Introduction to the R Package
WINS
A Brief Overview
An inherent limitation of the commonly used composite endpoint approach to analyze prioritized multiple outcomes in clinical trials is the convention to investigate only the event that occurs first. However, the “first-occurred” event is often of less clinical importance. To overcome this limitation, the win statistics (win ratio (Pocock et al., 2012), net benefit (Buyse, 2010) and win odds (Dong et al., 2020)), a class of generalized pairwise comparison methods (Buyse, 2010), were introduced in the literature to analyze prioritized multiple endpoints to determine wins or losses based on the hierarchical order of clinical importance. Specifically, this method compares each subject in the experimental arm with every subject in the control arm by the most important endpoint first, and evaluates lower-priority endpoints if and only if higher-priority endpoints can not determine a win (i.e., a tie) for each pairwise comparison. The win statistics, in particular the win ratio and net benefit, have recently received much attention in methodological research as well as in designs and analyses of Phase III clinical trials. The pioneering work of Finkelstein and Schoenfeld (1999) is closely related to the win statistics.
This document introduces the basic usage of the R package WINS. With the main function win.stat, one can calculate the win statistics and make statistical inference on prioritized multiple endpoints. Although recent methodology research has focused more on time-to-event endpoints, this function can deal with various type of outcomes (time-to-event, continuous, binary, ordinal, or a mixture of them). One can use the functions stat_t.plot, and partition_t.plot to plot the win statistics and win proportions over study time, respectively. A simulation function sim.data can be used to simulate data of multiple endpoints to evaluate the operating characteristics of the methods.
Introduction of the Win Statistics
In this section, we will provide a brief introduction of the win statistics. Denote Pt as the win probability/proportion in the treatment group, and Pc as the win probability/proportion in the control group. The question of interest can then be formulated as testing the null hypothesis H0 : Pt = Pc versus its alternative. The win statistics are defined based on Pt and Pc as
- Win ratio: WR = Pt/Pc.
- Net benefit: NB = Pt − Pc.
- Win odds: $WO = \frac{P_t+0.5(1-P_t-P_c)}{P_c+0.5(1-P_t-P_c)}$.
To obtain the win statistics for a composite of J prioritized outcomes, the counting approach is the most straightforward and intuitive especially when J > 2. The basic idea is to compare each subject in the treatment group with every subject in the control group (a total of NtNc pairs, where Nt and Nc are the number of subjects in the treatment group and the control group, respectively.). For each pair of subjects, the comparison starts from the most important outcome. The next lower-priority outcome is considered only if the higher-priority outcome results in a tie (i.e., a winner cannot be determined).
First we introduce some necessary notations. Suppose all the J outcomes are time-to-event endpoints. For the treatment group, the observed data are noted as Y = {Y1(t), …, YJ(t)} and the event status δ = {δ1(t), …, δJ(t)} with δj(t) = 1 as the event and δj(t) = 0 as censoring. Continuous/binary endpoints can be viewed as a special case of the time-to-event outcome with δj(t) = 1 for all the subjects. The notations are the same for the control group except that the superscript is (c).
Consider a randomized clinical trial with Nt and Nc subjects in the treatment group and the control group, respectively. We use i = 1, …, Nt for subjects in the treatment group and l = 1, …, Nc for subjects in the Control group. Using the counting approach, we define the kernel functions K and L as follows: Kil = 1 if subject i in the treatment group wins over subject l in the control group, otherwise Kil = 0; and Lil = 1 if subject l wins over subject i, otherwise Lil = 0. Following the counting approach, we then define the win proportion Pt and Pc as $$ P_t = \frac{\sum_{i=1}^{N_t}\sum_{l=1}^{N_c}K_{il}}{N_tN_c} \text{ and } P_c = \frac{\sum_{i=1}^{N_t}\sum_{l=1}^{N_c}L_{il}}{N_tN_c}, $$ where Kil and Lil are the kernel functions as defined above.
Since the pioneering work by Pocock et.al (2012), the win statistics have received much attention in methodological research. Methods have been developed to correct for bias in the presence of censoring and to extend the use to stratified analysis. For example, Dong et.al (2018, 2023) considered a clinical trial with patients randomized into two groups with M strata and proposed the stratified win ratio in a similar way as the Mantel-Haenszel stratified odds ratio. To estimate the win proportions and win statistics in the presence of independent or dependent censoring, Dong et.al (2020, 2021) introduced the inverse-probability-of-censoring weighting (IPCW) adjusted method and CovIPCW-adjusted method using baseline covariates and/or time-dependent covariates that can predict the dependent censoring. Moreover, Wang et.al (2023) proposes an adjusted win ratio to control for baseline imbalances in covariates by inverse probability of treatment weighting (IPTW) method.
