Aalen additive hazards model with time-varying coefficients.
Usage
The additive hazards model is an alternative to Cox regression when the proportional hazards assumption fails. Instead of modeling the log-hazard ratio as constant over time, it models the hazard itself as a sum of covariate-specific, time-varying functions:
h(t \mid x) = \beta_0(t) + \sum_{j=1}^{p} \beta_j(t)\,x_j
Estimation proceeds by ordinary least squares at each event time: for each death, the coefficient increment d\hat{B}(t) is obtained by regressing the event indicator on the at-risk covariate matrix. Cumulative coefficients \hat{B}(t) are obtained by summing these increments.
nmin: int | None = NoneMinimum risk-set size. Event times where the risk set has fewer than nmin subjects are dropped (late-tail instability). Defaults to 3 * (p + 1) where p is the number of covariates.
test: str = "aalen"Weighting scheme for the summary test statistics. "aalen" (default) uses equal weights, "variance" uses inverse-variance weights, and "nrisk" uses the number at risk.
qrtol: float = 1e-071e-7).
Fit the Aalen additive model on the lung dataset:
import greenwood as gw
lung = gw.load_dataset("lung", backend="polars")
aalen = gw.AalenAdditive().fit("Surv(time, status) ~ age + sex", data=lung)
aalenAalenAdditive (additive hazards, test='aalen')
slope coef se(coef) z p
Intercept 0.00483 0.005448 0.004703 1.158 0.247
age 8.986e-05 0.0001263 6.903e-05 1.829 0.0674
sex -0.003151 -0.003938 0.001222 -3.222 0.00127
n = 228, events = 165, event times used = 164| Name | Description |
|---|---|
| cumulative_coefficients() | Return the cumulative regression coefficients \hat{B}(t) as a DataFrame. |
| fit() | Fit the Aalen additive hazards model. |
| predict() | Predict survival or cumulative hazard from the fitted Aalen additive model. |
| to_frame() | Return the summary test table as a DataFrame. |
Return the cumulative regression coefficients \hat{B}(t) as a DataFrame.
Usage
Each row corresponds to an event time. The cumulative coefficient for covariate j at time t is \hat{B}_j(t) = \sum_{s \le t} d\hat{B}_j(s), the running total of OLS increments up to time t.
format: str | None = NoneNone (default), "pandas", "polars", or "pyarrow".
DataFrametime and one column per term (Intercept plus covariates).
Fit the Aalen additive hazards model.
Usage
At each event time, an OLS regression of the event indicator on the at-risk design matrix produces one coefficient increment per covariate. The cumulative sum of these increments gives the cumulative regression function \hat{B}(t).
surv: Surv | Outcome | strA right-censored or counting-process Surv response. An Outcome or a formula string such as 'Surv(time, status == 2) ~ age + sex' is also accepted, with its columns read from data. The right-hand side sets covariates.
covariates: Any = NoneA dataframe (pandas or polars), 2-D array, or formula string evaluated against data. A list of column names in data also works.
data: Any = Nonecovariates. When surv is an Outcome or a formula, rows with a missing value in any column used are dropped before fitting. Covariate formula strings and lists of column names are also resolved here.
AalenAdditiveFit the Aalen additive model on the lung dataset with two covariates. The summary shows the cumulative-regression test for each covariate:
import greenwood as gw
lung = gw.load_dataset("lung", backend="polars")
aalen = gw.AalenAdditive().fit("Surv(time, status) ~ age + sex", data=lung)
aalenAalenAdditive (additive hazards, test='aalen')
slope coef se(coef) z p
Intercept 0.00483 0.005448 0.004703 1.158 0.247
age 8.986e-05 0.0001263 6.903e-05 1.829 0.0674
sex -0.003151 -0.003938 0.001222 -3.222 0.00127
n = 228, events = 165, event times used = 164The formula interface combines the response and covariates in a single string:
AalenAdditive (additive hazards, test='aalen')
slope coef se(coef) z p
Intercept 0.00483 0.005448 0.004703 1.158 0.247
age 8.986e-05 0.0001263 6.903e-05 1.829 0.0674
sex -0.003151 -0.003938 0.001222 -3.222 0.00127
n = 228, events = 165, event times used = 164Predict survival or cumulative hazard from the fitted Aalen additive model.
Usage
The cumulative hazard for a subject with covariates x at time t is
\hat{H}(t \mid x) = \hat{B}_0(t) + \sum_{j=1}^{p} \hat{B}_j(t)\,x_j
where \hat{B}_j(t) are the estimated cumulative regression coefficients. Survival follows as \hat{S}(t \mid x) = \exp(-\hat{H}(t \mid x)).
Because the additive hazard model can produce negative cumulative hazard increments for some covariate patterns, predicted cumulative hazards are clipped at zero.
newdata: Any = NoneCovariate values for prediction. A DataFrame, 2-D array, or None. When None, predictions are made for the training data subjects. A data frame may also hold other columns. The covariates are picked out by name and coded as they were at fit time, so the frame the model was fit on can be passed as is.
type: str = "survival"Prediction type: "survival" (default) or "cumhaz".
times: Any = NoneQuery times at which to evaluate predictions. An array-like of positive floats. When None, the model’s fitted event times are used.
format: str | None = NoneNone (auto-detect), "pandas", "polars", or "pyarrow".
DataFrametime and subject_1, subject_2, etc., one row per query time.
import greenwood as gw
lung = gw.load_dataset("lung", backend="polars")
aalen = gw.AalenAdditive().fit("Surv(time, status) ~ age + sex", data=lung)
aalen.predict(lung[:3], times=[180, 365], format="polars")| time | subject_1 | subject_2 | subject_3 |
|---|---|---|---|
| f64 | f64 | f64 | f64 |
| 180.0 | 0.602005 | 0.626021 | 0.676966 |
| 365.0 | 0.300138 | 0.320896 | 0.366818 |
Return the summary test table as a DataFrame.
Usage
One row per term with the slope (weighted average of increments), its standard error, z-statistic, and p-value.
format: str | None = NoneNone (default), "pandas", "polars", or "pyarrow".
DataFrameterm, slope, coef, se, z, p.