm{obs}})\\ &=\sum_z F_k(t\mid x_{\mathrm{obs}},z) P(Z=z\mid x_{\mathrm{obs}}). \end{aligned}

For continuous predictors, the sum becomes an integral. The equality is the law of total probability; it requires that the conditional distribution of the missing predictor used in the average corresponds to the target population and available information. In practice, that distribution is estimated, so its adequacy is an additional modeling concern.

The competing-risk interpretation is preserved because each term is a cumulative incidence for cause $k$, with competing events accounted for in the model. Averaging cause-specific risks separately also preserves coherence if each completed profile has valid cause probabilities: nonnegative weighted averages remain nonnegative, and if the cause risks across all causes sum to at most 1 for every profile, their averages also sum to at most 1.

Worked Example: Averaging an Individual Risk Over a Missing Predictor

Suppose a hypothetical model predicts 5-year risks for two causes. A person's binary predictor $Z$ is missing, but among people with the same observed predictors the estimated probability of $Z=1$ is 0.25. The model gives the following risks:

Value of $Z$	Cause 1 risk by 5 years	Cause 2 risk by 5 years
0	0.08	0.04
1	0.28	0.10

The probability of $Z=0$ is $1-0.25=0.75$. The averaged cause 1 risk is

$$ 0.75(0.08)+0.25(0.28)=0.06+0.07=0.13. $$

For cause 2, the corresponding calculation is

$$ 0.75(0.04)+0.25(0.10)=0.03+0.025=0.055. $$

Thus the predictions are 13% for cause 1 and 5.5% for cause 2 by 5 years. Their sum is 18.5%, below 100%, consistent with the supplied profiles. This calculation assumes that the probability 0.25 adequately represents the distribution of $Z$ for this person's observed information and prediction target. It does not claim that the person's unobserved value is literally 0.25.

Complete-Case Prediction and Cohort Standardization

A complete-case prediction analysis excludes records with any missing predictor. This may be practical, but it changes the population over which a standardized risk is averaged. If the goal is the mean predicted risk in a target cohort of $n$ people, the target is

\overline F_k(t)=\frac{1}{n}\sum_{i=1}^{n}F_k(t\mid x_i),

where every person's risk is defined for their profile. When some covariates are missing, each such contribution can instead be averaged over that person's appropriate conditional distribution of missing values. Averaging only the complete cases estimates a different quantity: the mean risk among people with complete predictor records. It equals the full-cohort target only under conditions that make the complete cases representative for this prediction target, or after suitable adjustment.

Worked Example: Complete-Case Versus Full-Cohort Risk

Consider a hypothetical cohort of four people and the predicted 3-year risk of one cause. Three people have complete predictors, with risks 0.10, 0.20, and 0.15. For the fourth person, $Z$ is missing. Given that person's observed predictors, the estimated probabilities are 0.75 for $Z=0$ and 0.25 for $Z=1$; the corresponding risks are 0.12 and 0.32.

The fourth person's averaged risk is

$$ 0.75(0.12)+0.25(0.32)=0.09+0.08=0.17. $$

The complete-case mean is

\frac{0.10+0.20+0.15}{3} =\frac{0.45}{3}=0.15.

Using the imputation-averaged risk for the fourth person, the full-cohort mean is

\frac{0.10+0.20+0.15+0.17}{4} =\frac{0.62}{4}=0.155.

The complete-case mean is 15.0%, whereas the full-cohort estimate is 15.5%. The difference is small in this hypothetical example, but neither number is automatically the right target: the choice depends on whether the intended population is the complete cases or the full cohort. The calculation also assumes the risk model and the conditional distribution used for the missing predictor are adequate.

Multiple Imputation for Prediction

Multiple imputation represents uncertainty about missing values by creating $M$ plausible completed data sets, rather than filling each blank once. For a prediction, calculate the cause-specific risk in each completed data set and average the risks:

\overline F_k^{\,\mathrm{MI}}(t) = \frac{1}{M}\sum_{m=1}^{M}\widehat F_{k,m}(t).

The predictions $\widehat F_{k,m}(t)$ may also reflect model-estimation variation if the prediction model is refitted in each imputed development data set. When predicting a new person, the missing predictor is drawn from an imputation distribution appropriate to that person's observed baseline information; future outcomes for that person cannot be supplied to that imputation.

The imputation models should be compatible with the variables and relationships relevant to the prediction model. They should preserve important nonlinearities, interactions, and associations with event type and time in model development. The imputation procedure, fitted prediction model, and target population must also be sufficiently aligned. A large number of imputations cannot repair a badly specified imputation model or an unsupported MAR assumption.

