find.shapeMLE: Compute the MLE of a shape-constrained hazard function

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 a shape-constrained hazard function 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.shapeMLE(x, delta=rep(1, length(x)), type = "increasing", plot = FALSE)

Arguments

x	vector of length n containing the data
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, assuming that no observations were censored.
type	string indicating type of shape constraint. Options are "increasing", "decreasing", "unimodal", and "ushaped".
plot	logical, if TRUE, the graphical representation of the MLE is shown

Value

A list containing the following elements:

h.range	endpoints for the values of the hazard MLE
h.val	values of the hazard MLE in between the endpoints
phi	the criterion function Phi (negative log-likelihood) 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

The MLE can be found in different ways. We use the graphical representation via convex minorants or concave majorants of appropriate functions. Also, 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).

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/

Examples

# random sample from the uniform density
n	<-	500
x	<-	runif(n)

# compute MLE of increasing hazard
mle	<-	find.shapeMLE(x, type="increasing")


# plot the fitted hazard
plot(mle)
rug(x)

# add true hazard to the plot
h.true	<-	function(x) 1/(1-x)
plot(h.true, col="red", add=TRUE)

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