In Pocock et.al (2012), the confidence interval was constructed based on bootstrap resampling, and the hypothesis testing was based on the non-parametric method. To provide a more efficient inference procedure, Luo et al. (2015) developed a close-form variance estimator for the win ratio and the win difference only for cases with two time-to-event endpoints. Later on, Dong et al. (2016) and Bebu and Lachin (2016) developed a more generalized close-form variance estimator for the win statistics which has the flexibility to allow for any number of endpoints. Moreover, Dong et al. (2023) studied the relationship among the three estimated win statistics (WR, WO and NB) and justify that WR, WO, and NB can complement one another to show the strength of the treatment effect. Meanwhile, though it should be enough to use and present only one of WR, WO, or NB for clinical trial design and analysis, presenting all together can give a more detailed picture of an analysis.
In the next sections, we will provide some simulated examples to
illustrate the use of our R package WINS to analyze
prioritized multiple outcomes.
Estimate Win Statistics using win.stat
In this section, we show how the main function win.stat works. We
first provide three examples with three build-in simulated datasets in
this R package. The number of endpoints in all these three example
datasets are 3. Priority order of multiple endpoints can be specified in
the option priority. For example, suppose we input endpoint
1, 2 and 3 as Y_1, Y_2 and Y_3,
respectively. priority=c(2,1,3) specifies that
Y_2 is the most important and Y_3 is the least
important.
An example with three binary endpoints
We consider the values 1 vs 0 as the default setting for binary variables since oftentimes the value 1 represents response to the drug, and the default win strategy is that the value 1 wins over the value 0.
The example dataset data_binary contains 3 binary
outcomes.
head(data_binary)
#> id arm Y_1 Y_2 Y_3
#> 1 1 A 0 0 0
#> 2 2 A 0 0 1
#> 3 3 A 1 1 0
#> 4 4 A 1 1 0
#> 5 5 A 1 1 1
#> 6 6 A 0 0 0
res_binary <- win.stat(data = data_binary, ep_type = "binary", priority = c(1:3),
stratum.weight = "unstratified", arm.name = c("A","B"), alpha=0.05,
digit = 3,pvalue = "two-sided")
#> The outcome type for all the endpoints: binary
#> This analysis is unstratified.
#>
#> Win Ratio : 2.299
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.511
#> Upper limit of 95 % CI of the win ratio: 3.497
#>
#>
#> Net Benefit : 0.278
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.130
#> Upper limit of 95 % CI of the net benefit: 0.426
#>
#>
#> Win Odds : 1.769
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.316
#> Upper limit of 95 % CI of the win odds: 2.379
#> An example with three continuous endpoints
The default win strategy for continuous outcomes is “the larger value wins”.
The example code is presented below.
head(data_continuous)
#> id arm Y_1 Y_2 Y_3
#> 1 1 A 0.4621177 0.2069386 -0.52627613
#> 2 2 A 0.3258938 0.2595162 -0.11247766
#> 3 3 A 2.4887816 0.5406375 -0.34632514
#> 4 4 A 3.9419121 0.4956837 -0.44683113
#> 5 5 A 2.3304126 0.5481473 -0.02682927
#> 6 6 A 0.3168587 0.1186624 -1.85656323
res_continuous <- win.stat(data = data_continuous, ep_type = "continuous",
arm.name = c("A","B"),tau = 0, priority = c(1:3),
stratum.weight = "unstratified", alpha=0.05, digit = 3,
pvalue = "two-sided")
#> The outcome type for all the endpoints: continuous
#> This analysis is unstratified.
#>
#> Win Ratio : 1.695
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.241
#> Upper limit of 95 % CI of the win ratio: 2.316
#>
#>
#> Net Benefit : 0.258
#>
#> two-sided p-value is: 0.001
#> Lower limit of 95 % CI of the net benefit: 0.102
#> Upper limit of 95 % CI of the net benefit: 0.414
#>
#>
#> Win Odds : 1.695
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.241
#> Upper limit of 95 % CI of the win odds: 2.316
#> Note that in this example, the WR = WO because there are no ties for continuous outcomes.
An example with three time-to-event endpoints
For time-to-event (TTE) outcomes, the adjustment for censoring can be
specified in the option method.
For illustration, we first calculate the win statistics without using the IPCW approach to dealing with the censoring. The example code is presented below.
head(data_tte)
#> id arm Delta_1 Delta_2 Delta_3 Y_1 Y_2 Y_3
#> 1 1 A 0 0 0 0.1268054 2.9063267 0.2605294
#> 2 2 A 0 1 1 1.0137019 1.7945845 1.5220050
#> 3 3 A 1 0 0 1.4323415 1.5009229 0.6054241
#> 4 4 A 1 1 0 1.6961593 5.1323268 1.9331923
#> 5 5 A 1 0 1 3.5786849 0.4329275 0.6391820
#> 6 6 A 1 1 0 0.5615477 8.2259638 0.5401791
### Without using the IPCW approach to dealing with the censoring
res_tte <- win.stat(data = data_tte, ep_type = "tte", arm.name = c("A","B"),
tau = 0.1, priority = c(1:3), alpha = 0.05, digit = 3,
stratum.weight = "unstratified", method = "unadjusted",
pvalue = "two-sided")
#> The outcome type for all the endpoints: tte
#> This analysis is unstratified.
