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Tests for mutual and serial independence of categorical variablesDescriptionThe tests are constructed from the Mobius transformation applied to the probability cells in a multi-way contingency table. The Pearson chi-squared test of mutual independence is partitioned into A-dependence statistics over all subsets A of variables. The goal of the partition is to identify subsets of dependent variables when the mutual independence hypothesis is rejected by the Pearson chi-squared test. The methodology can be directly adapted to test for serial independence of d successive observations of a stationary categorical time series. For categorical time series, especially those of a nominal (non ordinal) nature, the user should be aware that tests of serial independence obtained by methods suited to quantitative sequences by quantification of the labels are not invariant to permutation of the labels contrary to the test described here. UsageA.dep.tests(Xmat,choice=1,d=0,m=d,freqname="",type="text") Arguments
ValueReturns an object of class list containing the following components:
Author(s)M. Bilodeau, P. Lafaye de Micheaux ReferencesBilodeau, M., Lafaye de Micheaux, P (2009). A-dependence statistics for mutual and serial independence of categorical variables, Journal of Statistical Planning and Inference, 139, 2407-2419. Agresti A. (2002). Categorical data analysis, Wiley, p. 322 Whisenant, E.C., Rasheed, B.K.A., Ostrer, H., Bhatnagar, Y.M. (1991). Evolution and sequence analysis of a human Y-chromosomal DNA fragment, J. Mol. Evol., 33, 133-141. Examples# Test of mutual independence between 3 independent Bernoulli variables. n <- 100 data <- data.frame(X1=rbinom(n,1,0.3),X2=rbinom(n,1,0.3),X3=rbinom(n,1,0.3)) X <- table(data) A.dep.tests(X) # Test of mutual independence between 4 variables which are # 2-independent and 3-independent but 4-dependent. n <- 100 W <- sample(x=1:8,size=n,TRUE) X1 <- W %in% c(1,2,3,5) X2 <- W %in% c(1,2,4,6) X3 <- W %in% c(1,3,4,7) X4 <- W %in% c(2,3,4,8) data <- data.frame(X1,X2,X3,X4) X <- table(data) A.dep.tests(X) # Test of serial independence of a nucleotide sequence of length # 4156 described in Whisenant et al. (1991). data(dna) x2 <- dna[1] for (i in 2:length(dna)) x2 <- paste(x2, dna[i], sep = "") x <- unlist(strsplit(x2, "")) x[x=="a"|x=="g"] <- "r" x[x=="c"|x=="t"] <- "y" out <- A.dep.tests(x,choice=2,d=1501,m=2)$TA[[1]] plot(100:1500,out[100:1500],xlab="lag j",ylab="T(1,j+1)",pch=19) abline(h=qchisq(.995,df=1)) # Analysis of a contingency table in Agresti (2002) p. 322 data(highschool) A.dep.tests(highschool,freqname="count") Results
R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
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Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
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> library(IndependenceTests)
Loading required package: xtable
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/IndependenceTests/A-dep-tests.Rd_%03d_medium.png", width=480, height=480)
> ### Name: A.dep.tests
> ### Title: Tests for mutual and serial independence of categorical
> ### variables
> ### Aliases: A.dep.tests
>
> ### ** Examples
>
> # Test of mutual independence between 3 independent Bernoulli variables.
>
> n <- 100
> data <- data.frame(X1=rbinom(n,1,0.3),X2=rbinom(n,1,0.3),X3=rbinom(n,1,0.3))
> X <- table(data)
> A.dep.tests(X)
TA fA pvalA
A
1,2 0.06 1 0.81
1,3 1.49 1 0.22
2,3 0.04 1 0.84
1,2,3 0.11 1 0.74
X2 1.69 4 0.79
>
> # Test of mutual independence between 4 variables which are
> # 2-independent and 3-independent but 4-dependent.
