# Average treatment effect (ATE) for Competing risks and binary outcomes

#### Klaus Holst & Thomas Scheike

#### 2026-05-23

The `binregATE` function estimates the average treatment effect for binary outcomes with IPCW adjustment at a specific time-point, and can thus be used for survival or competing risks data. Computation is linear in data size and influence functions are computed and available.

Key features:

- the censoring weights can be stratum-dependent
- predictions can be computed with standard errors
- computation time is linear in data size, including standard errors
- cluster-corrected standard errors are available via the `clusters` argument

# Average treatment effect

First we simulate some data that mimics that of Kumar et al. (2012). This is data from multiple myeloma patients treated with allogeneic stem cell transplantation from the Center for International Blood and Marrow Transplant Research (CIBMTR). The data consisted of patients transplanted from 1995 to 2005, comparing outcomes between transplant periods: 2001–2005 (\(N=488\)) versus 1995–2000 (\(N=375\)). The two competing events were relapse (cause 2) and treatment-related mortality (TRM, cause 1) defined as death without relapse. Kumar et al. (2012) considered the following risk covariates: transplant time period (`gp`, main interest of the study: 1 for transplanted in 2001–2005 versus 0 for transplanted in 1995–2000), donor type (`dnr`: 1 for unrelated or other related donor (\(N=280\)) versus 0 for HLA-identical sibling (\(N=584\))), prior autologous transplant (`preauto`: 1 for Auto+Allo transplant (\(N=399\)) versus 0 for allogeneic transplant alone (\(N=465\))) and time to transplant (`ttt24`: 1 for more than 24 months (\(N=289\)) versus 0 for less than or equal to 24 months (\(N=575\))).

We generate similar data by assuming that the two cumulative incidence curves are logistic and that censoring depends on covariates via a Cox model, all wrapped in the `kumarsim` function. The simulation does not enforce the constraint \(F_1+F_2 < 1\), but as we increase the sample size we still recover the parameters of cause 2.

We start by computing the average treatment effects for the transplant periods (`gp`), then consider the possible interaction between `gp` and donor type (`dnr`) and estimate all contrasts between the risk estimates for all combinations of `gp` and `dnr`. The key components are:

- an outcome model (fitted by IPCW): `Event(...) ~ .....`
- a propensity score model (using logistic regression): `treat.model=...`
- an IPCW weight estimated using a stratified Kaplan-Meier model: `cens.model=~strata(..)`

    library(mets)
    set.seed(100)
    ###
    n <- 400
    kumar <- kumarsim(n,depcens=1)
    kumar$cause <- kumar$status
    kumar$ttt24 <- kumar[,6]
    dtable(kumar,~cause)
    #>
    #> cause
    #>   0   1   2
    #> 182  72 146
    dfactor(kumar) <- gp.f~gp
    kumar$id <- 1:n
    kumar$idc <- sample(100,n,TRUE)
    kumar$ids <- sample(n,n)
    kumar$id2 <- sample(n,n)
    kumar2 <- kumar[order(kumar$id2),]
    kumar$int <- interaction(kumar$gp,kumar$dnr)
    kumar2$int <- interaction(kumar2$gp,kumar2$dnr)
    clust <- 0

