Fits a proportional hazards model to survival data as a generalized linear model (R.W. Payne).

### Options

`PRINT` = string tokens |
Controls printed output (`model` , `deviance` , `summary` , `estimates` , `correlations` , `fittedvalues` , `accumulated` , `monitoring` , `loglikelihood` ); default `mode` , `summ` , `esti` |
---|---|

`MAXIMALMODEL` = formula |
Defines the full model to explore (using `RPHCHANGE` ); default uses the model defined by the `TERMS` parameter |

`SUBJECTS` = factor |
Subject corresponding to each observation |

`TIMES` = factor or variate |
Time of each observation |

`CENSORED` = variate |
Contains the value 1 for censored observations, otherwise 0; if unset it is assumed that there is no censoring |

`OFFSET` = variate |
Offset to include in the model |

`POOL` = string token |
Whether to pool terms in the accumulated summary generated by the fit |

### Parameter

`TERMS` = formula |
Model to fit |
---|

### Description

The data for `RPHFIT`

consist of a set of subjects observed at one or more times. The final time is usually at the time of death (or failure), otherwise (if the subject survives the trial) the observation is said to be censored. The `CENSORED`

option can be used to specify a variate with an entry for each subject containing one when there is censoring, otherwise zero. If this is not specified, it is assumed that there is no censoring. The `SUBJECTS`

option can specify a factor to indicate the subject corresponding to each observation; this can be omitted if there is only one observation per subject. The time at which each observation was made is specified by the `TIME`

option, in either a factor or a variate.

The model to be fitted is specified using the `TERMS`

parameter. You can modify the model later by using procedure `RPHCHANGE`

. If you intend to use `RPHCHANGE`

to include additional model terms, you should use option `MAXIMALMODEL`

of `RPHFIT`

to define the largest model that you may want to consider (this option acts similarly to the `TERMS`

directive in ordinary generalized linear modelling). You can display further output using procedure `RPHDISPLAY`

, and save information using procedure `RPHKEEP`

.

The proportional hazards model (Cox 1972) makes the assumption that the subjects have a baseline hazard function which is modified proportionally by treatments and covariates. In `RPHFIT`

it is assumed that the survival times follow a piecewise exponential distribution (Breslow 1974). This partitions the time axis using a set of discrete cut-points *a _{i}*, and assumes a constant baseline hazard γ

*between each one. This corresponds to an exponential distribution with mean 1/γ*

_{i}*for the survival times (in the absence of treatments) within each time interval. A cut-point is defined at every time that a death (or failure) occurs and, if the covariates or treatments vary with time, also at every time when the subjects are observed.*

_{i}To fit a proportional hazards model as a generalized linear model, the x-variates (i.e. covariates) and factors must be expanded so that, for each subject, there is a unit for every time interval up to the last one during which the subject was observed. If (as usually happens) the subject was not observed at every cutpoint, the covariates and treatments are taken to be constant during the intervals between the times of the observations. `RPHFIT`

automatically produces the expanded sets of values (using procedure `RPHVECTORS`

). These replace the original values while `RPHFIT`

is fitting and displaying the model. The original values are then reinstated before exit from the procedure, unless a fault is generated e.g. from the regression directives `FIT`

&c. You can call `RPHVECTORS`

directly if you do want to obtain the expanded values. Alternatively, procedure `RPHKEEP`

can save the index variate that is used to construct them.

The y-variate used within the generalized linear model is an indicator that takes the value 0 if the subject was still surviving within the time interval concerned, otherwise it has the value 1. The model also contains an offset representing the log of the exposure time within each interval. Any additional offset can be specified, if required, using the `OFFSET`

option. (These two variates are also obtainable from `RPHKEEP`

.)

The `PRINT`

option controls printed output with similar settings to those of the `FIT`

directive, except that there is an extra setting `loglikelihood`

to print -2 times the log-likelihood. The deviance produced for the terms in the regression model can be assessed using chi-square distributions as usual, but the residual deviance is not usable as the maximal model assumed by the generalized linear models method is inappropriate. So, the residual line is suppressed in the summary and accumulated analysis of deviance. By default the terms in the model are fitted individually so that they will all have their own lines in an accumulated analysis of deviance. However, you can set option `POOL=yes`

to fit them all at once.

Options: `PRINT`

, `MAXIMALMODEL`

, `SUBJECTS`

, `TIMES`

, `CENSORED`

, `OFFSET`

, `POOL`

.

Parameter: `TERMS`

.

### Method

The expanded sets of values for the variates and factors in the model are formed using procedure `RPHVECTORS`

, together with the response and offset variates that are needed. Further details of the method can be found in Aitkin *et al*. (1989).

### Action with `RESTRICT`

None of the vectors must be restricted, and any restrictions will be cancelled.

### References

Aitkin, M., Anderson, A., Francis, B. & Hinde, J. (1989). *Statistical Modelling in GLIM*. Oxford University Press.

Breslow. N. (1974). Covariance analysis of censored survival data. *Biometrics*, 30, 89-99.

Cox, D.R. (1972). Regression models and life tables (with discussion). *Journal of the Royal Statistical Society Series B*, 34, 187-220.

### See also

Procedures: `KAPLANMEIER`

, `RLIFETABLE`

, `RPHCHANGE`

, `RPHDISPLAY`

, `RPHKEEP`

, `RPHVECTORS`

, `RPROPORTIONAL`

, `RSTEST`

, `RSURVIVAL`

.

Commands for: Survival analysis.

### Example

CAPTION 'RPHFIT example',\ 'Data from Gehan (1965, Biometrika, 52, 203-223).';\ STYLE=meta,plain VARIATE [VALUES=1,1,2,2,3,4,4,5,5,8,8,8,8,11,11,12,12,15,17,22,23,\ 6,6,6,6,7,9,10,10,11,13,16,17,19,20,22,23,25,32,32,34,35] Time & [VALUES=24(0),1,0,1,0,1,1,0,0,1,1,1,0,0,1,1,1,1,1] Censor FACTOR [LABELS=!t(control,'6-mercaptopurine'); VALUES=21(1,2)] Treat FACTOR [LEVELS=42; VALUES=1...42] Subject RPHFIT [TIMES=Time; SUBJECTS=Subject; CENSORED=Censor] Treat