Author: Klaus Holst & Thomas Scheike
Date: 2021-09-05

# A practical guide to Human Genetics with Lifetime Data

#### Klaus Holst & Thomas Scheike

#### 2021-09-05

This vignette demonstrates how to analyze familial resemblance for twins using the {}-package and is accompanying the review by Scheike and Holst (2020).

We consider a data-set in that resembles the data of that were based on the NorTwinCan a collaborative research project studying the genetic and environmental components of prostate cancer. The data comprises around 18,000 DZ twins and 11,000 MZ twins. It was a population based register study based on the Danish, Finnish, Norwegian, and Swedish twin registries.

We first illustrate a hazards based analysis to show how one would study dependence in survival data. This needs to be done under assumptions about independent competing risks when the outcome of interest is observed subject to competing risks (here death).

This seems reasonable here since the occurrence of cancer prior to death only contains weak association with the risk of death for the other twin, and vice-versa.

First looking at the data

     library(mets)
     data(prt)

     dtable(prt,~status+cancer)
    #>
    #>        cancer     0     1
    #> status
    #> 0             21283     0
    #> 1              6997     0
    #> 2                 0   942
     dtable(prt,~zyg+country,level=1)
    #>
    #> zyg
    #>    DZ    MZ
    #> 17991 11231
    #>
    #> country
    #> Denmark Finland  Norway  Sweden
    #>    9671    3926    4107   11518

we see that there are 21283 censorings and 6997 deaths (prior to cancer) and a total of 942 prostate cancers. Approximately half the data consist of DZ twins. In addition we see that there are around 10000 twins from Denmark and Sweden, and only 4000 from Norway and Finland, respectively.

# Survival

Under assumption of random effects acting independently on different cause specific hazards we can analyse competing risks data considering the cause-specific hazard. Typically, this can be questionable and the cumulative incidence modelling below does not rely on this assumption.

We consider the cause specific hazard of cancer in the competing risks model with death and cancer.

First estimating the marginal hazards for each country.

     # Marginal Cox model here stratified on country without covariates
     margph <- phreg(Surv(time,cancer)~strata(country)+cluster(id),data=prt)
     plot(margph)

We see that the marginal of Denmark in particular is quite different.

Then we fit a two-stage random effects models with country specific marginals and random-effects variances that differ for MZ and DZ twins.

     # Clayton-Oakes, MLE , overall variance
     fitco1<-twostageMLE(margph,data=prt,theta=2.7)
     summary(fitco1)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                Coef.        SE        z        P-val Kendall tau        SE
    #> dependence1 2.782962 0.4225572 6.586001 4.518319e-11   0.5818491 0.0369421
    #>
    #> $type
    #> NULL
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

     fitco2 <- survival.twostage(margph,data=prt,theta=2.7,clusters=prt$id,var.link=0)
     summary(fitco2)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                Coef.        SE        z        P-val Kendall tau         SE
    #> dependence1 2.782962 0.4225529 6.586069 4.516254e-11   0.5818491 0.03694172
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

     ## now with different random effects for MZ and DZ
     mm <- model.matrix(~-1+factor(zyg),prt)
     fitco3<-twostageMLE(margph,data=prt,theta=1,theta.des=mm)
     summary(fitco3)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                  Coef.        SE        z        P-val Kendall tau         SE
    #> factor(zyg)DZ 1.318966 0.3861577 3.415614 6.363831e-04   0.3974027 0.07011148
    #> factor(zyg)MZ 5.421921 0.9626267 5.632423 1.776956e-08   0.7305280 0.03495065
    #>
    #> $type
    #> NULL
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

     fitco4 <- survival.twostage(margph,data=prt,theta=1,clusters=prt$id,var.link=0,theta.des=mm)
     summary(fitco4)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                  Coef.        SE        z        P-val Kendall tau         SE
    #> factor(zyg)DZ 1.318966 0.3861745 3.415466 6.367306e-04   0.3974027 0.07011454
    #> factor(zyg)MZ 5.421920 0.9625027 5.633148 1.769493e-08   0.7305280 0.03494615
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"
     round(estimate(coef=fitco4$coef,vcov=fitco4$var.theta)$coefmat[,c(1,3:4)],2)
    #>               Estimate 2.5% 97.5%
    #> factor(zyg)DZ     1.32 0.56  2.08
    #> factor(zyg)MZ     5.42 3.54  7.31

     ## mz kendalls tau
     kendall.ClaytonOakes.twin.ace(fitco4$theta[2],0,K=10000)$mz.kendall
    #> [1] 0.7327442
     ## dz kendalls tau
     kendall.ClaytonOakes.twin.ace(fitco4$theta[1],0,K=10000)$mz.kendall
    #> [1] 0.3977883

The dependence of MZ twins is much stronger, and is summarized by a variance at \(5.42\) in contrast to the \(DZ\) variance at \(1.32\).

Now we look at the polygenic modelling for survival data, here applied to the cause specific hazards.

     ### setting up design for random effects and parameters of random effects
     desace <- twin.polygen.design(prt,type="ace")

     ### ace model
     fitace <- survival.twostage(margph,data=prt,theta=1,
           clusters=prt$id,var.link=0,model="clayton.oakes",
           numDeriv=1,random.design=desace$des.rv,theta.des=desace$pardes)
     summary(fitace)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                 Coef.       SE         z        P-val Kendall tau          SE
    #> dependence1  7.223855 1.841525  3.922757 8.754161e-05   0.7831709  0.04328951
    #> dependence2 -1.582396 1.088423 -1.453843 1.459898e-01  -3.7892257 12.48240540
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>      Estimate Std.Err    2.5%   97.5%   P-value
    #> [1,]   1.2805  0.1698  0.9477 1.61324 4.615e-14
    #> [2,]  -0.2805  0.1698 -0.6132 0.05226 9.850e-02
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5%   P-value
    #> p1    5.641  0.9894 3.702 7.581 1.184e-08
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

     ### ace model with positive random effects variances
     # fitacee <- survival.twostage(margph,data=prt,theta=1,
     #      clusters=prt$id,var.link=1,model="clayton.oakes",
     #      numDeriv=1,random.design=desace$des.rv,theta.des=desace$pardes)
     #summary(fitacee)

