# Introduction

#### Samvel B. Gasparyan

## `hce` package intro

### Background

The purpose of the package is to simulate and analyze hierarchical composite endpoints (HCEs). The primary analysis method is win odds, but other win statistics (win ratio, net benefit)(Dong et al. 2023) are also implemented, provided there is no censoring. Win odds uses the DeLong-DeLong-Clarke-Pearson fromula (DeLong, DeLong, and Clarke-Pearson 1988) for the variance of the win proportion and is based on the Brunner-Munzel test (Brunner and Munzel 2000).

The power and sample size formulas consider different alternative classes (“shifted”, “ordered”, and “max” for the maximum value of the standard deviation). All formulas are derived from Bamber (1975). By default, Noether’s formula (Noether 1987) is used for shifted distributions. For more information on the power calculations for shifted distributions see also Gasparyan, Kowalewski, et al. (2021), Gasparyan, Kowalewski, and Koch (2022).

For a review of designing HCEs in clinical trials, see Gasparyan et al. (2022), also Gasparyan et al. (2023) and Little et al. (2023). The Basic Data Structure (BDS), which conforms to the Analysis Data Model (ADaM) (CDISC 2009) principles for hierarchical composite endpoints, is detailed in Gasparyan et al. (2024). To visualize HCEs, the maraca plot (Karpefors, Lindholm, and Gasparyan 2023) can be utilized.

The stratified and adjusted win odds are calculated based on the randomization-based covariate adjustment theory developed in Koch et al. (1998) (for a review, see Gasparyan, Folkvaljon, et al. (2021)).

All implementations are rank-based. For the review of methodology see (Brunner, Bathke, and Konietschke 2018).

### Setup

Load the package `hce` and check the version

    library(hce)
    packageVersion("hce")
    #> [1] '0.8.8'

For citing the package, run `citation("hce")` (Gasparyan 2025).

### Contents

List the functions and the datasets in the package

    ls("package:hce")
    #>  [1] "ADET"        "ADLB"        "ADSL"        "COVID19"     "COVID19b"
    #>  [6] "COVID19plus" "HCE1"        "HCE2"        "HCE3"        "HCE4"
    #> [11] "IWP"         "KHCE"        "as_hce"      "calcWINS"    "calcWO"
    #> [16] "hce"         "minWO"       "powerWO"     "propWINS"    "regWO"
    #> [21] "simHCE"      "simKHCE"     "simORD"      "simTTE"      "sizeWO"
    #> [26] "sizeWR"      "stratWO"     "summaryWO"

In brief, the package contains the following:

1. Datasets: use `data(package = "hce")` for the list of all datasets included in the package.

- Simulated datasets - `HCE1 - HCE4` that contain two treatment groups and analysis values `AVAL` of a hierarchical composite endpoint.

- The datasets `COVID19`, `COVID19b`, `COVID19plus` of COVID-19 ordinal scale outcomes (Beigel et al. 2020; Kalil et al. 2021).

- The datasets `ADET` (event-time), `ADLB` (laboratory), `ADSL` (subject-level baseline characteristics) for kidney events and their timing, kidney-related laboratory measurements of eGFR (estimated glomerular filtration rate), and, based on this, the derived kidney hierarchical composite endpoint dataset `KHCE` for the same patients (Heerspink et al. 2023).

2. Functions to create `hce` objects:

- `hce(), as_hce(), simHCE()` (see Gasparyan et al. (2024)).

- The function `simKHCE()` simulates continuous eGFR (estimated Glomerular Filtration Rate) values over time and derives an `hce` object for the kidney HCE (Heerspink et al. 2023).

- The function `simORD()` simulates ordinal outcomes by categorization of beta distributions.

3. Win odds (win ratio with ties) calculation generic functions for hierarchical composite endpoints:

- `calcWO(), summaryWO()` (see Gasparyan, Kowalewski, et al. (2021)). Win odds uses the DeLong-DeLong-Clarke-Pearson formula (DeLong, DeLong, and Clarke-Pearson 1988) for the variance of the win proportion and is based on the Brunner-Munzel test (Brunner and Munzel 2000).

4. Win statistics (win odds, win ratio, net benefit, Goodman Kruskal’s `gamma`) and their confidence intervals calculation function:

- `calcWINS()` (see Gasparyan et al. (2023)). The argument `SE_WP_Type` by default is set to `biased` since it produces DeLong-DeLong-Clarke-Pearson variance (the same as in `calcWO()`) which is biased for small samples. The `SE_WP_Type` can be changed to `unbiased` to produce Brunner-Konietschke version (Brunner and Konietschke 2025) of the Bamber (Bamber 1975) estimator for the variance of win proportion.