    b2 <- binregATE(Event(time,cause)~gp.f+dnr+preauto+ttt24,kumar,cause=2,
        treat.model=gp.f~dnr+preauto+ttt24,time=40,cens.model=~strata(gp,dnr))
    summary(b2)
    #>    n events
    #>  400    137
    #>
    #>  400 clusters
    #> coeffients:
    #>              Estimate   Std.Err      2.5%     97.5% P-value
    #> (Intercept) -1.310227  0.221121 -1.743616 -0.876838  0.0000
    #> gp.f1        0.898030  0.271691  0.365526  1.430535  0.0009
    #> dnr          0.323059  0.271545 -0.209160  0.855278  0.2342
    #> preauto      0.269177  0.278181 -0.276049  0.814402  0.3332
    #> ttt24        0.496202  0.276045 -0.044836  1.037240  0.0722
    #>
    #> exp(coeffients):
    #>             Estimate    2.5%  97.5%
    #> (Intercept)  0.26976 0.17489 0.4161
    #> gp.f1        2.45476 1.44127 4.1809
    #> dnr          1.38135 0.81127 2.3520
    #> preauto      1.30889 0.75878 2.2578
    #> ttt24        1.64247 0.95615 2.8214
    #>
    #> Average Treatment effects (G-formula) :
    #>           Estimate  Std.Err     2.5%    97.5% P-value
    #> treat0    0.293021 0.039148 0.216292 0.369750   0e+00
    #> treat1    0.496481 0.041744 0.414665 0.578297   0e+00
    #> treat:1-0 0.203460 0.060294 0.085285 0.321635   7e-04
    #>
    #> Average Treatment effects (double robust) :
    #>           Estimate  Std.Err     2.5%    97.5% P-value
    #> treat0    0.311984 0.042914 0.227875 0.396094   0e+00
    #> treat1    0.512310 0.042454 0.429103 0.595518   0e+00
    #> treat:1-0 0.200326 0.060585 0.081582 0.319070   9e-04

    b5 <- binregATE(Event(time,cause)~int+preauto+ttt24,kumar,cause=2,
            treat.model=int~preauto+ttt24,cens.code=0,time=60)
    summary(b5)
    #>    n events
    #>  400    142
    #>
    #>  400 clusters
    #> coeffients:
    #>              Estimate   Std.Err      2.5%     97.5% P-value
    #> (Intercept) -1.115768  0.285147 -1.674646 -0.556890  0.0001
    #> int1.0       0.674582  0.333967  0.020018  1.329146  0.0434
    #> int0.1       0.034751  0.494229 -0.933920  1.003421  0.9439
    #> int1.1       1.340278  0.431989  0.493594  2.186961  0.0019
    #> preauto      0.298967  0.272324 -0.234779  0.832713  0.2723
    #> ttt24        0.476755  0.286955 -0.085667  1.039177  0.0966
    #>
    #> exp(coeffients):
    #>             Estimate    2.5%  97.5%
    #> (Intercept)  0.32766 0.18737 0.5730
    #> int1.0       1.96321 1.02022 3.7778
    #> int0.1       1.03536 0.39301 2.7276
    #> int1.1       3.82010 1.63819 8.9081
    #> preauto      1.34846 0.79075 2.2995
    #> ttt24        1.61084 0.91790 2.8269
    #>
    #> Average Treatment effects (G-formula) :
    #>                 Estimate    Std.Err       2.5%      97.5% P-value
    #> treat0.0       0.3124362  0.0586931  0.1973999  0.4274724  0.0000
    #> treat1.0       0.4676985  0.0422018  0.3849846  0.5504125  0.0000
    #> treat0.1       0.3197899  0.0896402  0.1440983  0.4954815  0.0004
    #> treat1.1       0.6274521  0.0726051  0.4851488  0.7697554  0.0000
    #> treat:1.0-0.0  0.1552624  0.0738341  0.0105502  0.2999745  0.0355
    #> treat:0.1-0.0  0.0073537  0.1048541 -0.1981565  0.2128640  0.9441
    #> treat:1.1-0.0  0.3150160  0.0978046  0.1233225  0.5067094  0.0013
    #> treat:0.1-1.0 -0.1479086  0.1003040 -0.3445009  0.0486837  0.1403
    #> treat:1.1-1.0  0.1597536  0.0845518 -0.0059649  0.3254721  0.0588
    #> treat:1.1-0.1  0.3076622  0.1178093  0.0767602  0.5385643  0.0090
    #>
    #> Average Treatment effects (double robust) :
    #>                 Estimate    Std.Err       2.5%      97.5% P-value
    #> treat0.0       0.3580588  0.0674640  0.2258318  0.4902858  0.0000
    #> treat1.0       0.4871368  0.0501492  0.3888461  0.5854274  0.0000
    #> treat0.1       0.3102783  0.1210473  0.0730299  0.5475266  0.0104
    #> treat1.1       0.7645588  0.1887893  0.3945387  1.1345790  0.0001
    #> treat:1.0-0.0  0.1290779  0.0831680 -0.0339284  0.2920843  0.1207
    #> treat:0.1-0.0 -0.0477806  0.1400295 -0.3222333  0.2266722  0.7329
    #> treat:1.1-0.0  0.4065000  0.2040550  0.0065596  0.8064404  0.0464
    #> treat:0.1-1.0 -0.1768585  0.1315359 -0.4346641  0.0809471  0.1788
    #> treat:1.1-1.0  0.2774221  0.1937486 -0.1023182  0.6571624  0.1522
    #> treat:1.1-0.1  0.4542806  0.2102654  0.0421680  0.8663931  0.0307