     ### ae model
     #desae <- twin.polygen.design(prt,type="ae")
     #fitae <- survival.twostage(margph,data=prt,theta=1,
     #      clusters=prt$id,var.link=0,model="clayton.oakes",
     #      numDeriv=1,random.design=desae$des.rv,theta.des=desae$pardes)
     #summary(fitae)

     ### de model
     desde <- twin.polygen.design(prt,type="de")
     fitde <- survival.twostage(margph,data=prt,theta=1,
           clusters=prt$id,var.link=0,model="clayton.oakes",
           numDeriv=1,random.design=desde$des.rv,theta.des=desde$pardes)
     summary(fitde)
    #> Dependence parameter for Clayton-Oakes model
    #> Variance of Gamma distributed random effects
    #> $estimates
    #>                Coef.        SE        z        P-val Kendall tau         SE
    #> dependence1 5.940643 0.9837336 6.038873 1.551941e-09   0.7481312 0.03120299
    #>
    #> $type
    #> [1] "clayton.oakes"
    #>
    #> $h
    #>      Estimate Std.Err 2.5% 97.5% P-value
    #> [1,]        1       0    1     1       0
    #>
    #> $vare
    #> NULL
    #>
    #> $vartot
    #>    Estimate Std.Err  2.5% 97.5%   P-value
    #> p1    5.941  0.9837 4.013 7.869 1.552e-09
    #>
    #> attr(,"class")
    #> [1] "summary.mets.twostage"

The DE model fits quite well. In summary all shared variance is due to genes and there is no suggestion of a shared environmental effect.

# Concordance and Casewise

First we estimate the concordance of joint prostate cancer. The two-twins are censored at the same time, otherwise we would enforce this in the data by artificially censor both twins at the first censoring time. Given, however, that we have the same-censoring assumption satisfied we can do the stanadar Aalen-Johansen product-limit estimator of the concordance probabilities for MZ and DZ twins.

For simplicity we do not do this for each country even though as we show there are big differences between the countries.

     prt <-  force.same.cens(prt,cause="status")

     dtable(prt,~status+cancer)
    #>
    #>        cancer     0     1
    #> status
    #> 0             21580    77
    #> 1              6700     0
    #> 2                 0   865
     dtable(prt,~status+country)
    #>
    #>        country Denmark Finland Norway Sweden
    #> status
    #> 0                 7394    2622   3154   8487
    #> 1                 2140    1138    830   2592
    #> 2                  137     166    123    439
     dtable(prt,~zyg+country)
    #>
    #>     country Denmark Finland Norway Sweden
    #> zyg
    #> DZ             6191    2833   2393   6574
    #> MZ             3480    1093   1714   4944

     ## cumulative incidence with cluster standard errors.
     cif1 <- cif(Event(time,status)~strata(country)+cluster(id),prt,cause=2)
     plot(cif1,se=1)

     cifa <- cif(Event(time,status)~+1,prt,cause=2)

     ### concordance estimator, ignoring country differences.
     p11 <- bicomprisk(Event(time,status)~strata(zyg)+id(id),data=prt,cause=c(2,2))
    #> Strata 'DZ'
    #> Strata 'MZ'

     p11mz <- p11$model$"MZ"
     p11dz <- p11$model$"DZ"
     par(mfrow=c(1,2))
     ## Concordance
     plot(p11mz,ylim=c(0,0.1));
     plot(p11dz,ylim=c(0,0.1));

Now we compare the concordance to the marginals to get a measure that takes the marginals into account when evaluating the strength of the association.

     library(prodlim)
     outm <- prodlim(Hist(time,status)~+1,data=prt)

     cifzyg <- cif(Event(time,status)~+strata(zyg)+cluster(id),data=prt,cause=2)
     cifprt <- cif(Event(time,status)~country+cluster(id),data=prt,cause=2)

     times <- 70:100
     cifmz <- predict(outm,cause=2,time=times,newdata=data.frame(zyg="MZ")) ## cause is 2 (second cause)
     cifdz <- predict(outm,cause=2,time=times,newdata=data.frame(zyg="DZ"))

     ### concordance for MZ and DZ twins<
     cc <- bicomprisk(Event(time,status)~strata(zyg)+id(id),data=prt,cause=c(2,2),prodlim=TRUE)
    #> Strata 'DZ'
    #> Strata 'MZ'
     ccdz <- cc$model$"DZ"
     ccmz <- cc$model$"MZ"

     cdz <- casewise(ccdz,outm,cause.marg=2)
     cmz <- casewise(ccmz,outm,cause.marg=2)

     dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
     conczyg <- cif(Event(time,status)~strata(zyg)+cluster(id),data=dd,cause=1)

     par(mfrow=c(1,2))
     plot(conczyg,se=TRUE,col=cols[2:1], lty=ltys[2:1], legend=FALSE,xlab="Age",ylab="Concordance")
     legend("topleft",c("concordance-MZ","concordance-DZ"),col=cols[1:2],lty=ltys[1:2])

     plot(cmz,ci=NULL,ylim=c(0,.8),xlim=c(70,97),legend=FALSE,col=cols[c(1,3,3)],lty=ltys[c(1,3,3)],
          ylab="Casewise",xlab="Age")
      plot(cdz,ci=NULL,ylim=c(0,.8),xlim=c(70,97),legend=FALSE,ylab="Casewise",xlab="Age",
          col=c(cols[2],NA,NA), lty=ltys[c(2,3,3)], add=TRUE)
     with(data.frame(cmz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=cols[1]))
     with(data.frame(cdz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=cols[2]))
     legend("topleft",c("casewise-MZ","casewise-DZ","marginal"),col=cols, lty=ltys, bg="white")