- Back-calculation of number of wins, losses, and ties given the win odds and win ratio using the function `propWINS()`.

5. Adjusted win odds calculation for a single, numeric covariate:

- `regWO()` (see Gasparyan, Folkvaljon, et al. (2021); Gasparyan et al. (2023)).

6. Stratified win odds with a possible adjustment after stratification with a single, numeric covariate:

- `stratWO()` (see Gasparyan, Folkvaljon, et al. (2021); Gasparyan et al. (2023)).

7. Power, sample size, and minimum detectable win odds calculation functions:

- `powerWO(), sizeWO(), minWO()` provide tools for the win odds power, sample size, and minimum detectable treatment effect calculation for different alternative classes (“shifted”, “ordered”, and “max” for the maximum value of the standard deviation) based on the win odds. All formulas are from Bamber (1975). By default, uses Noether’s formula (Noether 1987) for shifted distributions. For shifted distributions see also Gasparyan, Kowalewski, et al. (2021), Gasparyan, Kowalewski, and Koch (2022).

- Win ratio sample size calculation formula `sizeWR()` (Yu and Ganju 2022).

8. Print and plot methods for `hce_results` objects, generated by the functions `powerWO(), sizeWO(), minWO()`.

9. A plot method for `hce` objects (created by `as_hce()`) to provide the ordinal dominance graph (Bamber 1975).

10. The function `IWP()` to calculate patient-level individual win proportions (Gasparyan, Folkvaljon, et al. 2021; Gasparyan, Kowalewski, et al. 2021).

## `hce` Objects

### `hce()` Function

The main objects in the package are called `hce` objects, which are data frames with a specific structure matching the design of hierarchical composite endpoints (HCE). These are complex endpoints that combine different clinical events into a composite, using a hierarchy to prioritize the clinically most important event for a patient. These endpoints are implemented in clinical trials across different therapeutic areas. See, for example, Gasparyan et al. (2022) for an implementation in a COVID-19 setting and some practical considerations for constructing hierarchical composite endpoints. For the Chronic Kidney Disease (CKD) outcomes see (Little et al. 2023; Heerspink et al. 2023; Kondo et al. 2024).

HCEs are ordinal endpoints that can be thought of as having ‘greater’, ‘less’, or ‘equal’ defined for them, but without a definition of how much greater or less. In this sense, the ordinal outcomes can be represented as numeric vectors as long as numeric operations (e.g., sum or division) are not performed on them.

`hce` objects can be constructed using the helper function `hce()`, which has the following arguments:

    args("hce")
    #> function (GROUP, TRTP, AVAL0 = NULL, PADY = NULL)
    #> NULL

We see that the required arguments are `GROUP`, which specifies the clinically most important event of a patient to be included in the analysis, and `TRTP`, which specifies the (planned) treatment group of a patient (exactly two treatment groups should be present). Note that:

- The `hce` structure assumes that only one event per patient is present for the analysis, meaning that the resulting `hce` object created by the `hce()` function is a patient-level dataset. The function `hce()` does not select the clinically most important event of the patient but requires it to be already done when calling it.

- The argument `TRTP` should have exactly two levels.

Consider the following example of ordinal outcomes ‘I’, ‘II’, and ‘III’:

    set.seed(2022)
    n <- 100
    dat <- hce(GROUP = rep(x = c("I", "II", "III"), each = 100),
               TRTP = sample(x = c("Active", "Control"), size = n*3, replace = TRUE))
    class(dat)
    #> [1] "hce"        "data.frame"

This dataset has the appropriate structure of `hce` objects, but its class inherits from an object of class `data.frame`. This means that all functions available for data frames can be applied to `hce` objects, for example, the function `head()`:

    head(dat)
    #>      TRTP GROUP PARAMN AVAL
    #> 1 Control     I      1    1
    #> 2  Active     I      1    1
    #> 3 Control     I      1    1
    #> 4  Active     I      1    1
    #> 5  Active     I      1    1
    #> 6 Control     I      1    1

We see that the dataset has a very specific structure. The column `PARAMN` shows how the function `hce()` generated the order of given events (it uses usual alphabetic order for the unique values in the `GROUP` column to determine the clinical importance of events):

    unique(dat[, c("GROUP", "PARAMN")])
    #>     GROUP PARAMN
    #> 1       I      1
    #> 101    II      2
    #> 201   III      3

