Functional mediation analysis under historical influence regression model with point-wise bootstrap confidence interval

This function performs functional mediation regression under the historical influence model with given tuning parameter. Point-wise confidence bands are obtained from bootstrap.

FMA.historical.boot(Z, M, Y, delta.grid1 = 1, delta.grid2 = 1, delta.grid3 = 1, 
    intercept = TRUE, basis1 = NULL, Ld2.basis1 = NULL, basis2 = NULL, Ld2.basis2 = NULL, 
    basis.type = c("fourier"), nbasis1 = 3, nbasis2 = 3, 
    timeinv = c(0, 1), timegrids = NULL, 
    lambda1.m = 0.01, lambda2.m = 0.01, lambda1.y = 0.01, lambda2.y = 0.01, 
    sims = 1000, boot = TRUE, boot.ci.type = c("bca", "perc"), 
    conf.level = 0.95, verbose = TRUE)

Arguments

Z	a data matrix. `Z` is the treatment trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
M	a data matrix. `M` is the mediator trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
Y	a data matrix. `Y` is the outcome trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
delta.grid1	a number indicates the width of treatment-mediator time interval in the mediator model.
delta.grid2	a number indicates the width of treatment-outcome time interval in the outcome model.
delta.grid3	a number indicates the width of mediator-outcome time interval in the outcome model.
intercept	a logic variable. Default is `TRUE`, an intercept term is included in the regression model.
basis1	a data matrix. Basis function on the $s$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
Ld2.basis1	a data matrix. The second derivative of the basis function on the $s$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
basis2	a data matrix. Basis function on the $t$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
Ld2.basis2	a data matrix. The second derivative of the basis function on the $t$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
basis.type	a character of basis function type. Default is Fourier basis (`basis.type = "fourier"`).
nbasis1	an integer, the number of basis function on the $s$ domain included. If `basis1` is provided, this argument will be ignored.
nbasis2	an integer, the number of basis function on the $t$ domain included. If `basis2` is provided, this argument will be ignored.
timeinv	a numeric vector of length two, the time interval considered in the analysis. Default is (0,1).
timegrids	a numeric vector of time grids of measurement. If `timegrids = NULL`, it is assumed the between measurement time interval is constant.
lambda1.m	a numeric vector of tuning parameter values on the $s$ domain in the mediator model.
lambda2.m	a numeric vector of tuning parameter values on the $t$ domain in the mediator model.
lambda1.y	a numeric vector of tuning parameter values on the $s$ domain in the outcome model.
lambda2.y	a numeric vector of tuning parameter values on the $t$ domain in the outcome model.
sims	an integer indicating the number of simulations for inference.
boot	a logical value, indicating whether or not bootstrap should be used. Default is `TRUE`.
boot.ci.type	a character of confidence interval method. `boot.ci.type = "bca"` bias corrected confidence interval; `boot.ci.type = "perc"` percentile confidence interval.
conf.level	a number of significance level. Default is 0.95.
verbose	a logical value, indicating whether print out bootstrap replications.

Details

The historical influence mediation model is $$M(t)=\int_{\Omega_{t}^{1}}Z(s)\alpha(s,t)ds+\epsilon_{1}(t),$$ $$Y(t)=\int_{\Omega_{t}^{2}}Z(s)\gamma(s,t)ds+\int_{\Omega_{t}^{3}}M(s)\beta(s,t)ds+\epsilon_{2}(t),$$ where $\alpha(s,t)$, $\beta(s,t)$, $\gamma(s,t)$ are coefficient curves; $\Omega_{t}^{j}=[(t-\delta_{j})\vee 0,t]$ for $j=1,2,3$. The model coefficient curves are estimated by minimizing the penalized $L_{2}$-loss.

