We consider two while-alive estimands for recurrent events data\[\begin{align*}\frac{E(N(D \wedge t))}{E(D \wedge t)}\end{align*}\] and the mean of the subject specific events pertime-unit\[\begin{align*}E( \frac{N(D \wedge t)}{D \wedge t} )\end{align*}\] for two treatment-groups in the case of an RCT.For the mean of events per time-unit it has been seen that when thesample size is small one can improve the finite sample properties byemploying a transformation such as square or cube-root, and thusconsider\[\begin{align*}E( (\frac{N(D \wedge t)}{D \wedge t})^.33 )\end{align*}\]
data(hfactioncpx12)dtable(hfactioncpx12,~status)#>#> status#> 0 1 2#> 617 1391 124dd<-WA_recurrent(Event(entry,time,status)~treatment+cluster(id),hfactioncpx12,time=2,death.code=2)summary(dd)#> While-Alive summaries:#>#> RMST, E(min(D,t))#> Estimate Std.Err 2.5% 97.5% P-value#> treatment0 1.859 0.02108 1.817 1.900 0#> treatment1 1.924 0.01502 1.894 1.953 0#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treatment0] - [treat.... -0.06517 0.02588 -0.1159 -0.01444 0.0118#>#> Null Hypothesis:#> [treatment0] - [treatment1] = 0#> mean events, E(N(min(D,t))):#> Estimate Std.Err 2.5% 97.5% P-value#> treatment0 1.572 0.09573 1.384 1.759 1.375e-60#> treatment1 1.453 0.10315 1.251 1.656 4.376e-45#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treatment0] - [treat.... 0.1185 0.1407 -0.1574 0.3943 0.4#>#> Null Hypothesis:#> [treatment0] - [treatment1] = 0#> _______________________________________________________#> Ratio of means E(N(min(D,t)))/E(min(D,t))#> Estimate Std.Err 2.5% 97.5% P-value#> treatment0 0.8457 0.05264 0.7425 0.9488 4.411e-58#> treatment1 0.7555 0.05433 0.6490 0.8619 5.963e-44#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treatment0] - [treat.... 0.09022 0.07565 -0.05805 0.2385 0.233#>#> Null Hypothesis:#> [treatment0] - [treatment1] = 0#> _______________________________________________________#> Mean of Events per time-unit E(N(min(D,t))/min(D,t))#> Estimate Std.Err 2.5% 97.5% P-value#> treat0 1.0725 0.1222 0.8331 1.3119 1.645e-18#> treat1 0.7552 0.0643 0.6291 0.8812 7.508e-32#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treat0] - [treat1] 0.3173 0.1381 0.04675 0.5879 0.02153#>#> Null Hypothesis:#> [treat0] - [treat1] = 0dd<-WA_recurrent(Event(entry,time,status)~treatment+cluster(id),hfactioncpx12,time=2,death.code=2,trans=.333)summary(dd,type="log")#> While-Alive summaries, log-scale:#>#> RMST, E(min(D,t))#> Estimate Std.Err 2.5% 97.5% P-value#> treatment0 0.6199 0.011340 0.5977 0.6421 0#> treatment1 0.6543 0.007807 0.6390 0.6696 0#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treatment0] - [treat.... -0.03446 0.01377 -0.06145 -0.007478 0.01231#>#> Null Hypothesis:#> [treatment0] - [treatment1] = 0#> mean events, E(N(min(D,t))):#> Estimate Std.Err 2.5% 97.5% P-value#> treatment0 0.4523 0.06090 0.3329 0.5716 1.119e-13#> treatment1 0.3739 0.07097 0.2348 0.5130 1.376e-07#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treatment0] - [treat.... 0.07835 0.09352 -0.1049 0.2616 0.4022#>#> Null Hypothesis:#> [treatment0] - [treatment1] = 0#> _______________________________________________________#> Ratio of means E(N(min(D,t)))/E(min(D,t))#> Estimate Std.Err 2.5% 97.5% P-value#> treatment0 -0.1676 0.06224 -0.2896 -0.04563 7.081e-03#> treatment1 -0.2804 0.07192 -0.4214 -0.13947 9.651e-05#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treatment0] - [treat.... 0.1128 0.09511 -0.07361 0.2992 0.2356#>#> Null Hypothesis:#> [treatment0] - [treatment1] = 0#> _______________________________________________________#> Mean of Events per time-unit E(N(min(D,t))/min(D,t))#> Estimate Std.Err 2.5% 97.5% P-value#> treat0 -0.3833 0.04939 -0.4801 -0.2865 8.487e-15#> treat1 -0.5380 0.05666 -0.6491 -0.4270 2.191e-21#>#> Estimate Std.Err 2.5% 97.5% P-value#> [treat0] - [treat1] 0.1548 0.07517 0.007459 0.3021 0.03948#>#> Null Hypothesis:#> [treat0] - [treat1] = 0We see that the ratio of means are not very different, but that thesubject specific mean of events per time-unit shows that those on theactive treatment has fewer events per time-unit on average.
