Package 'ICsurv'

A statistical package for regression analysis of interval-censored data under the semiparametric proportional hazards (PH) model

Description

This package allows for semiparametric regression analysis of interval-censored data under the proportional hazards model (PH). Monotone splines are used to estimate the unknown nondecreasing cumulative baseline hazard function. Model fitting is completed through the implementation of an expectation maximization (EM) algorithm.

Details

Package:	ICsurv
Type:	Package
Version:	1.0
Date:	2013-10-18

Author(s)

Christopher S. McMahan and Lianming Wang

Maintainers: Christopher S. McMahan <[email protected]> and Lianming Wang <[email protected]>

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EM algorithm for general interval-censored data under the proportional hazards model

Description

This function is equivalent to the function "PH.ICsurv.EM" but runs much faster.

Usage

fast.PH.ICsurv.EM(d1, d2, d3, Li, Ri, Xp, n.int, order, g0, b0,  tol, t.seq, equal = TRUE)

Arguments

d1	vector indicating whether an observation is left-censored (1) or not (0).
d2	vector indicating whether an observation is interval-censored (1) or not (0).
d3	vector indicating whether an observation is right-censored (1) or not (0).
Li	the left endpoint of the observed interval, if an observation is left-censored its corresponding entry should be 0.
Ri	the right endpoint of the observed interval, if an observation is right-censored its corresponding entry should be Inf.
Xp	design matrix of predictor variables (in columns), should be specified without an intercept term.
n.int	the number of interior knots to be used.
order	the order of the basis functions.
g0	initial estimate of the spline coefficients; should be of length n.int+order.
b0	initial estimate of regression coefficients; should be of length dim(Xp)[2].
tol	the convergence criterion of the EM algorithm, see details for further description.
t.seq	an increasing sequence of points at which the cumulative baseline hazard function is evaluated.
equal	logical, if TRUE knots are spaced evenly across the range of the endpoints of the observed intervals and if FALSE knots are placed at quantiles.

Details

The above function fits the semiparametric proportional hazards model (PH), proposed in Wang et al. (2014+), to interval censored data via an EM algorithm. For a discussion of starting values, number of interior knots, order, and further details please see Wang et al. (2014+). The EM algorithm converges when the maximum of the absolute difference in the parameter estimates (to include the regression and spline coefficients) is less than tol. The Hessian matrix of the observed likelihood is given in the output. The variance-covariance matrix of the estimated regression and spline coefficients can be obtained by taking the inverse of the Hessian matrix. When the Hessian matrix is singular, the variance matrix of the regression parameters is obtained by using the inverse of blocked matrix, which only involves taking inverse of lower dimensional matrices. To further provide robustness, the generalized inverse function "ginv" in the supporting package "MASS" is used in this case. A function in the supporting R package "matrixcalc" is used to check whether the Hessian matrix is singular.

Value

b	estimates of the regression coefficients.
g	estimates of the spline coefficients.
hz	estimated cumulative baseline hazard function evaluated at the points t.seq.
Hessian	the Hessian matrix. Its inverse is the variance covariance matrix of b and g.
var.b	the variance covariance matrix of b
flag	the indicator whether the Hessian matrix is non-singular. When flag=0, the variance estimate may not be accurate.
AIC	the Akaike information criterion.
BIC	the Bayesian information/Schwarz criterion.
ll	the value of the maximized log-likelihood.

References

Wang, L., McMahan, C., and Hudgens, M. (2014+). A flexible and computationally efficient method for fitting the proportional hazards model to interval censored data. Submitted.

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Calculating the Hessian matrix using Louis's Method (1982)

Description

This function is a fast version of PH.Louis.ICsurv. It calculates the negative of the Hessian of the log of the observed data likelihood, obtained via Louis's method, evaluated at the last step of the EM algorithm described in Wang et al. (2014+). This is a support function for fast.PH.ICsurv.EM.

