Win Ratio
By
Wei Yann Tsai
Department of Biostatistics, Columbia U.
Department of Statistics, National Cheng Kung U.
2/20/2014
Outline



INTRODUCTION
METHOD
SIMULATION
INTRODUCTION


Cancer Clinical Trials
Primary End Points
Over All Survival
Progression Free Survival
INTRODUCTION



Cancer Clinical Trials
Primary End Point
Over All Survival
Progression Free Survival
Secondary End Points
Response Rate
Duration of Response
INTRODUCTION




Cancer Clinical Trials
Primary End Point
Over All Survival
Progression Free Survival
Secondary End Points
Response Rate
Duration of Response
Probability of Being in Response Function (PBRF)
PBRF(t)=Prob (Patient response before time t and
still alive at time t)
INTRODUCTION



Cardiovascular (CV) Trials
Primary End Points
CV Death
Myocardial Infarction,
Stroke
Secondary End Points
Hospitalization Rate
Duration of Hospitalization




Composite Endpoints
Non-Fatal Events, Fatal Events
Traditional analysis of composite endpoints only
evaluates “time to first event”
An Informative Censoring Scheme:
 Semi-Competing Risks Data
Win Ratio
Pocock et al. (2012)

Matched Treatment and Control Trials
 “winner” or “loser” of new treatment is
determined by comparing “time to component
events” sequentially according to their order of
the clinical priorities
 Win Ratio= total number of Winner/ Total
number of Loser
Win Ratio
Pocock et al. (2012)

UnMatched Treatment and Control Trials
 Subjects from treatment arm are matched to
every subject from Control arm.
 Win Ratio= total number of Winner/ Total
number of Loser
METHOD

Formulation of the statistical test
 Let TH and TD be two random variables denoting time
to hospitalization and time to death.
 These two variables are usually correlated to each
other.
 TD can right-censor TH but not vice versa.
 Let Z = 1 denote the new treatment group and Z = 0
the control group.
 TC is a random variable for censoring, which is
assumed to be independent of (TH; TD) given Z but
can censor both TH and TD.
Y1 = min(TH; TD; TC)
Y2 = min(TD; TC).
I1 be the corresponding event indicator for Y1
I2 be the corresponding event indicator for Y2







(a) patient in new treatment arm has death first
Na
(b) patient in standard treatment arm has death first
Nb
(c) patient in new treatment arm has hospitalization first
Nc
(d) patient in control arm has hospitalization first
Nd
(e) none of the above
Win Ratio:
Win Difference :
WR = (Nb+Nd)/(Na+Nc)
WD= (Nb+Nd)-(Na+Nc)
METHOD

Null Hypothesis
 H01: hD1(y2) = hD0(y2) for all y2
 H02: hH1(y1|y2) = hH0(y1|y2) for all y1,y2
 H0: H01 and H02
 hDk(y2) = pr(TD = y2 |TD >= y2;Z = k)
 hHk(y1|y2) = pr(TH = y1 | TH >= y1;TD >= y2;Z = k)
 Nb-Na is to test against H01
 Nd-Nc is to test against H02
METHOD

Variance Estimation
 The variance of WD can be calculated by using the
technique of U-statistics, and a consistent estimator
of the variance can be constructed using Hajek’s
projection method.
Discussion and Future Research

A Proportional Hazards Model
 hD(y2|z) = exp(bDz)hD(y2) for all y2
hH(y1|y2,z) = exp(bHz)hH0(y1|y2) for all y1,y2
 bD=bH=b then WR=exp(-b)
 Win Product WP=(Nb/Na)x(Nd/Nc)=exp(-bD-bH)

(Weighted) Log Rank Type Testing Statistics and
Estimation