     summary(cdz)
    #> Casewise concordance and standard errors
    #>        time casewise conc se casewise
    #>  [1,]  59.5        0.0866      0.0865
    #>  [2,]  60.5        0.0659      0.0659
    #>  [3,]  61.6        0.0593      0.0593
    #>  [4,]  62.7        0.0483      0.0483
    #>  [5,]  63.7        0.0358      0.0358
    #>  [6,]  64.8        0.0279      0.0279
    #>  [7,]  65.8        0.0223      0.0223
    #>  [8,]  66.9        0.0197      0.0197
    #>  [9,]  68.0        0.0415      0.0297
    #> [10,]  69.0        0.0335      0.0240
    #> [11,]  70.1        0.0452      0.0264
    #> [12,]  71.1        0.0855      0.0352
    #> [13,]  72.2        0.0728      0.0300
    #> [14,]  73.2        0.0888      0.0317
    #> [15,]  74.3        0.1010      0.0321
    #> [16,]  75.4        0.1020      0.0310
    #> [17,]  76.4        0.1130      0.0318
    #> [18,]  77.5        0.1230      0.0320
    #> [19,]  78.5        0.1400      0.0334
    #> [20,]  79.6        0.1470      0.0332
    #> [21,]  80.7        0.1530      0.0329
    #> [22,]  81.7        0.1460      0.0307
    #> [23,]  82.8        0.1470      0.0298
    #> [24,]  83.8        0.1600      0.0307
    #> [25,]  84.9        0.1470      0.0282
    #> [26,]  86.0        0.1620      0.0297
    #> [27,]  87.0        0.1680      0.0300
    #> [28,]  88.1        0.1820      0.0311
    #> [29,]  89.1        0.1760      0.0301
    #> [30,]  90.2        0.1950      0.0323
    #> [31,]  91.2        0.2040      0.0332
    #> [32,]  92.3        0.1970      0.0321
    #> [33,]  93.4        0.1940      0.0315
    #> [34,]  94.4        0.1970      0.0318
    #> [35,]  95.5        0.1940      0.0314
    #> [36,]  96.5        0.1930      0.0312
    #> [37,]  97.6        0.2040      0.0330
    #> [38,]  98.7        0.2010      0.0325
    #> [39,]  99.7        0.1990      0.0322
    #> [40,] 101.0        0.1980      0.0321
    #> [41,] 102.0        0.1950      0.0316
    #> [42,] 103.0        0.1940      0.0314
    #> [43,] 104.0        0.1940      0.0314
    #> [44,] 105.0        0.1930      0.0312
    #> [45,] 106.0        0.1920      0.0311
    #> [46,] 107.0        0.1920      0.0311
    #> [47,] 108.0        0.1910      0.0309
     summary(cmz)
    #> Casewise concordance and standard errors
    #>        time casewise conc se casewise
    #>  [1,]  60.6         0.519      0.2590
    #>  [2,]  61.6         0.466      0.2330
    #>  [3,]  62.7         0.380      0.1900
    #>  [4,]  63.7         0.285      0.1420
    #>  [5,]  64.8         0.286      0.1280
    #>  [6,]  65.8         0.228      0.1020
    #>  [7,]  66.9         0.295      0.1120
    #>  [8,]  67.9         0.306      0.1090
    #>  [9,]  68.9         0.327      0.1040
    #> [10,]  70.0         0.338      0.0981
    #> [11,]  71.0         0.345      0.0926
    #> [12,]  72.1         0.399      0.0946
    #> [13,]  73.1         0.414      0.0909
    #> [14,]  74.2         0.426      0.0874
    #> [15,]  75.2         0.388      0.0798
    #> [16,]  76.3         0.391      0.0773
    #> [17,]  77.3         0.410      0.0769
    #> [18,]  78.4         0.392      0.0723
    #> [19,]  79.4         0.410      0.0721
    #> [20,]  80.5         0.423      0.0714
    #> [21,]  81.5         0.400      0.0666
    #> [22,]  82.6         0.442      0.0685
    #> [23,]  83.6         0.446      0.0676
    #> [24,]  84.7         0.433      0.0643
    #> [25,]  85.7         0.413      0.0612
    #> [26,]  86.8         0.389      0.0578
    #> [27,]  87.8         0.396      0.0578
    #> [28,]  88.9         0.396      0.0573
    #> [29,]  89.9         0.399      0.0574
    #> [30,]  91.0         0.386      0.0556
    #> [31,]  92.0         0.400      0.0570
    #> [32,]  93.1         0.393      0.0560
    #> [33,]  94.1         0.415      0.0590
    #> [34,]  95.2         0.477      0.0669
    #> [35,]  96.2         0.493      0.0690
    #> [36,]  97.3         0.511      0.0714
    #> [37,]  98.3         0.507      0.0708
    #> [38,]  99.4         0.500      0.0699
    #> [39,] 100.0         0.525      0.0739
    #> [40,] 101.0         0.520      0.0731
    #> [41,] 103.0         0.514      0.0723
    #> [42,] 104.0         0.541      0.0767
    #> [43,] 105.0         0.541      0.0767

     Cpred(cmz$casewise,80)
    #>      xval casewise conc se casewise
    #> [1,]   80     0.4096715  0.07211013
     Cpred(cdz$casewise,80)
    #>      xval casewise conc se casewise
    #> [1,]   80     0.1469487  0.03321234

     dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
     conczyg <- cif(Event(time,status)~strata(zyg)+cluster(id),data=dd,cause=1)

     par(mfrow=c(1,2))
     plot(conczyg,se=TRUE,legend=FALSE,xlab="Age",ylab="Concordance")
     legend("topleft",c("concordance-DZ","concordance-MZ"),col=c(1,2),lty=1)
     plot(cmz,ci=NULL,ylim=c(0,0.6),xlim=c(70,100),legend=FALSE,col=c(2,3,3),ylab="Casewise",xlab="Age",lty=c(1,3))
     plot(cdz,ci=NULL,ylim=c(0,0.6),xlim=c(70,100),legend=FALSE,ylab="Casewise",xlab="Age",
          col=c(1,3,3), add=TRUE, lty=c(2,3))
     legend("topleft",c("casewise-MZ","casewise-DZ","marginal"),col=c(2,1,3),lty=1)
     with(data.frame(cmz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=2))
     with(data.frame(cdz$casewise),plotConfRegionSE(time,casewise.conc,se.casewise,col=1))

The standard errors above are slightly off since they only reflect the uncertainty from the concordance estimation. This can be improved by doing specific calculations for a specific time-point uisng the binomial regression function that gives and iid decomposition for the paramters. We thus apply the binomial regression to estimate the concordance as well as the marginal, and combine the iid decompositions when estimating the standard error. We also do this ignoring country differences.

