/Statsmodels

# Methods for Survival and Duration Analysis

`statsmodels.duration` implements several standard methods for working with censored data. These methods are most commonly used when the data consist of durations between an origin time point and the time at which some event of interest occurred. A typical example is a medical study in which the origin is the time at which a subject is diagnosed with some condition, and the event of interest is death (or disease progression, recovery, etc.).

Currently only right-censoring is handled. Right censoring occurs when we know that an event occurred after a given time `t`, but we do not know the exact event time.

## Survival function estimation and inference

The `statsmodels.api.SurvfuncRight` class can be used to estimate a survival function using data that may be right censored. `SurvfuncRight` implements several inference procedures including confidence intervals for survival distribution quantiles, pointwise and simultaneous confidence bands for the survival function, and plotting procedures. The `duration.survdiff` function provides testing procedures for comparing survival distributions.

## Examples

Here we create a `SurvfuncRight` object using data from the `flchain` study, which is available through the R datasets repository. We fit the survival distribution only for the female subjects.

```import statsmodels.api as sm

data = sm.datasets.get_rdataset("flchain", "survival").data
df = data.loc[data.sex == "F", :]
sf = sm.SurvfuncRight(df["futime"], df["death"])
```

The main features of the fitted survival distribution can be seen by calling the `summary` method:

```sf.summary().head()
```

We can obtain point estimates and confidence intervals for quantiles of the survival distribution. Since only around 30% of the subjects died during this study, we can only estimate quantiles below the 0.3 probability point:

```sf.quantile(0.25)
sf.quantile_ci(0.25)
```

To plot a single survival function, call the `plot` method:

```sf.plot()
```

Since this is a large dataset with a lot of censoring, we may wish to not plot the censoring symbols:

```fig = sf.plot()
ax = fig.get_axes()[0]
pt = ax.get_lines()[1]
pt.set_visible(False)
```

We can also add a 95% simultaneous confidence band to the plot. Typically these bands only plotted for central part of the distribution.

```fig = sf.plot()
lcb, ucb = sf.simultaneous_cb()
ax = fig.get_axes()[0]
ax.fill_between(sf.surv_times, lcb, ucb, color='lightgrey')
ax.set_xlim(365, 365*10)
ax.set_ylim(0.7, 1)
ax.set_ylabel("Proportion alive")
ax.set_xlabel("Days since enrollment")
```

Here we plot survival functions for two groups (females and males) on the same axes:

```gb = data.groupby("sex")
ax = plt.axes()
sexes = []
for g in gb:
sexes.append(g[0])
sf = sm.SurvfuncRight(g[1]["futime"], g[1]["death"])
sf.plot(ax)
li = ax.get_lines()
li[1].set_visible(False)
li[3].set_visible(False)
plt.figlegend((li[0], li[2]), sexes, "center right")
plt.ylim(0.6, 1)
ax.set_ylabel("Proportion alive")
ax.set_xlabel("Days since enrollment")
```

We can formally compare two survival distributions with `survdiff`, which implements several standard nonparametric procedures. The default procedure is the logrank test:

```stat, pv = sm.duration.survdiff(data.futime, data.death, data.sex)
```

Here are some of the other testing procedures implemented by survdiff:

```# Fleming-Harrington with p=1, i.e. weight by pooled survival time
stat, pv = sm.duration.survdiff(data.futime, data.death, data.sex, weight_type='fh', fh_p=1)

# Gehan-Breslow, weight by number at risk
stat, pv = sm.duration.survdiff(data.futime, data.death, data.sex, weight_type='gb')

# Tarone-Ware, weight by the square root of the number at risk
stat, pv = sm.duration.survdiff(data.futime, data.death, data.sex, weight_type='tw')
```

## Regression methods

Proportional hazard regression models (“Cox models”) are a regression technique for censored data. They allow variation in the time to an event to be explained in terms of covariates, similar to what is done in a linear or generalized linear regression model. These models express the covariate effects in terms of “hazard ratios”, meaning the the hazard (instantaneous event rate) is multiplied by a given factor depending on the value of the covariates.

## Examples

```import statsmodels.api as sm
import statsmodels.formula.api as smf

data = sm.datasets.get_rdataset("flchain", "survival").data
del data["chapter"]
data = data.dropna()
data["lam"] = data["lambda"]
data["female"] = (data["sex"] == "F").astype(int)
data["year"] = data["sample.yr"] - min(data["sample.yr"])
status = data["death"].values

mod = smf.phreg("futime ~ 0 + age + female + creatinine + "
"np.sqrt(kappa) + np.sqrt(lam) + year + mgus",
data, status=status, ties="efron")
rslt = mod.fit()
print(rslt.summary())
```

See Statsmodels Examples for more detailed examples.

There are some notebook examples on the Wiki: Wiki notebooks for PHReg and Survival Analysis

### References

References for Cox proportional hazards regression model:

```T Therneau (1996). Extending the Cox model. Technical report.
http://www.mayo.edu/research/documents/biostat-58pdf/DOC-10027288

G Rodriguez (2005). Non-parametric estimation in survival models.
http://data.princeton.edu/pop509/NonParametricSurvival.pdf

B Gillespie (2006). Checking the assumptions in the Cox proportional
hazards model.
http://www.mwsug.org/proceedings/2006/stats/MWSUG-2006-SD08.pdf
```

## Module Reference

The class for working with survival distributions is:

 `SurvfuncRight`(time, status[, entry, title, …]) Estimation and inference for a survival function.

The proportional hazards regression model class is:

 `PHReg`(endog, exog[, status, entry, strata, …]) Fit the Cox proportional hazards regression model for right censored data.

The proportional hazards regression result class is:

 `PHRegResults`(model, params, cov_params[, …]) Class to contain results of fitting a Cox proportional hazards survival model.

© 2009–2012 Statsmodels Developers
© 2006–2008 Scipy Developers
© 2006 Jonathan E. Taylor