perm.path: Perform Permutation Based Pathway Analysis

Description Usage Arguments Value References Examples

View source: R/perm.path.R

Description

This is the package main function.

Usage

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perm.path(expr, y, local.test, global.test="wilcoxon", B, gset, min.num=2, max.num, 
imputeval=NULL, transfun=function(x){x}, sort="pval", anno=NULL)

Arguments

expr

An K \times n matrix of gene expression data, where K is the number of genes and n is the number of samples.

y

An outcome vector of length n.

local.test

Local test statistic of each gene. Current possible choices are t-test, Wilcoxon test, Pearson, Spearman and JT test.

global.test

Global test statictic, used to compute the score. Current possible choices are mean, meanabs (mean of absolute values) and maxmean.

B

specifies the number of random permutations to be performed.

gset

A list of pathways. Each element is a vector of gene names. The list element names are the pathway names.

min.num

Specifies the minimum number of genes that a pathway should have. Pathways with smaller number of genes will be excluded.

max.num

Specifies the maximum number of genes that a pathway should have. Pathways with larger number of genes will be excluded.

imputeval

The gene expression value to be imputed in case of missing values. The default choice is NULL in which case no imputation is done.

transfun

Specifies transformation of the gene expression data. The default option is untransformed gene expression data.

sort

Specifies sorting of the results. If sort="pval" sorting is done in order of increasing p-values. If sort="score" sorting is done in order of decreasing scores.

anno

If TRUE the output contains annotation of each pathway.

Value

This function returns a list consisting of the following elements:

res

Data frame consisting of the pathway names (Pathway), the genes involved in each pathway (Genes), the number of genes in each pathway (Size), the score for each pathway (Score), the permutation raw p-value (pval), the FWER-adjusted permutation p-value (pfwer), the FDR-adjusted permutation p-value, the Bonferroni-adjusted permutation p-value (bonferroni)

stats

The individual test statistic for each gene

scores

A matrix of scores. The matrix is of dimension (B+1)\times K, where K is the number of pathways. The first column contains the unpermuted scores, the remaining B columns contain the scores computed after each permutation.

References

B. Efron, R. Tibshirani (2007) On Testing the Significance of Sets of Genes. The Annals of Applied Statistics. Vol. 1, No 1, 107–129.

A. Subramanian, P. Tamayo, V. K. Mootha, S. Mukherjee, B. L. Ebert, M. A. Gillette, A. Paulovich, S. L. Pomeroy, T. R. Golub, E. S. Lander and J. P. Mesirov (2005), Gene Set Enrichment Analysis: A knowledge-based Approach for Interpreting Genome-Wide Expression Profiles. Proc. Natl. Acad. Sci. USA. Vol. 102, No 43, 15545–15550.

Examples

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set.seed(1234)

## Generate toy phenotype and gene expression data sets
## This example consists of 40 genes grouped into 5 pathways and 100 patients
## grp is a binary trait (e.g., case vs control)
## bp is a continuous trait (e.g., blood pressure)
## g is a group indicator

n = 100
K = 40
grp = rep(1:0,each=n/2)
bp = rnorm(n)
g = rep(1:(n/20), rep(20,n/20))

pdat = data.frame(grp, bp, g)
rm(grp, bp)
expdat = matrix(rnorm(K*n),K,n)

## Assign marker names g1,...,gK to the expression data set and
## patient ids id1,...,idn to the expression and phenotype data
gnames = paste("g",1:K,sep="")
rownames(expdat) = gnames
patid = paste("id",1:n,sep="")
rownames(pdat) = patid
colnames(expdat) = patid

#Group the K genes into M pathways of sizes n1,...,nM
M = 5
p = runif(M)
p = p/sum(p)
nM = rmultinom(1, size=K, prob=p)
gset = lapply(nM, function(x){gnames[sample(x)]})
names(gset) = paste("pathway",1:M,sep="")

## Carry out permutation analysis with grp as the outcome
## using the two-sample Wilcoxon with B=100 random permutations
perm.path(expdat, y=pdat[["grp"]], local.test="wilcoxon", global.test="maxmean", B=100, 
gset=gset, min.num=2, max.num=50, sort="score")

## Carry out permutation analysis with g as the outcome
## using the JT test with B=100 random permutations
perm.path(expdat, y=pdat[["g"]], local.test="jt", global.test="maxmean", B=100, 
gset=gset, min.num=2, max.num=50, sort="score")