Value

alpha

a list of output for $\alpha$ estimate coefficients: the result of the coefficient estimates corresponding to the basis function curve: the point-wise estimate of the coefficient curve

gamma

a list of output for $\gamma$ estimate coefficients: the result of the coefficient estimates corresponding to the basis function curve: the point-wise estimate of the coefficient curve

beta

a list of output for $\beta$ estimate coefficients: the result of the coefficient estimates corresponding to the basis function curve: the point-wise estimate of the coefficient curve

a list of output for indirect effect estimate curve: the point-wise estimate of the coefficient curve

a list of output for direct effect estimate curve: the point-wise estimate of the coefficient curve

References

Zhao et al. (2017). Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data. arXiv preprint arXiv:1805.06923.

Author

Yi Zhao, Johns Hopkins University, zhaoyi1026@gmail.com;

Xi Luo, Brown University xi.rossi.luo@gmail.com;

Martin Lindquist, Johns Hopkins University, mal2053@gmail.com;

Brian Caffo, Johns Hopkins University, bcaffo@gmail.com

Examples


##################################################
# Historical influence functional mediation model
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)

# \donttest{
# consider Fourier basis
fit.boot<-FMA.historical.boot(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3,
    intercept=FALSE,timeinv=c(0,300))
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#> [1] "Bootstrap sample 504"
#> [1] "Bootstrap sample 505"
#> [1] "Bootstrap sample 506"
#> [1] "Bootstrap sample 507"
#> [1] "Bootstrap sample 508"
#> [1] "Bootstrap sample 509"
#> [1] "Bootstrap sample 510"
#> [1] "Bootstrap sample 511"
#> [1] "Bootstrap sample 512"
#> [1] "Bootstrap sample 513"
#> [1] "Bootstrap sample 514"
#> [1] "Bootstrap sample 515"
#> [1] "Bootstrap sample 516"
#> [1] "Bootstrap sample 517"
#> [1] "Bootstrap sample 518"
#> [1] "Bootstrap sample 519"
#> [1] "Bootstrap sample 520"
#> [1] "Bootstrap sample 521"
#> [1] "Bootstrap sample 522"
#> [1] "Bootstrap sample 523"
#> [1] "Bootstrap sample 524"
#> [1] "Bootstrap sample 525"
#> [1] "Bootstrap sample 526"
#> [1] "Bootstrap sample 527"
#> [1] "Bootstrap sample 528"
#> [1] "Bootstrap sample 529"
#> [1] "Bootstrap sample 530"
#> [1] "Bootstrap sample 531"
#> [1] "Bootstrap sample 532"
#> [1] "Bootstrap sample 533"
#> [1] "Bootstrap sample 534"
#> [1] "Bootstrap sample 535"
#> [1] "Bootstrap sample 536"
#> [1] "Bootstrap sample 537"
#> [1] "Bootstrap sample 538"
#> [1] "Bootstrap sample 539"
#> [1] "Bootstrap sample 540"
#> [1] "Bootstrap sample 541"
#> [1] "Bootstrap sample 542"
#> [1] "Bootstrap sample 543"
#> [1] "Bootstrap sample 544"
