Calculates the non-parametric maximum likelihood estimator (NPMLE) of the survival function for interval censored data, also known as the Turnbull estimator (V.M. Cave).
Options
PRINT = string tokens |
What output to print and whether to plot the Turnbull estimator of the survival function (estimate, summary, graph); default esti, grap |
PLOT = string tokens |
Information to be plotted on the graph (linearapproximation, lowerboundary, upperboundary); default line, lower, upper |
SUPERIMPOSE = string token |
Whether to superimpose the plots for different groups, supplied by GROUPS, onto a single graph; default no (i.e. a separate graph is plotted for each group) |
COLOURS = variate, text |
The colours used for plotting, and to distinguish the different groups when GROUPS is set and SUPERIMPOSE=yes; defaults are the colours of pens 1…n, where n is the number of groups |
TITLE = text |
Title for the graph(s); default generates the title automatically |
YTITLE = text |
Y-axis title for the graph(s); default generates the title automatically |
XTITLE = text |
X-axis title for the graph(s); default generates the title automatically |
WINDOW = scalar |
Window to use for the graph(s); default 1 |
KEYWINDOW = scalar |
Window to use for the key, with zero suppressing the key; default 2 |
MAXCYCLE = scalar |
Sets a limit on the number of iterations performed by the Turnbull algorithm; default 1000 |
TOLERANCES = scalar |
Sets a tolerance limit for convergence of the Turnbull algorithm; default 1e-6 |
Parameters
LOWER = variate |
Identifier of the variate holding the lower limit of the censoring interval |
UPPER = variate |
Identifier of the variate holding the upper limit of the censoring interval, with right censored values ( i.e., observations yet to fail) represented by a missing value |
GROUPS = factor |
Factor specifying the different groups for which the survival function is to be estimated |
SAVE = pointer |
Pointer to save the Turnbull estimates of the survival function and the Turnbull intervals |
Description
TURNBULL calculates the Turnbull estimator of the survival function for interval censored data. Survival analysis is used to model time-to-event data. For example, the survival time of patients following treatment or the lifetime of components from their commission. The survival function S(t) is a key element in the analysis of survival data. It is the probability that the time-to-event occurs after time t has elapsed. That is, the probability a patient or component will survive past time t. Often in practice the exact time-to-event, t, is unknown, but it can be narrowed to between two observational times. That is, LOWER < t < UPPER. This is interval censored survival data. The Turnbull estimator is a generalization of the Kaplan-Meier estimator that models interval censored data. Note, observational units may also be right censored if the event of interest has not occurred before the end of the study. Here, the UPPER time limit is not observed. If the exact time-to-event is observed, the observation is uncensored, in which case t = LOWER = UPPER.
Variates containing the observed lower and upper limits of the interval censored survival data must be supplied using the LOWER and UPPER parameters, respectively, with right censored observational units given a missing value (*) in UPPER. Uncensored observations are allowed. However, if all observational units are either uncensored or right censored, the KAPLANMEIER procedure must be used. For an uncensored observational unit, its values in LOWER and UPPER must both be set to the observed time-to-event.
The GROUPS parameter can be used to supply a factor indicating the different groups whose survival functions are to be estimated.
Printed output is controlled by the PRINT option, with the following settings:
| estimate | Turnbull intervals with the Turnbull estimates of the survival function; |
| summary | number and percentage of interval censored, right censored and uncensored observations, and the number of iterations for convergence of the Turnbull algorithm; and |
| graph | plot of the Turnbull estimator of the survival function. The output can be suppressed by setting option PRINT=*. The default is esti, grap. The PLOT option controls how the survival function is plotted, with the following settings: |
| linearapproximation | plots the survival function using a linear approximation between the lower and upper boundaries of the Turnbull intervals (see Methods); |
| lowerboundary | plots the survival function at the lower boundaries of the Turnbull intervals; and |
| upperboundary | plots the survival function at the upper boundaries of the Turnbull intervals. |
The default is to plot all the graphs.
The options SUPERIMPOSE, COLOURS, TITLE, YTITLE, and XTITLE are used to control the appearance of the graph(s). With the default SUPERIMPOSE=no, a separate graph is plotted for each group supplied by parameter GROUPS. Whereas if SUPERIMPOSE=yes, the survival function for each group is plotted on the same graph.
The colours used for plotting are controlled using the COLOURS option. This can supply either a variate, defining the colours using the RGB system, or in a text, containing the names of pre-defined colours; see PEN for details. When GROUPS is supplied and SUPERIMPOSE=yes, the colours are used to distinguish the different groups. Thus, COLOURS must supply n colours, where n is the number of groups. The defaults are the default colours of pens 1…n. Otherwise (i.e., when GROUPS isn’t supplied or SUPERIMPOSE=no), the colours are used to plot the items supplied by the PLOT option, given in the order linear approximation, upper boundary and lower boundary, and COLOURS must specify a colour for every item supplied. By default, the linear approximation is plotted in blue, and the upper and lower boundaries in black.