>
> n <- 100
> W <- sample(x=1:8,size=n,TRUE)
> X1 <- W %in% c(1,2,3,5)
> X2 <- W %in% c(1,2,4,6)
> X3 <- W %in% c(1,3,4,7)
> X4 <- W %in% c(2,3,4,8)
> data <- data.frame(X1,X2,X3,X4)
> X <- table(data)
> A.dep.tests(X)
TA fA pvalA
A
1,2 2.58 1 0.11
1,3 2.93 1 0.09
1,4 0.03 1 0.85
2,3 0.04 1 0.84
2,4 3.24 1 0.07
3,4 2.52 1 0.11
1,2,3 0 1 0.97
1,2,4 1.51 1 0.22
1,3,4 0.04 1 0.85
2,3,4 0.3 1 0.58
1,2,3,4 97.85 1 0
X2 111.03 11 0
>
> # Test of serial independence of a nucleotide sequence of length
> # 4156 described in Whisenant et al. (1991).
>
> data(dna)
> x2 <- dna[1]
> for (i in 2:length(dna)) x2 <- paste(x2, dna[i], sep = "")
> x <- unlist(strsplit(x2, ""))
> x[x=="a"|x=="g"] <- "r"
> x[x=="c"|x=="t"] <- "y"
> ## No test:
> out <- A.dep.tests(x,choice=2,d=1501,m=2)$TA[[1]]
TA fA pvalA
A
1,2 55.76 1 0
1,3 21.83 1 0
1,4 18.61 1 0
1,5 24.43 1 0
1,6 7.76 1 0.01
1,7 4.5 1 0.03
1,8 20.46 1 0
1,9 7.8 1 0.01
1,10 10.15 1 0
1,11 12.26 1 0
1,12 18.01 1 0
1,13 18.68 1 0
1,14 11.95 1 0
1,15 12.81 1 0
1,16 4.35 1 0.04
1,17 6.54 1 0.01
1,18 2.72 1 0.1
1,19 7.13 1 0.01
1,20 7.13 1 0.01
1,21 6.5 1 0.01
1,22 0.89 1 0.35
1,23 13.64 1 0
1,24 6.95 1 0.01
1,25 1.39 1 0.24
1,26 7.13 1 0.01
1,27 12.5 1 0
1,28 13.64 1 0
1,29 4.84 1 0.03
1,30 3.26 1 0.07
1,31 2.86 1 0.09
1,32 11.71 1 0
1,33 4.68 1 0.03
1,34 5.39 1 0.02
1,35 5.76 1 0.02
1,36 8.71 1 0
1,37 1.3 1 0.25
1,38 4.35 1 0.04
1,39 0.97 1 0.32
1,40 4.03 1 0.04
1,41 12.81 1 0
1,42 3.86 1 0.05
1,43 0.61 1 0.43
1,44 0.95 1 0.33
1,45 8.2 1 0
1,46 13.64 1 0
1,47 2.72 1 0.1
1,48 5.19 1 0.02
1,49 1.04 1 0.31
1,50 1.67 1 0.2
1,51 1.28 1 0.26
1,52 19.99 1 0
1,53 3.82 1 0.05
1,54 3.69 1 0.05
1,55 3.39 1 0.07
1,56 4.99 1 0.03
1,57 0.5 1 0.48
1,58 3.99 1 0.05
1,59 2.33 1 0.13
1,60 6.5 1 0.01
1,61 3.86 1 0.05
1,62 11.12 1 0
1,63 3.55 1 0.06
1,64 6.72 1 0.01
1,65 0.25 1 0.61
1,66 1.28 1 0.26
1,67 7.56 1 0.01
1,68 8.71 1 0
1,69 1.69 1 0.19
1,70 0.76 1 0.38
1,71 1.23 1 0.27
1,72 1.8 1 0.18
1,73 0.76 1 0.38
1,74 1.06 1 0.3
1,75 0.18 1 0.67