#> [1] "Bootstrap sample 545"
#> [1] "Bootstrap sample 546"
#> [1] "Bootstrap sample 547"
#> [1] "Bootstrap sample 548"
#> [1] "Bootstrap sample 549"
#> [1] "Bootstrap sample 550"
#> [1] "Bootstrap sample 551"
#> [1] "Bootstrap sample 552"
#> [1] "Bootstrap sample 553"
#> [1] "Bootstrap sample 554"
#> [1] "Bootstrap sample 555"
#> [1] "Bootstrap sample 556"
#> [1] "Bootstrap sample 557"
#> [1] "Bootstrap sample 558"
#> [1] "Bootstrap sample 559"
#> [1] "Bootstrap sample 560"
#> [1] "Bootstrap sample 561"
#> [1] "Bootstrap sample 562"
#> [1] "Bootstrap sample 563"
#> [1] "Bootstrap sample 564"
#> [1] "Bootstrap sample 565"
#> [1] "Bootstrap sample 566"
#> [1] "Bootstrap sample 567"
#> [1] "Bootstrap sample 568"
#> [1] "Bootstrap sample 569"
#> [1] "Bootstrap sample 570"
#> [1] "Bootstrap sample 571"
#> [1] "Bootstrap sample 572"
#> [1] "Bootstrap sample 573"
#> [1] "Bootstrap sample 574"
#> [1] "Bootstrap sample 575"
#> [1] "Bootstrap sample 576"
#> [1] "Bootstrap sample 577"
#> [1] "Bootstrap sample 578"
#> [1] "Bootstrap sample 579"
#> [1] "Bootstrap sample 580"
#> [1] "Bootstrap sample 581"
#> [1] "Bootstrap sample 582"
#> [1] "Bootstrap sample 583"
#> [1] "Bootstrap sample 584"
#> [1] "Bootstrap sample 585"
#> [1] "Bootstrap sample 586"
#> [1] "Bootstrap sample 587"
#> [1] "Bootstrap sample 588"
#> [1] "Bootstrap sample 589"
#> [1] "Bootstrap sample 590"
#> [1] "Bootstrap sample 591"
#> [1] "Bootstrap sample 592"
#> [1] "Bootstrap sample 593"
#> [1] "Bootstrap sample 594"
#> [1] "Bootstrap sample 595"
#> [1] "Bootstrap sample 596"
#> [1] "Bootstrap sample 597"
#> [1] "Bootstrap sample 598"
#> [1] "Bootstrap sample 599"
#> [1] "Bootstrap sample 600"
#> [1] "Bootstrap sample 601"
#> [1] "Bootstrap sample 602"
#> [1] "Bootstrap sample 603"
#> [1] "Bootstrap sample 604"
#> [1] "Bootstrap sample 605"
#> [1] "Bootstrap sample 606"
#> [1] "Bootstrap sample 607"
#> [1] "Bootstrap sample 608"
#> [1] "Bootstrap sample 609"
#> [1] "Bootstrap sample 610"
#> [1] "Bootstrap sample 611"
#> [1] "Bootstrap sample 612"
#> [1] "Bootstrap sample 613"
#> [1] "Bootstrap sample 614"
#> [1] "Bootstrap sample 615"
#> [1] "Bootstrap sample 616"
#> [1] "Bootstrap sample 617"
#> [1] "Bootstrap sample 618"
#> [1] "Bootstrap sample 619"
#> [1] "Bootstrap sample 620"
#> [1] "Bootstrap sample 621"
#> [1] "Bootstrap sample 622"
#> [1] "Bootstrap sample 623"
#> [1] "Bootstrap sample 624"
#> [1] "Bootstrap sample 625"
#> [1] "Bootstrap sample 626"
#> [1] "Bootstrap sample 627"
#> [1] "Bootstrap sample 628"
#> [1] "Bootstrap sample 629"
#> [1] "Bootstrap sample 630"
#> [1] "Bootstrap sample 631"
#> [1] "Bootstrap sample 632"
#> [1] "Bootstrap sample 633"
#> [1] "Bootstrap sample 634"
#> [1] "Bootstrap sample 635"
