isoph {isoSurv}R Documentation

Fit Isotonic Proportional Hazards Model

Description

Nonparametric estimation of a monotone covariate effect under the proportional hazards model.

Usage

  isoph(formula, data, maxiter, eps)

Arguments

formula

a formula object: response ~ iso(zz,shape="increasing")+x1+x2+...+xpx_1+x_2+...+x_p. The response must be right-censored survival outcome using the Surv function in the survival package. The iso function attributes the covariate zz' name, shape and anchor point.

data

data.frame includes variables named in the formula argument.

maxiter

maximum number of iteration (default is 10410^4).

eps

stopping convergence criteria (default is 10310^-3).

Details

The isoph function estimates (ψ\psi, β\beta) in the isotonic proportional hazards model, defined as

λ(tz,x)=λ0(t)exp(ψ(z)+β1x1+β2x2+...+βpxp),\lambda(t|z,x)=\lambda0(t)exp(\psi(z)+\beta_1x_1+\beta_2x_2+...+\beta_px_p),

based on the partial likelihood with unspecified baseline hazard function λ0\lambda0, where ψ\psi is a monotone increasing (or decreasing) covariate effect function, zz is a univariate variable, x=(x1,x2,...,xp)x=(x_1,x_2,...,x_p) is a set of covariates, and β=(β1,β2,...,βp)\beta=(\beta_1,\beta_2,...,\beta_p) is a set of corresponding regression parameters. It allows to estimate ψ\psi only if xx is removed in the formula object. Using the nonparametric maximum likelihood approaches, estimated ψ\psi is a right continuous increasing (or left continuos decreasing) step function.

For the anchor constraint, one point has to be fixed with ψ(K)=0\psi(K)=0 to solve the identifiability problem, e.g. λ0(t)exp(ψ(z))=(λ0(t)exp(c))(exp(ψ(z)+c))\lambda0(t)exp(\psi(z))=(\lambda0(t)exp(-c))(exp(\psi(z)+c)) for any constant cc. KK is called an anchor point. By default, we set KK as a median of values of zz's. The choice of anchor points are not important because, for example, different anchor points results in the same hazard ratios.

Value

A list of class isoph:

iso.cov

data.frame with zz and estimated ψ\psi.

beta

estimated β1,β2,...,βp\beta_1,\beta_2,...,\beta_p.

conv

algorithm convergence status.

iter

total number of iterations.

Zk

anchor point satisfying ψ(Zk)\psi(Zk)=0.

shape

Order-restriction imposed on ψ\psi.

Author(s)

Yunro Chung [aut, cre]

References

Yunro Chung, Anastasia Ivanova, Michael G. Hudgens, Jason P. Fine, Partial likelihood estimation of isotonic proportional hazards models, Biometrika. 2018, 105 (1), 133-148. doi:10.1093/biomet/asx064

Examples

# test1
test1=data.frame(
  time=  c(2, 5, 1, 7, 9, 5, 3, 6, 8, 9, 7, 4, 5, 2, 8),
  status=c(0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1),
  z=     c(2, 1, 1, 3, 5, 6, 7, 9, 3, 0, 2, 7, 3, 9, 4)
)
isoph.fit1=isoph(Surv(time, status)~iso(z,shape="inc"),data=test1)
print(isoph.fit1)
plot(isoph.fit1)

# test2
test2=data.frame(
  time=  c(2, 5, 1, 7, 9, 5, 3, 6, 8, 9, 7, 4, 5, 2, 8),
  status=c(0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1),
  z=     c(2, 1, 1, 3, 5, 6, 7, 9, 3, 0, 2, 7, 3, 9, 4),
  trt=   c(1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0)
)
isoph.fit2=isoph(Surv(time, status)~iso(z,shape="inc")+trt, data=test2)
print(isoph.fit2)
plot(isoph.fit2)

[Package isoSurv version 0.3.0 Index]