find.shapeCPH: Compute the MLE of the effect parameters and...

In CPHshape: Find the maximum likelihood estimator of the shape constrained hazard baseline and the effect parameters in the Cox proportional hazards model

Description

Compute the maximum likelihood estimator (MLE) of the shape-constrained hazard baseline and the effect parameters in the Cox proportional hazards model under IID sampling. We assume that the data are continuous and allow for right censoring. The function 'find.shapeMLE' allows for four different shape constraints: increasing, decreasing, unimodal, and u-shaped.

Usage

find.shapeCPH(x, z, delta, type="increasing", 
beta0=rep(1, length(as.matrix(z)[1,])), max.loop=250, eps=1e-5, 
eps.beta=1e-5, print=FALSE)

Arguments

x	vector of length n containing the data
z	matrix of size n x p containing the covariate values
delta	logical vector containing the (right) censoring information. If delta_i=1 then the observation was not censored. The default is delta_i=1 for all i, that is, no observations were censored.
type	string indicating type of shape constraint. Options are "increasing", "decreasing", "unimodal", and "ushaped".
beta0	vector of length p containing the starting value of beta for the algorithm. The default is all elements equal to one.
max.loop	maximum number of iterations. The default is 250.
eps	precision for stopping criterion. The default is 1e-05.
eps.beta	precision for second stopping criterion. The default is 1e-05.
print	logical, if TRUE, output from each iteration of the algorithm is shown. The default is FALSE.

Value

A list containing the following elements:

beta	MLE of the effect parameters
h.range	endpoints for the values of the hazard MLE
h.val	values of the hazard MLE
phi	the criterion function Phi evaluated at the MLE
H	the cumulative hazard MLE evaluated at the data points
mode	location of the mode (for unimodal) or antimode (for u-shaped). Note that the antimode is not unique, and the midpoint of all possible values is returned.
type	string indicating type of shape constraint used

Note

For the increasing and unimodal setting, the MLE takes the value of infinity at the mode (the largest observation for type="increasing"). In such situations, this value is removed from the likelihood in the maximization process. A similar approach was taken in Gernander (1956).
We note also that for the shapes type="unimodal" and type="ushaped" the algorithm can take some time, especially for larger sample sizes.

Author(s)

Rihong Hui and Hanna Jankowski <[e-mail]>

References

Hui, R. and Jankowski, H. (2012). Maximum likelihood estimation of a shape-constrained hazard in the proportional hazard model. Technical Report. http://www.math.yorku.ca/~hkj/

Mykytyn, S. and Santner, T. (1981) Maximum likelihood estimation of the survival function based on censored data under hazard rate assumptions. Comm. Statist. A - Theory and Methods 10 (14): 1369-1387.

Examples

# random sample from the proportional hazard model
n	<-	200
beta1	<-	1
beta2	<-	2
z1	<-	rbinom(n,1,0.5)
z2	<-	runif(n,-1,1)
w	<-	exp(beta1*z1+beta2*z2)
x	<-	rexp(n, rate=0.3*w)
delta	<-	1*(x<=2.5)
x	<-	pmin(x,2.5)

# compute MLE
mle	<-	find.shapeCPH(x, cbind(z1,z2) , delta, print=TRUE, type="decreasing")

# estimates of the effect parameter
mle$beta

# plot resulting estimate of baseline hazard

plot(mle)
abline(h=0.3, col="red") # add true baseline
rug(x)

CPHshape documentation built on May 30, 2017, 4:32 a.m.