Estimation of survival functions under the stochastic order
constraint.‘Gary’ Thuc Nguyen
University of California, Berkeley
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
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 tini = number of individuals alive prior to ti
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
Kaplan Meier estimators for 2 groups
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))
Rojo and Ma (1996))’s Kaplan-Meier plot
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.
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
• Ti1 = min(xi , c*i ) for group 1
• Ti2 = 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)
Simulation steps continued• Pair up survival times of two groups with its respective indicator
• (T1i , δ*i ) to come up with Kaplan-Meier estimator for group 1
• (T2i , 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
Transformation applied
• Stochastically ordered estimators:
• Sˆ*1 (t) = max(KM1, KM2) and
• Sˆ*2 (t)= min(KM1, KM2)
• These estimators will be point-wise
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
1/ Untransformed Kaplan Meiers vs Transformed Kaplan Meiers
2nd pair
3rd pair
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
Different censoring rates
• High censoring rate, low sample size
• Stochastic order Applied to one Curve only
• Plot provided
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
Bias plots for transformed and untransformed Kaplan-Meier estimators (λ1 = 0.25, λ2 =0.5)
Bias plot for S1 and S2 (λ1 = 0.55, λ2 =0.6)
MSE plots with λ1 = 0.25, λ2 =0.5
MSE plots (λ1 = 0.55, λ2 =0.6)
ResultsBIAS
• 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
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,
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
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