In the class `hce`, higher values for the ordering mean clinically less important events. For example, death, which is the most important event, should always get the lowest ordinal value. If there is a need to specify the order of outcomes, then the `GROUP` argument can be provided as a factor with the levels specifying the necessary order:

    set.seed(2022)
    n <- 100
    GROUP = rep(x = c("I", "II", "III"), each = 100)
    GROUP <- factor(GROUP, levels = c("III", "II", "I"))
    dat <- hce(GROUP = GROUP,
               TRTP = sample(x = c("Active", "Control"), size = n*3, replace = TRUE))
    unique(dat[, c("GROUP", "PARAMN")])
    #>     GROUP PARAMN
    #> 1       I      3
    #> 101    II      2
    #> 201   III      1

This means that the clinically most important event is ‘III’ instead of ‘I’. The argument `AVAL0` is meant to help in cases where we want to introduce sub-ordering within each `GROUP` category. For example, if two events in the group ‘I’ can be compared based on other parameters, then the `AVAL0` argument can be specified to take that into account.

Below we use the built-in data frame `HCE1` to construct an `hce` object. Before specifying the order of events, it is a good idea to check what are the unique events included in the `GROUP` column:

    data(HCE1)
    unique(HCE1$GROUP)
    #> [1] "C"    "TTE4" "TTE1" "TTE2" "TTE3"

Therefore, we can construct the following object using the `hce()` function:

    HCE <- hce(GROUP = factor(HCE1$GROUP, levels = c("TTE1", "TTE2", "TTE3", "TTE4", "C")),
               TRTP = HCE1$TRTP, AVAL0 = HCE1$AVAL0, PADY = 1080)
    class(HCE)
    #> [1] "adhce"      "hce"        "data.frame"
    head(HCE)
    #>   TRTP GROUP  AVAL0 PARAMN    AVAL PADY GROUPN
    #> 1    A     C  -2.21      5 5446.32 1080   5400
    #> 2    P     C  17.06      5 5465.59 1080   5400
    #> 3    A  TTE4 966.00      4 5286.00 1080   4320
    #> 4    P  TTE4 352.00      4 4672.00 1080   4320
    #> 5    A     C -12.45      5 5436.08 1080   5400
    #> 6    P     C  44.62      5 5493.15 1080   5400

The object’s class is `adhce`, which inherits from `hce`. This design indicates that the object includes additional columns.

### Create an `hce` Object from a Data Frame

Consider the dataset `HCE1`, which is part of the package `hce`:

    data(HCE1, package = "hce")
    class(HCE1)
    #> [1] "data.frame"
    head(HCE1)
    #>   ID TRTP GROUP GROUPN AVALT  AVAL0    AVAL PADY
    #> 1  1    A     C   5400  1080  -2.21 5446.32 1080
    #> 2  2    P     C   5400  1080  17.06 5465.59 1080
    #> 3  3    A  TTE4   4320   966 966.00 5286.00 1080
    #> 4  4    P  TTE4   4320   352 352.00 4672.00 1080
    #> 5  5    A     C   5400  1080 -12.45 5436.08 1080
    #> 6  6    P     C   5400  1080  44.62 5493.15 1080

This dataset has the appropriate structure of `hce` objects, but its class is `data.frame.` A generic way of coercing data structures to an `hce` object is to use the function `as_hce()`. This function performs checks (using an internal validator function) and creates an `hce` object from the given data structure (using an internal constructor function). If coercion is not possible, it will throw an error explaining the issue.

    dat1 <- as_hce(HCE2)
    str(dat1)
    #> Classes 'adhce', 'hce' and 'data.frame': 1000 obs. of  9 variables:
    #>  $ ID    : int  1 2 3 4 5 6 7 8 9 10 ...
    #>  $ TRTP  : chr  "A" "P" "A" "P" ...
    #>  $ GROUP : Factor w/ 5 levels "TTE1","TTE2",..: 1 5 5 1 2 3 5 2 1 5 ...
    #>  $ GROUPN: num  1080 5400 5400 1080 2160 3240 5400 2160 1080 5400 ...
    #>  $ AVALT : num  120 1080 1080 577 782 985 1080 293 313 1080 ...
    #>  $ AVAL0 : num  120 3.35 22.8 577 782 ...
    #>  $ AVAL  : num  1200 5461 5480 1657 2942 ...
    #>  $ PADY  : num  1080 1080 1080 1080 1080 1080 1080 1080 1080 1080 ...
    #>  $ PARAMN: num  1 5 5 1 2 3 5 2 1 5 ...

