calculate_pseudostates()

Compute Aalen Johansen pseudo-observations at a fixed time horizon.

Usage

Source

calculate_pseudostates(
    times_and_reals,
    fixed_time_horizon,
)

This function computes jackknife pseudo-observations for state occupation probabilities (or related Aalen Johansen estimands) evaluated at a fixed time horizon. Pseudo-observations are constructed using a leave-one-out scheme:

.. math::

\text{pseudo}_i = n \cdot \theta_{\text{full}} - (n - 1) \cdot \theta_{(-i)}

where :math:\theta_{\text{full}} is the Aalen Johansen estimate based on the full sample, and :math:\theta_{(-i)} is the estimate obtained after removing individual :math:i.

The resulting pseudo-observations can be used as individual-level outcomes in regression models (e.g., GEE or GLM) to assess covariate effects on cumulative incidence or state occupation probabilities.

Parameters

times_and_reals: pl.DataFrame

Input data containing individual event times and realized states. Must be compatible with prepare_event_table and predict_aj_estimates. Each row corresponds to a single individual.

fixed_time_horizon: int

Time point at which the Aalen Johansen estimates are evaluated.

Returns

pl.DataFrame

A Polars DataFrame containing pseudo-observations for each individual. The output includes:

• Identifier columns (e.g., state, time, horizon) copied from the full-sample Aalen Johansen estimate.
• Numeric columns containing the pseudo-observations.
• A row_id column indicating the index of the left-out observation.

For each numeric column, the mean of the pseudo-observations equals the corresponding full-sample Aalen Johansen estimate.

See Also

Convert raw data into an event table.

Predict Aalen-Johansen estimates.

Notes

• This implementation uses an explicit leave-one-out loop and therefore has time complexity O(n²). It is intended for methodological work, simulations, or moderate sample sizes.
• In the absence of censoring, the pseudo-observations reduce to the empirical individual contributions.
• Non-numeric columns (e.g., state labels) are treated as identifiers and are not transformed.

References

.. [1] Andersen, P. K., & Pohar Perme, M. (2010). Pseudo-observations in survival analysis. Statistical Methods in Medical Research, 19(1), 71–99.