#>
#> Win Ratio : 2.320
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.805
#> Upper limit of 95 % CI of the win ratio: 2.983
#>
#>
#> Net Benefit : 0.349
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.239
#> Upper limit of 95 % CI of the net benefit: 0.459
#>
#>
#> Win Odds : 2.071
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.661
#> Upper limit of 95 % CI of the win odds: 2.581
#> When censoring is assumed to be independent of the outcomes, we
provide the option method = "ipcw" to calculate
IPCW-adjusted win statistics (Dong et al., 2020). The example code is
presented below.
### IPCW adjustment for independent censoring
res_tte_ipcw <- win.stat(data = data_tte, ep_type = "tte", arm.name = c("A","B"),
tau = 0.1, priority = c(1:3), alpha = 0.05, digit = 3,
stratum.weight = "unstratified", method = "ipcw",
pvalue = "two-sided")
#> The outcome type for all the endpoints: tte
#> This analysis is unstratified.
#>
#> Win Ratio : 2.287
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.796
#> Upper limit of 95 % CI of the win ratio: 2.912
#>
#>
#> Net Benefit : 0.389
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.269
#> Upper limit of 95 % CI of the net benefit: 0.509
#>
#>
#> Win Odds : 2.275
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.789
#> Upper limit of 95 % CI of the win odds: 2.893
#> When censoring is suspected to be dependent on baseline or time
dependent covariates which correlate with the outcomes, we provide the
option method = "covipcw" to calculate CovIPCW-adjusted win
statistics (Dong et al., 2021). The example code is presented below.
head(Z_t_trt)
#> id time Z1 Z2
#> 1 1 0.0 0.9257546 0.74457231
#> 2 2 0.0 0.1970530 0.56999281
#> 3 2 0.2 0.7914286 0.65883216
#> 4 2 0.4 0.2435229 0.01919255
#> 5 2 0.6 0.3393664 0.69654040
#> 6 2 0.8 0.1932459 0.98218826
### CovIPCW adjustment for dependent censoring
res_tte_covipcw <- win.stat(data = data_tte, ep_type = "tte", tau = 0.1,
arm.name = c("A","B"), stratum.weight = "unstratified",
Z_t_trt = Z_t_trt, Z_t_con = Z_t_con,
priority = c(1:3), alpha = 0.05, digit = 3,
method = "covipcw", pvalue = "two-sided")
#> The outcome type for all the endpoints: tte
#> This analysis is unstratified.
#>
#> Win Ratio : 2.254
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.779
#> Upper limit of 95 % CI of the win ratio: 2.855
#>
#>
#> Net Benefit : 0.385
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.267
#> Upper limit of 95 % CI of the net benefit: 0.504
#>
#>
#> Win Odds : 2.254
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.779
#> Upper limit of 95 % CI of the win odds: 2.855
#> In the last two examples, there is WR = WO because we adjusted the win proportions so that the tie proportion is not negative (i.e. equal to zero after adjustment) after IPCW adjustment.
These three examples presented above show the basic usage of the R
function win.stat with various types of endpoints. In the
next example, we provide a more complex data example.
An example with a mixture of endpoint types: two continuous and one TTE
This example is for a simulated dataset with three endpoints as a mixture of two continuous endpoints and one TTE endpoint.
The example code is presented below.
head(data_mix)
#> id arm Delta_1 Y_1 Y_2 Y_3
#> 1 1 A 0 0.7252257 0.5557491 0.3580795
#> 2 2 A 1 0.4242003 0.1810918 -1.4360364
#> 3 3 A 1 0.6900970 0.2399882 -1.8876155
#> 4 4 A 0 2.8391707 0.7336988 0.7722108
#> 5 5 A 0 0.6134876 0.3533288 -0.2674180
#> 6 6 A 1 0.4434930 0.2379743 -1.2457559
res_mix <- win.stat(data = data_mix, ep_type = c("tte","continuous","continuous"),
arm.name = c("A","B"), tau = 0.1, priority = c(1:3),
alpha = 0.05, digit = 3, method = "unadjusted",
stratum.weight = "unstratified", pvalue = "two-sided")
#> This analysis is unstratified.
