# Analysis of multivariate binomial data: family analysis

#### Klaus Holst & Thomas Scheike

#### 2026-07-29

# Overview

When analysing multivariate binomial data with the aim of learning about the dependence present, possibly after adjusting for covariates, many models and methods are available:

- Random-effects logistic regression models covered elsewhere (`glmer` in `lme4`).

In the mets package you can fit the

- Pairwise odds ratio model

- Bivariate Probit model

  - With random effects
  - Special functionality for polygenic random effects modelling such as ACE, ADE ,AE and so forth.

- Additive gamma random effects model

  - Special functionality for polygenic random effects modelling such as ACE, ADE ,AE and so forth.

These last three models are all fitted in the mets package using composite likelihoods based on pairs within clusters. The models can be fitted specifically based on specifying which pairs one wants to use for the composite score.

The models are described in further details in the binomial-twin vignette.

# Simulated family data

We start by simulating family data with an additive gamma structure on ACE form. Here 10000 families consisting of two parents and two children. The response is ybin and there is one covariate x.

     library(mets)
     library(timereg)
    #> Loading required package: survival
    #>
    #> Attaching package: 'timereg'
    #> The following objects are masked from 'package:mets':
    #>
    #>     Event, kmplot, plotConfregion
     set.seed(100)
     data <- sim_binClaytonOakes_family_ace(500,2,1,beta=NULL,alpha=NULL)
     data$number <- c(1,2,3,4)
     data$child <- 1*(data$number==3)
     head(data)
    #>   ybin x   type cluster number child
    #> 1    1 1 mother       1      1     0
    #> 2    1 0 father       1      2     0
    #> 3    1 0  child       1      3     1
    #> 4    1 1  child       1      4     0
    #> 5    1 1 mother       2      1     0
    #> 6    1 0 father       2      2     0

We fit the marginal models, and here find a covariate effect at 0.3 for x. The marginals can be specified as desired.

     aa <- margbin <- glm(ybin~x,data=data,family=binomial())
     summary(aa)
    #>
    #> Call:
    #> glm(formula = ybin ~ x, family = binomial(), data = data)
    #>
    #> Coefficients:
    #>             Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)  0.49908    0.06387   7.814 5.52e-15 ***
    #> x            0.17531    0.09355   1.874   0.0609 .
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for binomial family taken to be 1)
    #>
    #>     Null deviance: 2610.2  on 1999  degrees of freedom
    #> Residual deviance: 2606.7  on 1998  degrees of freedom
    #> AIC: 2610.7
    #>
    #> Number of Fisher Scoring iterations: 4

# Additive Gamma model

For the Additive Gamma model we set-up the random effects included in such a family to make the ACE valid using some special functions for this.

The model is constructed with one environmental effect shared by all in the family and 8 genetic random effects with size 1/4 genetic variance. Looking at the first family we see that the mother and father both share half the genes with the children and that the two children also share half their genes with this specification. Below we also show an alternative specification of this model using all pairs.

    # make ace random effects design
    out <- ace_family_design(data,member="type",id="cluster")
    out$pardes
    #>       [,1] [,2]
    #>  [1,] 0.25    0
    #>  [2,] 0.25    0
    #>  [3,] 0.25    0
    #>  [4,] 0.25    0
    #>  [5,] 0.25    0
    #>  [6,] 0.25    0
    #>  [7,] 0.25    0
    #>  [8,] 0.25    0
    #>  [9,] 0.00    1
    head(out$des.rv,4)
    #>      m1 m2 m3 m4 f1 f2 f3 f4 env
    #> [1,]  1  1  1  1  0  0  0  0   1
    #> [2,]  0  0  0  0  1  1  1  1   1
    #> [3,]  1  1  0  0  1  1  0  0   1
    #> [4,]  1  0  1  0  1  0  1  0   1

We can now fit the model calling the two-stage function

    # fitting ace model for family structure
    ts <- binomial_twostage(margbin,data=data,clusters=data$cluster,
    theta=c(2,1),random.design=out$des.rv,theta.des=out$pardes)
    summary(ts)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                theta        se
    #> dependence1 2.817486 0.8981749
    #> dependence2 1.137637 0.2806719
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>             Estimate Std.Err   2.5%  97.5%   P-value
    #> dependence1   0.7124 0.09262 0.5308 0.8939 1.457e-14
    #> dependence2   0.2876 0.09262 0.1061 0.4692 1.899e-03
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5%   P-value
    #> p1    3.955  0.8668 2.256 5.654 5.051e-06
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"
    # true variance parameters
    c(2,1)
    #> [1] 2 1
    # total variance
    3
    #> [1] 3

