Trace Missingness in Renewal Applications

Why Missingness Needs Its Own Investigation

A renewal model may have a fully recorded outcome but an incomplete application predictor, such as annual mileage, vehicle use, or a telematics score. The first response should not be to delete every incomplete record or fill every gap with an average. Missingness can differ across renewal outcomes and policy groups, and either pattern may change what the modeling data represent.

This investigation has two distinct purposes. First, describe where and for whom values are missing. Second, choose a handling rule that can be reproduced when the model is fitted and when new renewal applications are scored. A difference in missingness rates is evidence of an association, not by itself proof of why values are missing.

Calculate rates using the relevant subset as the denominator. For example, to compare missingness by renewal outcome, divide the number of missing predictor values among renewed policies by the total number of renewed policies, and do the analogous calculation for nonrenewed policies. The share of all missing records that belong to a group answers a different question and should not be substituted for that rate.

Compare Outcomes and Groups Before Choosing a Rule

For each important predictor, summarize its missingness against the observed outcome and policy groups. A compact table can reveal whether a single overall rate hides important differences. If the outcome is binary, report the rate for each outcome level; for group comparisons, report rates within each group. Where practical, examine outcome and group together, because the overall outcome comparison can be affected by differences in group composition.

The usual missingness labels describe assumptions about the process, not conclusions that can be read directly from a table:

• Missing completely at random (MCAR): whether \(X\) is missing is independent of the observed data and the unobserved value of \(X\). This is a strong condition.
• Missing at random (MAR): after conditioning on observed information, whether \(X\) is missing does not depend on the unobserved value of \(X\). For instance, missingness might vary by an observed application channel.
• Missing not at random (MNAR): missingness still depends on the unobserved value of \(X\), even after conditioning on observed information. A missingness-rate table alone generally cannot establish or rule out this dependence.

A difference in rates by \(Y\) or \(G\) rules out the simplest interpretation that the observed missingness pattern is unrelated to those variables, but it does not establish MNAR. For instance, rates may differ by outcome because outcome groups have different mixes of observed policy groups. Conversely, similar rates across the groups inspected do not prove MCAR: unmeasured information or the unobserved values themselves may still affect missingness.

Why Group Comparisons Can Change the Interpretation

A marginal comparison combines records from different groups. If groups have different sizes or different outcome rates, the combined rates can obscure the pattern within each group. The appropriate next step is to show the outcome-specific missingness rates within important policy groups, not to infer a mechanism from a single portfolio-wide percentage.

A Justification for Stratified Missingness Rates

The effect of group composition follows from the law of total expectation. Let \(G\) be a discrete policy group and \(C\) the missingness indicator. Within a fixed outcome level, the overall missingness probability is a weighted average of the group-specific probabilities:

The weights \(P(G=g\mid Y=y)\) can differ by outcome. Consequently, the overall rates can differ even when \(P(C=1\mid G=g,Y=y)\) is the same across outcome levels within every group. The preceding example has exactly this structure: the within-group rates match, while the mixes of groups differ.

More generally, for a response \(Y\) and observed information \(Z\), the conditional-expectation identity is

This follows by conditioning the expectation of \(Y\), among records with \(C=1\), on the observed information \(Z\). If \(Z\) is discrete, the right-hand side can be written as \(\sum_z E[Y\mid Z=z,C=1]P(Z=z\mid C=1)\). If \(Z\) is continuous, the corresponding expression is an integral over the conditional distribution of \(Z\) given \(C=1\); the displayed sum is not valid for a general continuous \(Z\).

Choose a Handling Strategy That Can Be Reproduced

The handling rule depends on the modeling purpose, the extent and pattern of missingness, and what information will be available when a renewal application is scored. No single method is correct for every case.

• Complete-case analysis uses only records with all required predictors observed. It is simple, but it loses information and can change the modeled population when inclusion is selective. Do not assume it is harmless merely because the number of deleted rows seems small.
• Imputation replaces missing values with estimates based on observed data. A median or mode is simple and reproducible; a conditional rule can reflect known group differences. A single imputed value does not restore the uncertainty in the missing value, so it should not be presented as if it were observed.
• A missingness indicator records whether the value was missing and can be used alongside an imputed value when that pattern may carry predictive information. It does not, by itself, correct selection bias or establish a causal interpretation.
• A separate missing category may be operationally useful for a categorical predictor when “not supplied” is a meaningful recorded state. It should not be confused with an actual value of the predictor.

For a prediction task, any imputation or encoding rule used at scoring time must be computable from information available then. The renewal outcome is useful for diagnosing training data, but it is not available when predicting that outcome for a new application. Using it to impute a predictor for deployment would make the fitted procedure impossible to reproduce at prediction time and can leak target information into model fitting.

A reproducible procedure separates diagnosis from transformation. First describe missingness in the training data by outcome and relevant groups. Then define the handling rule using only permitted information. Estimate quantities such as group medians on the training set, store them with the model, and apply those same values to validation, test, and scoring records. Do not calculate a new median on each later data set. Specify a fallback for a group with no observed training values, as well as how an unseen group will be treated. Record which variables receive indicators and how missing and novel categories are encoded.

Common Traps in a Renewal Case

• Using the wrong denominator. The percentage of missing records that renewed is not the missingness rate among renewed policies.
• Interpreting association as mechanism. Different rates by outcome or group do not prove that the outcome or the missing value caused missingness.
• Relying only on a portfolio-wide rate. A single rate can conceal group patterns or differences in group composition.
• Imputing with information unavailable at scoring. A model cannot use a future renewal outcome to fill in a predictor for a new application.
• Refitting preprocessing on validation or test data. Store training-derived values and apply them consistently, including a defined rule for groups not represented in training.