Example output

Loading required package: R2HTML
Loading required package: xtable
$res
          Pathway                              Genes Size      Score pval pfwer
pathway4 pathway4                              g2;g1    2 -0.9754755 0.43  0.65
pathway1 pathway1 g5;g3;g8;g4;g10;g6;g1;g9;g2;g7;g11   11 -0.8134707 0.68  0.83
pathway2 pathway2 g3;g9;g10;g2;g7;g6;g8;g5;g4;g11;g1   11 -0.8134707 0.68  0.83
pathway3 pathway3            g7;g1;g8;g5;g3;g6;g4;g2    8 -0.6783519 0.85  0.95
pathway5 pathway5            g6;g5;g4;g8;g2;g3;g1;g7    8 -0.6783519 0.85  0.95
          fdr bonferroni
pathway4 0.85          1
pathway1 0.85          1
pathway2 0.85          1
pathway3 0.85          1
pathway5 0.85          1

$stats
         g5          g3          g8          g4         g10          g6 
 0.28264661 -1.12369264  0.08961966 -0.30332807  1.06854208 -0.01378764 
         g1          g9          g2          g7         g11 
-0.42741683 -1.46838363 -1.52353419  1.17884320 -0.83415220 

$scores
             [,1]       [,2]       [,3]       [,4]       [,5]
  [1,] -0.8134707 -0.8134707 -0.6783519 -0.9754755 -0.6783519
  [2,]  0.9799073  0.9799073  1.0919811  1.6683044  1.0919811
  [3,] -1.5125041 -1.5125041 -1.6441760 -2.0336769 -1.6441760
  [4,] -0.7776229 -0.7776229 -1.1053091  0.2688590 -1.1053091
  [5,] -0.8203646 -0.8203646  0.7335024  0.4308637  0.7335024
  [6,]  1.7923932  1.7923932  1.7992870 -0.3481379  1.7992870
  [7,] -0.6014858 -0.6014858 -0.6163075 -0.4377576 -0.6163075
  [8,] -0.9858162 -0.9858162 -1.3580825  0.9306657 -1.3580825
  [9,]  0.8146197  0.8146197  0.7445325 -0.5859747  0.7445325
 [10,] -0.5830202 -0.5830202 -0.6290611  0.1723455 -0.6290611
 [11,]  1.1857370  1.1857370  1.3351031 -2.2611729  1.3351031
 [12,] -0.7573354 -0.7573354  0.8272584 -0.5170365  0.8272584
 [13,]  0.7399367  0.7399367 -0.8019810  0.4549921 -0.8019810
 [14,]  0.6350932  0.6350932  0.6460494  0.8893028  0.6460494
 [15,] -0.9013669 -0.9013669  0.6233983  0.3309034  0.6233983
 [16,] -1.2428572 -1.2428572 -1.2822505 -0.5583994 -1.2822505
 [17,] -1.1512679 -1.1512679  1.2696118 -0.9858162  1.2696118
 [18,]  0.9892632  0.9892632 -1.4086372  2.2887482 -1.4086372
 [19,]  1.6269415  1.6269415  2.4783282  1.8406499  2.4783282
 [20,] -1.0971022 -1.0971022 -0.9237719  0.4894612 -0.9237719
 [21,]  1.2241454  1.2241454  1.1903329 -1.2477814  1.1903329
 [22,] -1.0252095 -1.0252095 -1.1351823 -0.8031300 -1.1351823