The TITLE, YTITLE and XTITLE options can supply an overall title, a y-axis title and a x-axis title, respectively. If no titles are supplied, suitable titles are generated automatically. To omit a title, a blank string can be supplied, e.g.. TITLE=’ ‘.
The WINDOW option defines the window to use for the graphs; default 1.The KEYWINDOW option specifies the window in which the key appears; by default this is window 2. You can set KEYWINDOW=0 to suppress the key.
The MAXCYCLE option specifies the maximum number of iterations performed by the Turnbull algorithm (default 1000). By default, the Turnbull algorithm is deemed to have converged if the maximum absolute change in the estimated probabilities of an event occurring at time t is less than 1E-6. However, you can use the TOLERANCE option to supply a scalar to change this tolerance limit.
The SAVE parameter saves the Turnbull intervals and the Turnbull estimates in a pointer.
Options: PRINT, PLOT,SUPERIMPOSE, COLOURS, TITLE, YTITLE, XTITLE, WINDOW, KEYWINDOW, MAXCYCLE, TOLERANCE.
Parameters: LOWER, UPPER, GROUPS, SAVE.
Method
TURNBULL estimates the survival function for interval censored data using the iterative Turnbull algorithm (Turnbull 1974), as described in Rodrigues et al. (2018).
The Turnbull estimator is unidentifiable inside the Turnbull intervals, resulting in a survival function with rectangular boxes representing areas of equal likelihood within the Turnbull intervals. A linear approximation through these rectangles is given by the diagonal line running from the upper left corner to the lower right corner.
Action with RESTRICT
TURNBULL will work with restricted LOWER and UPPER variates, and a restricted GROUPS factor. However, if more than one is restricted, they must all be restricted in the same way. Note that LOWER,UPPER and GROUPS must all have the be same unrestricted length.
Reference
Rodrigues, A.S., Calsavara, V.F., Silva, F.I.B., Alves, F.A. & Vivas, A.P.M. (2018). Use of interval-censored survival data as an alternative to Kaplan-Meier survival curves: studies of oral lesion occurrence in liver transplants and cancer recurrence. Applied Cancer Research, 38(16), 1-10. https://doi.org/10.1186/s41241-018-0067-7.
Turnbull, B.W. (1974). Nonparametric estimation of a survivorship function with doubly censored data. Journal of the American Statistical Association, 69(345), 169-173.
See also
Procedures: KAPLANMEIER, RLIFETABLE, RPHFIT,RPROPORTIONAL, RSTEST, RSURVIVAL
GenStat Reference Manual 1 Summary section on: Survival analysis.
Example
CAPTION 'TURNBULL example',\
!t('The data are time to cosmetic deterioration in breast cancer patients',\
'for two treatments: radiotherapy alone versus radiotherapy plus chemotherapy'),\
!t('Presented in Klein and Moeschberger (1997). ~i{Survival Analysis: Techniques}',\
'~i{for Censored and Truncated Data} (1st ed.). Springer New York.'); \
STYLE=meta,plain,plain
"Data"
VARIATE [VALUES=0,0,4,4,5,8,8,10,10,11,11,11,11,11,12,13,13,13,13,14,14,15,16,16,16,16,17,17,\
17,18,18,19,21,22,23,24,24,30,30,31,32,33,34,34,35,35,44,48,0,0,0,4,5,5,6,7,7,\
11,11,15,17,17,17,18,19,18,22,24,24,25,26,27,32,33,34,36,36,36,36,37,37,37,37,\
38,40,45,46,46,46,46,46,46,46,46] lower
VARIATE [VALUES=22,5,9,8,8,12,21,35,17,13,*,17,*,20,20,*,39,*,*,17,19,22,24,20,24,60,27,23,26,\
25,24,32,*,32,*,31,30,34,36,*,*,40,*,*,*,39,48,*,7,8,5,11,12,11,10,16,14,15,18,\
*,*,25,25,*,35,26,*,*,*,37,40,34,*,*,*,44,48,*,*,44,*,*,*,*,*,*,*,*,*,*,*,*,*,\
*] upper
FACTOR [LEVELS=!(0,1); LABELS=!T('Radiotherapy+Chemotherapy','Radiotherapy')] Trt; \
VALUES=!(48(0),46(1))
"Treatment groups plotted in separate graphs"
TURNBULL LOWER=lower; UPPER=upper; GROUP=Trt
"Treatment groups plotted in a single graphs with bespoke colours"
TURNBULL [COLOURS=!T('blue','red'); SUPERIMPOSE=yes] LOWER=lower; UPPER=upper; GROUP=Trt