#> [1] "Bootstrap sample 636"
#> [1] "Bootstrap sample 637"
#> [1] "Bootstrap sample 638"
#> [1] "Bootstrap sample 639"
#> [1] "Bootstrap sample 640"
#> [1] "Bootstrap sample 641"
#> [1] "Bootstrap sample 642"
#> [1] "Bootstrap sample 643"
#> [1] "Bootstrap sample 644"
#> [1] "Bootstrap sample 645"
#> [1] "Bootstrap sample 646"
#> [1] "Bootstrap sample 647"
#> [1] "Bootstrap sample 648"
#> [1] "Bootstrap sample 649"
#> [1] "Bootstrap sample 650"
#> [1] "Bootstrap sample 651"
#> [1] "Bootstrap sample 652"
#> [1] "Bootstrap sample 653"
#> [1] "Bootstrap sample 654"
#> [1] "Bootstrap sample 655"
#> [1] "Bootstrap sample 656"
#> [1] "Bootstrap sample 657"
#> [1] "Bootstrap sample 658"
#> [1] "Bootstrap sample 659"
#> [1] "Bootstrap sample 660"
#> [1] "Bootstrap sample 661"
#> [1] "Bootstrap sample 662"
#> [1] "Bootstrap sample 663"
#> [1] "Bootstrap sample 664"
#> [1] "Bootstrap sample 665"
#> [1] "Bootstrap sample 666"
#> [1] "Bootstrap sample 667"
#> [1] "Bootstrap sample 668"
#> [1] "Bootstrap sample 669"
#> [1] "Bootstrap sample 670"
#> [1] "Bootstrap sample 671"
#> [1] "Bootstrap sample 672"
#> [1] "Bootstrap sample 673"
#> [1] "Bootstrap sample 674"
#> [1] "Bootstrap sample 675"
#> [1] "Bootstrap sample 676"
#> [1] "Bootstrap sample 677"
#> [1] "Bootstrap sample 678"
#> [1] "Bootstrap sample 679"
#> [1] "Bootstrap sample 680"
#> [1] "Bootstrap sample 681"
#> [1] "Bootstrap sample 682"
#> [1] "Bootstrap sample 683"
#> [1] "Bootstrap sample 684"
#> [1] "Bootstrap sample 685"
#> [1] "Bootstrap sample 686"
#> [1] "Bootstrap sample 687"
#> [1] "Bootstrap sample 688"
#> [1] "Bootstrap sample 689"
#> [1] "Bootstrap sample 690"
#> [1] "Bootstrap sample 691"
#> [1] "Bootstrap sample 692"
#> [1] "Bootstrap sample 693"
#> [1] "Bootstrap sample 694"
#> [1] "Bootstrap sample 695"
#> [1] "Bootstrap sample 696"
#> [1] "Bootstrap sample 697"
#> [1] "Bootstrap sample 698"
#> [1] "Bootstrap sample 699"
#> [1] "Bootstrap sample 700"
#> [1] "Bootstrap sample 701"
#> [1] "Bootstrap sample 702"
#> [1] "Bootstrap sample 703"
#> [1] "Bootstrap sample 704"
#> [1] "Bootstrap sample 705"
#> [1] "Bootstrap sample 706"
#> [1] "Bootstrap sample 707"
#> [1] "Bootstrap sample 708"
#> [1] "Bootstrap sample 709"
#> [1] "Bootstrap sample 710"
#> [1] "Bootstrap sample 711"
#> [1] "Bootstrap sample 712"
#> [1] "Bootstrap sample 713"
#> [1] "Bootstrap sample 714"
#> [1] "Bootstrap sample 715"
#> [1] "Bootstrap sample 716"
#> [1] "Bootstrap sample 717"
#> [1] "Bootstrap sample 718"
#> [1] "Bootstrap sample 719"
#> [1] "Bootstrap sample 720"
#> [1] "Bootstrap sample 721"
#> [1] "Bootstrap sample 722"
#> [1] "Bootstrap sample 723"
#> [1] "Bootstrap sample 724"
#> [1] "Bootstrap sample 725"
#> [1] "Bootstrap sample 726"
#> [1] "Bootstrap sample 727"