 [23,] -1.0120128 -1.0120128 -1.0120128 -1.2684629 -1.0120128
 [24,]  0.4770523  0.4770523  0.4325872  0.1103011  0.4325872
 [25,]  1.0823297  1.0823297 -0.9918483 -0.5342710 -0.9918483
 [26,]  0.4653328  0.4653328  0.5135896  0.1447702  0.5135896
 [27,]  1.0995643  1.0995643  1.1116285  0.8686213  1.1116285
 [28,] -0.5790809 -0.5790809 -0.5790809 -0.7169573 -0.5790809
 [29,]  0.9444533  0.9444533 -0.9973059  0.9099842 -0.9973059
 [30,]  0.4398257  0.4398257  0.5222069 -0.4894612  0.5222069
 [31,]  1.1719494  1.1719494  1.4011689  0.9582410  1.4011689
 [32,] -1.4201269 -1.4201269  1.5442157  1.2856974  1.5442157
 [33,]  0.7100634  0.7100634 -0.5968078 -1.0926705 -0.5968078
 [34,] -0.9389383 -0.9389383  0.8934391  1.8751190  0.8934391
 [35,] -0.7376387 -0.7376387 -0.7652140 -0.9547941 -0.7652140
 [36,]  0.7085315  0.7085315  0.7356691  1.5511095  0.7356691
 [37,] -0.8755151 -0.8755151 -0.8755151  0.8134707 -0.8755151
 [38,]  0.8712065  0.8712065  0.7279874  0.7686609  0.7279874
 [39,] -0.6847861 -0.6847861 -0.7100634  0.4067354 -0.7100634
 [40,]  0.7845167  0.7845167  0.6893820 -0.7962362  0.6893820
 [41,]  1.0124067  1.0124067  1.1259906  1.6510699  1.1259906
 [42,] -0.9086055 -0.9086055 -0.9398574 -1.0788828 -0.9398574
 [43,]  0.6457211  0.6457211 -0.6997227 -0.6135500 -0.6997227
 [44,]  1.1636768  1.1636768  1.2822505  1.5959193  1.2822505
 [45,]  0.7073059  0.7073059  0.7135104 -0.6342314  0.7135104
 [46,]  1.2865591  1.2865591  1.4339145  1.9750794  1.4339145
 [47,] -1.0498303 -1.0498303  1.1374803 -0.7238511  1.1374803
 [48,]  0.9901249  0.9901249  0.7624565 -0.6893820  0.7624565
 [49,]  0.9444533  0.9444533  0.9706498  1.0168384  0.9706498
 [50,] -0.9927101 -0.9927101 -1.2087164 -1.6131539 -1.2087164
 [51,] -0.7486688 -0.7486688 -0.8238115 -0.5790809 -0.8238115
 [52,]  1.1719494  1.1719494  1.2489304  0.7858955  1.2489304
 [53,] -0.9099842 -0.9099842 -0.8520761 -1.0754359 -0.8520761
 [54,] -0.8499095 -0.8499095 -0.3860539  0.3136688 -0.3860539
 [55,]  0.5712022  0.5712022 -0.5549525  0.7790016 -0.5549525
 [56,] -0.6985737 -0.6985737 -0.6935183 -1.1685025 -0.6935183
 [57,] -0.8548337 -0.8548337 -0.8307053 -1.1857370 -0.8307053
 [58,] -1.2202061 -1.2202061 -1.3845088 -0.9961570 -1.3845088
 [59,] -1.3029320 -1.3029320 -1.3029320  0.4722267 -1.3029320
 [60,] -1.0513075 -1.0513075 -1.1953884 -2.4404122 -1.1953884
 [61,]  0.8375991  0.8375991 -0.7031696 -1.0064977 -0.7031696
 [62,] -0.8696061 -0.8696061 -1.0289026 -1.9233757 -1.0289026
