The Kaplan-Meier Mean Covariate (KMMC) plot:
a new diagnostic for covariates in time-to-event models.
Andrew C. Hooker and Mats O. Karlsson
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Background
Results
KMMC can be used for continuous or categorical covariates and can easily
handle both time-constant and time-varying covariates.
80
2.0
60
40
mean DOSE
150
mean CONC
Survival (%)
When building time-to-event (TTE)
models it is often difficult to screen for
covariates (or predictors) and it can be
especially challenging to understand,
and illustrate, how covariates should be
included in the model. Additionally, it
can be challenging to illustrate the
importance of included covariates.
100
1.5
20
100
1.0
Stratification can work for timeconstant categorical predictors but
cannot be used for continuous and/or
time-varying covariates without loss of
information.
0
Arm 3 (DOSE = 50)
100
200
300
400
0
0
100
200
300
400
500
0.5
Figure 1. Model with constant
hazard. Data is simulated with
concentration dependence.
0
200
300
400
500
0
100
200
300
400
500
Time
Figure 5. A KMMC plot using
concentration (continuous and timevarying).
Study design for investigations:
100
100
Time
500
Arm 4 (DOSE = 200)
50
Time
Figure 6. A KMMC plot using
dose (categorical – only 4
allowed dose levels in study).
80
• A (simulated) proof-of-concept
clinical trial evaluating a new
agent for inhibition/
amelioration of reflux events
Survival (%)
40
20
0
Arm 1 (Dose = 0)
Arm 2 (DOSE = 10)
From a base TTE model all covariates can be screened right away using the
KMMC plot.
0
100 200 300 400 500
Max Score per hour
Avg. Conc.
3.0
3
2.0
60
40
400
500
2.5
2
1
Time
2.0
300
• Time to first moderate or
severe reflux event recorded.
Conc.
Dose
Max Score
2.0
1.0
50
0.5
In this work we present a new graphical tool that can overcome these
problems; the Kaplan-Meier Mean Covariate (KMMC) plot.
1.5
1.0
Objective
100
1.5
2.5
150
2.0
Figure 2. Kaplan-Meier VPC using
categorical covariate stratification.
3.0
200
0.5
100
Mean
0
1.0
0
1.5
• Randomized, double-blind, 4
arm, single-dose (placebo, 10
50, 200 mg) parallel trial
20
Figure 7. KMMC
plots for all
covariates of
interest using a
constant hazard
model. Data is
simulated with a
concentration
dependent hazard.
All covariates are
shown to be
potentially
interesting.
3.5
• 48 patients with frequent reflux
episodes
4
2.5
100
80
ARM
4.0
60
Methods
0
100 200 300 400 500
0
100 200 300 400 500
Time
The KMMC plot computes the mean (or any other function) of a covariate
for all of the individuals still in a study at every inflection point of a KaplanMeier survival curve.
Below, KMMC plots for both before and after a covariate is included in a TTE
model identify the covariate effect and show when a model adequately
describes that effect.
This “running” mean of a covariate that is influential on survival would be
expected to increase or decrease as a study progresses.
Hazard = f(dose)
(ΔOFV=-31)
Hazard = Constant
Run 51
Run 52
Run 53
Dose
Dose
Dose
mean DOSE
150
100
PsN command:
vpc run51.mod -tte=RTTE -flip_comments
-samples=100 -stratify_on=DOSE
50
Xpose command:
0
100
200
300
400
Time
500
kaplan.plot(x="TIME", y="DV",
object=xpdb,VPC=T, cov="DOSE",
cov.fun="mean")
References
[1] Perl Speaks NONMEM (PsN), www.psn.sf.net.
[2] Xpose, www.xpose.sf.net.
Acknowledgement: The research leading to these results has received support from the Innovative
Medicines Initiative Joint Undertaking under grant agreement n° 115156, resources of which are
composed of financial contributions from the European Union's Seventh Framework Programme
(FP7/2007-2013) and EFPIA companies’ in kind contribution. The DDMoRe project is also supported
by financial contribution from Academic and SME partners. This work does not necessarily represent
the view of all DDMoRe partners.
Concentration
1.0
1.0
1.0
1.2
1.4
1.5
1.6
2.0
1.8
Concentration
0.6
0.8
0.5
Figure 4. A KMMC plot of mean dose
for the model and data shown in figures
1-3. Mean dose increases as no/low
dose individuals have events. Indicates
that dose or some other correlated
covariate should be in TTE model.
Mean
Mean
Concentration
1.5
The plot is easily created using the latest versions of PsN [1] and Xpose [2].
Mean
50
100
100
100
150
150
150
200
200
Simulating from a model numerous times will give numerous simulated
KMMC curves, and a VPC of the KMMC can thus be created, allowing for
comparison between model predictions and the true data.
Hazard = f(conc)
(ΔOFV=-39)
0
100
200
300
Time
400
500
0
100
200
300
Time
400
500
0
100
200
300
400
Time
Figure 8. Performance of the KMMC plot. Data is simulated with a
concentration dependent hazard.
Conclusions
• KMMC plots can
- Identify potential covariates that may then be investigated in further
model building.
- Show when a model adequately incorporates covariates into the
model.
- Give hints as to the structure or form the covariate inclusion should
take.
- Handle both continuous and categorical data
- Handle time-varying covariates
- Use other functions instead of the mean to summarize covariates
- Filter out or stratify individuals that are censored in the study so that
covariates influencing both censoring and event can be visualized.
500