Visualising and analysing
time-to-event data: lifting the veil of
censoring
Patrick Royston
Cancer Group, MRC Clinical
Trials Unit, London
A poet writes about censored observations:
Last night I saw upon the stair
A little man who wasn't there
He wasn't there again today
Oh, how I wish he'd go away!
From
Antigonish
(1899)
Outline
•
Why is censoring of time-to-event data an
issue?
•
Example in breast cancer
•
Visualisation of censored data using
model-based imputation
•
Multiple imputation and analysis of
survival data with missing covariate
observations
Why is censoring an issue?
•
You can’t picture the raw data easily
•
Reliance on Kaplan-Meier plots
Exaggerates differences between groups
Attracts attention to unreliable survival
estimates at extreme times
•
Data will be analysed using Cox model
Still the almost-automatic choice – although
decent alternatives exist
•
Time is “forgotten about” in the Cox model
•
Results of Cox regression models are
usually expressed as (log) hazard ratios
Indirect – not dealing directly with time
Can be hard to interpret – different effect on
survival curves at high and low survival probs
Particularly difficult for interactions – ‘ratio of
hazard ratios’
•
Non-proportional hazards
Data with long-term follow-up typically have it
Modelling and interpretation may be complex
Example: Primary
node-positive breast cancer
•
GBSG trial BMFT-2
•
686 patients, 299 events for
recurrence-free survival (RFS)
•
Patients assigned to hormonal therapy
(TAM) or not
•
Visualise the effect of TAM on RFS
•
Visualise interaction between TAM and ER
(estrogen receptor status)
Traditional visualisation:
Kaplan-Meier by TAM group
0.00
0.25
0.50
0.75
1.00
S(t)
0
2
4
6
8
Recurrence-free survival time, yr
hormone = No TAM
hormone = TAM
Dot plot by TAM therapy –
unhelpful with censored data
000000 1011000 01111110 11111111110 11100011110111111111 1111111111111111111 11111111010011111111111 1011110011111111111110000 00110010101111111011011 000110110010110 1010011001011010010111011100 0100101011111110101 010001000111 1011000000111011 0000101010001 101011010110100 1000111001001100000000 11010101 0001010001110000 10000000000101 00100010 01001000001000 0001100000000100 0100010000000 11000000010000 00000000 0000000001 0000001001 00000001 00000000 010 0000 100 0 0 000 11100011 1010 011011 10111111 1000111 1111111111111111 11110 0111101 10011010110000 10100111000 00101001 000101 0101100 1001001100 0101 10000 100001100 11010 0000000111 0000 000000110 0001000000000 00000000000010000 0000001 110000100 010000011 000 000000 0 00010 000 00 000 0
0
2
4
6
8
R
e
cu
rre
n
ce
-f
re
e
su
rvi
va
l t
ime
,
yr
No TAM
TAM
Hormone therapy, 1=no, 2=yes
How better to visualise
survival times?
•
To make progress with visualisation, aim to
impute the “missing” part of censored times
•
Assume a parametric distribution of survival time
•
Survival times are sometimes approximately
lognormally distributed (Royston 2001a)
Can check by using modified Normal Q-Q plot
•
If lognormal approximation is not good, can
consider Box-Cox transformation of time
Assessing lognormality:
modified Normal Q-Q plot
•
Simple transformation of Kaplan-Meier survival curve
.2
.5
1
2
5
10
Recurrence-free survival time (log scale)
-3
-2
-1
0
1
Normal Q-Q plot by TAM group
.2
.5
1
2
5
10
Recurrence-free survival time (log scale)
-3
-2
-1
0
1
Normal equivalent deviates
Visualisation of censored data
using imputation
•
Create m (
≥
1) copies of the data with
censored survival times imputed
•
Need an imputation model to reflect
Distribution of times (e.g. lognormal)
Effects of covariates (prognostic factors)
•
Creating an imputation model:
Use
mfp
with cnreg (censored normal regrn.)
to model poss. non-linear effects of covariates
E.g.
mfp cnreg lnt x1 x2 x3 x4a x4b x5 x6
x7 hormone, censored(c) select(1)
Creating the imputed
dataset(s)
•
Can use the
ice
multiple imputation
command to create the imputations
Royston (2004, 2005a, 2005b)
Stata J
•
ice
varlist
using
filename
[.dta]
[if
exp
] [in
range
] [
weight
],
[m(
#
) cmd(
cmdlist
) cycles(
#
)
boot[(
varlist
)] seed(
#
) dryrun
eq(
eqlist
) passive(
passivelist
)
substitute(
sublist
) dropmissing
Interval censoring with
ice
•
gen ll = lnt
•
gen ul = cond(_d==1, lnt, ln(50))
// chose upper limit of 50 years for
RFS: can use . for
+
∞
•
(generate FP transformations of
x1
,
x5
,
x6
)
•
ice x1_1 x2 x3 x4a x4b x5_1 x6_1 x7
hormone ll ul lnt using imputed.dta,
interval(lnt:ll ul) m(10)
How
interval()
works
Censored obs Upper limit Complete obs -4 -3 -2 -1 0 1 2 3 4Code fragment from uvis.ado
`cmd' `yvarlist' `xvars' `wgt',
`options'
...