#>
#> Win Ratio : 2.274
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.761
#> Upper limit of 95 % CI of the win ratio: 2.937
#>
#>
#> Net Benefit : 0.384
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.258
#> Upper limit of 95 % CI of the net benefit: 0.510
#>
#>
#> Win Odds : 2.245
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.745
#> Upper limit of 95 % CI of the win odds: 2.889
#> When there is at least one TTE endpoints among all, the option censoring_adjust to calculate IPCW-adjusted and CovIPCW-adjusted win statistic is available.
The example code is presented below.
### IPCW adjustment for independent censoring
res_mix_ipcw <- win.stat(data = data_mix,
ep_type = c("tte","continuous","continuous"),
arm.name = c("A","B"), tau = 0.1, priority = c(1:3),
alpha = 0.05, digit = 3, method = "ipcw",
stratum.weight="unstratified", pvalue = "two-sided")
#> This analysis is unstratified.
#>
#> Win Ratio : 2.274
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.761
#> Upper limit of 95 % CI of the win ratio: 2.937
#>
#>
#> Net Benefit : 0.384
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.258
#> Upper limit of 95 % CI of the net benefit: 0.510
#>
#>
#> Win Odds : 2.245
#>
#> two-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.745
#> Upper limit of 95 % CI of the win odds: 2.889
#> An example with a mixture of endpoint types: two continuous and one TTE with three strata
In this example, besides a mixture of endpoint types, there are three strata in the data. By default, the stratified win statistics proposed in Dong et al. (2018) are calculated.
Let Nt(m) and Nc(m) denote the number of subjects in the treatment group and the control group of the mth stratum, respectively, and let N(m) = Nt(m) + Nc(m) denote the total number of subjects in the mth stratum for m = 1, …, M. Within each stratum, define the kernal functions Kil(m)(Yi(m), δi(m), Yl(m), δl(m)) and Lil(m)(Yi(m), δi(m), Yl(m), δl(m)) in a similar way as the case without stratification. The stratified win ratio is then defined as $$ WR_{MH} = \frac{\sum_{m=1}^M w^{(m)}n_t^{(m)}}{\sum_{m=1}^M w^{(m)}n_c^{(m)}}, $$ where $n_t^{(m)} = \sum_{i=1}^{N_t^{(m)}}\sum_{l=1}^{N_c^{(m)}}K_{il}^{(m)}$, $n_c^{(m)} = \sum_{i=1}^{N_t^{(m)}}\sum_{l=1}^{N_c^{(m)}}L_{il}^{(m)}$, and {w(m), m = 1, …, M} represent the weight assigned to the M strata.
Firstly, we provide the option to assign equal weights for all the strata, e.g., $w(m)=\frac{1}{M}$. The example code is presented below.
res_mix_equal <- win.stat(data = data_mix_stratum,
ep_type = c("tte","continuous","continuous"),
arm.name = c("A","B"), tau = 0.1, priority = c(1:3),
alpha = 0.05, digit = 3, method = "unadjusted",
stratum.weight = "equal", pvalue = "one-sided")
#>
#> Win Ratio : 2.296
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.776
#> Upper limit of 95 % CI of the win ratio: 2.966
#>
#>
#> Net Benefit : 0.388
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.261
#> Upper limit of 95 % CI of the net benefit: 0.514
#>
#>
#> Win Odds : 2.266
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.760
#> Upper limit of 95 % CI of the win odds: 2.917
#> Then we consider the option to assign the weight for the m-th stratum as w(m) = 1/N(m). The example code is presented below.
head(data_mix_stratum)
#> id arm stratum Delta_1 Y_1 Y_2 Y_3
#> 1 1 A 2 0 0.7252257 0.5557491 0.3580795
#> 2 2 A 3 1 0.4242003 0.1810918 -1.4360364
#> 3 3 A 2 1 0.6900970 0.2399882 -1.8876155
#> 4 4 A 2 0 2.8391707 0.7336988 0.7722108
#> 5 5 A 1 0 0.6134876 0.3533288 -0.2674180
#> 6 6 A 3 1 0.4434930 0.2379743 -1.2457559
res_mix_MH <- win.stat(data = data_mix_stratum,
ep_type = c("tte","continuous","continuous"),
arm.name = c("A","B"), tau = 0.1, priority = c(1:3),
alpha = 0.05, digit = 3, method = "unadjusted",
stratum.weight = "MH-type", pvalue = "one-sided")
#>
#> Win Ratio : 2.294
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.776
#> Upper limit of 95 % CI of the win ratio: 2.964
#>
#>
#> Net Benefit : 0.387
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.261
#> Upper limit of 95 % CI of the net benefit: 0.514
#>
#>
#> Win Odds : 2.265
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.759
#> Upper limit of 95 % CI of the win odds: 2.915
#> In the above example, the stratified win ratio is proposed in a similar way as the Mantel-Haenszel stratified odds ratio. Specifically, given M stratum, the stratum-specific wins are weighted to estimate the stratified win ratio.