The number of events can be generalized in various ways by usingother outcomes than\(N(D \wedge t)\),for example,
\[\begin{align*}\tilde N(D \wedge t) = \int_0^t I(D \geq s) M(s) dN(s) + \sum_j M_jI(D \leq t,\epsilon=j) )\end{align*}\] where\(M(s)\)are the marks related to\(N(s)\) andare\(M_j\) marks associated with thedifferent causes of the terminal event. This provides an extension ofthe weighted composite outcomes measure of Mao & Lin (2022).
The marks (or here weights) can be stochastic if we are coutinghosptial expenses, for example, and is vector on the data-frame. Themarks for the event times (defined through the causes) will then beused.
Here weighting death with weight 2 and otherwise couting therecurrent of events as before (with weight 1)
hfactioncpx12$marks<-runif(nrow(hfactioncpx12))##ddmg <- WA_recurrent(Event(entry,time,status)~treatment+cluster(id),hfactioncpx12,time=2,##cause=1:2,death.code=2,marks=hfactioncpx12$marks)##summary(ddmg)ddm<-WA_recurrent(Event(entry,time,status)~treatment+cluster(id),hfactioncpx12,time=2,cause=1:2,death.code=2,marks=hfactioncpx12$status)sessionInfo()#> R version 4.5.1 (2025-06-13)#> Platform: aarch64-apple-darwin25.0.0#> Running under: macOS Tahoe 26.0.1#>#> Matrix products: default#> BLAS: /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRblas.dylib#> LAPACK: /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRlapack.dylib; LAPACK version 3.12.1#>#> locale:#> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8#>#> time zone: Europe/Copenhagen#> tzcode source: internal#>#> attached base packages:#> [1] splines stats graphics grDevices utils datasets methods#> [8] base#>#> other attached packages:#> [1] prodlim_2025.04.28 timereg_2.0.7 survival_3.8-3 mets_1.3.8#>#> loaded via a namespace (and not attached):#> [1] cli_3.6.5 knitr_1.50 rlang_1.1.6#> [4] xfun_0.53 KernSmooth_2.23-26 data.table_1.17.8#> [7] jsonlite_2.0.0 future.apply_1.20.0 listenv_0.9.1#> [10] lava_1.8.1 htmltools_0.5.8.1 sass_0.4.10#> [13] rmarkdown_2.30 grid_4.5.1 evaluate_1.0.5#> [16] jquerylib_0.1.4 fastmap_1.2.0 numDeriv_2016.8-1.1#> [19] yaml_2.3.10 mvtnorm_1.3-3 lifecycle_1.0.4#> [22] compiler_4.5.1 codetools_0.2-20 ucminf_1.2.2#> [25] Rcpp_1.1.0 future_1.67.0 lattice_0.22-7#> [28] digest_0.6.37 R6_2.6.1 parallelly_1.45.1#> [31] parallel_4.5.1 Matrix_1.7-4 bslib_0.9.0#> [34] tools_4.5.1 globals_0.18.0 cachem_1.1.0