# Estimate a Survival Curve with Cox PH

Use the Cox proportional hazards family when your outcome is the time until an event (death, equipment failure, customer churn, or relapse) and some observations may be right-censored (the event had not yet occurred when follow-up ended). The GAM extension allows the log-hazard to be a smooth function of continuous covariates like age, avoiding the need to categorize or assume linearity. Parametric terms like [treatment](../reference/CausalGAM.md#whittaker.CausalGAM.treatment) enter as log-hazard ratios, directly interpretable as relative hazard multipliers.


# Load data

Load the survival dataset and inspect its columns.


``` python
import whittaker as wk

data = wk.load_dataset("survival", as_frame=True)
data.columns.tolist()
```


    ['time', 'event', 'age', 'treatment']


The dataset has four columns: `time`, `event`, `age`, and [treatment](../reference/CausalGAM.md#whittaker.CausalGAM.treatment).


``` python
data.head()
```


|     | time | event | age       | treatment |
|-----|------|-------|-----------|-----------|
| 0   | 5.0  | 0.0   | 58.129296 | 0.0       |
| 1   | 5.0  | 0.0   | 70.374621 | 1.0       |
| 2   | 5.0  | 0.0   | 64.905856 | 0.0       |
| 3   | 5.0  | 0.0   | 40.134324 | 0.0       |
| 4   | 5.0  | 0.0   | 43.507483 | 0.0       |


The `event` column is a 0/1 indicator where 1 means the event occurred and 0 means the observation was censored. The [CoxPH](../reference/CoxPH.md#whittaker.CoxPH) family reads the `event` column automatically from the data by name, alongside the `time` response in the formula.


# Fit

Fit a Cox PH GAM with a smooth over age and a linear parametric term for treatment.


``` python
model = wk.GAM(
    "time ~ s(age) + treatment",
    family=wk.CoxPH(),
).fit(data)

model.summary()
```


    GAM fit summary
    ============================================================
    Formula:    time ~ s(age) + treatment
    Family:     CoxPH(ties='breslow')
    Inference:  GCV
    Observations: 250
    Coefficients: 11

    Parametric coefficients:
      Term                       Estimate    Std.Err    z value    p-value
      ------------------------ ---------- ---------- ---------- ----------
      (Intercept)                  0.3934     0.1156      3.402  0.0006686
      treatment                   -0.6502     0.1770     -3.674  0.0002384

    Approximate significance of smooth terms:
      Term                        EDF Ref.df     Chi.sq    p-value
      ------------------------ ------ ------ ---------- ----------
      s(age)                     1.09      2     13.872  0.0009724

    Total EDF:  3.09
    Scale est:  1.000000
    Deviance:   1356.2592
    Null dev:   1381.1403
    Dev. expl:  1.8%
    GCV score:  5.561854
    AIC:        1362.45
    BIC:        1373.34


The summary reports the coefficient for [treatment](../reference/CausalGAM.md#whittaker.CausalGAM.treatment) as a log-hazard ratio: exponentiate it to obtain the hazard ratio. The EDF for `s(age)` indicates how non-linearly age affects the log-hazard (an EDF near 1 implies a roughly log-linear age effect).


# Partial effects


``` python
wk.partial_effects(model)
```


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Partial effects are on the log-hazard scale. For `s(age)`, an upward slope at a given age means the instantaneous risk of the event increases with age in that region; a downward slope means decreasing risk. The [treatment](../reference/CausalGAM.md#whittaker.CausalGAM.treatment) panel shows the linear log-hazard contrast between treatment levels. Centered partial effects are relative to the mean log-hazard in the data, so positive values indicate above-average hazard.


# Diagnostics


``` python
wk.check(model)
```


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For Cox models, the diagnostic plots check the proportional hazards assumption and the distribution of martingale or deviance residuals. Systematic trends in residuals over time can indicate time-varying effects not captured by the model.