## calculate_pseudostates()


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


Usage

``` python
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

[](%60~polarstate.prepare_event_table%60)  
Convert raw data into an event table.

[](%60~polarstate.predict_aj_estimates%60)  
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.