Usage

fast.PH.Louis.ICsurv(b, g, bLi, bRi, d1, d2, d3, Xp)

Arguments

b	estimates of the regression coefficients obtained at the convergence of the EM algorithm.
g	estimates of the spline coefficients obtained from at the convergence of the EM algorithm.
bLi	an I-spline basis matrix of dimension c(length(knots)+order-2, length(x)), corresponding to the left end points of the observed intervals.
bRi	an I-spline basis matrix of dimension c(length(knots)+order-2, length(x),), corresponding to the left end points of the observed intervals.
d1	vector indicating whether an observation is left-censored (1) or not (0).
d2	vector indicating whether an observation is interval-censored (1) or not (0).
d3	vector indicating whether an observation is right-censored (1) or not (0).
Xp	design matrix of predictor variables (in columns), should be specified without an intercept term.

Details

To obtain the Hessian matrix of the observed likelihood evaluated at the last step of the EM algorithm.

Value

Hessian matrix.

References

Louis, T. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B 44, 226-233.

Wang, L., McMahan, C., and Hudgens, M. (2014+). A flexible and computationally efficient method for fitting the proportional hazards model to interval censored data. Submitted.

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Hemophilia data

Description

This data were collected as a part of a multi-center prospective study aimed at assessing the HIV-1 infection rate among people with hemophilia. In particular, the individuals in the study were at risk from contracting HIV-1 from blood products made from donor's plasma. In this study, 544 patients were categorized into one of four groups based on their average annual dose of blood products. Specifically, patients were classified into high, medium, low, or no dose group. The goal of the study was to compare the HIV-1 infection rate between these dose groups. The time at which patients contracted HIV-1 is not known exactly, but were known to occur before, between, or after certain observation times. For more details pertaining to this data set see Goedert et al. (1989) and Kroner et al. (1994).

Usage

data(Hemophilia)

Format

A data frame with 544 observations on the following 8 variables.

d1

Censoring indicator, 1 if patient's infection time was left censored, 0 otherwise.

d2

Censoring indicator, 1 if patient's infection time was interval censored, 0 otherwise.

d3

Censoring indicator, 1 if patient's infection time was right censored and, otherwise.

L

Left endpoint of the observation interval

R

Right endpoint of the observation interval

Low

Binary variable indicating the patient was a member of the low dose group.

Medium

Binary variable indicating the patient was a member of the medium dose group.

High

Binary variable indicating the patient was a member of the high dose group.

References

Goedert, J., Kessler, C., Adedort, L., Biggar, R., et al., 1989. A progressive-study of human immunodeficiency virus type-1 infection and the development of AIDS in subjects with hemophilia. New England Journal of Medicine 321, 1141-1148.

Kroner, B., Rosenberg, P., Adedort, L., Alvord, W., Goedert, J., 1994. HIV-1 infection incidence among people with hemophilia in the United States and Western Europe, 1978?1990. Journal of Acquired Immune Deficiency Syndromes 7, 279-286.

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Ispline

Description

Generates the I-spline basis matrix associated with integrated spline basis functions. Created by Cai and Wang in October, 2009.

Usage

Ispline(x, order, knots)

Arguments

x	the predictor variables.
order	the order of the basis functions.
knots	a sequence of increasing points specifying the placement of the knots.

Value

An I-spline basis matrix of dimension c(length(knots)+order-2, length(x)).

References

Wang, L., McMahan, C., and Hudgens, M. (2014+). A flexible and computationally efficient method for fitting the proportional hazards model to interval censored data. Submitted.

Ramsay, J. (1988). Monotone regression splines in action. Statistical Science 3, 425-441.

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EM algorithm for general interval-censored data under the proportional hazards model

Description

Fits the semiparametric proportional hazards model (PH), proposed in Wang et al. (2014+), to interval censored data via an EM algorithm.

Usage

PH.ICsurv.EM(d1, d2, d3, Li, Ri, Xp, n.int, order, g0, b0,  tol, t.seq, equal = TRUE)

Arguments

d1	vector indicating whether an observation is left-censored (1) or not (0).
d2	vector indicating whether an observation is interval-censored (1) or not (0).
d3	vector indicating whether an observation is right-censored (1) or not (0).
Li	the left endpoint of the observed interval, if an observation is left-censored its corresponding entry should be 0.
Ri	the right endpoint of the observed interval, if an observation is right-censored its corresponding entry should be Inf.
Xp	design matrix of predictor variables (in columns), should be specified without an intercept term.
n.int	the number of interior knots to be used.
order	the order of the basis functions.
g0	initial estimate of the spline coefficients; should be of length n.int+order.
b0	initial estimate of regression coefficients; should be of length dim(Xp)[2].
tol	the convergence criterion of the EM algorithm, see details for further description.
t.seq	an increasing sequence of points at which the cumulative baseline hazard function is evaluated.
equal	logical, if TRUE knots are spaced evenly across the range of the endpoints of the observed intervals and if FALSE knots are placed at quantiles.