# Pairwise fitting

We now specify the same model by extracting all pairs. The random effects structure is typically simpler when looking at pairs only, but we start by specifying the random effects jointly for the whole family. A special function writes up all combinations of pairs. There are 6 pairs within each family, and we keep track of which family each belongs to. We first simply supply the pairs and should get the same result as before.

    mm <- familycluster_index(data$cluster)
    head(mm$familypairindex,n=20)
    #>  [1] 1 2 1 3 1 4 2 3 2 4 3 4 5 6 5 7 5 8 6 7
    pairs <- mm$pairs
    dim(pairs)
    #> [1] 3000    2
    head(pairs,12)
    #>       [,1] [,2]
    #>  [1,]    1    2
    #>  [2,]    1    3
    #>  [3,]    1    4
    #>  [4,]    2    3
    #>  [5,]    2    4
    #>  [6,]    3    4
    #>  [7,]    5    6
    #>  [8,]    5    7
    #>  [9,]    5    8
    #> [10,]    6    7
    #> [11,]    6    8
    #> [12,]    7    8

Now with the pairs we fit the model

    tsp <- binomial_twostage(margbin,data=data,clusters=data$cluster,theta=c(2,1),detail=0,
            random.design=out$des.rv,theta.des=out$pardes,pairs=pairs)
    summary(tsp)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                theta        se
    #> dependence1 2.817486 0.8981749
    #> dependence2 1.137637 0.2806719
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>             Estimate Std.Err   2.5%  97.5%   P-value
    #> dependence1   0.7124 0.09262 0.5308 0.8939 1.457e-14
    #> dependence2   0.2876 0.09262 0.1061 0.4692 1.899e-03
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5%   P-value
    #> p1    3.955  0.8668 2.256 5.654 5.051e-06
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

Here a random sample of pairs are given instead and we get other estimates.

    set.seed(100)
    ssid <- sort(sample(1:nrow(pairs),nrow(pairs)/2))
    tsd <- binomial_twostage(aa,data=data,clusters=data$cluster,
                   theta=c(2,1),step=1.0,
                   random.design=out$des.rv,iid=1,Nit=10,
                   theta.des=out$pardes,pairs=pairs[ssid,])
    summary(tsd)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                 theta        se
    #> dependence1 3.0275402 1.6045129
    #> dependence2 0.8135103 0.3895686
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>             Estimate Std.Err     2.5%  97.5%   P-value
    #> dependence1   0.7882  0.1511  0.49210 1.0843 1.816e-07
    #> dependence2   0.2118  0.1511 -0.08431 0.5079 1.609e-01
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5% P-value
    #> p1    3.841   1.402 1.093 6.589 0.00615
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

To specify such a model when only the pairs are available we show how to specify the model. We here use the same marginal “aa” to make the results comparable. The marginal can also be fitted based on available data.

We start by selecting the data related to the pairs, setting up new ids, and specifying the model using the full design with 9 random effects. Below we also show how to use only the random effects needed for each pair, which is typically simpler.

    head(pairs[ssid,])
    #>      [,1] [,2]
    #> [1,]    1    2
    #> [2,]    1    3
    #> [3,]    2    3
    #> [4,]    2    4
    #> [5,]    3    4
    #> [6,]    6    8
    ids <- sort(unique(c(pairs[ssid,])))

    pairsids <- c(pairs[ssid,])
    pair.new <- matrix(fast.approx(ids,c(pairs[ssid,])),ncol=2)
    head(pair.new)
    #>      [,1] [,2]
    #> [1,]    1    2
    #> [2,]    1    3
    #> [3,]    2    3
    #> [4,]    2    4
    #> [5,]    3    4
    #> [6,]    5    7