#> [1] "Bootstrap sample 728"
#> [1] "Bootstrap sample 729"
#> [1] "Bootstrap sample 730"
#> [1] "Bootstrap sample 731"
#> [1] "Bootstrap sample 732"
#> [1] "Bootstrap sample 733"
#> [1] "Bootstrap sample 734"
#> [1] "Bootstrap sample 735"
#> [1] "Bootstrap sample 736"
#> [1] "Bootstrap sample 737"
#> [1] "Bootstrap sample 738"
#> [1] "Bootstrap sample 739"
#> [1] "Bootstrap sample 740"
#> [1] "Bootstrap sample 741"
#> [1] "Bootstrap sample 742"
#> [1] "Bootstrap sample 743"
#> [1] "Bootstrap sample 744"
#> [1] "Bootstrap sample 745"
#> [1] "Bootstrap sample 746"
#> [1] "Bootstrap sample 747"
#> [1] "Bootstrap sample 748"
#> [1] "Bootstrap sample 749"
#> [1] "Bootstrap sample 750"
#> [1] "Bootstrap sample 751"
#> [1] "Bootstrap sample 752"
#> [1] "Bootstrap sample 753"
#> [1] "Bootstrap sample 754"
#> [1] "Bootstrap sample 755"
#> [1] "Bootstrap sample 756"
#> [1] "Bootstrap sample 757"
#> [1] "Bootstrap sample 758"
#> [1] "Bootstrap sample 759"
#> [1] "Bootstrap sample 760"
#> [1] "Bootstrap sample 761"
#> [1] "Bootstrap sample 762"
#> [1] "Bootstrap sample 763"
#> [1] "Bootstrap sample 764"
#> [1] "Bootstrap sample 765"
#> [1] "Bootstrap sample 766"
#> [1] "Bootstrap sample 767"
#> [1] "Bootstrap sample 768"
#> [1] "Bootstrap sample 769"
#> [1] "Bootstrap sample 770"
#> [1] "Bootstrap sample 771"
#> [1] "Bootstrap sample 772"
#> [1] "Bootstrap sample 773"
#> [1] "Bootstrap sample 774"
#> [1] "Bootstrap sample 775"
#> [1] "Bootstrap sample 776"
#> [1] "Bootstrap sample 777"
#> [1] "Bootstrap sample 778"
#> [1] "Bootstrap sample 779"
#> [1] "Bootstrap sample 780"
#> [1] "Bootstrap sample 781"
#> [1] "Bootstrap sample 782"
#> [1] "Bootstrap sample 783"
#> [1] "Bootstrap sample 784"
#> [1] "Bootstrap sample 785"
#> [1] "Bootstrap sample 786"
#> [1] "Bootstrap sample 787"
#> [1] "Bootstrap sample 788"
#> [1] "Bootstrap sample 789"
#> [1] "Bootstrap sample 790"
#> [1] "Bootstrap sample 791"
#> [1] "Bootstrap sample 792"
#> [1] "Bootstrap sample 793"
#> [1] "Bootstrap sample 794"
#> [1] "Bootstrap sample 795"
#> [1] "Bootstrap sample 796"
#> [1] "Bootstrap sample 797"
#> [1] "Bootstrap sample 798"
#> [1] "Bootstrap sample 799"
#> [1] "Bootstrap sample 800"
#> [1] "Bootstrap sample 801"
#> [1] "Bootstrap sample 802"
#> [1] "Bootstrap sample 803"
#> [1] "Bootstrap sample 804"
#> [1] "Bootstrap sample 805"
#> [1] "Bootstrap sample 806"
#> [1] "Bootstrap sample 807"
#> [1] "Bootstrap sample 808"
#> [1] "Bootstrap sample 809"
#> [1] "Bootstrap sample 810"
#> [1] "Bootstrap sample 811"
#> [1] "Bootstrap sample 812"
#> [1] "Bootstrap sample 813"
#> [1] "Bootstrap sample 814"
#> [1] "Bootstrap sample 815"
#> [1] "Bootstrap sample 816"
#> [1] "Bootstrap sample 817"
#> [1] "Bootstrap sample 818"
#> [1] "Bootstrap sample 819"