     ### new version of Casewise for specific time-point based on binreg
     dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
     newdata <- data.frame(zyg=c("DZ","MZ"),id=1)

     ## concordance
     bcif1 <- binreg(Event(time,status)~-1+factor(zyg)+cluster(id),dd,time=80,cause=1,cens.model=~strata(zyg))
     pconc <- predict(bcif1,newdata)

     ## marginal estimates
     mbcif1 <- binreg(Event(time,status)~cluster(id),prt,time=80,cause=2)
     mc <- predict(mbcif1,newdata)

     ### casewise with improved se's from log-scale
     cse <- binregCasewise(bcif1,mbcif1)
     cse
    #> $coef
    #>     Estimate      2.5%     97.5%
    #> p1 0.1586277 0.1029195 0.2444898
    #> p2 0.4041311 0.2843829 0.5743030
    #>
    #> $logcoef
    #>    Estimate Std.Err   2.5%   97.5%   P-value
    #> p1   -1.841  0.2207 -2.274 -1.4086 7.331e-17
    #> p2   -0.906  0.1793 -1.257 -0.5546 4.346e-07

It can be useful also to simply model the concordance given covariates, and in this case we might find it important to adjust for country, or to see if the differences between MZ and DZ are comparable across contries even though clearly DK has a much lower cumulative incidence of prostate cancer.

     ### semi-parametric modelling of concordance
     dd <- bicompriskData(Event(time,status)~country+strata(zyg)+id(id),data=prt,cause=c(2,2))
     regconc <- cifreg(Event(time,status)~country*zyg,data=dd,prop=NULL)
     regconc
    #> Call:
    #> cifreg(formula = Event(time, status) ~ country * zyg, data = dd,
    #>     propodds = NULL)
    #>
    #>      n events
    #>  14222    106
    #>
    #>  14222 clusters
    #> coeffients:
    #>                      Estimate     S.E.  dU^-1/2 P-value
    #> countryFinland        2.20738  0.76281  0.76379  0.0038
    #> countryNorway         2.31658  0.78992  0.79059  0.0034
    #> countrySweden         2.02237  0.73973  0.74002  0.0063
    #> zygMZ                 2.30872  0.77397  0.77460  0.0029
    #> countryFinland:zygMZ -1.26737  0.87061  0.87193  0.1455
    #> countryNorway:zygMZ  -2.14613  0.94349  0.94430  0.0229
    #> countrySweden:zygMZ  -1.32560  0.82195  0.82282  0.1068
    #>
    #> exp(coeffients):
    #>                       Estimate      2.5%   97.5%
    #> countryFinland        9.091833  2.038674 40.5467
    #> countryNorway        10.140896  2.156230 47.6933
    #> countrySweden         7.556217  1.772747 32.2079
    #> zygMZ                10.061559  2.207303 45.8637
    #> countryFinland:zygMZ  0.281571  0.051112  1.5511
    #> countryNorway:zygMZ   0.116936  0.018401  0.7431
    #> countrySweden:zygMZ   0.265643  0.053046  1.3303
     ### interaction test
     wald.test(regconc,coef.null=5:7)
    #>       lin.comb        se     lower      upper       pval
    #> [1,] -1.267372 0.8706072 -2.973731  0.4389867 0.14546630
    #> [2,] -2.146126 0.9434944 -3.995341 -0.2969109 0.02292653
    #> [3,] -1.325604 0.8219473 -2.936591  0.2853834 0.10679668
    #>
    #>  Wald test
    #>
    #> data:
    #> chisq = 5.2661, df = 3, p-value = 0.1533

     regconc <- cifreg(Event(time,status)~country+zyg,data=dd,prop=NULL)
     regconc
    #> Call:
    #> cifreg(formula = Event(time, status) ~ country + zyg, data = dd,
    #>     propodds = NULL)
    #>
    #>      n events
    #>  14222    106
    #>
    #>  14222 clusters
    #> coeffients:
    #>                Estimate    S.E. dU^-1/2 P-value
    #> countryFinland  1.29246 0.34863 0.35151  0.0002
    #> countryNorway   0.93336 0.39545 0.39366  0.0183
    #> countrySweden   1.07439 0.31940 0.31876  0.0008
    #> zygMZ           1.01055 0.19995 0.19877  0.0000
    #>
    #> exp(coeffients):
    #>                Estimate   2.5%  97.5%
    #> countryFinland   3.6417 1.8389 7.2121
    #> countryNorway    2.5430 1.1715 5.5203
    #> countrySweden    2.9282 1.5658 5.4762
    #> zygMZ            2.7471 1.8564 4.0651

     ## logistic link
     logitregconc <- cifreg(Event(time,status)~country+zyg,data=dd)
     summary(logitregconc)
    #>
    #>      n events
    #>  14222    106
    #>
    #>  14222 clusters
    #> coeffients:
    #>                Estimate    S.E. dU^-1/2 P-value
    #> countryFinland  1.30427 0.35262 0.35146  0.0002
    #> countryNorway   0.94077 0.39999 0.39365  0.0187
    #> countrySweden   1.08494 0.32247 0.31871  0.0008
    #> zygMZ           1.02335 0.20283 0.19873  0.0000
    #>
    #> exp(coeffients):
    #>                Estimate   2.5%  97.5%
    #> countryFinland   3.6850 1.8462 7.3552
    #> countryNorway    2.5620 1.1698 5.6111
    #> countrySweden    2.9593 1.5729 5.5676
    #> zygMZ            2.7825 1.8697 4.1408