We note that the estimates found using the large censoring model are very different from those using the simple Kaplan-Meier weights that are severely biased for these data. This is due to a strong censoring dependence.

The average treatment is around \(0.17 = E(Y(1) - Y(0))\) at time 60 for the transplant period, under the standard causal assumptions. The 1/0 treatment variable used for the causal computation is found as the right hand side (rhs) of the treat.model or as the first argument on the rhs of the response model.

The binregATE default uses binreg with its default to fit the working model and is recommended, but the logitIPCW and logitIPCWATE can also be used and are GLM-type IPCW weighted models (see binreg help page/vignette). We can compare the logitIPCWATE/logitIPCW with that of wglm and its ate function in the riskRegression package, and they agree.

    ib2 <- logitIPCWATE(Event(time,cause)~gp.f+dnr+preauto+ttt24,kumar,cause=2,
        treat.model=gp.f~dnr+preauto+ttt24,time=40,cens.model=~strata(gp,dnr))
    summary(ib2)
    #>    n events
    #>  400    137
    #>
    #>  400 clusters
    #> coeffients:
    #>             Estimate  Std.Err     2.5%    97.5% P-value
    #> (Intercept) -1.27557  0.21917 -1.70514 -0.84601  0.0000
    #> gp.f1        0.83337  0.25904  0.32566  1.34108  0.0013
    #> dnr          0.36815  0.27656 -0.17390  0.91020  0.1831
    #> preauto      0.48521  0.30772 -0.11791  1.08832  0.1148
    #> ttt24        0.18958  0.30947 -0.41698  0.79614  0.5401
    #>
    #> exp(coeffients):
    #>             Estimate    2.5%  97.5%
    #> (Intercept)  0.27927 0.18175 0.4291
    #> gp.f1        2.30105 1.38494 3.8232
    #> dnr          1.44506 0.84038 2.4848
    #> preauto      1.62452 0.88878 2.9693
    #> ttt24        1.20874 0.65904 2.2170
    #>
    #> Average Treatment effects (G-formula) :
    #>         Estimate  Std.Err     2.5%    97.5% P-value
    #> treat-1 0.493059 0.040018 0.414625 0.571494  0.0000
    #> treat-0 0.302695 0.040274 0.223759 0.381632  0.0000
    #> p1      0.190364 0.058248 0.076200 0.304527  0.0011
    #>
    #> Average Treatment effects (double robust) :
    #>         Estimate  Std.Err     2.5%    97.5% P-value
    #> treat-1 0.524459 0.043164 0.439860 0.609058   0e+00
    #> treat-0 0.309211 0.043507 0.223938 0.394483   0e+00
    #> p1      0.215248 0.061268 0.095165 0.335331   4e-04

    ib5 <- logitIPCW(Event(time,cause)~gp.f+dnr+preauto+ttt24,kumar,cause=2,cens.code=0,
         time=60,cens.model=~strata(gp,dnr))
    summary(ib5)
    #>    n events
    #>  400    142
    #>
    #>  400 clusters
    #> coeffients:
    #>              Estimate   Std.Err      2.5%     97.5% P-value
    #> (Intercept) -1.142446  0.225267 -1.583960 -0.700931  0.0000
    #> gp.f1        0.716372  0.267369  0.192339  1.240405  0.0074
    #> dnr          0.551548  0.325499 -0.086418  1.189515  0.0902
    #> preauto      0.748078  0.343802  0.074239  1.421917  0.0296
    #> ttt24       -0.184844  0.373708 -0.917298  0.547610  0.6209
    #>
    #> exp(coeffients):
    #>             Estimate    2.5%  97.5%
    #> (Intercept)  0.31904 0.20516 0.4961
    #> gp.f1        2.04699 1.21208 3.4570
    #> dnr          1.73594 0.91721 3.2855
    #> preauto      2.11294 1.07706 4.1451
    #> ttt24        0.83123 0.39960 1.7291

