# concordance_index_ipcw()


IPCW concordance index (Uno et al., 2011) for right-censored survival data.


Usage

``` python
concordance_index_ipcw(
    surv,
    risk,
    *,
    tau=None,
)
```


An inverse-probability-of-censoring-weighted concordance index that corrects for censoring bias. Unlike Harrell's C, the IPCW concordance is a consistent estimator of the true concordance probability even under informative censoring patterns. It is more robust when censoring is heavy or differs across risk groups.

**When to use this instead of** [concordance_index](concordance_index.md#greenwood.concordance_index) (Harrell's C):

- When censoring exceeds 30-40% of subjects.
- When censoring patterns differ across risk groups (informative censoring).
- When comparing models across datasets with different censoring rates.

**Interpretation**: Same scale as Harrell's C (0.5 = random, 1.0 = perfect discrimination). Values are comparable to Harrell's C under light censoring and diverge when censoring is heavy.


## Parameters


`surv: Surv`  
A right-censored [Surv](Surv.md#greenwood.Surv) response (time-to-event data).

`risk: Any`  
Risk score for each subject, one per observation. Higher values indicate higher risk (earlier expected failure). Accepts a 1-D array, Pandas/Polars Series, or Python sequence.

`tau: float | None = None`  
Truncation time. Only pairs with an event before `tau` contribute. This avoids instability from low censoring survival in the tail. Defaults to the largest observed event time.


## Returns


`float`  
IPCW concordance index between 0 and 1.


## Details

**Uno estimator**: For each subject i with an observed event at T_i \le \tau, compare against all subjects j with T_j \> T_i:

 \hat{C}\_{\mathrm{IPCW}} = \frac{\displaystyle\sum\_{i:\Delta_i=1,\\T_i \le \tau} \hat{G}(T_i^-)^{-2} \Bigl\[\\\\j: T_j \> T_i,\\ \eta_j \< \eta_i\\ + \tfrac{1}{2}\\\\j: T_j \> T_i,\\ \eta_j = \eta_i\\\Bigr\]} {\displaystyle\sum\_{i:\Delta_i=1,\\T_i \le \tau} \hat{G}(T_i^-)^{-2} \cdot \\\\j: T_j \> T_i\\} 

where \hat{G} is the Kaplan-Meier estimate of the censoring distribution.

**Truncation**: Restricting to events before \tau avoids instability from dividing by very small censoring probabilities in the tail. The default \tau is the largest event time. Set it explicitly to the maximum follow-up of interest.


## References

Uno H., Cai T., Pencina M.J., D'Agostino R.B., Wei L.J. (2011). On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. *Statistics in Medicine*, 30(10), 1105-1117.


## Examples

Fit a Cox model on the `lung` dataset and compute the IPCW concordance:


``` python
import greenwood as gw

lung = gw.load_dataset("lung", backend="polars")
y = gw.Surv.right(lung["time"], event=(lung["status"] == 2))
cox = gw.CoxPH().fit(y, lung[["age", "sex"]])

lp = cox.predict(type="lp")

# Compare Harrell's C with IPCW C
c_harrell = gw.concordance_index(y, lp)
c_ipcw = gw.concordance_index_ipcw(y, lp)
print(f"Harrell C: {c_harrell:.4f}")
print(f"IPCW C:    {c_ipcw:.4f}")
```


    Harrell C: 0.6029
    IPCW C:    0.5959


Truncate at 1 year to focus on short-term discrimination:


``` python
c_1yr = gw.concordance_index_ipcw(y, lp, tau=365.0)
print(f"IPCW C (1-year): {c_1yr:.4f}")
```


    IPCW C (1-year): 0.5991