1. Estimation of survival functions
under the stochastic order
constraint.
‘Gary’ Thuc Nguyen
University of California, Berkeley

2. Motivational example
• History: Clinical trial conducted by Embury et al. (1977) at Stanford University.
• Aim: evaluate the efficacy of maintenance chemotherapy for the disease
• Method:
• After reaching a state of remission through treatment by chemotherapy, the
patients who entered the study were randomized into two groups.
• first group received maintenance chemotherapy (Maintained) ;
• the second group did not (Nonmaintained).
• Kaplan-Meier estimators used to determine survival probabilities for patients
with Acute Myelogenous Leukaemia (AML) maintenance chemotherapy

3. Overview of Kaplan-Meier estimators
• Definition of Kaplan–Meier estimator: a non-parametric method used to
estimate the survival function from lifetime data.
• Expression of Kaplan-Meier estimation as function of time t
With:
ti = time of death
di = number of deaths at ti
ni = number of individuals alive prior to ti

4. Kaplan-Meier estimators in AML study
• In context: Kaplan-Meier estimators showcase the differences
between 2 groups’ survival probabilities
• Significance: They are example of stochastically ordered
survival curve estimators

5. Kaplan Meier estimators for 2 groups

6. Objective
_ In clinical trial, survival probabilities of one group are observed
to be larger than those of the other (e.g maintained vs non-
maintained in AML trial )
_ Survival functions do not always satisfy this criteria (squamous
carcinoma study) due to small samples (Rojo and Ma (1996))

7. Rojo and Ma (1996))’s Kaplan-Meier plot

8. Objective continued
_ Impose stochastic order constraint to come up with better estimation
for survival curves
_ Compare Bias and MSE of Kaplan- Meier estimators with and without
stochastic order constraint.

9. Simulation steps
• Generate survival times and censoring times for two groups :
• Exponential distribution are used
• x1, ….,xn and C*
1,…., C*
n for S1
• y1 , …,ym and C1,…., Cm for S2
• Incorporate censoring by taking min(survival time, censoring time) pairwise
• Ti
1 = min(xi , c*i ) for group 1
• Ti
2 = min (yi ,ci ) for group 2
• δ*i stands for indicator for censoring for group 1( 1 if Ti = xi , 0 if other)
• δi as indicator group 2 ( 1 if Ti = yi , 0 if other)

10. Simulation steps continued
• Pair up survival times of two groups with its respective indicator
• (T1
i , δ*i ) to come up with Kaplan-Meier estimator for group 1
• (T2
i , δi ) to come up with Kaplan-Meier estimator for group 2
• In the Kaplan-Meier plot, y-axis (survival probability) is partitioned into
equally spaced sections, ui = 0.05, 0.10, …,0.95
 find corresponding 19 ti ‘s on the x-axis of the Kaplan-Meier plot
 Average Bias and Mean squared errors will be evaluated across these 19
points for different estimators

11. Transformation applied
•Stochastically ordered estimators:
• Sˆ*
1 (t) = max(KM1, KM2) and
• Sˆ*
2 (t)= min(KM1, KM2)
•These estimators will be point-wise

12. Pointwise constrained estimators
• These estimators resulted from imposing stochastic ordering
• preserves the relationship between two survival curves (satisfies
stochastic ordering constraint)
• Graphs comparing untransformed Kaplan-Meier estimators provided
• Later, Average Bias and MSE of the estimators are to be presented

13. 1/ Untransformed Kaplan Meiers vs Transformed Kaplan
Meiers

14. 2nd pair

15. 3rd pair

16. Another way of imposing stochastic order
• In the case of high censoring and small sample size, we may only need
to impose stochastic ordering on one curve
(one curve stay constant)
• Plot provided on next slide

18. Different censoring rates
• High censoring rate, low sample size
• Stochastic order Applied to one Curve only
• Plot provided

20. Bias and Mean Squared Error plots
• Average Bias and Mean Squared errors are plotted pair-wise for clear
comparison
• Original Kaplan Meier 1 with transformed Kaplan Meier 1
• Original Kaplan Meier 2 with transformed Kaplan Meier 2
• Plots are produced through 3000 simulations and with
different values of parameters λ1 and λ2

21. Bias plots for transformed and untransformed
Kaplan-Meier estimators (λ1 = 0.25, λ2 =0.5)

22. Bias plot for S1 and S2 (λ1 = 0.55, λ2 =0.6)

23. MSE plots with λ1 = 0.25, λ2 =0.5

24. MSE plots (λ1 = 0.55, λ2 =0.6)

25. Results
BIAS
• First group(S1): Original Kaplan Meier 1 estimator fair well
with transformed one for λ1 = 0.25, λ2 =0.5, but becomes
inferior when λ1 = 0.55, λ2 =0.6
• Second group (S2): Transformed estimators (stochastically
ordered) yield average biases significantly close to 0 for
both sets of chosen λ1 and λ2

26. Results continued
MEAN SQUARED ERROR
• First group (S1): all erratic for λ1 = 0.25, λ2 =0.5, comparable for λ1 =
0.55, λ2 =0.6,
• Second group (S2):
_ Original Kaplan-Meier 2 yield almost the same MSE with
transformed one for λ1 = 0.25, λ2 =0.5
_transformed Kaplan-Meier 2 yield MSE’s much closer to 0 for λ1
= 0.55, λ2 =0.6,

27. Conclusion
• Stochastic ordering is useful for estimating survival curves, provided
that we know for sure the behavior of one curve to the other
• Stochastic ordering should not be confused with manipulation of data
• Censoring time can affect Stochastic ordered estimators in terms of
estimating Bias and MSE

28. Acknowledgement
• This research was supported by The National Security Agency
through Grant H98230-15-1-0048 to The University of Nevada at
Reno, Javier Rojo PI.
• Thank you for listening to the presentation of my research project.
• I am thankful for the opportunity to conduct research at RUSIS 2015
this summer