    ibs <- logitIPCW(Event(time,cause)~gp.f+dnr+preauto+ttt24,kumar,cause=2,cens.code=0,time=60)
    summary(ibs)
    #>    n events
    #>  400    142
    #>
    #>  400 clusters
    #> coeffients:
    #>              Estimate   Std.Err      2.5%     97.5% P-value
    #> (Intercept) -1.294250  0.220215 -1.725864 -0.862636  0.0000
    #> gp.f1        1.633772  0.294651  1.056267  2.211277  0.0000
    #> dnr          0.020551  0.309235 -0.585538  0.626639  0.9470
    #> preauto      0.547275  0.305442 -0.051381  1.145931  0.0732
    #> ttt24        0.288357  0.309124 -0.317515  0.894228  0.3509
    #>
    #> exp(coeffients):
    #>             Estimate    2.5%  97.5%
    #> (Intercept)  0.27410 0.17802 0.4220
    #> gp.f1        5.12316 2.87562 9.1274
    #> dnr          1.02076 0.55681 1.8713
    #> preauto      1.72854 0.94992 3.1454
    #> ttt24        1.33423 0.72796 2.4454

    check <- 0
    if (check==1) {
    require(riskRegression)
    e.wglm <- wglm( regressor.event=~gp.f+dnr+preauto+ttt24, formula.censor = Surv(time,cause==0)~+1, times = 60, data = kumar, product.limit=TRUE,cause=2)
    summary(e.wglm)$coef
    estimate(ibs)

    es.wglm <- wglm( regressor.event=~gp.f+dnr+preauto+ttt24,
    formula.censor = Surv(time,cause==0)~strata(gp,dnr), times = 60,
    data = kumar, product.limit=TRUE,cause=2)
    summary(es.wglm)$coef
    estimate(ib5)
    }

The cluster argument should not be used for the logitIPCWATE, but works for binregATE.

# Average treatment for Competing risks data

The binreg function does direct binomial regression for one time-point, \(t\), fitting the model \[\begin{align*} P(T \leq t, \epsilon=1 | X ) & = \mbox{expit}( X^T \beta), \end{align*}\] for possible right censored data. The estimation procedure is based on IPCW adjusted estimating equation (EE) \[\begin{align*} U(\beta) = & X \left( \Delta(t) I(T \leq t, \epsilon=1 )/G_c(T \wedge t) - \mbox{expit}( X^T beta) \right) = 0 \end{align*}\] where \(G_c(t)=P(C > t)\), the censoring survival distribution, and with \(\Delta(t) = I( C > T \wedge t)\) the indicator of being uncensored at time \(t\).

The function logitIPCW instead considers the EE \[\begin{align*} U(\beta) = & X \frac{\Delta(t)}{G_c(T \wedge t)} \left( I(T \leq t, \epsilon=1 ) - \mbox{expit}( X^T beta) \right) = 0. \end{align*}\] The two score equations are quite similar, and exactly the same when the censoring model is fully-nonparametric given \(X\).

Additional functions logitATE, and binregATE computes the average treatment effect. We demonstrate their use below.

The functions binregATE (recommended) and logitATE also works when there is no censoring and we thus have simple binary outcome.

Variance is based on sandwich formula with IPCW adjustment, and naive.var is variance under a known censoring model. The influence functions are stored in the output. Further, the standard errors can be cluster corrected by specifying the relevant cluster for the working outcome model.

- We estimate the average treatment effect of our binary response \(I(T \leq t, \epsilon=1)\)
  - Using a working logistic model for the response (possibly with a cluster specification)
  - Using a working logistic model for treatment given covariates
    - The binregATE can also handle a factor with more than two levels and then uses the mlogit multinomial regression function (of mets).
  - Using a working model for censoring given covariates, this must be a stratified Kaplan-Meier.

If there are no censoring then the censoring weights are simply set to 1.

The average treatment effect is \[\begin{align*} E(Y(1) - Y(0)) \end{align*}\] using counterfactual outcomes.