### Simulate `hce` Objects Using `simHCE()`

To simulate values from a hierarchical composite endpoint, we use the function `simHCE()`, which has the following arguments:

    args("simHCE")
    #> function (n, n0 = n, TTE_A, TTE_P, CM_A, CM_P, CSD_A = 1, CSD_P = CSD_A,
    #>     fixedfy = 1, yeardays = 360, pat = 100, shape = 1, theta = 1,
    #>     logC = FALSE, seed = NULL, dec = 2, all_data = FALSE)
    #> NULL

- The vector arguments `TTE_A` and `TTE_P` specify the event rates per year for time-to-event outcomes in the active and control groups, respectively. The function assumes a Weibull survival function with the same shape parameter for simulating all time-to-event outcomes in both treatment groups (by default `shape = 1`, which assumes an exponential survival function). These two vectors should have the same length, which indicates the number of time-to-event outcomes.

- By default, the event rates are presented per 100 patient-years (`pat = 100`), which can be changed using the argument `pat`. The function simulates event times in days, and the `yeardays = 360` argument can be used to change the number of days in a year (e.g., 365 or 365.25).

- The function simulates events during a fixed follow-up period only, and the `fixedfy` argument can be used to change the length of the follow-up (in years).

- The function simulates the continuous outcome from a normal (default) or log-normal (if `logC = TRUE`) distribution with given means and standard deviations for two treatment groups.

    Rates_A <- c(1.72, 1.74, 0.58, 1.5, 1)
    Rates_P <- c(2.47, 2.24, 2.9, 4, 6)
    dat3 <- simHCE(n = 2500, n0 = 1500, TTE_A = Rates_A,
                   TTE_P = Rates_P,
                   CM_A = -3, CM_P = -6,
                   CSD_A = 16, CSD_P = 15,
                   fixedfy = 3, seed = 2023)

    class(dat3)
    #> [1] "adhce"      "hce"        "data.frame"
    head(dat3)
    #>   ID TRTP GROUP GROUPN AVALT  AVAL0    AVAL seed PADY PARAMN
    #> 1  1    A     C   6480  1080  -9.32 6526.85 2023 1080      6
    #> 2  2    A     C   6480  1080   0.11 6536.28 2023 1080      6
    #> 3  3    A     C   6480  1080  27.06 6563.23 2023 1080      6
    #> 4  4    A     C   6480  1080  -4.78 6531.39 2023 1080      6
    #> 5  5    A  TTE2   2160   390 390.00 2550.00 2023 1080      2
    #> 6  6    A     C   6480  1080   4.56 6540.73 2023 1080      6

## Generics for `hce` Objects

As we see, the function `simHCE()` creates an object of type `hce`, which inherits from the built-in class `data.frame`. We can check all implemented methods for this new class as follows:

    methods(class = "hce")
    #> [1] calcWINS  calcWO    plot      summaryWO
    #> see '?methods' for accessing help and source code

The function `calcWO()` calculates the win odds and its confidence interval, while `summaryWO()` provides a more detailed calculation of win odds, including the number of wins, losses, and ties by `GROUP` categories.