#> [1] "Bootstrap sample 820"
#> [1] "Bootstrap sample 821"
#> [1] "Bootstrap sample 822"
#> [1] "Bootstrap sample 823"
#> [1] "Bootstrap sample 824"
#> [1] "Bootstrap sample 825"
#> [1] "Bootstrap sample 826"
#> [1] "Bootstrap sample 827"
#> [1] "Bootstrap sample 828"
#> [1] "Bootstrap sample 829"
#> [1] "Bootstrap sample 830"
#> [1] "Bootstrap sample 831"
#> [1] "Bootstrap sample 832"
#> [1] "Bootstrap sample 833"
#> [1] "Bootstrap sample 834"
#> [1] "Bootstrap sample 835"
#> [1] "Bootstrap sample 836"
#> [1] "Bootstrap sample 837"
#> [1] "Bootstrap sample 838"
#> [1] "Bootstrap sample 839"
#> [1] "Bootstrap sample 840"
#> [1] "Bootstrap sample 841"
#> [1] "Bootstrap sample 842"
#> [1] "Bootstrap sample 843"
#> [1] "Bootstrap sample 844"
#> [1] "Bootstrap sample 845"
#> [1] "Bootstrap sample 846"
#> [1] "Bootstrap sample 847"
#> [1] "Bootstrap sample 848"
#> [1] "Bootstrap sample 849"
#> [1] "Bootstrap sample 850"
#> [1] "Bootstrap sample 851"
#> [1] "Bootstrap sample 852"
#> [1] "Bootstrap sample 853"
#> [1] "Bootstrap sample 854"
#> [1] "Bootstrap sample 855"
#> [1] "Bootstrap sample 856"
#> [1] "Bootstrap sample 857"
#> [1] "Bootstrap sample 858"
#> [1] "Bootstrap sample 859"
#> [1] "Bootstrap sample 860"
#> [1] "Bootstrap sample 861"
#> [1] "Bootstrap sample 862"
#> [1] "Bootstrap sample 863"
#> [1] "Bootstrap sample 864"
#> [1] "Bootstrap sample 865"
#> [1] "Bootstrap sample 866"
#> [1] "Bootstrap sample 867"
#> [1] "Bootstrap sample 868"
#> [1] "Bootstrap sample 869"
#> [1] "Bootstrap sample 870"
#> [1] "Bootstrap sample 871"
#> [1] "Bootstrap sample 872"
#> [1] "Bootstrap sample 873"
#> [1] "Bootstrap sample 874"
#> [1] "Bootstrap sample 875"
#> [1] "Bootstrap sample 876"
#> [1] "Bootstrap sample 877"
#> [1] "Bootstrap sample 878"
#> [1] "Bootstrap sample 879"
#> [1] "Bootstrap sample 880"
#> [1] "Bootstrap sample 881"
#> [1] "Bootstrap sample 882"
#> [1] "Bootstrap sample 883"
#> [1] "Bootstrap sample 884"
#> [1] "Bootstrap sample 885"
#> [1] "Bootstrap sample 886"
#> [1] "Bootstrap sample 887"
#> [1] "Bootstrap sample 888"
#> [1] "Bootstrap sample 889"
#> [1] "Bootstrap sample 890"
#> [1] "Bootstrap sample 891"
#> [1] "Bootstrap sample 892"
#> [1] "Bootstrap sample 893"
#> [1] "Bootstrap sample 894"
#> [1] "Bootstrap sample 895"
#> [1] "Bootstrap sample 896"
#> [1] "Bootstrap sample 897"
#> [1] "Bootstrap sample 898"
#> [1] "Bootstrap sample 899"
#> [1] "Bootstrap sample 900"
#> [1] "Bootstrap sample 901"
#> [1] "Bootstrap sample 902"
#> [1] "Bootstrap sample 903"
#> [1] "Bootstrap sample 904"
#> [1] "Bootstrap sample 905"
#> [1] "Bootstrap sample 906"
#> [1] "Bootstrap sample 907"
#> [1] "Bootstrap sample 908"
#> [1] "Bootstrap sample 909"
#> [1] "Bootstrap sample 910"