 [63,] -0.8636971 -0.8636971  0.7324684  1.1305865  0.7324684
 [64,]  0.8981662  0.8981662 -1.4201269  0.7858955 -1.4201269
 [65,]  0.9685817  0.9685817  1.3442949 -0.5721870  1.3442949
 [66,]  0.8732172  0.8732172  0.8355310 -1.9440572  0.8355310
 [67,]  0.7297601  0.7297601  0.7491284 -0.6273376  0.7491284
 [68,] -1.0023614 -1.0023614  0.9771990  1.2822505  0.9771990
 [69,] -0.9188477 -0.9188477 -0.9387085 -1.9681856 -0.9387085
 [70,] -0.9869652 -0.9869652  0.9609985  1.5821317  0.9609985
 [71,]  0.8456419  0.8456419  0.6618067  0.7893424  0.6618067
 [72,]  1.5979874  1.5979874  1.5979874 -0.6687005  1.5979874
 [73,]  0.8164252  0.8164252  1.0926705  1.0685421  1.0926705
 [74,] -0.7399367 -0.7399367 -0.9565175 -1.2029716 -0.9565175
 [75,]  0.8562124  0.8562124  0.8203646 -0.2826466  0.8203646
 [76,] -0.5388669 -0.5388669 -0.7255745  0.3102219 -0.7255745
 [77,]  0.9044692  0.9044692  0.8565571  2.3025358  0.8565571
 [78,] -1.0685421 -1.0685421 -1.2184827 -1.8475437 -1.2184827
 [79,]  0.7640650  0.7640650  0.7548733  1.2753567  0.7548733
 [80,]  0.8341522  0.8341522  0.5790809 -0.6893820  0.5790809
 [81,] -1.0151150 -1.0151150 -1.2064185 -1.6545168 -1.2064185
 [82,] -1.4184034 -1.4184034 -1.4184034 -2.4197308 -1.4184034
 [83,] -0.8778131 -0.8778131 -0.9996039 -0.3584786 -0.9996039
 [84,]  0.7362600  0.7362600 -0.9186015 -0.5583994 -0.9186015
 [85,] -0.9186015 -0.9186015 -1.0501586 -1.1719494 -1.0501586
 [86,] -0.9392830 -0.9392830  0.8904517  0.8961966  0.8904517
 [87,]  0.7927893  0.7927893  0.7927893 -0.4860143  0.7927893
 [88,] -1.1967671 -1.1967671 -1.1967671 -1.0961174 -1.1967671
 [89,] -0.8046073 -0.8046073  0.7996831 -0.7858955  0.7996831
 [90,]  1.7694138  1.7694138  1.7694138 -0.6962758  1.7694138
 [91,]  1.1443741  1.1443741  1.0530310 -0.4998019  1.0530310
 [92,]  1.2684629  1.2684629  1.2684629 -1.0133915  1.2684629
 [93,] -1.2932806 -1.2932806 -1.3098258 -0.4136292 -1.3098258
 [94,] -1.1478210 -1.1478210 -2.0612521  0.8238115 -2.0612521
 [95,]  0.6863181  0.6863181  0.8144556  0.8479398  0.8144556
 [96,]  0.5902833  0.5902833  0.6659430  1.2202061  0.6659430
 [97,]  0.7268056  0.7268056  0.7617671 -0.4446514  0.7617671
 [98,]  0.6174893  0.6174893  0.5273772 -0.8961966  0.5273772
 [99,] -0.8075618 -0.8075618 -0.8796514 -0.2964343 -0.8796514
[100,] -1.3672743 -1.3672743 -1.4518385 -1.8889066 -1.4518385
[101,]  1.0330881  1.0330881  1.1179478  1.5235342  1.1179478