    dataid <- dsort(data[ids,],"cluster")
    outid <- ace_family_design(dataid,member="type",id="cluster")
    outid$pardes
    #>       [,1] [,2]
    #>  [1,] 0.25    0
    #>  [2,] 0.25    0
    #>  [3,] 0.25    0
    #>  [4,] 0.25    0
    #>  [5,] 0.25    0
    #>  [6,] 0.25    0
    #>  [7,] 0.25    0
    #>  [8,] 0.25    0
    #>  [9,] 0.00    1
    head(outid$des.rv)
    #>      m1 m2 m3 m4 f1 f2 f3 f4 env
    #> [1,]  1  1  1  1  0  0  0  0   1
    #> [2,]  0  0  0  0  1  1  1  1   1
    #> [3,]  1  1  0  0  1  1  0  0   1
    #> [4,]  1  0  1  0  1  0  1  0   1
    #> [5,]  0  0  0  0  1  1  1  1   1
    #> [6,]  1  1  0  0  1  1  0  0   1

Now fitting the model with the data set up

    aa <- glm(ybin~x,data=dataid,family=binomial())
    tsdid <- binomial_twostage(aa,data=dataid,clusters=dataid$cluster,
             theta=c(2,1),random.design=outid$des.rv,theta.des=outid$pardes,pairs=pair.new)
    summary(tsdid)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                 theta       se
    #> dependence1 3.1389859 1.649758
    #> dependence2 0.8010544 0.399870
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>             Estimate Std.Err     2.5%  97.5%   P-value
    #> dependence1   0.7967  0.1492  0.50421 1.0892 9.358e-08
    #> dependence2   0.2033  0.1492 -0.08917 0.4958 1.731e-01
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5%  P-value
    #> p1     3.94   1.438 1.121 6.759 0.006153
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

We now specify the design specifically using the pairs. The random.design and design on the parameters are now given for each pair, as a 3 dimensional matrix. with a direct specification of random.design and the design on the parameters theta.design. In addition we need also to give the number of random effects for each pair. These basic things are constructed by certain functions for the ACE design.

    pair.types <-  matrix(dataid[c(t(pair.new)),"type"],byrow=T,ncol=2)
    head(pair.new,7)
    #>      [,1] [,2]
    #> [1,]    1    2
    #> [2,]    1    3
    #> [3,]    2    3
    #> [4,]    2    4
    #> [5,]    3    4
    #> [6,]    5    7
    #> [7,]    6    7
    head(pair.types,7)
    #>      [,1]     [,2]
    #> [1,] "mother" "father"
    #> [2,] "mother" "child"
    #> [3,] "father" "child"
    #> [4,] "father" "child"
    #> [5,] "child"  "child"
    #> [6,] "father" "child"
    #> [7,] "child"  "child"
    ###
    theta.des  <- rbind( c(rbind(c(1,0),  c(1,0),  c(0,1),  c(0,0))),
                 c(rbind(c(0.5,0),c(0.5,0),c(0.5,0),c(0,1))))
    random.des <- rbind(
            c(1,0,1,0),c(0,1,1,0),
            c(1,1,0,1),c(1,0,1,1))
    mf <- 1*(pair.types[,1]=="mother" & pair.types[,2]=="father")
    ##          pair, rv related to pairs,  theta.des related to pair
    pairs.new <- cbind(pair.new,(mf==1)*1+(mf==0)*3,(mf==1)*2+(mf==0)*4,(mf==1)*1+(mf==0)*2,(mf==1)*3+(mf==0)*4)

`pairs.new` is a matrix with

- columns 1:2 giving the indices of the data points

- columns 3:4 giving the indices of the `random.design` for the different pairs

- column 5 giving the indices of `theta.des` written as rows

- column 6 giving the number of random variables for this pair

The length of all rows of theta.des are the maximum number of random effects \(\times\) the number of parameters. These two numbers are given in the call. In this case 4 \(\times\) 2. So theta.des has rows of length \(8\), possibly including some 0’s for rows not relevant due to fewer random effects, as is the case here for pairs that do not share genetic effects.

For pair 1 that is a mother/father pair, we see that they share 1 environmental random effect of size 1. There are also two genetic effects that are unshared between the two. So a total of 3 random effects are needed here. The theta.des relates the 3 random effects to possible relationships in the parameters. Here the genetic effects are full and so is the environmental effect. In contrast we also consider a mother/child pair that share half the genes, now with random effects with (1/2) gene variance. We there need 4 random effects, 2 non-shared half-gene, 1 shared half-gene, and one shared full environmental effect.