We compute the simple G-estimator \[\begin{align*} \sum m_a(X_i) \end{align*}\] to estimate the risk \(E(Y(a))\).

The DR-estimator instead uses the estimating equations that are double robust wrt

- A working logistic model for the response
- A working logistic model for treatment given covariates

This is estimated using the estimator \[\begin{align*} \sum \left[ \frac{A_i Y_i}{\pi_A(X_i)}-\frac{A_i - \pi_A(X_i)}{\pi_A(X_i)} m_1(X_i) \right] - \left[ \frac{(1-A_i) Y_i}{1-\pi_A(X_i)}+\frac{A_i - \pi_A(X_i)}{1-\pi_A(X_i)} m_0(X_i) \right] \end{align*}\] where

- \(A_i\) is treatment indicator
- \(\pi_A(X_i) = P(A_i=1|X_i)\) is treatment model
- \(Y_i\) outcome, that in case of censoring is censoring adjusted \(\tilde Y_i \Delta(t) /G_c(T_i- \wedge t)\)
- \(\tilde Y_i = I(T_i \leq t, \epsilon_i=1)\) outcome before censoring.
- \(m_j(X_i)=P(Y_i=1| A_i=j,X_i)\) is outcome model, using binomial regression.

The standard errors are then based on an iid decomposition using taylor-expansions for the parameters of the treatment-model and the outcome-model, and the censoring probability.

We need that the censoring model is correct, so it can be important to use a sufficiently large censoring model as we also illustrate below.

- The censoring model can be specified by strata (used for `phreg`).

We also compute standard marginalization for average treatment effect (called differenceG) \[\begin{align*} \sum \left[ m_1(X_i) - m_0(X_i) \right] \end{align*}\] and again standard errors are based on the related influence functions and are also returned.

For large data where there are more than 2 treatment groups the computations can be memory extensive when there are many covariates due to the multinomial-regression model used for the propensity scores. Otherwise the function (binregATE) will run for large data.

The ATE functions require that the treatment variable, given as the first variable on the right-hand side of the outcome model, is a factor. The variable is also identified from the left-hand side of the treatment model (`treat.model`), which by default assumes that treatment does not depend on any covariates.

# Average treatment effect for binary or continuous responses

In the binary case a binary outcome is specified instead of the survival outcome, and as a consequence no censoring adjustment is done.

- the binary/numeric outcome must be a variable in the data frame

The following code runs the analysis (one can also use `binregATE` by coding `cause` without censoring values, i.e. setting `cens.code=2` and `time` large):

    kumar$cause2 <- 1*(kumar$cause==2)

    b3 <- logitATE(cause2~gp.f+dnr+preauto+ttt24,kumar,treat.model=gp.f~dnr+preauto+ttt24)
    summary(b3)
    #>    n events
    #>  400    400
    #>
    #>  400 clusters
    #> coeffients:
    #>              Estimate   Std.Err      2.5%     97.5% P-value
    #> (Intercept) -1.219960  0.200830 -1.613580 -0.826341  0.0000
    #> gp.f1        0.387595  0.243738 -0.090123  0.865314  0.1118
    #> dnr          0.633992  0.241410  0.160837  1.107147  0.0086
    #> preauto      0.139356  0.248680 -0.348049  0.626761  0.5752
    #> ttt24        0.449527  0.243475 -0.027675  0.926730  0.0648
    #>
    #> exp(coeffients):
    #>             Estimate    2.5%  97.5%
    #> (Intercept)  0.29524 0.19917 0.4376
    #> gp.f1        1.47343 0.91382 2.3758
    #> dnr          1.88512 1.17449 3.0257
    #> preauto      1.14953 0.70606 1.8715
    #> ttt24        1.56757 0.97270 2.5262
    #>
    #> Average Treatment effects (G-formula) :
    #>            Estimate   Std.Err      2.5%     97.5% P-value
    #> treat0     0.316084  0.037954  0.241695  0.390472  0.0000
    #> treat1     0.400809  0.033395  0.335356  0.466262  0.0000
    #> treat:1-0  0.084726  0.052651 -0.018468  0.187919  0.1076
    #>
    #> Average Treatment effects (double robust) :
    #>            Estimate   Std.Err      2.5%     97.5% P-value
    #> treat0     0.343451  0.042284  0.260577  0.426325  0.0000
    #> treat1     0.421615  0.033616  0.355729  0.487500  0.0000
    #> treat:1-0  0.078164  0.053961 -0.027599  0.183926  0.1475