$res
          Pathway                              Genes Size     Score pval pfwer
pathway1 pathway1 g5;g3;g8;g4;g10;g6;g1;g9;g2;g7;g11   11 0.6674101 0.86  0.97
pathway2 pathway2 g3;g9;g10;g2;g7;g6;g8;g5;g4;g11;g1   11 0.6674101 0.86  0.97
pathway4 pathway4                              g2;g1    2 0.5416368 0.81  0.99
pathway3 pathway3            g7;g1;g8;g5;g3;g6;g4;g2    8 0.4944718 0.97  1.00
pathway5 pathway5            g6;g5;g4;g8;g2;g3;g1;g7    8 0.4944718 0.97  1.00
          fdr bonferroni
pathway1 0.97          1
pathway2 0.97          1
pathway4 0.97          1
pathway3 0.97          1
pathway5 0.97          1

$stats
         g5          g3          g8          g4         g10          g6 
 0.45643546  0.43817805 -0.31037612 -0.27386128 -0.06694387 -0.39557740 
         g1          g9          g2          g7         g11 
 0.76681158  1.51536574  0.31646192 -0.35297676  0.51120772 

$scores
             [,1]       [,2]       [,3]        [,4]       [,5]
  [1,]  0.6674101  0.6674101  0.4944718  0.54163675  0.4944718
  [2,]  0.7137781  0.7137781 -0.7089964 -0.71203932 -0.7089964
  [3,]  0.9111321  0.9111321  0.9019165  1.94745798  0.9019165
  [4,] -0.9945946 -0.9945946 -0.7546400 -0.59945191 -0.7546400
  [5,] -0.6094500 -0.6094500 -0.8447099 -1.14413156 -0.8447099
  [6,] -0.9798148 -0.9798148 -0.7957192 -0.54772256 -0.7957192
  [7,] -0.7414541 -0.7414541 -0.8727046 -1.24150446 -0.8727046
  [8,] -1.0634946 -1.0634946 -1.1593461  0.60249481 -1.1593461
  [9,] -1.1928180 -1.1928180 -1.5092799 -2.95770181 -1.5092799
 [10,] -0.7911548 -0.7911548  0.8479557 -1.12587415  0.8479557
 [11,]  1.2399830  1.2399830  1.0487873 -0.79419771  1.0487873
 [12,]  0.9387356  0.9387356  1.0670447  1.03154415  1.0670447
 [13,]  0.9691646  0.9691646  1.1299314 -0.60249481  1.1299314
 [14,] -0.5251181 -0.5251181  0.4909217 -0.27690418  0.4909217
 [15,]  1.1876016  1.1876016 -1.2009324 -1.00415802 -1.2009324
 [16,]  0.5842374  0.5842374  0.5842374 -0.63900965  0.5842374
 [17,] -0.7229938 -0.7229938 -0.7229938 -2.36129280 -0.7229938
 [18,] -0.7077793 -0.7077793 -0.7295360 -0.52033643 -0.7295360
 [19,]  1.1989038  1.1989038  1.0893593  0.82158384  1.0893593
 [20,]  0.4468721  0.4468721  0.4533926  0.68465320  0.4533926
 [21,]  1.2736723  1.2736723  1.1514345  1.58230961  1.1514345
 [22,] -0.8573379 -0.8573379 -0.7400340  0.68769610 -0.7400340
 [23,] -0.9696718 -0.9696718  0.6014805  1.05893028  0.6014805
 [24,] -1.0056795 -1.0056795 -1.2962767  0.10954451 -1.2962767
 [25,] -0.7050841 -0.7050841 -0.7692459 -0.91895674 -0.7692459
 [26,] -0.8885277 -0.8885277  0.8598375  0.38949160  0.8598375
 [27,]  1.1806464  1.1806464  1.1806464  1.80139863  1.1806464
 [28,]  1.2623701  1.2623701  1.2942481  2.10873185  1.2942481
 [29,] -0.6420526 -0.6420526 -0.7290796  1.01024383 -0.7290796
 [30,]  0.9408656  0.9408656  0.9311283 -0.08520129  0.9311283
 [31,]  0.8175266  0.8175266  0.8175266  1.20194672  0.8175266
 [32,] -1.3206199 -1.3206199 -1.9109431  0.89765641 -1.9109431
 [33,] -0.7759403 -0.7759403 -0.7759403 -0.85201287 -0.7759403
 [34,]  1.0315441  1.0315441  0.8322340  0.83984125  0.8322340
 [35,] -0.7083878 -0.7083878 -0.7262395  0.96155738 -0.7262395
 [36,]  0.9737290  0.9737290  0.9737290 -2.12394636  0.9737290
 [37,] -0.9311283 -0.9311283  0.8933963  1.67359670  0.8933963
 [38,]  0.8114408  0.8114408  0.9980722 -0.30124741  0.9980722
 [39,] -1.0087224 -1.0087224  0.9818434  2.16654701  0.9818434
 [40,]  0.8763561  0.8763561  1.0528445  0.89461351  1.0528445
 [41,] -1.1319600 -1.1319600 -0.8154980  0.34993386 -0.8154980
 [42,] -0.6207522 -0.6207522 -0.7333396  0.76681158 -0.7333396
 [43,] -0.8276696 -0.8276696 -0.8276696 -1.89268573 -0.8276696