#> [1] "Bootstrap sample 911"
#> [1] "Bootstrap sample 912"
#> [1] "Bootstrap sample 913"
#> [1] "Bootstrap sample 914"
#> [1] "Bootstrap sample 915"
#> [1] "Bootstrap sample 916"
#> [1] "Bootstrap sample 917"
#> [1] "Bootstrap sample 918"
#> [1] "Bootstrap sample 919"
#> [1] "Bootstrap sample 920"
#> [1] "Bootstrap sample 921"
#> [1] "Bootstrap sample 922"
#> [1] "Bootstrap sample 923"
#> [1] "Bootstrap sample 924"
#> [1] "Bootstrap sample 925"
#> [1] "Bootstrap sample 926"
#> [1] "Bootstrap sample 927"
#> [1] "Bootstrap sample 928"
#> [1] "Bootstrap sample 929"
#> [1] "Bootstrap sample 930"
#> [1] "Bootstrap sample 931"
#> [1] "Bootstrap sample 932"
#> [1] "Bootstrap sample 933"
#> [1] "Bootstrap sample 934"
#> [1] "Bootstrap sample 935"
#> [1] "Bootstrap sample 936"
#> [1] "Bootstrap sample 937"
#> [1] "Bootstrap sample 938"
#> [1] "Bootstrap sample 939"
#> [1] "Bootstrap sample 940"
#> [1] "Bootstrap sample 941"
#> [1] "Bootstrap sample 942"
#> [1] "Bootstrap sample 943"
#> [1] "Bootstrap sample 944"
#> [1] "Bootstrap sample 945"
#> [1] "Bootstrap sample 946"
#> [1] "Bootstrap sample 947"
#> [1] "Bootstrap sample 948"
#> [1] "Bootstrap sample 949"
#> [1] "Bootstrap sample 950"
#> [1] "Bootstrap sample 951"
#> [1] "Bootstrap sample 952"
#> [1] "Bootstrap sample 953"
#> [1] "Bootstrap sample 954"
#> [1] "Bootstrap sample 955"
#> [1] "Bootstrap sample 956"
#> [1] "Bootstrap sample 957"
#> [1] "Bootstrap sample 958"
#> [1] "Bootstrap sample 959"
#> [1] "Bootstrap sample 960"
#> [1] "Bootstrap sample 961"
#> [1] "Bootstrap sample 962"
#> [1] "Bootstrap sample 963"
#> [1] "Bootstrap sample 964"
#> [1] "Bootstrap sample 965"
#> [1] "Bootstrap sample 966"
#> [1] "Bootstrap sample 967"
#> [1] "Bootstrap sample 968"
#> [1] "Bootstrap sample 969"
#> [1] "Bootstrap sample 970"
#> [1] "Bootstrap sample 971"
#> [1] "Bootstrap sample 972"
#> [1] "Bootstrap sample 973"
#> [1] "Bootstrap sample 974"
#> [1] "Bootstrap sample 975"
#> [1] "Bootstrap sample 976"
#> [1] "Bootstrap sample 977"
#> [1] "Bootstrap sample 978"
#> [1] "Bootstrap sample 979"
#> [1] "Bootstrap sample 980"
#> [1] "Bootstrap sample 981"
#> [1] "Bootstrap sample 982"
#> [1] "Bootstrap sample 983"
#> [1] "Bootstrap sample 984"
#> [1] "Bootstrap sample 985"
#> [1] "Bootstrap sample 986"
#> [1] "Bootstrap sample 987"
#> [1] "Bootstrap sample 988"
#> [1] "Bootstrap sample 989"
#> [1] "Bootstrap sample 990"
#> [1] "Bootstrap sample 991"
#> [1] "Bootstrap sample 992"
#> [1] "Bootstrap sample 993"
#> [1] "Bootstrap sample 994"
#> [1] "Bootstrap sample 995"
#> [1] "Bootstrap sample 996"
#> [1] "Bootstrap sample 997"
#> [1] "Bootstrap sample 998"
#> [1] "Bootstrap sample 999"
#> [1] "Bootstrap sample 1000"
# }
##################################################