We then provide another option to define the stratified win ratio, i.e., the weight is applied directly to the stratum-specific win statistics with the number of subjects. As an example, the stratified win ratio is defined as $$ WR_{wt,1} = \sum_{m=1}^M w_1^{(m)}WR^{(m)} = \sum_{m=1}^M \frac{w_1^{(m)}n_t^{(m)}}{n_c^{(m)}} $$ with $w_1^{(m)} =\frac{N^{(m)}}{\sum_{m=1}^M N^{(m)}}$.
The example code is presented below.
res_mix_wt1 <- win.stat(data = data_mix_stratum,
ep_type = c("tte","continuous","continuous"),
arm.name = c("A","B"), tau = 0.1, priority = c(1:3),
alpha = 0.05, digit = 3, method = "unadjusted",
stratum.weight = "wt.stratum1", pvalue = "one-sided")
#>
#> Win Ratio : 2.299
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 2.057
#> Upper limit of 95 % CI of the win ratio: 2.570
#>
#>
#> Net Benefit : 0.387
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.261
#> Upper limit of 95 % CI of the net benefit: 0.514
#>
#>
#> Win Odds : 2.270
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 2.031
#> Upper limit of 95 % CI of the win odds: 2.537
#> When there exist at least one TTE endpoints, we further provide a third option to set weight directly to the win statistics in each stratum relative to the number of events, i.e., the stratified win ratio is defined as $$ WR_{wt,2} = \sum_{m=1}^M w_2^{(m)}WR^{(m)} = \sum_{m=1}^M \frac{w_2^{(m)}n_t^{(m)}}{n_c^{(m)}} $$ with $w_2^{(m)} =\frac{N_{event}^{(m)}}{\sum_{m=1}^M N_{event}^{(m)}}$.
The example code is presented below.
res_mix_wt2 <- win.stat(data = data_mix_stratum,
ep_type = c("tte","continuous","continuous"),
arm.name = c("A","B"), tau = 0.1, priority = c(1:3),
alpha = 0.05, digit = 3, method = "unadjusted",
stratum.weight = "wt.stratum2", pvalue = "one-sided")
#>
#> Win Ratio : 2.303
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 2.060
#> Upper limit of 95 % CI of the win ratio: 2.574
#>
#>
#> Net Benefit : 0.388
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.261
#> Upper limit of 95 % CI of the net benefit: 0.515
#>
#>
#> Win Odds : 2.273
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 2.034
#> Upper limit of 95 % CI of the win odds: 2.541
#> An example to conduct IPTW-adjusted win statistics analysis
In this example, the IPTW-adjusted win statistics proposed in Wang et al. (2023) are calculated.
individual.weight <- rep(1,nrow(data_tte))
res_iptw <- win.stat(data = data_tte,
ep_type = c("tte","tte","tte"), iptw.weight = individual.weight,
arm.name = c("A","B"), tau = 0, priority = c(1:3),
alpha = 0.05, digit = 3, method = "iptw",
stratum.weight = "equal", pvalue = "one-sided")
#> This analysis is unstratified.
#>
#> Win Ratio : 2.196
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.716
#> Upper limit of 95 % CI of the win ratio: 2.811
#>
#>
#> Net Benefit : 0.337
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.226
#> Upper limit of 95 % CI of the net benefit: 0.448
#>
#>
#> Win Odds : 2.015
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.614
#> Upper limit of 95 % CI of the win odds: 2.515
#> Data simulation with the function sim.data
In this section, we provide two examples to show how to use
sim.data to simulate data. We provide two options. One is
based on exponential distributions, the other simulates correlated
endpoints using the existing R package copula. The accrual
time is assumed to follow a certain distribution, for example, uniform
distribution.
The first option focuses on generating two TTE endpoints Y1 and Y2 with the more important TTE endpoint Y2 expected to occur later. Also, the two endpoints are in a semi-competing risk setting with Y2 as the terminal endpoint (i.e., Y1 cannot occur later than Y2). A real example for those two endpoints is the composite of progression (Y1) and death (Y2) in oncology clinical trials. If death is the event that happened first, then Y1 is censored by death; while the reverse is not true.
The example code is presented below.
#### Generate two TTE endpoints with the more important TTE endpoint expected to occur
#### later with exponential distributions.
sim.data_tte <- sim.data(n_trt = 150, n_con = 100, n_ep = 2,
arm.name = c("A","B"), ep_type = c("tte","tte"),
cdist.rate = 0.5, sim_method = "tte_exponential",
rate_trt = c(0.3,0.2), rate_con = c(0.5,0.4),
max_accrual_time = 5)
win_stat_sim_tte <- win.stat(data = sim.data_tte, ep_type = c("tte","tte"),
arm.name = c("A","B"), digit = 3, priority = c(2,1))
#> This analysis is unstratified.
