# Fit additive hazards model

### Description

Fits both the additive hazards model of Aalen and the semi-parametric additive hazards model of McKeague and Sasieni. Estimates are un-weighted. Time dependent variables and counting process data (multiple events per subject) are possible.

Resampling is used for computing p-values for tests of time-varying effects.

The modelling formula uses the standard survival modelling given in the
**survival** package.

### Usage

1 2 3 4 |

### Arguments

`formula` |
a formula object with the response on the left of a '~' operator, and the independent terms on the right as regressors.The response must be a survival object as returned by the ‘Surv’ function. Time- invariant regressors are specified by the wrapper const(), and cluster variables (for computing robust variances) by the wrapper cluster(). |

`data` |
a data.frame with the variables. |

`start.time` |
start of observation period where estimates are computed. |

`max.time` |
end of observation period where estimates are computed. Estimates thus computed from [start.time, max.time]. Default is max of data. |

`robust` |
to compute robust variances and construct processes for resampling. May be set to 0 to save memory. |

`id` |
For timevarying covariates the variable must associate each record with the id of a subject. |

`clusters` |
cluster variable for computation of robust variances. |

`n.sim` |
number of simulations in resampling. |

`weighted.test` |
to compute a variance weighted version of the test-processes used for testing time-varying effects. |

`residuals` |
to returns residuals that can be used for model validation in the function cum.residuals |

`covariance` |
to compute covariance estimates for nonparametric terms rather than just the variances. |

`resample.iid` |
to return i.i.d. representation for nonparametric and parametric terms. |

`deltaweight` |
uses weights to estimate semiparametric model, under construction, default=1 is standard least squares estimates |

`silent` |
set to 0 to print warnings for non-inverible design-matrices for different timepoints, default is 1. |

`weights` |
weights for estimating equations. |

`max.clust` |
sets the total number of i.i.d. terms in i.i.d. decompostition. This can limit the amount of memory used by coarsening the clusters. When NULL then all clusters are used. Default is 1000 to save memory and time. |

`gamma` |
fixes gamme at this value for estimation. |

`offsets` |
offsets for the additive model, to make excess risk modelling. |

`caseweight` |
caseweight: mutiplied onto dN for score equations. |

### Details

The data for a subject is presented as multiple rows or 'observations', each of which applies to an interval of observation (start, stop]. For counting process data with the )start,stop] notation is used the 'id' variable is needed to identify the records for each subject. The program assumes that there are no ties, and if such are present random noise is added to break the ties.

### Value

returns an object of type "aalen". With the following arguments:

`cum` |
cumulative timevarying regression coefficient estimates are computed within the estimation interval. |

`var.cum` |
the martingale based pointwise variance estimates for cumulatives. |

`robvar.cum` |
robust pointwise variances estimates for cumulatives. |

`gamma` |
estimate of parametric components of model. |

`var.gamma` |
variance for gamma. |

`robvar.gamma` |
robust variance for gamma. |

`residuals` |
list with residuals. Estimated martingale increments (dM) and corresponding time vector (time). |

`obs.testBeq0` |
observed absolute value of supremum of cumulative components scaled with the variance. |

`pval.testBeq0` |
p-value for covariate effects based on supremum test. |

`sim.testBeq0` |
resampled supremum values. |

`obs.testBeqC` |
observed absolute value of supremum of difference between observed cumulative process and estimate under null of constant effect. |

`pval.testBeqC` |
p-value based on resampling. |

`sim.testBeqC` |
resampled supremum values. |

`obs.testBeqC.is` |
observed integrated squared differences between observed cumulative and estimate under null of constant effect. |

`pval.testBeqC.is` |
p-value based on resampling. |

`sim.testBeqC.is` |
resampled supremum values. |

`conf.band` |
resampling based constant to construct robust 95% uniform confidence bands. |

`test.procBeqC` |
observed test-process of difference between observed cumulative process and estimate under null of constant effect over time. |

`sim.test.procBeqC` |
list of 50 random realizations of test-processes under null based on resampling. |

`covariance` |
covariances for nonparametric terms of model. |

`B.iid` |
Resample processes for nonparametric terms of model. |

`gamma.iid` |
Resample processes for parametric terms of model. |

`deviance` |
Least squares of increments. |

### Author(s)

Thomas Scheike

### References

Martinussen and Scheike, Dynamic Regression Models for Survival Data, Springer (2006).

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | ```
data(sTRACE)
# Fits Aalen model
out<-aalen(Surv(time,status==9)~age+sex+diabetes+chf+vf,
sTRACE,max.time=7,n.sim=100)
summary(out)
par(mfrow=c(2,3))
plot(out)
# Fits semi-parametric additive hazards model
out<-aalen(Surv(time,status==9)~const(age)+const(sex)+const(diabetes)+chf+vf,
sTRACE,max.time=7,n.sim=100)
summary(out)
par(mfrow=c(2,3))
plot(out)
## Excess risk additive modelling
data(mela.pop)
dummy<-rnorm(nrow(mela.pop));
# Fits Aalen model with offsets
out<-aalen(Surv(start,stop,status==1)~age+sex+const(dummy),
mela.pop,max.time=7,n.sim=100,offsets=mela.pop$rate,id=mela.pop$id,
gamma=0)
summary(out)
par(mfrow=c(2,3))
plot(out,main="Additive excess riks model")
# Fits semi-parametric additive hazards model with offsets
out<-aalen(Surv(start,stop,status==1)~age+const(sex),
mela.pop,max.time=7,n.sim=100,offsets=mela.pop$rate,id=mela.pop$id)
summary(out)
plot(out,main="Additive excess riks model")
``` |