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