Worked Example: Combining Risks Across Imputations

For a hypothetical new person with one missing predictor, suppose three imputed values lead to the following 5-year risks from a fitted competing-risk model. The imputations are equally weighted for this illustration.

Imputation	Cause 1 risk	Cause 2 risk
1	0.12	0.05
2	0.18	0.07
3	0.15	0.06

The averaged risks are

\overline F_1(5)=\frac{0.12+0.18+0.15}{3} =\frac{0.45}{3}=0.15, \qquad \overline F_2(5)=\frac{0.05+0.07+0.06}{3} =\frac{0.18}{3}=0.06.

The resulting prediction is 15% for cause 1 and 6% for cause 2 by 5 years. The sum, 21%, is coherent with the imputation-specific predictions, whose cause risks each sum to less than 1. The spread across the three predictions also signals that the missing value matters for prediction. With only three imputations this example is arithmetic, not a recommendation for the number of imputations in an analysis. Uncertainty intervals require accounting for both model and imputation uncertainty; the simple average alone is a point prediction.

Why Predicting at Average Covariates Can Fail

A tempting shortcut is to replace a missing predictor by its conditional mean and evaluate the risk model once. This is generally not equivalent to averaging risks. The risk function is often nonlinear in the predictors, and competing-risk prediction itself combines event processes in a nonlinear way. In general,

F_k(t\mid x_{\mathrm{obs}},E[Z\mid x_{\mathrm{obs}}]) \ne E\{F_k(t\mid x_{\mathrm{obs}},Z)\mid x_{\mathrm{obs}}\}.

For a categorical predictor, its mean may not even represent a possible category. For continuous predictors, a mean-profile prediction may describe no typical person and does not account for uncertainty about the missing value. Use risk averaging or multiple imputation when the desired target is an average over plausible missing values.

Diagnostics, Alternatives, and Limitations

Before interpreting predictions, check the data and the modeling steps. Report how many records have each missingness pattern, whether missingness differs across observed predictors, and how many complete cases remain. For imputation, inspect whether imputed values are plausible, compare their distributions with observed values where meaningful, and check that important associations and predictor ranges are represented. For prediction, compare risks across imputations and assess whether any profiles lead to incoherent cause probabilities.

These checks can identify errors or instability, but they cannot establish MAR from observed data alone or prove that an imputation distribution is correct. If missingness may be MNAR, examine how conclusions change under plausible alternative assumptions. If the target population differs from the population used to estimate the imputation distribution, transportability also needs justification.

Approach	What it estimates or does	Main concern
Complete-case prediction	Uses records with all required predictors	May target a selected subset and discard information
Single imputation	Fills each missing value once, then predicts	Typically treats the filled values as known and understates missing-value uncertainty
Multiple imputation and risk averaging	Averages predictions across plausible completed data sets	Depends on adequate imputation models and assumptions about missingness
Prediction at a mean covariate value	Evaluates one profile using a substituted value	Generally differs from average risk and may be meaningless for categorical predictors

No approach removes the need to specify the target. A complete-case analysis can be appropriate if the intended population is explicitly the complete cases, but it should not be described as a full-cohort prediction without justification. Imputation can use more of the available information, but its output is conditional on assumptions and models that should be reported.

Common Mistake

Common mistake: “The missing predictor is probably 0.25, so I will enter 0.25 and report the resulting risk as the person's risk.” This substitutes a mean covariate value for an average over plausible covariate values. The correction is to define the prediction target and, when it is the risk conditional on observed information, average the cause-specific risks over an appropriate conditional distribution or over multiple imputations. Also distinguish this prediction from the mean risk among complete cases.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

An individual has missing binary predictor $Z$, with $P(Z=1\mid X_{\mathrm{obs}})=0.40$. The cause 1 risks at the horizon are 0.10 for $Z=0$ and 0.25 for $Z=1$. What is the risk averaged over $Z$?
Why does the mean risk among complete cases not automatically estimate the standardized risk for the full cohort?
In multiple imputation, should the final point prediction be the risk from the average imputed covariate value or the average of the risks? Explain.
What does MAR mean, and can it be confirmed from observed data alone?

Solutions. (1) $0.60(0.10)+0.40(0.25)=0.06+0.10=0.16$, or 16% by the stated horizon. (2) Complete cases may be a selected subset whose predictor profiles and risks differ from those in the full cohort; equality requires representativeness for the target or an appropriate adjustment. (3) Average the risks across imputations: a nonlinear risk function generally makes the risk at an average covariate value different from the average risk. (4) MAR means missingness does not depend on the missing value after conditioning on the relevant observed information. MAR is an assumption that cannot generally be verified from observed data alone.