While Alive estimands for Recurrent Events

We consider two while-alive estimands for recurrent events data E(N(Dt))E(Dt)\begin{align*} \frac{E(N(D \wedge t))}{E(D \wedge t)} \end{align*} and the mean of the subject specific events per time-unit E(N(Dt)Dt)\begin{align*} E( \frac{N(D \wedge t)}{D \wedge t} ) \end{align*} for two treatment-groups in the case of an RCT.

data(hfaction_cpx12)

dtable(hfaction_cpx12,~status)
#> 
#> status
#>    0    1    2 
#>  617 1391  124
dd <- WA_recurrent(Event(entry,time,status)~treatment+cluster(id),hfaction_cpx12,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] = 0

SessionInfo

sessionInfo()
#> R version 4.4.2 (2024-10-31)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.1 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
#>  [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
#>  [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
#> [10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   
#> 
#> time zone: UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] mets_1.3.5     timereg_2.0.6  survival_3.7-0
#> 
#> loaded via a namespace (and not attached):
#>  [1] cli_3.6.3           knitr_1.49          rlang_1.1.4        
#>  [4] xfun_0.50           textshaping_0.4.1   jsonlite_1.8.9     
#>  [7] listenv_0.9.1       future.apply_1.11.3 lava_1.8.0         
#> [10] htmltools_0.5.8.1   ragg_1.3.3          sass_0.4.9         
#> [13] rmarkdown_2.29      grid_4.4.2          evaluate_1.0.3     
#> [16] jquerylib_0.1.4     fastmap_1.2.0       mvtnorm_1.3-3      
#> [19] numDeriv_2016.8-1.1 yaml_2.3.10         lifecycle_1.0.4    
#> [22] compiler_4.4.2      codetools_0.2-20    fs_1.6.5           
#> [25] Rcpp_1.0.13-1       future_1.34.0       systemfonts_1.1.0  
#> [28] lattice_0.22-6      digest_0.6.37       R6_2.5.1           
#> [31] parallelly_1.41.0   parallel_4.4.2      splines_4.4.2      
#> [34] bslib_0.8.0         Matrix_1.7-1        tools_4.4.2        
#> [37] globals_0.16.3      pkgdown_2.1.1       cachem_1.1.0       
#> [40] desc_1.4.3