    ### library(Publish)
    ### publish(round(summary(logitregconc)$exp.coef[,-c(2,5)],2),latex=TRUE,digits=2)

# Competing risk using additive Gamma

Here we do the cumulative incidence random effects modelling

      times <- seq(50,90,length.out=5)
      cif1 <- comp.risk(Event(time,status)~-1+factor(country)+cluster(id),prt,
               cause=2,times=times,max.clust=NULL)

      mm <- model.matrix(~-1+factor(zyg),prt)
      out1<-random.cif(cif1,data=prt,cause1=2,cause2=2,theta=1,
              theta.des=mm,same.cens=TRUE,step=0.5)
      summary(out1)
    #> Random effect variance for variation due to clusters
    #>
    #> Cause 2 and cause 2
    #>                  Coef.        SE        z        P-val Cross odds ratio
    #> factor(zyg)DZ 1.034939 0.4379824 2.362969 0.0181292052         2.034939
    #> factor(zyg)MZ 3.877446 1.1338630 3.419678 0.0006269539         4.877446
    #>                      SE
    #> factor(zyg)DZ 0.4379824
    #> factor(zyg)MZ 1.1338630
      round(estimate(coef=out1$theta,vcov=out1$var.theta)$coefmat[,c(1,3:4)],2)
    #>    Estimate 2.5% 97.5%
    #> p1     1.03 0.18  1.89
    #> p2     3.88 1.66  6.10

      desace <- twin.polygen.design(prt,type="ace")

      outacem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
         same.cens=TRUE,theta=c(0.45,0.15),var.link=0,
             step=0.5,theta.des=desace$pardes,random.design=desace$des.rv)
      outacem$score
    #>              [,1]
    #> [1,] 4.494699e-07
    #> [2,] 2.722100e-08
      summary(outacem)
    #> Random effect parameters for additive gamma random effects
    #>
    #> Cause 2 and cause 2
    #>
    #>
    #>        Coef.     SE     z    P-val
    #> [1,]  0.3360 0.0905  3.71 0.000206
    #> [2,] -0.0798 0.0673 -1.19 0.236000
    #> $estimate
    #>        Coef.     SE     z    P-val
    #> [1,]  0.3360 0.0905  3.71 0.000206
    #> [2,] -0.0798 0.0673 -1.19 0.236000
    #>
    #> $h
    #>    Estimate Std.Err    2.5%  97.5%   P-value
    #> p1   1.3118  0.2941  0.7354 1.8881 8.157e-06
    #> p2  -0.3118  0.2941 -0.8881 0.2646 2.890e-01

     ###  variances
     estimate(coef=outacem$theta,vcov=outacem$var.theta,f=function(p) p/sum(p)^2)
    #>    Estimate Std.Err    2.5% 97.5% P-value
    #> p1    5.125   2.278  0.6612 9.590 0.02443
    #> p2   -1.218   1.355 -3.8737 1.437 0.36862

     ## AE polygenic model
     # desae <- twin.polygen.design(prt,type="ae")
     # outaem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
     #    same.cens=TRUE,theta=c(0.45,0.15),var.link=0,
     #        step=0.5,theta.des=desae$pardes,random.design=desae$des.rv)
     # outaem$score
     # summary(outaem)
     # estimate(coef=outaem$theta,vcov=outaem$var.theta,f=function(p)     p/sum(p)^2)

     ## AE polygenic model
     # desde <- twin.polygen.design(prt,type="de")
     # outaem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
     #   same.cens=TRUE,theta=c(0.35),var.link=0,
     #   step=0.5,theta.des=desde$pardes,random.design=desde$des.rv)
     # outaem$score
     # summary(outaem)
     # estimate(coef=outaem$theta,vcov=outaem$var.theta,f=function(p) p/sum(p)^2)

      times <- 90
      cif1 <- comp.risk(Event(time,status)~-1+factor(country)+cluster(id),prt,
               cause=2,times=times,max.clust=NULL)

      mm <- model.matrix(~-1+factor(zyg),prt)
      out1<-random.cif(cif1,data=prt,cause1=2,cause2=2,theta=1,
              theta.des=mm,same.cens=TRUE,step=0.5)
      summary(out1)
    #> Random effect variance for variation due to clusters
    #>
    #> Cause 2 and cause 2
    #>                   Coef.        SE        z       P-val Cross odds ratio
    #> factor(zyg)DZ 0.8737827 0.4091680 2.135511 0.032719303         1.873783
    #> factor(zyg)MZ 3.1990114 0.9791185 3.267236 0.001086031         4.199011
    #>                      SE
    #> factor(zyg)DZ 0.4091680
    #> factor(zyg)MZ 0.9791185
      round(estimate(coef=out1$theta,vcov=out1$var.theta)$coefmat[,c(1,3:4)],2)
    #>    Estimate 2.5% 97.5%
    #> p1     0.87 0.07  1.68
    #> p2     3.20 1.28  5.12

     desde <- twin.polygen.design(prt,type="de")
     outaem <- Grandom.cif(cif1,data=prt,cause1=2,cause2=2,
        same.cens=TRUE,theta=c(0.35),var.link=0,
            step=0.5,theta.des=desde$pardes,random.design=desde$des.rv)
     outaem$score
    #>              [,1]
    #> [1,] 4.250025e-07
     summary(outaem)
    #> Random effect parameters for additive gamma random effects
    #>
    #> Cause 2 and cause 2
    #>
    #>
    #>      Coef.     SE    z    P-val
    #> [1,] 0.299 0.0842 3.55 0.000384
    #> $estimate
    #>      Coef.     SE    z    P-val
    #> [1,] 0.299 0.0842 3.55 0.000384
    #>
    #> $h
    #>    Estimate Std.Err 2.5% 97.5% P-value
    #> p1        1       0    1     1       0
     estimate(coef=outaem$theta,vcov=outaem$var.theta,f=function(p) p/sum(p)^2)
    #>    Estimate Std.Err  2.5% 97.5%   P-value
    #> p1    3.346  0.9422 1.499 5.192 0.0003842

