R: Fit Proportional Hazards Regression Model

coxph {survival} | R Documentation

## Fit Proportional Hazards Regression Model

### Description

Fits a Cox proportional hazards regression model. Time dependent variables, time dependent strata, multiple events per subject, and other extensions are incorporated using the counting process formulation of Andersen and Gill.

### Usage

    coxph(formula, data=parent.frame(), weights, subset,
           na.action, init, control, method=c("efron","breslow","exact"),
           singular.ok=TRUE, robust=FALSE,
           model=FALSE, x=FALSE, y=TRUE,...
           )

### Arguments

`formula` | a formula object, with the response on the left of a `~` operator, and the terms on the right. The response must be a survival object as returned by the `Surv` function.
`data` | a data.frame in which to interpret the variables named in the `formula`, or in the `subset` and the `weights` argument.
`subset` | expression saying that only a subset of the rows of the data should be used in the fit.
`na.action` | a missing-data filter function, applied to the model.frame, after any subset argument has been used. Default is `options()$na.action`.
`weights` | case weights.
`init` | vector of initial values of the iteration. Default initial value is zero for all variables.
`control` | Object of class `coxph.control` specifying iteration limit and other control options. Default is `coxph.control(...)`.
`method` | a character string specifying the method for tie handling. If there are no tied death times all the methods are equivalent. Nearly all Cox regression programs use the Breslow method by default, but not this one. The Efron approximation is used as the default here, as it is much more accurate when dealing with tied death times, and is as efficient computationally. The exact method computes the exact partial likelihood, which is equivalent to a conditional logistic model. If there are a large number of ties the computational time will be excessive.
`singular.ok` | logical value indicating how to handle collinearity in the model matrix. If `TRUE`, the program will automatically skip over columns of the X matrix that are linear combinations of earlier columns. In this case the coefficients for such columns will be NA, and the variance matrix will contain zeros. For ancillary calculations, such as the linear predictor, the missing coefficients are treated as zeros.
`robust` | if TRUE a robust variance estimate is returned. Default is `TRUE` if the model includes a `cluster()` operative, `FALSE` otherwise.
`model` | flags to control what is returned. If these are true, then the model frame, the model matrix, and/or the response is returned as components of the fitted model, with the same names as the flag arguments.
`x` | Return the design matrix in the model object?
`y` | return the response in the model object?
`...` | Other arguments will be passed to `coxph.control`

### Details

The proportional hazards model is usually expressed in terms of a single survival time value for each person, with possible censoring. Andersen and Gill reformulated the same problem as a counting process; as time marches onward we observe the events for a subject, rather like watching a Geiger counter. The data for a subject is presented as multiple rows or "observations", each of which applies to an interval of observation (start, stop].

### Value

an object of class `"coxph"`. See `coxph.object` for details.

### Side Effects

Depending on the call, the `predict`, `residuals`, and `survfit` routines may need to reconstruct the x matrix created by `coxph`. Differences in the environment, such as which data frames are attached or the value of `options()$contrasts`, may cause this computation to fail or worse, to be incorrect. See the survival overview document for details.

### SPECIAL TERMS

There are two special terms that may be used in the model equation. A 'strata' term identifies a stratified Cox model; separate baseline hazard functions are fit for each strata. The `cluster` term is used to compute a robust variance for the model. The term `+ cluster(id)`, where `id == unique(id)`, is equivalent to specifying the `robust=T` argument, and produces an approximate jackknife estimate of the variance. If the `id` variable were not unique, but instead identifies clusters of correlated observations, then the variance estimate is based on a grouped jackknife.

### CONVERGENCE

In certain data cases the actual MLE estimate of a coefficient is infinity, e.g., a dichotomous variable where one of the groups has no events. When this happens the associated coefficient grows at a steady pace and a race condition will exist in the fitting routine: either the log likelihood converges, the information matrix becomes effectively singular, an argument to exp becomes too large for the computer hardware, or the maximum number of interactions is exceeded. The routine attempts to detect when this has happened, not always successfully.

### PENALISED REGRESSION

`coxph` can now maximise a penalised partial likelihood with arbitrary user-defined penalty. Supplied penalty functions include ridge regression (ridge), smoothing splines (pspline), and frailty models (frailty).

### References

P. Andersen and R. Gill. "Cox's regression model for counting processes, a large sample study", Annals of Statistics, 10:1100-1120, 1982.

T. Therneau, P. Grambsch, and T. Fleming. "Martingale based residuals for survival models", Biometrika, March 1990.

### See Also

`cluster`, `survfit`, `Surv`, `strata`,`ridge`, `pspline`,`frailty`.

### Examples

    # Create the simplest test data set
    #
     test1 <- list(time=  c(4, 3,1,1,2,2,3),
                    status=c(1,NA,1,0,1,1,0),
                    x=     c(0, 2,1,1,1,0,0),
                    sex=   c(0, 0,0,0,1,1,1))
     coxph( Surv(time, status) ~ x + strata(sex), test1)  #stratified model

    #
    # Create a simple data set for a time-dependent model
    #
    test2 <- list(start=c(1, 2, 5, 2, 1, 7, 3, 4, 8, 8),
                    stop =c(2, 3, 6, 7, 8, 9, 9, 9,14,17),
                    event=c(1, 1, 1, 1, 1, 1, 1, 0, 0, 0),
                    x    =c(1, 0, 0, 1, 0, 1, 1, 1, 0, 0) )

    summary( coxph( Surv(start, stop, event) ~ x, test2))

[Package survival version 2.34-1 Index]