Introduction to the win-ratio

The win-ratio is a statistical methodology for comparing two groups when considering multiple endpoints. The win-ratio is calculated by the number of wins divided by the number of losses of a group. For instance, in a clinical trial with a treatment arm and a control arm, one would calculate the win-ratio of the treatment arm as the number of times treatment “won” over control divided by the number of times treatment “lost” to control. For example, if the win-ratio was 1.5 this would indicate that the treatment group had 50% more wins than the control group. A “win” or a “loss” is determined in pairs of patients for instance, patient T vs. patient C. In this example there will be two outcomes of interest, 1. Death and 2. Metastasis; where death is the first priority and metastasis is the second. Patient T is in the treatment group, they did not die during the study but they did develop metastasis. Patient C is in the control group, they died during the study and also developed metastasis prior to death. Patient T will “win” over patient C (“loss”) as they did not die during the study and death is the 1st priority outcome. If metastasis was the first priority and death second; patient T and patient C would tie on metastasis and the second priority death would have to be consider. As patient C died and patient T did not, patient T would “win” here also.

The win-ratio is a particularly useful method for clinical trials as it allows for the determination of the holistic effect of a treatment while considering the hierarchy of all endpoints of interest at once.

Sample size calculation

An important aspect of a clinical trial design is the sample size determination. It is necessary to calculate the required sample size to ensure the clinical trial has sufficient statistical power.

The WRestimates package contains the function wr.ss() which allows for the calculation of a sample size. For example, in a study of hormone therapy vs. placebo treatment for prostate cancer, the study designers deem a win-ratio of 1.25 to be clinically significant (25% more wins for the hormone therapy over the placebo), patients will be allocated in 1:1 and the expected number of tied pairs is 10%.

wr.ss(WR.true = 1.25, k = 0.5, p.tie = 0.1)
#> $N
#> [1] 1376
#> 
#> $alpha
#> [1] 0.025
#> 
#> $beta
#> [1] 0.1
#> 
#> $WR.true
#> [1] 1.25
#> 
#> $sigma.sqr
#> [1] 6.518519
#> 
#> $k
#> [1] 0.5
#> 
#> $p.tie
#> [1] 0.1

As the power and level of significance (alpha) and beta (1 - power) were not specified they are assumed to be the default; defined as follows:

• alpha = 0.025; equivalent to a 5% level of significance in a two-sided test
• beta = 0.1; equivalent to a power of 90%

The required sample size is calculated to be 1376.

Power calculation

In some circumstances a sample size may be restricted. This may be due to economic or ethical considerations or simply due to the population of interest being rare.

In these circumstances, rather than calculating a sample size, the power of the study is determined based on a set sample size.

The WRestimates package contains the function wr.power() which allows for the calculation of power. For example, in a study of hormone therapy vs. placebo treatment for prostate cancer, the study designers deem a win-ratio of 1.25 to be clinically significant (25% more wins for the hormone therapy over the placebo), there will be a sample size of 1000 patients who will be allocated in 1:1 and the expected number of tied pairs is 10%.

wr.power(N = 1000, WR.true = 1.25, k = 0.5, p.tie = 0.1)
#> $power
#> [1] 0.7892602
#> 
#> $N
#> [1] 1000
#> 
#> $alpha
#> [1] 0.025
#> 
#> $WR.true
#> [1] 1.25
#> 
#> $sigma.sqr
#> [1] 6.518519
#> 
#> $k
#> [1] 0.5
#> 
#> $p.tie
#> [1] 0.1

As the level of significance (alpha) was not specified it is assumed to be the default; defined as follows:

• alpha = 0.025; equivalent to a 5% level of significance in a two-sided test

The power is calculated to be 0.789.

Confidence Interval calculation

Confidence intervals are important because they allow us to:

• Assess how precise our estimates are.
• Determine the clinical significance of our results.

The WRestimates package contains the function wr.ci() which allows for the calculation of the confidence interval (CI) from summary statistics. For example, in a study of hormone therapy vs. placebo treatment for prostate cancer, the win-ratio is calculated to be of 1.22 (22% more wins for the hormone therapy over the placebo), the sample size is 1200 patients allocated in 1:1, and 4% of pairs are tied. The 95% confidence interval of this win ratio is:

wr.ci(WR = 1.22, Z = 1.96, N = 1200, k = 0.5, p.tie = 0.04)
#> $ci
#> lower.CI upper.CI 
#> 1.064866 1.397735 
#> 
#> $WR
#> [1] 1.22
#> 
#> $Z
#> [1] 1.96
#> 
#> $var.ln.WR
#> [1] 0.004814815
#> 
#> $N
#> [1] 1200
#> 
#> $sigma.sqr
#> [1] 5.777778
#> 
#> $k
#> [1] 0.5
#> 
#> $p.tie
#> [1] 0.04

As the Z-score (Z) was not specified it is assumed to be the default; defined as follows:

• Z = 1.96; equivalent to a 5% level of significance in a two-sided test, or 95% CI.

The confidence interval is calculated to be (1.065, 1.398).