# Competing risk modeling using the Liabilty Threshold model

First we fit the bivariate probit model (same marginals in MZ and DZ twins but different correlation parameter). Here we evaluate the risk of getting cancer before the last double cancer event (95 years)

    rm(prt)
    data(prt)
    prt0 <-  force.same.cens(prt, cause="status", cens.code=0, time="time", id="id")
    prt0$country <- relevel(prt0$country, ref="Sweden")
    prt_wide <- fast.reshape(prt0, id="id", num="num", varying=c("time","status","cancer"))
    prt_time <- subset(prt_wide,  cancer1 & cancer2, select=c(time1, time2, zyg))
    tau <- 95
    tt <- seq(70, tau, length.out=5) ## Time points to evaluate model in

    b0 <- bptwin.time(cancer ~ 1, data=prt0, id="id", zyg="zyg", DZ="DZ", type="cor",
                  cens.formula=Surv(time,status==0)~zyg, breaks=tau)
    summary(b0)
    #>
    #>                Estimate   Std.Err        Z   p-value
    #> (Intercept)   -1.348188  0.026276 -51.3086 < 2.2e-16 ***
    #> atanh(rho) MZ  0.735992  0.087838   8.3789 < 2.2e-16 ***
    #> atanh(rho) DZ  0.353023  0.068234   5.1737 2.295e-07 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #>  Total MZ/DZ Complete pairs MZ/DZ
    #>  1994/3618   997/1809
    #>
    #>                            Estimate 2.5%    97.5%
    #> Tetrachoric correlation MZ 0.62672  0.51081 0.72024
    #> Tetrachoric correlation DZ 0.33905  0.21584 0.45164
    #>
    #> MZ:
    #>                      Estimate 2.5%    97.5%
    #> Concordance          0.03504  0.02779 0.04409
    #> Casewise Concordance 0.39458  0.31876 0.47584
    #> Marginal             0.08880  0.08086 0.09743
    #> Rel.Recur.Risk       4.44351  3.50520 5.38182
    #> log(OR)              2.34131  1.87104 2.81157
    #> DZ:
    #>                      Estimate 2.5%    97.5%
    #> Concordance          0.01952  0.01449 0.02625
    #> Casewise Concordance 0.21983  0.16667 0.28415
    #> Marginal             0.08880  0.08086 0.09743
    #> Rel.Recur.Risk       2.47556  1.81096 3.14017
    #> log(OR)              1.23088  0.81020 1.65156
    #>
    #>                          Estimate 2.5%    97.5%
    #> Broad-sense heritability 0.57533  0.25790 0.89276
    #>
    #>
    #> Event of interest before time 95

Liability threshold model with ACE random effects structure

    b1 <- bptwin.time(cancer ~ 1, data=prt0, id="id", zyg="zyg", DZ="DZ", type="ace",
                  cens.formula=Surv(time,status==0)~zyg, breaks=tau)
    summary(b1)
    #>
    #>             Estimate  Std.Err        Z p-value
    #> (Intercept) -2.20664  0.16463 -13.4034  <2e-16 ***
    #> log(var(A))  0.43260  0.39149   1.1050  0.2691
    #> log(var(C)) -1.98290  2.52345  -0.7858  0.4320
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #>  Total MZ/DZ Complete pairs MZ/DZ
    #>  1994/3618   997/1809
    #>
    #>                    Estimate 2.5%     97.5%
    #> A                   0.57533  0.25790  0.89276
    #> C                   0.05139 -0.20836  0.31114
    #> E                   0.37328  0.26874  0.47782
    #> MZ Tetrachoric Cor  0.62672  0.51081  0.72024
    #> DZ Tetrachoric Cor  0.33905  0.21584  0.45164
    #>
    #> MZ:
    #>                      Estimate 2.5%    97.5%
    #> Concordance          0.03504  0.02779 0.04409
    #> Casewise Concordance 0.39458  0.31876 0.47584
    #> Marginal             0.08880  0.08086 0.09743
    #> Rel.Recur.Risk       4.44351  3.50521 5.38182
    #> log(OR)              2.34131  1.87105 2.81157
    #> DZ:
    #>                      Estimate 2.5%    97.5%
    #> Concordance          0.01952  0.01449 0.02625
    #> Casewise Concordance 0.21983  0.16667 0.28415
    #> Marginal             0.08880  0.08086 0.09743
    #> Rel.Recur.Risk       2.47556  1.81095 3.14016
    #> log(OR)              1.23088  0.81020 1.65156
    #>
    #>                          Estimate 2.5%    97.5%
    #> Broad-sense heritability 0.57533  0.25790 0.89276
    #>
    #>
    #> Event of interest before time 95

In this case the ACE model fits the data well - it is in fact indistinguishable from the flexible bivariate Probit model as seen by the IPCW weighted AIC measure

    AIC(b0, b1)
    #>    df      AIC
    #> b0  3 17340.13
    #> b1  3 17340.12

ACE model with marginal adjusted for country

    b2 <- bptwin.time(cancer ~ country, data=prt0, id="id", zyg="zyg", DZ="DZ", type="ace",
                  cens.formula=Surv(time,status==0)~zyg+country, breaks=95)
    summary(b2)
    #>
    #>                Estimate  Std.Err        Z   p-value
    #> (Intercept)    -1.97165  0.15371 -12.8267 < 2.2e-16 ***
    #> countryDenmark -0.72489  0.11920  -6.0812 1.193e-09 ***
    #> countryFinland  0.18968  0.12518   1.5152    0.1297
    #> countryNorway  -0.11611  0.16621  -0.6986    0.4848
    #> log(var(A))     0.40388  0.40524   0.9966    0.3189
    #> log(var(C))    -3.88761 17.56421  -0.2213    0.8248
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #>  Total MZ/DZ Complete pairs MZ/DZ
    #>  1994/3618   997/1809
    #>
    #>                    Estimate 2.5%     97.5%
    #> A                   0.59474  0.25169  0.93779
    #> C                   0.00814 -0.27297  0.28925
    #> E                   0.39712  0.28435  0.50989
    #> MZ Tetrachoric Cor  0.60288  0.47809  0.70381
    #> DZ Tetrachoric Cor  0.30551  0.17238  0.42767
    #>
    #> MZ:
    #>                      Estimate 2.5%    97.5%
    #> Concordance          0.04295  0.03307 0.05561
    #> Casewise Concordance 0.40128  0.32263 0.48535
    #> Marginal             0.10703  0.09453 0.12096
    #> Rel.Recur.Risk       3.74923  2.94155 4.55690
    #> log(OR)              2.15979  1.67935 2.64023
    #> DZ:
    #>                      Estimate 2.5%    97.5%
    #> Concordance          0.02439  0.01747 0.03396
    #> Casewise Concordance 0.22788  0.17060 0.29749
    #> Marginal             0.10703  0.09453 0.12096
    #> Rel.Recur.Risk       2.12912  1.54508 2.71315
    #> log(OR)              1.06262  0.62460 1.50064
    #>
    #>                          Estimate 2.5%    97.5%
    #> Broad-sense heritability 0.59474  0.25169 0.93779
    #>
    #>
    #> Event of interest before time 95