#>
#> Win Ratio : 1.893
#>
#> one-sided p-value is: 0.001
#> Lower limit of 95 % CI of the win ratio: 1.251
#> Upper limit of 95 % CI of the win ratio: 2.866
#>
#>
#> Net Benefit : 0.161
#>
#> one-sided p-value is: 0.002
#> Lower limit of 95 % CI of the net benefit: 0.053
#> Upper limit of 95 % CI of the net benefit: 0.269
#>
#>
#> Win Odds : 1.383
#>
#> one-sided p-value is: 0.002
#> Lower limit of 95 % CI of the win odds: 1.115
#> Upper limit of 95 % CI of the win odds: 1.717
#> The second option is to generate the data with the R package
copula. This option works for any number of endpoints and
is able to simulate the correlation between the endpoints, however there
will not be competing risk among the simulated endpoints.
The below is an example with three endpoints, noted as Y1, Y2, and Y3, with endpoint type as TTE, TTE and continuous. For both the treatment group and the control group, the correlation coefficients cor(Y1, Y2), cor(Y1, Y3) and cor(Y2, Y3) are 0.9, 0.8 and 0.95, respectively. For each treatment group, the marginal distribution for Y1, Y2, and Y3 are Gamma, Beta and Student t specified as a vector in “margins_trt”/“margins_con”. The parameters are specified as a list corresponding to the marginal distributions in “paramMargins_trt” or “paramMargins_con”.
The example code is presented below.
sim.data <- sim.data(n_trt = 150, n_con = 100, n_ep = 3, arm.name = c("A","B"),
ep_type = c("tte","tte","continuous"),
cdist.rate = 0.5, sim_method = "copula",
copula_trt=copula::normalCopula(param=c(0.9,0.8,0.95), dim = 3,
dispstr = "un"),
margins_trt=c("gamma", "beta", "t"),
paramMargins_trt=list(list(shape=2, scale=1),
list(shape1=2, shape2=2), list(df=5)),
copula_con=copula::normalCopula(param=c(0.9,0.8,0.95), dim = 3,
dispstr = "un"),
margins_con=c("gamma", "beta", "t"),
paramMargins_con=list(list(shape=1, scale=1),
list(shape1=1, shape2=2), list(df=2)),
max_accrual_time = 5)
win_stat <- win.stat(data = sim.data, ep_type = c("tte","tte","continuous"),
arm.name = c("A","B"), digit = 3, priority = c(1,2,3))
#> This analysis is unstratified.
#>
#> Win Ratio : 2.333
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win ratio: 1.671
#> Upper limit of 95 % CI of the win ratio: 3.256
#>
#>
#> Net Benefit : 0.400
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the net benefit: 0.233
#> Upper limit of 95 % CI of the net benefit: 0.567
#>
#>
#> Win Odds : 2.333
#>
#> one-sided p-value is: < 0.001
#> Lower limit of 95 % CI of the win odds: 1.671
#> Upper limit of 95 % CI of the win odds: 3.256
#> Plot the Win Statistics/Win Proportions over Time
Win statistics and win proportions are dependent on follow-up time
(e.g., Oakes, 2016), and they can be graphically presented (Finkelstein
and Schoenfeld, 2019). In this section, we show the usage of the two
plot functions stat_t.plot and
partition_t.plot for TTE endpoints only. The two functions
are able to plot the win statistics of stated with its point-wise 95%
confidence interval, as well as the win proportion in each treatment
group over study time. If the study entry time for each individual is
not provided in the dataset, then by default all the individuals are
assumed to enter the study at the same time.
The example code is presented below.
#### An simulated example with three TTE endpoints.
data <- sim.data(n_trt = 200, n_con = 200, n_ep = 3, arm.name = c("A","B"),
ep_type = "tte", cdist.rate = 1, sim_method = "copula",
copula_trt=copula::normalCopula(param=c(0.9,0.8,0.95), dim = 3,
dispstr = "un"),
margins_trt=c("gamma", "beta", "gamma"),
paramMargins_trt=list(list(shape=2, scale=2),
list(shape1=2, shape2=2),
list(shape=2, scale=3)),
copula_con=copula::normalCopula(param=c(0.9,0.8,0.95), dim = 3,
dispstr = "un"),
margins_con=c("gamma", "beta", "gamma"),
paramMargins_con=list(list(shape=2, scale=1),
list(shape1=2, shape2=1),
list(shape=2, scale=2)),
max_accrual_time = 5)
#> [1] "The outcome type for all the endpoints: tte"
stat_t.plot(data, arm.name = c("A","B"),priority = c(3,2,1),
Ctime = seq(2,8,0.2),plotTimeUnit = "years",
statistic = "WR", tau = 0, plot_CI = TRUE)
#> This analysis is unstratified.