Details

The above function fits the semiparametric proportional hazards model (PH), proposed in Wang et al. (2014+), to interval censored data via an EM algorithm. For a discussion of starting values, number of interior knots, order, and further details please see Wang et al. (2014+). The EM algorithm converges when the maximum of the absolute difference in the parameter estimates (to include the regression and spline coefficients) is less than tol. The Hessian matrix of the observed likelihood is given in the output. The variance-covariance matrix of the estimated regression and spline coefficients can be obtained by taking the inverse of the Hessian matrix. When the Hessian matrix is singular, the variance matrix of the regression parameters is obtained by using the inverse of blocked matrix, which only involves taking inverse of lower dimensional matrices. To further provide robustness, the generalized inverse function "ginv" in the supporting package "MASS" is used in this case. A function in the supporting R package "matrixcalc" is used to check whether the Hessian matrix is singular.

Value

b	estimates of the regression coefficients.
g	estimates of the spline coefficients.
hz	estimated cumulative baseline hazard function evaluated at the points t.seq.
Hessian	the Hessian matrix. Its inverse is the variance covariance matrix of b and g.
var.b	the variance covariance matrix of b
flag	the indicator whether the Hessian matrix is non-singular. When flag=0, the variance estimate may not be accurate.
AIC	the Akaike information criterion.
BIC	the Bayesian information/Schwarz criterion.
ll	the value of the maximized log-likelihood.

References

Wang, L., McMahan, C., and Hudgens, M. (2014+). A flexible and computationally efficient method for fitting the proportional hazards model to interval censored data. Submitted.

Examples

data(Hemophilia)

d1<-Hemophilia[,1]
d2<-Hemophilia[,2]
d3<-Hemophilia[,3]
Li<-Hemophilia[,4]
Ri<-Hemophilia[,5]
Xp<-as.matrix(Hemophilia[,c(6,7,8)])

fit <- PH.ICsurv.EM(d1, d2, d3, Li, Ri, Xp, n.int=8, order=3, g0=rep(1,11), b0=rep(0,3),  tol=0.001,
 t.seq=seq(0,57,1), equal = TRUE)

fit$b  

# [1] 1.837590 3.018500 3.418981

fit$var.b

#             [,1]        [,2]        [,3]
#  [1,]  0.008526765 -0.01090578  0.01199610
#  [2,] -0.010905780  0.01265952 -0.01462116
#  [3,]  0.011996095 -0.01462116  0.08624411

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Calculating the Hessian matrix using Louis's Method (1982)

Description

Calculates the negative of the Hessian of the log of the observed data likelihood, obtained via Louis's method, evaluated at the last step of the EM algorithm described in Wang et al. (2014+). This is a support function for PH.ICsurv.EM.

Usage

PH.Louis.ICsurv(b, g, bLi, bRi, d1, d2, d3, Xp)

Arguments

b	estimates of the regression coefficients obtained at the convergence of the EM algorithm.
g	estimates of the spline coefficients obtained from at the convergence of the EM algorithm.
bLi	an I-spline basis matrix of dimension c(length(knots)+order-2, length(x)), corresponding to the left end points of the observed intervals.
bRi	an I-spline basis matrix of dimension c(length(knots)+order-2, length(x),), corresponding to the left end points of the observed intervals.
d1	vector indicating whether an observation is left-censored (1) or not (0).
d2	vector indicating whether an observation is interval-censored (1) or not (0).
d3	vector indicating whether an observation is right-censored (1) or not (0).
Xp	design matrix of predictor variables (in columns), should be specified without an intercept term.

Details

To obtain the Hessian matrix of the observed likelihood evaluated at the last step of the EM algorithm.

Value

Hessian matrix.

References

Louis, T. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B 44, 226-233.

Wang, L., McMahan, C., and Hudgens, M. (2014+). A flexible and computationally efficient method for fitting the proportional hazards model to interval censored data. Submitted.

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