if "`cmd'"=="intreg" {
tempvar PhiA PhiB
gen `PhiA‘ = cond(missing(`ll'), 0,
norm((`ll'-`xb')/`rmsestar'))
gen `PhiB‘ = cond(missing(`ul'), 1,
norm((`ul'-`xb')/`rmsestar'))
replace `yimp‘ = `xb‘
+`rmsestar'*invnorm(`u'*
(`PhiB'-`PhiA')+`PhiA')
}
Uses of the
interval()
option
•
Impute right-, left- or interval-censored
outcomes
Response variable in time-to-event studies
•
Impute when a covariate is sometimes
partly observed, sometimes complete
Some observations recorded exactly
Others known to be below or above a cutoff
E.g. D-dimer in DVT, PgR/ER in breast cancer
•
Interval censored covariates
Breast cancer data: visualisation
of time to recurrence
0 0 0000 0 0 0 0000 10 1 101 1010 11110111111010 11011011111 111010111111111000110 11111011111111111111111111111 111110101111111011111011001111111111111 1101111101100111001110111101011111111011111010110110011111111 0100011111010100000111101000110101110011001110110010010010100 100100110111110111111001111110100000010001001000101100110110101000000 0010100000100101011001001011100011100000110000100111001101110110101 10010000101000001000001000100010100000000110100000001011011100100010001011001100010 000100101010000001100000010000000010100000000010100000010000000000000001000000000001000000000000001000100000010 00100000000110010100000000000000011000010001000000000000010001001000000000000 00001000000000010000 1 1 11 11 11111111111 1111111011 1111111110111111 1111111111111111111111111111 11111111111111111111111111110111101 111111111101111111111111111111111111101111111111 100111111111111111111111111111101 01101011011111111010111111111101101011111011 10101011001111111010111111111111101111 001011110101100110001010011110001100111011011010 00101000000010001010100101110110100110000010001000 0000000010001100001000100001000000000000000000110001010000010000100 0000000000000000000000000000000000010000001000000000 00000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000 000000000000000000000000000000000 0000000000000000000000000 000000000000000000000000 000000000000000 0000000 00000000 0.0
5
.1
.2
.5
1
2
5
10
20
50
Recu
rr
en
ce-fr
e
e sur
v
ival tim
e
, yr
0
1
imputation number
fig_responseVisualisation: some plots using
the first imputed sample
.2 .5 1 2 5 10 20 50
Recurrence-free survival time (log scale)
No TAM TAM
Hormone therapy
.2 .5 1 2 5 10 20 50Recurrence-free survival time (log scale)
0 10 20 30 40 50
Visualisation: treatment by
covariate interaction
.2
.5
1
2
5
10
20
50
Recurrence-free survival time (log scale)
ER neg, no TAM
ER neg, TAM
ER pos, no TAM
ER pos, TAM
Limitations
•
Imputed times to event are helpful for
visualisation, but less so for analysis
Effectively, such imputations are
extrapolations into the future
We don’t
know
the future distribution
Estimates of means, SD’s, regression coeffs
etc. are heavily dependent on the distributional
assumptions
Potential for bias if assumed distr’n is wrong
•
Imputed times may be unrealistic
Other approaches
•
A reasonably large literature exists
•
Buckley-James estimation (Buckley & James
1979)
Estimates the mean of the censored part
Not so good for visualisation
•
Wei & Tanner (1991)
Two algorithms which give multiple imputations of the
censored part
Relaxes the normality assumption – samples taken from
the distribution of the residuals
•
stpm
(Royston 2001b, Royston & Parmar 2002)
Imputation of survival data with
missing covariate observations
•
So far, have assumed covariates have complete data
•
If covariates have
missing data
, need a suitable algorithm
for multiple imputation of all missing values
e.g. MICE (
ice
)
•
To reduce bias, must include the response (time-to-event)
in the imputation model
How?
•
“Standard” approach is to include (censored)
log time
and
the
censoring indicator
in the imputation model
No theoretical justification
•
May be better to
Include covariates as usual
Impute right-censored times using
ice
with
interval()
option
Analysis of survival data with
missing covariate observations
•
Disregard the imputed times in the MI
dataset
Except for visualisation purposes
•
Use original time and censoring indicator
•
Can analyse the MI dataset using
stcox
(Cox regression)
streg
(several models available)
stpm
(flexible parametric survival models)
Conclusions
•
Use of familiar graphical tools with imputed times
to event can give greater insight into censored
survival data
Scatter plots, smoothers, etc
•
Treatment or prognostic effects may be
depressingly small when displayed as scatter
plots of times
Much overlap between groups
Weak regression relationships
•
Imputation of times may be helpful in multiple
imputation with missing covariate values
Some references
• Buckley J, James I (1979) Linear regression with censored data. Biometrika 66: 429-436
• Faucett CL, Schenker N, Taylor JMG (2002) Survival analysis using auxiliary variables via multiple imputation, with application to AIDS clinical trial data. Biometrics 58: 37-47.
• Ma SG (2006) Multiple augmentation with partial missing regressors. Biometrical Journal 48: 83-92 • Pan W (2000) A multiple imputation approach to Cox regression with interval-censored data.
Biometrics 56: 199-203
• Royston P (2001a) The lognormal distribution as a model for survival time in cancer, with an emphasis on prognostic factors. Statistica Neerlandica 55: 89-104
• Royston P (2001b) Flexible alternatives to the Cox model, and more. The Stata Journal 1:1-28. • Royston P, Parmar MKB (2002) Flexible proportional-hazards and proportional-odds models for
censored survival data, with application to prognostic modelling and estimation of treatment effects. Statistics in Medicine 21: 2175-2197
• Royston P (2004) Multiple imputation of missing values. Stata Journal 4: 227-241
• Royston P (2005a) Multiple imputation of missing values: update. Stata Journal 5: 188-201
• Royston P. (2005b) Multiple imputation of missing values: update of ice. Stata Journal 5: 527-536 • Tanner MA, Wing HW (1987) The calculation of posterior distributions by data augmentation. JASA
82: 528-540. [Cited 642 times, WoS 10.9.2006]
• Wei GCG, Tanner MA (1990) Posterior computations for censored regression data. JASA 85: 829-39 • Wei GCG, Tanner MA (1991) Application of multiple imputation to the analysis of censored