 [44,] -0.9169281 -0.9169281 -1.1197883  0.16127386 -1.1197883
 [45,]  0.6108628  0.6108628  0.6795817  0.66335288  0.6795817
 [46,]  1.0589303  1.0589303  1.2841051  1.39669252  1.2841051
 [47,]  1.2493291  1.2493291 -1.2110754  1.61882445 -1.2110754
 [48,]  0.9554716  0.9554716  1.0619732 -0.20083160  1.0619732
 [49,] -1.1502174 -1.1502174 -1.3114912 -1.40582123 -1.3114912
 [50,] -0.7870976 -0.7870976 -0.9052637 -1.55796639 -0.9052637
 [51,]  1.0969666  1.0969666 -1.0479758  1.41190704 -1.0479758
 [52,] -0.8041958 -0.8041958 -0.8946135 -1.37234930 -0.8946135
 [53,]  1.0609589  1.0609589  1.4514648  0.97372899  1.4514648
 [54,]  0.8733132  0.8733132 -1.0345871  1.29627672 -1.0345871
 [55,]  0.7711586  0.7711586 -0.6648743  0.46860708 -0.6648743
 [56,] -1.1137025 -1.1137025 -1.1137025 -1.66142509 -1.1137025
 [57,]  0.6779588  0.6779588 -0.5818031 -0.31341902 -0.5818031
 [58,]  1.0224154  1.0224154 -0.7774617  0.41383482 -0.7774617
 [59,] -0.5553298 -0.5553298 -0.5343338 -0.67248158 -0.5343338
 [60,] -0.8611416 -0.8611416 -0.8611416 -1.93528637 -0.8611416
 [61,]  0.9806842  0.9806842  0.9092194 -0.87939900  0.9092194
 [62,]  0.9363448  0.9363448  1.2634134  1.03762996  1.2634134
 [63,] -1.3495275 -1.3495275 -1.4403075 -1.71619735 -1.4403075
 [64,] -0.9463429 -0.9463429 -1.1806464 -2.81772827 -1.1806464
 [65,]  0.7531185  0.7531185  0.9392428  0.69986771  0.9392428
 [66,]  1.4569420  1.4569420  1.3145341 -0.22821773  1.3145341
 [67,] -0.6613243 -0.6613243  0.4138348  0.73029674  0.4138348
 [68,] -0.9408656 -0.9408656 -1.1197883 -1.20498963 -1.1197883
 [69,]  1.0398035  1.0398035  1.1563032  0.87635609  1.1563032
 [70,] -0.7981100 -0.7981100 -0.7911548 -0.91895674 -0.7911548
 [71,] -0.8236124 -0.8236124 -0.9615574  0.57510869 -0.9615574
 [72,]  0.6095949  0.6095949  0.8383198 -0.99198641  0.8383198
 [73,] -0.6349524 -0.6349524  0.6582814 -0.27994708  0.6582814
 [74,]  0.8996850  0.8996850  0.9883349  0.99807222  0.9883349
 [75,] -1.1532603 -1.1532603 -0.9798148 -1.05284447 -0.9798148
 [76,] -0.8322340 -0.8322340 -0.8933963 -0.37731998 -0.8933963
 [77,] -1.0102438 -1.0102438 -1.2095540 -2.21827636 -1.2095540
 [78,] -0.8733132 -0.8733132  1.0747534  0.22517483  1.0747534
 [79,]  0.9235211  0.9235211  0.9235211  0.69986771  0.9235211
 [80,]  0.8276696  0.8276696  0.8383198 -0.87027029  0.8383198
 [81,]  1.0508159  1.0508159  0.9883349  1.78922702  0.9883349
 [82,]  0.7843083  0.7843083  0.7444970  1.07718770  0.7444970
 [83,] -0.9822491 -0.9822491 -1.0041580 -2.56821021 -1.0041580
 [84,] -0.7400340 -0.7400340 -0.9514144  0.15214515 -0.9514144
 [85,] -0.9846834 -0.9846834 -1.0893593  1.00720093 -1.0893593
 [86,]  1.2415045  1.2415045  1.2435331  0.72421094  1.2435331
 [87,] -0.8106294 -0.8106294 -0.9088137  0.75463997 -0.9088137
 [88,] -1.0406729 -1.0406729 -1.0406729  0.60858062 -1.0406729
 [89,] -0.9615574 -0.9615574  0.9676432 -0.21908902  0.9676432
 [90,]  1.1999181  1.1999181  1.3419203  2.11481765  1.3419203
 [91,] -0.8925849 -0.8925849 -1.3084483  0.84592706 -1.3084483
 [92,]  1.5104971  1.5104971  1.2278114  1.44537897  1.2278114
 [93,] -1.1776035 -1.1776035 -0.7911548  0.75768287 -0.7911548
 [94,] -0.6572671 -0.6572671 -0.7570743  0.15214515 -0.7570743
 [95,] -1.0406729 -1.0406729 -1.3145341 -1.57013800 -1.3145341
 [96,] -0.7343539 -0.7343539 -0.7343539 -1.30540543 -0.7343539
 [97,] -0.5616330 -0.5616330 -0.4975147  0.70595352 -0.4975147
 [98,] -0.5598942 -0.5598942  0.8246267  0.73029674  0.8246267
 [99,]  1.1715177  1.1715177  1.1715177  0.95547157  1.1715177
[100,]  0.7515971  0.7515971  0.8413627 -1.26584769  0.8413627
[101,]  1.8257419  1.8257419  2.6199396  2.92727278  2.6199396

permPATH documentation built on Feb. 10, 2021, 5:11 p.m.