    # 3 rvs here
    random.des
    #>      [,1] [,2] [,3] [,4]
    #> [1,]    1    0    1    0
    #> [2,]    0    1    1    0
    #> [3,]    1    1    0    1
    #> [4,]    1    0    1    1
    theta.des
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> [1,]  1.0  1.0  0.0    0    0    0    1    0
    #> [2,]  0.5  0.5  0.5    0    0    0    0    1

    head(pairs.new)
    #>      [,1] [,2] [,3] [,4] [,5] [,6]
    #> [1,]    1    2    1    2    1    3
    #> [2,]    1    3    3    4    2    4
    #> [3,]    2    3    3    4    2    4
    #> [4,]    2    4    3    4    2    4
    #> [5,]    3    4    3    4    2    4
    #> [6,]    5    7    3    4    2    4

Now fitting the model, and we see that it is a lot quicker due to the fewer random effects needed for pairs. We need to also specify the number of parameters in this case.

    tsdid2 <- binomial_twostage(aa,data=dataid,clusters=dataid$cluster, theta=c(2,1),
               random.design=random.des,theta.des=theta.des,pairs=pairs.new,dim.theta=2)
    summary(tsdid2)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                 theta       se
    #> dependence1 3.1389859 1.649758
    #> dependence2 0.8010544 0.399870
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>             Estimate Std.Err     2.5%  97.5%   P-value
    #> dependence1   0.7967  0.1492  0.50421 1.0892 9.358e-08
    #> dependence2   0.2033  0.1492 -0.08917 0.4958 1.731e-01
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5%  P-value
    #> p1     3.94   1.438 1.121 6.759 0.006153
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

The same model can be specifed even simpler via the kinship coefficient. For this specification there are 4 random effects for each pair, but some have variance 0. The mother-father pair, here shares a random effect with variance 0, and have two non-shared genetic effects with full variance, in addition to a fully shared environmental effect.

    kinship  <- rep(0.5,nrow(pair.types))
    kinship[pair.types[,1]=="mother" & pair.types[,2]=="father"] <- 0
    head(kinship,n=10)
    #>  [1] 0.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

    out <- make_pairwise_design(pair.new,kinship,type="ace")

    tsdid3 <- binomial_twostage(aa,data=dataid,clusters=dataid$cluster,
                 theta=c(2,1)/9,random.design=out$random.design,
                 theta.des=out$theta.des,pairs=out$new.pairs,dim.theta=2)
    summary(tsdid3)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                 theta       se
    #> dependence1 3.1389859 1.649758
    #> dependence2 0.8010544 0.399870
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>             Estimate Std.Err     2.5%  97.5%   P-value
    #> dependence1   0.7967  0.1492  0.50421 1.0892 9.358e-08
    #> dependence2   0.2033  0.1492 -0.08917 0.4958 1.731e-01
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5%  P-value
    #> p1     3.94   1.438 1.121 6.759 0.006153
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

## Pairwise dependence modelling

    library(mets)
    set.seed(1000)
    data <- sim_binClaytonOakes_family_ace(500,2,1,beta=NULL,alpha=NULL)
    head(data)
    #>   ybin x   type cluster
    #> 1    0 1 mother       1
    #> 2    0 0 father       1
    #> 3    0 1  child       1
    #> 4    0 0  child       1
    #> 5    1 0 mother       2
    #> 6    1 0 father       2
    data$number <- c(1,2,3,4)
    data$child <- 1*(data$number==3)

    mm <- familycluster_index(data$cluster)
    head(mm$familypairindex,n=20)
    #>  [1] 1 2 1 3 1 4 2 3 2 4 3 4 5 6 5 7 5 8 6 7
    pairs <- mm$pairs
    dim(pairs)
    #> [1] 3000    2
    head(pairs,12)
    #>       [,1] [,2]
    #>  [1,]    1    2
    #>  [2,]    1    3
    #>  [3,]    1    4
    #>  [4,]    2    3
    #>  [5,]    2    4
    #>  [6,]    3    4
    #>  [7,]    5    6
    #>  [8,]    5    7
    #>  [9,]    5    8
    #> [10,]    6    7
    #> [11,]    6    8
    #> [12,]    7    8

     dtypes <- interaction( data[pairs[,1],"type"], data[pairs[,2],"type"])
     dtypes <- droplevels(dtypes)
     table(dtypes)
    #> dtypes
    #>   child.child  father.child  mother.child mother.father
    #>           500          1000          1000           500
     dm <- model.matrix(~-1+factor(dtypes))