    HCE <- hce(GROUP = factor(HCE3$GROUP, levels = c("TTE1", "TTE2", "TTE3", "TTE4", "C")),
               TRTP = HCE3$TRTP, AVAL0 = HCE3$AVAL0, PADY = 1080)
    calcWO(HCE)
    #>         WO     LCL      UCL         SE WOnull alpha       Pvalue       WP
    #> 1 1.332829 1.15078 1.543678 0.07493202      1  0.05 0.0001014229 0.571336
    #>      LCL_WP    UCL_WP      SE_WP     SD_WP    N
    #> 1 0.5353673 0.6073047 0.01835169 0.5803314 1000
    calcWINS(HCE)
    #> $summary
    #>      WIN   LOSS TIE  TOTAL Pties
    #> 1 142824 107156  20 250000 8e-05
    #>
    #> $WP
    #>         WP       LCL       UCL       Pvalue
    #> 1 0.571336 0.5353673 0.6073047 0.0001014229
    #>
    #> $NetBenefit
    #>   NetBenefit       LCL       UCL       Pvalue
    #> 1   0.142672 0.0707347 0.2146093 0.0001014229
    #>
    #> $WO
    #>         WO     LCL      UCL       Pvalue
    #> 1 1.332829 1.15078 1.543678 0.0001014229
    #>
    #> $WR1
    #>         WR     LCL1     UCL1     Pvalue1
    #> 1 1.332861 1.150793 1.543733 0.000125974
    #>
    #> $WR2
    #>         WR     LCL2     UCL2      Pvalue2
    #> 1 1.332861 1.155092 1.537987 8.351682e-05
    #>
    #> $gamma
    #>       gamma        LCL       UCL      Pvalue
    #> 1 0.1426834 0.07074057 0.2146263 0.000101418
    #>
    #> $SE
    #>        WP_SE NetBenefit_SE  logWR_SE1  logWR_SE2   gamma_SE
    #> 1 0.01835169    0.03670338 0.07493804 0.07303552 0.03670621
    #>
    #> $ref
    #> [1] "A vs P"
    #>
    #> $Input
    #>   alpha WOnull
    #> 1  0.05      1
    HCE$TRTP <- factor(HCE$TRTP, levels = c("P", "A"))
    plot(HCE, fill = TRUE, col = "#865A4F", type = 'l', lwd = 2)
    abline(a = 0, b = 1, lwd = 2, col = "#999999", lty = 2)

To check the generic functions available for the `adhce` class by running the following command:

    methods(class = "adhce")
    #> [1] summaryWO
    #> see '?methods' for accessing help and source code

The main difference between `summaryWO.hce()` and `summaryWO.adhce()` is that `summaryWO.adhce()` also provides a group-wise summary, in addition to the overall summary:

    summaryWO(HCE)
    #> $summary
    #>   TRTP    WIN   LOSS TIE  TOTAL        WR        WO
    #> 1    A 142824 107156  20 250000 1.3328605 1.3328294
    #> 2    P 107156 142824  20 250000 0.7502661 0.7502835
    #>
    #> $summary_by_GROUP
    #>    TRTP GROUP    WIN  LOSS TIE  TOTAL
    #> 1     A  TTE1   2473 39525   2  42000
    #> 2     P  TTE1   2985 29513   2  32500
    #> 3     A  TTE2   6847 22648   5  29500
    #> 4     P  TTE2  13857 45138   5  59000
    #> 5     A  TTE3  13251 16745   4  30000
    #> 6     P  TTE3  13356 25140   4  38500
    #> 7     A  TTE4   9056  6443   1  15500
    #> 8     P  TTE4  15383 19616   1  35000
    #> 9     A     C 111197 21795   8 133000
    #> 10    P     C  61575 23417   8  85000
    #>
    #> $WO
    #>         WO         SE       WP      SE_WP
    #> 1 1.332829 0.07493202 0.571336 0.01835169
    #>
    #> $cumsummary_by_GROUP
    #>    GROUPN   WINS  COUNT  TOTAL     PROP
    #> 1       1 A wins  23417 250000 0.093668
    #> 6       1 P wins  21795 250000 0.087180
    #> 11      1   Ties 204788 250000 0.819152
    #> 2       2 A wins  52930 250000 0.211720
    #> 7       2 P wins  61320 250000 0.245280
    #> 12      2   Ties 135750 250000 0.543000
    #> 3       3 A wins  98068 250000 0.392272
    #> 8       3 P wins  83968 250000 0.335872
    #> 13      3   Ties  67964 250000 0.271856
    #> 4       4 A wins 123208 250000 0.492832
    #> 9       4 P wins 100713 250000 0.402852
    #> 14      4   Ties  26079 250000 0.104316
    #> 5       5 A wins 142824 250000 0.571296
    #> 10      5 P wins 107156 250000 0.428624
    #> 15      5   Ties     20 250000 0.000080

An important addition to `summaryWO.adhce()` is the `cumsummary_by_GROUP` output, which reports cumulative wins, losses, and ties as outcomes are added sequentially according to the priority order:

    res0 <- summaryWO(HCE, ref = "P")
    res <- res0$cumsummary_by_GROUP
    barplot(PROP ~ WINS + GROUPN, data = res,
    col = c("darkgreen", "darkred", "darkblue"),
    xlab = "Proportions", xlim = c(0, 1),
    ylab = "Cumulative components by prioritization",
    legend.text = unique(res$WINS), beside = TRUE, horiz = TRUE)
    grid()

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