#> $statistic
#> [1] "WR"
#>
#> $values
#> time win_stat lower_ci upper_ci
#> 1 2.0 0.6391489 0.3501887 1.166546
#> 2 2.2 0.6500593 0.3783346 1.116940
#> 3 2.4 0.7167798 0.4317604 1.189950
#> 4 2.6 0.7264924 0.4544961 1.161267
#> 5 2.8 0.6979066 0.4463499 1.091237
#> 6 3.0 0.6891560 0.4500785 1.055229
#> 7 3.2 0.7037225 0.4648179 1.065418
#> 8 3.4 0.7179833 0.4817856 1.069978
#> 9 3.6 0.7780083 0.5287173 1.144840
#> 10 3.8 0.7751268 0.5309108 1.131681
#> 11 4.0 0.7015848 0.4852616 1.014342
#> 12 4.2 0.7115139 0.4983674 1.015821
#> 13 4.4 0.7085259 0.4997771 1.004466
#> 14 4.6 0.7240062 0.5144887 1.018847
#> 15 4.8 0.7356322 0.5291556 1.022676
#> 16 5.0 0.7585466 0.5530718 1.040358
#> 17 5.2 0.7792752 0.5738776 1.058187
#> 18 5.4 0.7872996 0.5823905 1.064304
#> 19 5.6 0.7893421 0.5859536 1.063328
#> 20 5.8 0.7978184 0.5931881 1.073039
#> 21 6.0 0.8024024 0.5969874 1.078498
#> 22 6.2 0.8050622 0.5990741 1.081878
#> 23 6.4 0.8061084 0.5999812 1.083052
#> 24 6.6 0.8065872 0.6004244 1.083538
#> 25 6.8 0.8067061 0.6005489 1.083633
#> 26 7.0 0.8072372 0.6010138 1.084221
#> 27 7.2 0.8079595 0.6016086 1.085088
#> 28 7.4 0.8079595 0.6016086 1.085088
#> 29 7.6 0.8079595 0.6016086 1.085088
#> 30 7.8 0.8086986 0.6022261 1.085960
#> 31 8.0 0.8086986 0.6022261 1.085960partition_t.plot(data, Ctime = c(seq(0,9,0.5),seq(9.1,11,0.1)), arm.name = c("A","B"),
priority = c(3,2,1), tau = 0, plotTimeUnit = "years", trt_group = "trt")
#> $win_trt_t
#> time endpoint1 endpoint2 endpoint3
#> 1 0.0 0.000000 0.000000 0.000000
#> 2 0.5 0.000000 0.000000 0.000000
#> 3 1.0 0.000025 0.001625 0.000375
#> 4 1.5 0.000325 0.006625 0.001625
#> 5 2.0 0.001525 0.014625 0.002625
#> 6 2.5 0.004100 0.027125 0.010750
#> 7 3.0 0.008250 0.040075 0.014750
#> 8 3.5 0.013950 0.055500 0.016100
#> 9 4.0 0.017775 0.067875 0.025025
#> 10 4.5 0.024300 0.087000 0.031725
#> 11 5.0 0.031225 0.120375 0.042550
#> 12 5.5 0.035875 0.144450 0.045775
#> 13 6.0 0.041750 0.145450 0.046600
#> 14 6.5 0.042475 0.144550 0.048000
#> 15 7.0 0.042800 0.144525 0.048025
#> 16 7.5 0.042750 0.144450 0.048300
#> 17 8.0 0.042750 0.144425 0.048500
#> 18 8.5 0.042750 0.144425 0.048500
#>
#> $win_con_t
#> time endpoint1 endpoint2 endpoint3
#> 1 0.0 0.000000 0.000000 0.000000
#> 2 0.5 0.000000 0.000125 0.000000
#> 3 1.0 0.000000 0.004925 0.000000
#> 4 1.5 0.000000 0.017650 0.000000
#> 5 2.0 0.000000 0.029375 0.000000
#> 6 2.5 0.000575 0.057075 0.000150
#> 7 3.0 0.000925 0.090300 0.000300
#> 8 3.5 0.001375 0.113375 0.000375
#> 9 4.0 0.001975 0.155200 0.000575
#> 10 4.5 0.002525 0.200900 0.000675
#> 11 5.0 0.002925 0.250850 0.002175
#> 12 5.5 0.005475 0.280350 0.003225
#> 13 6.0 0.005800 0.281950 0.003625
#> 14 6.5 0.005850 0.281650 0.003950
#> 15 7.0 0.005900 0.281475 0.004175
#> 16 7.5 0.005900 0.281350 0.004225
#> 17 8.0 0.005900 0.281300 0.004225
#> 18 8.5 0.005900 0.281300 0.004225
#>
#> $win_tie_t
#> time proportion of ties
#> 1 0.0 1.000000
#> 2 0.5 0.999875
#> 3 1.0 0.993050
#> 4 1.5 0.973775
#> 5 2.0 0.951850
#> 6 2.5 0.900225
#> 7 3.0 0.845400
#> 8 3.5 0.799325
#> 9 4.0 0.731575
#> 10 4.5 0.652875
#> 11 5.0 0.549900
#> 12 5.5 0.484850
#> 13 6.0 0.474825
#> 14 6.5 0.473525
#> 15 7.0 0.473100
#> 16 7.5 0.473025
#> 17 8.0 0.472900
#> 18 8.5 0.472900
#>
#> $max_study_time
#> [1] 8.608721Acknowledgement
- Dr. Gaohong Dong for reviewing the vignette and providing constructive comments.