Now with the pairs we fit the model

    aa <- glm(ybin~x,data=data,family=binomial())

    tsp <- binomial_twostage(aa,data=data, clusters=data$cluster,
             theta.des=dm,pairs=cbind(pairs,1:nrow(dm)))
    summary(tsp)
    #> Dependence parameter for Odds-Ratio (Plackett) model
    #> $estimates
    #>                                theta        se
    #> factor(dtypes)child.child   3.520109 0.7143594
    #> factor(dtypes)father.child  3.871698 0.6055928
    #> factor(dtypes)mother.child  3.922180 0.6233304
    #> factor(dtypes)mother.father 1.328730 0.2657202
    #>
    #> $log.or
    #>                             Estimate Std.Err    2.5%  97.5%   P-value
    #> factor(dtypes)child.child     1.2585  0.2029  0.8607 1.6562 5.596e-10
    #> factor(dtypes)father.....     1.3537  0.1564  1.0471 1.6603 4.952e-18
    #> factor(dtypes)mother.....     1.3666  0.1589  1.0552 1.6781 8.017e-18
    #> factor(dtypes)mother......1   0.2842  0.2000 -0.1077 0.6762 1.552e-01
    #>
    #> $type
    #> [1] "plackett"
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

# Pairwise odds ratio model

To fit the pairwise odds-ratio model in the case of a pair-specification there are two options for fitting the model.

1. One option is to set up some artificial data similar to twin data with

- a pair-cluster-id (clusters)
- with a cluster-id to get GEE type standard errors (se.cluster)

2. We can also use the specify the design via the theta.des that is also a matrix of dimension pairs x design with the design for POR model.

Starting by the second option. We need to start by specify the design of the odds-ratio of each pair. We set up the data and find all combinations within the pairs. Subsequently, we remove all the empty groups, by grouping together the factor levels 4:9, and then we construct the design.

    tdp <-cbind( dataid[pair.new[,1],],dataid[pair.new[,2],])
    names(tdp) <- c(paste(names(dataid),"1",sep=""),
            paste(names(dataid),"2",sep=""))
    tdp <-transform(tdp,tt=interaction(type1,type2))
    dlevel(tdp)
    #> tt #levels=:6
    #> [1] "child.child"   "father.child"  "mother.child"  "child.father"
    #> [5] "father.father" "mother.father"
    #> -----------------------------------------
    drelevel(tdp,newlevels=list(mother.father=4:9)) <-  obs.types~tt
    dtable(tdp,~tt+obs.types)
    #>
    #>               obs.types mother.father child.child father.child mother.child
    #> tt
    #> child.child                         0         232            0            0
    #> father.child                        0           0          525            0
    #> mother.child                        0           0            0          510
    #> child.father                        0           0            0            0
    #> father.father                       0           0            0            0
    #> mother.father                     233           0            0            0
    tdp <- model.matrix(~-1+factor(obs.types),tdp)

We can then fit the pairwise model using the pairs and the pair-design for describing the OR. The results are consistent with the ACE model: mother-father pairs have a lower dependence since it is driven only by environmental effects. All other combinations appear to have the same dependence, as expected.

To fit the OR model it is generally recommended to use the var.link to use the parametrization with log-odd-ratio regression.

    ###porpair <- binomial.twostage(aa,data=dataid,clusters=dataid$cluster,
    ###           theta.des=tdp,pairs=pair.new,model="or",var.link=1)
    ###summary(porpair)

# 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_1.3.12
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] digest_0.6.39          numDeriv_2016.8-1.1    codetools_0.2-20
    #>  [4] RcppArmadillo_15.2.6-1 Matrix_1.7-5           xfun_0.57
    #>  [7] lattice_0.22-9         splines_4.6.0          lava_1.9.2.1
    #> [10] knitr_1.51             parallel_4.6.0         mvtnorm_1.3-7
    #> [13] cli_3.6.6              parallelly_1.47.0      grid_4.6.0
    #> [16] future_1.70.0          compiler_4.6.0         globals_0.19.1
    #> [19] tools_4.6.0            listenv_0.10.1         future.apply_1.20.2
    #> [22] evaluate_1.0.5         Rcpp_1.1.1-1.1         otel_0.2.0
    #> [25] rlang_1.2.0