Analyzes survival and reliability data using Cox's proportional hazards model.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute] public class ProportionalHazards

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class ProportionalHazards

Visual C++
[SerializableAttribute] public ref class ProportionalHazards

Remarks

Class ProportionalHazards computes parameter estimates and other statistics in Proportional Hazards Generalized Linear Models. These models were first proposed by Cox (1972). Two methods for handling ties are allowed. Time-dependent covariates are not allowed. The user is referred to Cox and Oakes (1984), Kalbfleisch and Prentice (1980), Elandt-Johnson and Johnson (1980), Lee (1980), or Lawless (1982), among other texts, for a thorough discussion of the Cox proportional hazards model.

Let $\lambda(t,z_i)$ represent the hazard rate at time t for observation number i with covariables contained as elements of row vector . The basic assumption in the proportional hazards model (the proportionality assumption) is that the hazard rate can be written as a product of a time varying function $\lambda_0(t)$ , which depends only on time, and a function , which depends only on the covariable values. The function used in ProportionalHazards is given as $f(z_i)= \textup{exp}(w_i+\beta z_i)$ where is a fixed constant assigned to the observation, and b is a vector of coefficients to be estimated. With this function one obtains a hazard rate $\lambda(t, z_i)=\lambda_0(t)\textup{exp}(w_i+ \beta z_i)$ . The form of $\lambda_0(t)$ is not important in proportional hazards models.

The constants may be known theoretically. For example, the hazard rate may be proportional to a known length or area, and the can then be determined from this known length or area. Alternatively, the may be used to fix a subset of the coefficients $\beta$ (say, $\beta_1$ ) at specified values. When is used in this way, constants $w_i= \beta_1z_{1i}$ are used, while the remaining coefficients in $\beta$ are free to vary in the optimization algorithm. Constants are defined as 0.0 by default. If user-specified constants are desired, use the ConstantColumn property to specify which column contains the constant.

With this definition of $\lambda(t,z_i)$ , the usual partial (or marginal, see Kalbfleisch and Prentice (1980)) likelihood becomes

$L=\prod_{i=1}^{n_d}\frac{\textup{exp} (w_i+\beta z_i)} {\sum_{j\in R(t_i)}^{}\textup{exp}(w_j+\beta z_j)}$

where denotes the set of indices of observations that have not yet failed at time (the risk set), denotes the time of failure for the i-th observation, is the total number of observations that fail. Right-censored observations (i.e., observations that are known to have survived to time , but for which no time of failure is known) are incorporated into the likelihood through the risk set . Such observations never appear in the numerator of the likelihood. When TiesOption is set to BreslowsApproximate (the default), all observations that are censored at time are not included in , while all observations that fail at time are included in .

If it can be assumed that the dependence of the hazard rate upon the covariate values remains the same from stratum to stratum, while the time-dependent term, $\lambda_0(t)$ , may be different in different strata, then ProportionalHazards allows the incorporation of strata into the likelihood as follows. Let k index the m strata (set with StratumColumn). Then, the likelihood is given by

$L_S=\prod_{k=1}^{m}\left[ \prod_{i=1}^{n_k}\frac{\textup{exp}(w_{ki}+\beta z_{ki})}{\sum_{j\in R(t_{ki})}^{}\textup{exp}(w_{kj}+\beta z_{kj})} \right ]$

In ProportionalHazards, the log of the likelihood is maximized with respect to the coefficients $\beta$ . A quasi-Newton algorithm approximating the Hessian via the matrix of sums of squares and cross products of the first partial derivatives is used in the initial iterations. When the change in the log-likelihood from one iteration to the next is less than 100 times the convergence tolerance, Newton-Raphson iteration is used. If, during any iteration, the initial step does not lead to an increase in the log-likelihood, then step halving is employed to find a step that will increase the log-likelihood.

Once the maximum likelihood estimates have been computed, the algorithm computes estimates of a probability associated with each failure. Within stratum k, an estimate of the probability that the i-th observation fails at time given the risk set $R(t_{ki})$ is given by

$p_{ki}=\frac{\textup{exp}(w_{ki}+\beta z_{ki})}{\sum_{j\in R(t_{ki})}^{}\textup{exp}(w_{kj}+\beta z_{kj})}$

A diagnostic "influence" or "leverage" statistic is computed for each noncensored observation as:

$l_{ki}=-{g}'_{ki}H^{-1}_s{g}'_{ki}$

where

is the matrix of second partial derivatives of the log-likelihood, and

${g}'_{ki}$

is computed as:

${g}'_{ki}=z_{ki}-\frac{z_{ki} \textup{exp}(w_{ki}+\beta z_{ki})}{\sum_{j\in R(t_{ki})}^{}\textup{exp} (w_{kj}+\beta z_{kj})}$

Influence statistics are not computed for censored observations.

A "residual" is computed for each of the input observations according to methods given in Cox and Oakes (1984, page 108). Residuals are computed as

$r_{ki}=\textup{exp}(w_{ki}+\hat{\beta} z_{ki})\sum_{j \in R(t_{ki})}^{}\frac{d_{kj}}{\sum_{l \in R(t_{kj})}^{} \textup{exp}(w_{kl}+\hat{\beta} z_{kl})}$

where $d_{kj}$ is the number of tied failures in group k at time $t_{kj}$ . Assuming that the proportional hazards assumption holds, the residuals should approximate a random sample (with censoring) from the unit exponential distribution. By subtracting the expected values, centered residuals can be obtained. (The j-th expected order statistic from the unit exponential with censoring is given as

$e_j=\sum_{l \le j}^{}\frac{1}{h-l+1}$

where h is the sample size, and censored observations are not included in the summation.)

An estimate of the cumulative baseline hazard within group k is given as

$\hat{H}_{k0}(t_{ik})=\sum_{t_{kj}\le t_{ki}}^{}\frac{d_{kj}}{\sum_{l \in R(t_{kj})}^{}\textup{exp}(w_{kl}+ \hat{\beta} z_{kl})}$

The observation proportionality constant is computed as

$\textup{exp}(w_{ki}+\hat{\beta}z_{ki})$

Note that one can use logging to generate intermediate output for this class. Accumulated levels of detail correspond to Fine, Finer, and Finest logging levels with Fine yielding the smallest amount of information and Finest yielding the most. The levels of output yield the following:

Level	Output
Fine	Logging is enabled, but observational statistics are not printed.
Finer	All output statistics are printed.
Finest	Tracks progress through internal methods.

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.ProportionalHazards

Syntax

Remarks

Inheritance Hierarchy

See Also