    ###library(targeted)
    ###b3a <- ate(cause2~gp.f|dnr+preauto+ttt24| dnr+preauto+ttt24,kumar,family=binomial)
    ###summary(b3a)

    ## calculate also relative risk
    estimate(coef=b3$riskDR,vcov=b3$var.riskDR,f=function(p) p[1]/p[2])
    #>        Estimate Std.Err   2.5% 97.5%   P-value
    #> treat0   0.8146  0.1194 0.5807 1.049 8.831e-12

Or with continuous response using normal estimating equations

    b3 <- normalATE(time~gp.f+dnr+preauto+ttt24,kumar,treat.model=gp.f~dnr+preauto+ttt24)
    summary(b3)
    #>    n events
    #>  400    400
    #>
    #>  400 clusters
    #> coeffients:
    #>             Estimate  Std.Err     2.5%    97.5% P-value
    #> (Intercept)  43.4758   4.0796  35.4800  51.4716  0.0000
    #> gp.f1       -27.2211   4.1379 -35.3313 -19.1109  0.0000
    #> dnr           2.9485   4.4768  -5.8259  11.7228  0.5101
    #> preauto      -2.8172   3.6191  -9.9105   4.2760  0.4363
    #> ttt24        -1.9787   4.1285 -10.0704   6.1131  0.6318
    #>
    #>
    #> Average Treatment effects (G-formula) :
    #>           Estimate  Std.Err     2.5%    97.5% P-value
    #> treat0     42.3170   3.8362  34.7982  49.8357       0
    #> treat1     15.0958   1.3284  12.4922  17.6994       0
    #> treat:1-0 -27.2211   4.1379 -35.3313 -19.1109       0
    #>
    #> Average Treatment effects (double robust) :
    #>           Estimate  Std.Err     2.5%    97.5% P-value
    #> treat0     42.3200   4.0417  34.3985  50.2415       0
    #> treat1     15.1545   1.2580  12.6888  17.6201       0
    #> treat:1-0 -27.1655   4.2405 -35.4768 -18.8543       0

# SessionInfo

    sessionInfo()
    #> R version 4.6.0 (2026-04-24)
    #> Platform: x86_64-pc-linux-gnu
    #> Running under: Ubuntu 24.04.4 LTS
    #>
    #> Matrix products: default
    #> BLAS:   /home/kkzh/.asdf/installs/r/4.6.0/lib/R/lib/libRblas.so
    #> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0  LAPACK version 3.12.0
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    #>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    #>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    #>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    #>
    #> time zone: Europe/Copenhagen
    #> tzcode source: system (glibc)
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] timereg_2.0.7  survival_3.8-6 mets_1.3.10
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] cli_3.6.6              knitr_1.51             rlang_1.2.0
    #>  [4] xfun_0.57              otel_0.2.0             jsonlite_2.0.0
    #>  [7] listenv_0.10.1         future.apply_1.20.2    lava_1.9.1
    #> [10] htmltools_0.5.9        stats4_4.6.0           sass_0.4.10
    #> [13] rmarkdown_2.31         grid_4.6.0             evaluate_1.0.5
    #> [16] jquerylib_0.1.4        fastmap_1.2.0          numDeriv_2016.8-1.1
    #> [19] yaml_2.3.12            mvtnorm_1.3-7          lifecycle_1.0.5
    #> [22] compiler_4.6.0         codetools_0.2-20       ucminf_1.2.3
    #> [25] Rcpp_1.1.1-1.1         future_1.70.0          lattice_0.22-9
    #> [28] digest_0.6.39          R6_2.6.1               parallelly_1.47.0
    #> [31] parallel_4.6.0         splines_4.6.0          Matrix_1.7-5
    #> [34] bslib_0.11.0           tools_4.6.0            RcppArmadillo_15.2.6-1
    #> [37] globals_0.19.1         cachem_1.1.0