    bt0 <- bptwin.time(cancer ~ 1, data=prt0, id="id", zyg="zyg", DZ="DZ", type="ace",
                  cens.formula=Surv(time,status==0)~zyg,
                  summary.function=function(x) x, breaks=tt)
    h2 <- Reduce(rbind, lapply(bt0$coef, function(x) x$heritability))[,c(1,3,4),drop=FALSE]
    concMZ <- Reduce(rbind, lapply(bt0$coef, function(x) x$probMZ["Concordance",,drop=TRUE]))

    par(mfrow=c(1,2))
    plot(tt, h2[,1], type="s", lty=1, col=cols[3], xlab="Age", ylab="Heritability", ylim=c(0,1))
    confband(tt, h2[,2], h2[,3],polygon=TRUE, step=TRUE, col=Col(cols[3], 0.1), border=NA)
    plot(tt, concMZ[,1], type="s", lty=1, col=cols[1], xlab="Age", ylab="Concordance", ylim=c(0,.1))
    confband(tt, concMZ[,2], concMZ[,3],polygon=TRUE, step=TRUE, col=Col(cols[1], 0.1), border=NA)

Bivariate probit model at time different time points

    system.time(a.mz <- biprobit.time(cancer~1, id="id", data=subset(prt0, zyg=="MZ"),
                                   cens.formula = Surv(time,status==0)~1, pairs.only=TRUE,
                                    breaks=tt))
    #>    user  system elapsed
    #>   0.196   0.009   0.204
    system.time(a.dz <- biprobit.time(cancer~1, id="id", data=subset(prt0, zyg=="DZ"),
                                   cens.formula = Event(time,status==0)~1, pairs.only=TRUE,
                                   breaks=tt))
    #>    user  system elapsed
    #>   0.283   0.017   0.301

    #system.time(a.zyg <- biprobit.time(cancer~1, rho=~1+zyg, id="id", data=prt,
    #                               cens.formula = Event(time,status==0)~1,
    #                               eqmarg=FALSE, fix.cens.weight
    #                               breaks=seq(75,100,by=10)))

    a.mz
    #>
    #>  1:Concordance
    #>  2:Casewise Concordance
    #>  3:Marginal
    #>  4:Rel.Recur.Risk
    #>  5:OR
    #>  6:Tetrachoric correlation
    #>
    #>       Time 1:Concor... 2:Casewi... 3:Marginal 4:Rel.Re...    5:OR 6:Tetrac...
    #> [1,] 70.00      0.0049      0.2976     0.0166     17.9648 35.3874      0.6973
    #> [2,] 76.25      0.0125      0.3468     0.0362      9.5939 21.1446      0.6834
    #> [3,] 82.50      0.0247      0.3759     0.0656      5.7307 13.1453      0.6481
    #> [4,] 88.75      0.0308      0.3675     0.0839      4.3781  9.4430      0.5993
    #> [5,] 95.00      0.0409      0.4144     0.0988      4.1952 10.3180      0.6352
    a.dz
    #>
    #>  1:Concordance
    #>  2:Casewise Concordance
    #>  3:Marginal
    #>  4:Rel.Recur.Risk
    #>  5:OR
    #>  6:Tetrachoric correlation
    #>
    #>       Time 1:Concor... 2:Casewi... 3:Marginal 4:Rel.Re...   5:OR 6:Tetrac...
    #> [1,] 70.00      0.0007      0.0767     0.0088      8.6698 9.9964      0.3855
    #> [2,] 76.25      0.0037      0.1612     0.0228      7.0632 9.6181      0.4682
    #> [3,] 82.50      0.0074      0.1660     0.0445      3.7329 4.9289      0.3752
    #> [4,] 88.75      0.0136      0.2001     0.0680      2.9417 4.0347      0.3614
    #> [5,] 95.00      0.0174      0.2091     0.0831      2.5163 3.4243      0.3335

    plot(conczyg,se=TRUE,legend=FALSE,xlab="Age",ylab="Concordance", ylim=c(0,0.07))
    plot(a.mz, ylim=c(0,.07), col=cols[1], lty=ltys[1], legend=FALSE, add=TRUE)
    plot(a.dz, col=cols[2], lty=ltys[2], add=TRUE)

Bivariate probit model adjusting for country

    system.time(a.mz_country <- biprobit.time(cancer~country, id="id", data=subset(prt0, zyg=="MZ"),
                                   cens.formula = Surv(time,status==0)~country, pairs.only=TRUE,
                                    breaks=tt))
    #>    user  system elapsed
    #>   0.365   0.014   0.382
    system.time(a.dz_country <- biprobit.time(cancer~country, id="id", data=subset(prt0, zyg=="DZ"),
                                   cens.formula = Event(time,status==0)~country, pairs.only=TRUE,
                                   breaks=tt))
    #>    user  system elapsed
    #>   0.476   0.024   0.501

