# While Alive estimands for Recurrent Events

#### Klaus Holst & Thomas Scheike

#### 2025-10-21

# While Alive estimands for Recurrent Events

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 per time-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 the sample size is small one can improve the finite sample properties by employing a transformation such as square or cube-root, and thus consider \[\begin{align*} E( (\frac{N(D \wedge t)}{D \wedge t})^.33 ) \end{align*}\]

    data(hfactioncpx12)

    dtable(hfactioncpx12,~status)
    #>
    #> status
    #>    0    1    2
    #>  617 1391  124
    dd <- 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] = 0

    dd <- 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] = 0

We see that the ratio of means are not very different, but that the subject specific mean of events per time-unit shows that those on the active treatment has fewer events per time-unit on average.

# Composite outcomes involving death and marks

The number of events can be generalized in various ways by using other outcomes than \(N(D \wedge t)\), for example,
\[\begin{align*} \tilde N(D \wedge t) = \int_0^t I(D \geq s) M(s) dN(s) + \sum_j M_j I(D \leq t,\epsilon=j) ) \end{align*}\] where \(M(s)\) are the marks related to \(N(s)\) and are \(M_j\) marks associated with the different causes of the terminal event. This provides an extension of the weighted composite outcomes measure of Mao & Lin (2022).

The marks (or here weights) can be stochastic if we are couting hosptial expenses, for example, and is vector on the data-frame. The marks for the event times (defined through the causes) will then be used.

Here weighting death with weight 2 and otherwise couting the recurrent 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

    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