Reference
Bebu, I. and Lachin, J.M., 2016. Large sample inference for a win ratio analysis of a composite outcome based on prioritized components. Biostatistics.
Buyse, M., 2010. Generalized pairwise comparisons of prioritized outcomes in the two‐sample problem. Statistics in medicine.
Cui, Y., Dong, G., Kuan, P.F. and Huang, B., 2022. Evidence synthesis analysis with prioritized benefit outcomes in oncology clinical trials. Journal of Biopharmaceutical Statistics.
Dong, G., Li, D., Ballerstedt, S. and Vandemeulebroecke, M., 2016. A generalized analytic solution to the win ratio to analyze a composite endpoint considering the clinical importance order among components. Pharmaceutical statistics.
Dong, G., Qiu, J., Wang, D. and Vandemeulebroecke, M., 2018. The stratified win ratio. Journal of biopharmaceutical statistics.
Dong, G., Huang, B., Chang, Y.W., Seifu, Y., Song, J. and Hoaglin, D.C., 2020. The win ratio: Impact of censoring and follow‐up time and use with nonproportional hazards. Pharmaceutical statistics.
Dong, G., Mao, L., Huang, B., Gamalo-Siebers, M., Wang, J., Yu, G. and Hoaglin, D.C., 2020. The inverse-probability-of-censoring weighting (IPCW) adjusted win ratio statistic: an unbiased estimator in the presence of independent censoring. Journal of biopharmaceutical statistics.
Dong, G., Hoaglin, DC., Qiu, J., Matsouaka, RA., Chang, Y., Wang, J., Vandemeulebroecke, M., 2020. The win ratio: on interpretation and handling of ties. Statistics in Biopharmaceutical Research.
Dong, G., Huang, B., Wang, D., Verbeeck, J., Wang, J. and Hoaglin, D.C., 2021. Adjusting win statistics for dependent censoring. Pharmaceutical Statistics.
Dong, G., Huang, B., Verbeeck, J., Cui, Y., Song, J., Gamalo‐Siebers, M., Wang, D., Hoaglin, D.C., Seifu, Y., Mütze, T. and Kolassa, J., 2022. Win statistics (win ratio, win odds, and net benefit) can complement one another to show the strength of the treatment effect on time‐to‐event outcomes. Pharmaceutical Statistics.
Dong, G., Hoaglin, D.C., Huang, B., Cui, Y., Wang, D., Cheng, Y. and Gamalo‐Siebers, M., 2023. The stratified win statistics (win ratio, win odds, and net benefit). Pharmaceutical Statistics.
Finkelstein, D.M. and Schoenfeld, D.A., 1999. Combining mortality and longitudinal measures in clinical trials. Statistics in medicine.
Finkelstein, D.M. and Schoenfeld, D.A., 2019. Graphing the Win Ratio and its components over time. Statistics in medicine.
Luo, X., Tian, H., Mohanty, S. and Tsai, W.Y., 2015. An alternative approach to confidence interval estimation for the win ratio statistic. Biometrics.
Pocock, S.J., Ariti, C.A., Collier, T.J. and Wang, D., 2012. The win ratio: a new approach to the analysis of composite endpoints in clinical trials based on clinical priorities. European heart journal.
Wang D, Pocock S., 2016. A win ratio approach to comparing continuous non-normal outcomes in clinical trials. Pharmaceutical Statistics.
Wang, D., Zheng S., Cui, Y., He, N., Chen, T., Huang, B., 2023. Adjusted win ratio using inverse probability treatment weighting (IPTW) propensity score analysis. Journal of Biopharmaceutical Statistics.
Zheng S, Wang D, Qiu J, Chen T, Gamalo M., 2023. A win ratio approach for comparing crossing survival curves in clinical trials. Journal of Biopharmaceutical Statistics.
Affiliation: Stanford University, Pfizer Inc.↩︎