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=, weights, subset, 
      na.action, init, control, 
      ties=c("efron","breslow","exact"), 
      singular.ok=TRUE, robust, 
      model=FALSE, x=FALSE, y=TRUE, tt, method=ties,
      id, cluster, istate, statedata, nocenter=c(-1, 0, 1), ...)

Arguments

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].

The routine internally scales and centers data to avoid overflow in the argument to the exponential function. These actions do not change the result, but lead to more numerical stability. Any column of the X matrix whose values lie within nocenter list are not recentered. The practical consequence of the default is to not recenter dummy variables corresponding to factors. However, arguments to offset are not scaled since there are situations where a large offset value is a purposefully used. In general, however, users should not avoid very large numeric values for an offset due to possible loss of precision in the estimates.

Value

an object of class coxph representing the fit. See coxph.object and coxphms.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. It is possible for this to fail, as in the example below in which the predict function is unable to find tform.

  tfun <- function(tform) coxph(tform, data=lung)
  fit <- tfun(Surv(time, status) ~ age)
  predict(fit)

In such a case add the model=TRUE option to the coxph call to obviate the need for reconstruction, at the expense of a larger fit object.

Case weights

Case weights are treated as replication weights, i.e., a case weight of 2 is equivalent to having 2 copies of that subject's observation. When computers were much smaller grouping like subjects together was a common trick to used to conserve memory. Setting all weights to 2 for instance will give the same coefficient estimate but halve the variance. When the Efron approximation for ties (default) is employed replication of the data will not give exactly the same coefficients as the weights option, and in this case the weighted fit is arguably the correct one.

When the model includes a cluster term or the robust=TRUE option the computed variance treats any weights as sampling weights; setting all weights to 2 will in this case give the same variance as weights of 1.

Special terms

There are three 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 each value of id is unique is equivalent to specifying the robust=TRUE argument. If the id variable is not unique, it is assumed that it identifies clusters of correlated observations. The robust estimate arises from many different arguments and thus has had many labels. It is variously known as the Huber sandwich estimator, White's estimate (linear models/econometrics), the Horvitz-Thompson estimate (survey sampling), the working independence variance (generalized estimating equations), the infinitesimal jackknife, and the Wei, Lin, Weissfeld (WLW) estimate.

A time-transform term allows variables to vary dynamically in time. In this case the tt argument will be a function or a list of functions (if there are more than one tt() term in the model) giving the appropriate transform. See the examples below.

One user mistake that has recently arisen is to slavishly follow the advice of some coding guides and prepend survival:: onto everthing, including the special terms, e.g., survival::coxph(survival:Surv(time, status) ~ age + survival::cluster(inst), data=lung) First, this is unnecessary: arguments within the coxph call will be evaluated within the survival namespace, so another package's Surv or cluster function would not be noticed. (Full qualification of the coxph call itself may be protective, however.) Second, and more importantly, the call just above will not give the correct answer. The specials are recognized by their name, and survival::cluster is not the same as cluster; the above model would treat inst as an ordinary variable. A similar issue arises from using stats::offset as a term, in either survival or glm models.

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. (Most often number 1 is the first to occur.) The routine attempts to detect when this has happened, not always successfully. The primary consequence for the user is that the Wald statistic = coefficient/se(coefficient) is not valid in this case and should be ignored; the likelihood ratio and score tests remain valid however.

Ties

There are three possible choices for handling tied event times. The Breslow approximation is the easiest to program and hence became the first option coded for almost all computer routines. It then ended up as the default option when other options were added in order to "maintain backwards compatability". The Efron option is more accurate if there are a large number of ties, and it is the default option here. In practice the number of ties is usually small, in which case all the methods are statistically indistinguishable.

Using the "exact partial likelihood" approach the Cox partial likelihood is equivalent to that for matched logistic regression. (The clogit function uses the coxph code to do the fit.) It is technically appropriate when the time scale is discrete and has only a few unique values, and some packages refer to this as the "discrete" option. There is also an "exact marginal likelihood" due to Prentice which is not implemented here.

The calculation of the exact partial likelihood is numerically intense. Say for instance 180 subjects are at risk on day 7 of which 15 had an event; then the code needs to compute sums over all 180-choose-15 > 10^43 different possible subsets of size 15. There is an efficient recursive algorithm for this task, but even with this the computation can be insufferably long. With (start, stop) data it is much worse since the recursion needs to start anew for each unique start time.

Multi state models are a more difficult case. First of all, a proper extension of the Efron argument is much more difficult to do, and this author is not yet fully convinced that the resulting algorithm is defensible. Secondly, the current code for Efron case does not consistently compute that extended logic (and extension would require major changes in the code). Due to this complexity, the default is ties='breslow' for the multistate case. If ties='efron' is selected the current code will, in effect, only apply to to tied transitions of the same type.

A separate issue is that of artificial ties due to floating-point imprecision. See the vignette on this topic for a full explanation or the timefix option in coxph.control. Users may need to add timefix=FALSE for simulated data sets.

Penalized regression

coxph can 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

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

Therneau, T., Grambsch, P., Modeling Survival Data: Extending the Cox Model. Springer-Verlag, 2000.

See Also

coxph.object, coxphms.object, coxph.control, cluster, strata, Surv, survfit, pspline.

Examples

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

#
# 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))

# Fit a stratified model, clustered on patients 

bladder1 <- bladder[bladder$enum < 5, ] 
coxph(Surv(stop, event) ~ (rx + size + number) * strata(enum),
      cluster = id, bladder1)

# Fit a time transform model using current age
coxph(Surv(time, status) ~ ph.ecog + tt(age), data=lung,
     tt=function(x,t,...) pspline(x + t/365.25))