    a.mz_country
    #>
    #>  1:Concordance
    #>  2:Casewise Concordance
    #>  3:Marginal
    #>  4:Rel.Recur.Risk
    #>  5:OR
    #>  6:Tetrachoric correlation
    #>
    #>       Time 1:Concor... 2:Casewi... 3:Marginal 4:Rel.Re...    5:OR 6:Tetrac...
    #> [1,] 70.00      0.0059      0.3061     0.0193     15.8281 31.7992      0.6944
    #> [2,] 76.25      0.0143      0.3502     0.0409      8.5640 18.9150      0.6744
    #> [3,] 82.50      0.0292      0.3867     0.0755      5.1197 11.9530      0.6417
    #> [4,] 88.75      0.0404      0.3907     0.1034      3.7803  8.4889      0.5946
    #> [5,] 95.00      0.0497      0.4190     0.1186      3.5322  8.5020      0.6079
    a.dz_country
    #>
    #>  1:Concordance
    #>  2:Casewise Concordance
    #>  3:Marginal
    #>  4:Rel.Recur.Risk
    #>  5:OR
    #>  6:Tetrachoric correlation
    #>
    #>       Time 1:Concor... 2:Casewi... 3:Marginal 4:Rel.Re...   5:OR 6:Tetrac...
    #> [1,] 70.00      0.0009      0.0824     0.0113      7.2623 8.4379      0.3735
    #> [2,] 76.25      0.0054      0.1783     0.0302      5.9038 8.2629      0.4623
    #> [3,] 82.50      0.0090      0.1649     0.0546      3.0215 3.8987      0.3333
    #> [4,] 88.75      0.0173      0.2076     0.0831      2.4985 3.3867      0.3305
    #> [5,] 95.00      0.0221      0.2184     0.1012      2.1574 2.8946      0.3013

    ## ACE model (time-varying) with and without adjustment for country
    a1 <- bptwin.time(cancer~1, id="id", data=prt0, type="ace",
                                  zyg="zyg", DZ="DZ",
                                  cens.formula=Surv(time,status==0)~zyg,
                                  breaks=tt)

    #a2 <- bptwin.time(cancer~country, id="id", data=prt0, #type="ace",
    #                              zyg="zyg", DZ="DZ",
    #                              #cens.formula=Surv(time,status==0)~country+zyg,
    #                              breaks=tt)

    plot(a.mz, which=c(6), xlab="Age", ylab="Correlation", ylim=c(0,1), col=cols[1], lty=ltys[1], legend=NULL, alpha=.1)
    plot(a.dz, which=c(6), col=cols[2], lty=ltys[2], legend=NULL, add=TRUE, alpha=.1)
    legend("topleft", c("MZ tetrachoric correlation", "DZ tetrachoric correlation"),
           col=cols, lty=ltys, lwd=2)

    plot(a.mz, which=c(4), xlab="Age", ylab="Relative Recurrence Risk",
         ylim=c(1,20), col=cols[1], lty=ltys[1], legend=NULL, lwd=2, alpha=.1)
    plot(a.dz, which=c(4), col=cols[2], lty=ltys[2], legend=NULL, add=TRUE, lwd=2, alpha=.1)
    legend("topright", c("MZ relative recurrence risk", "DZ relative recurrence risk"),
           col=cols, lty=ltys, lwd=2)

    plot(a1, which=c(5,6), xlab="Age", ylab="Correlation", ylim=c(0,1), col=cols[1:2], lty=ltys[1:2], lwd=2, alpha=0.1,
         legend=c("MZ tetrachoric correlation", "DZ tetrachoric correlation"))

    plot(a1, which=c(1), xlab="Age", ylim=c(0,1), col="black", lty=1, ylab="Heritability", legend=NULL, alpha=.1)

# SessionInfo

    sessionInfo()
    #> R version 4.1.1 (2021-08-10)
    #> Platform: x86_64-apple-darwin20.4.0 (64-bit)
    #> Running under: macOS Big Sur 11.5.2
    #>
    #> Matrix products: default
    #> BLAS:   /usr/local/Cellar/openblas/0.3.17/lib/libopenblasp-r0.3.17.dylib
    #> LAPACK: /usr/local/Cellar/r/4.1.1/lib/R/lib/libRlapack.dylib
    #>
    #> locale:
    #> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    #>
    #> attached base packages:
    #> [1] splines   stats     graphics  grDevices utils     datasets  methods
    #> [8] base
    #>
    #> other attached packages:
    #> [1] prodlim_2019.11.13 ggplot2_3.3.5      cowplot_1.1.1      mets_1.2.9
    #> [5] lava_1.6.11        timereg_2.0.0      survival_3.2-13
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] Rcpp_1.0.7          mvtnorm_1.1-2       lattice_0.20-44
    #>  [4] listenv_0.8.0       assertthat_0.2.1    digest_0.6.27
    #>  [7] utf8_1.2.2          parallelly_1.27.0   R6_2.5.1
    #> [10] evaluate_0.14       highr_0.9           pillar_1.6.2
    #> [13] rlang_0.4.11        jquerylib_0.1.4     Matrix_1.3-4
    #> [16] rmarkdown_2.10      labeling_0.4.2      stringr_1.4.0
    #> [19] munsell_0.5.0       compiler_4.1.1      numDeriv_2016.8-1.1
    #> [22] xfun_0.25           pkgconfig_2.0.3     globals_0.14.0
    #> [25] htmltools_0.5.2     tidyselect_1.1.1    tibble_3.1.4
    #> [28] codetools_0.2-18    fansi_0.5.0         future_1.22.1
    #> [31] crayon_1.4.1        ucminf_1.1-4        dplyr_1.0.7
    #> [34] withr_2.4.2         MASS_7.3-54         grid_4.1.1
    #> [37] jsonlite_1.7.2      gtable_0.3.0        lifecycle_1.0.0
    #> [40] DBI_1.1.1           magrittr_2.0.1      scales_1.1.1
    #> [43] future.apply_1.8.1  stringi_1.7.4       farver_2.1.0
    #> [46] bslib_0.2.5.1       ellipsis_0.3.2      vctrs_0.3.8
    #> [49] generics_0.1.0      tools_4.1.1         glue_1.4.2
    #> [52] purrr_0.3.4         parallel_4.1.1      fastmap_1.1.0
    #> [55] yaml_2.2.1          colorspace_2.0-2    isoband_0.2.5
    #> [58] knitr_1.33          sass_0.4.0