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.

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.

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 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).