1,76 0.7 1 0.4
1,77 0.2 1 0.66
1,78 2.89 1 0.09
1,79 3.75 1 0.05
1,80 2.15 1 0.14
1,81 0.24 1 0.63
1,82 0 1 0.99
1,83 0.01 1 0.92
1,84 0.11 1 0.74
1,85 4.76 1 0.03
1,86 0.52 1 0.47
1,87 3.92 1 0.05
1,88 1.83 1 0.18
1,89 7.04 1 0.01
1,90 8.34 1 0
1,91 2.52 1 0.11
1,92 4.76 1 0.03
1,93 3.96 1 0.05
1,94 5.88 1 0.02
1,95 2.69 1 0.1
1,96 1.02 1 0.31
1,97 2.95 1 0.09
1,98 4.31 1 0.04
1,99 3.26 1 0.07
1,100 0.89 1 0.35
1,101 4.83 1 0.03
1,102 3.1 1 0.08
1,103 0.29 1 0.59
1,104 0.07 1 0.79
1,105 0.33 1 0.57
1,106 0.14 1 0.71
1,107 0.49 1 0.49
1,108 1.97 1 0.16
1,109 0.03 1 0.87
1,110 0.49 1 0.49
1,111 1.19 1 0.28
1,112 0.13 1 0.72
1,113 2.69 1 0.1
1,114 1.33 1 0.25
1,115 0.07 1 0.79
1,116 0.17 1 0.68
1,117 1.01 1 0.31
1,118 0.11 1 0.74
1,119 6.01 1 0.01
1,120 0.47 1 0.49
1,121 0.66 1 0.42
1,122 0.43 1 0.51
1,123 0.28 1 0.6
1,124 1.25 1 0.26
1,125 0.03 1 0.85
1,126 6.08 1 0.01
1,127 0 1 0.95
1,128 0.08 1 0.77
1,129 0.06 1 0.8
1,130 0.59 1 0.44
1,131 0.42 1 0.52
1,132 2.17 1 0.14
1,133 2.17 1 0.14
1,134 1.73 1 0.19
1,135 0.92 1 0.34
1,136 0 1 0.95
1,137 0.03 1 0.86
1,138 1.17 1 0.28
1,139 3.65 1 0.06
1,140 1.02 1 0.31
1,141 1.87 1 0.17
1,142 2.33 1 0.13
1,143 1.14 1 0.29
1,144 0.83 1 0.36
1,145 0.08 1 0.78
1,146 0.15 1 0.7
1,147 0.69 1 0.41
1,148 0.49 1 0.49
1,149 1.76 1 0.19
1,150 0.63 1 0.43
1,151 0.08 1 0.77
1,152 0.17 1 0.68
1,153 2.06 1 0.15
1,154 0.72 1 0.4
1,155 0.46 1 0.5
1,156 0.27 1 0.6
1,157 1.2 1 0.27
1,158 0.63 1 0.43
1,159 0 1 0.99
1,160 0.25 1 0.62
1,161 1.32 1 0.25
1,162 0.17 1 0.68
1,163 0.71 1 0.4
1,164 2.3 1 0.13
1,165 1.76 1 0.19
1,166 0.15 1 0.7
1,167 5.34 1 0.02
1,168 1.11 1 0.29
1,169 0.69 1 0.41
1,170 0.15 1 0.7
1,171 0.11 1 0.75
1,172 0.55 1 0.46
1,173 0.22 1 0.64
1,174 0.45 1 0.5
1,175 0.06 1 0.81
1,176 1.21 1 0.27
1,177 0.44 1 0.51
1,178 0.34 1 0.56
1,179 0.05 1 0.81
1,180 0.68 1 0.41
1,181 0.55 1 0.46
1,182 0.08 1 0.78
1,183 0.34 1 0.56
1,184 0.05 1 0.82
1,185 0.49 1 0.48
1,186 0.73 1 0.39
1,187 3.24 1 0.07
1,188 0.29 1 0.59
1,189 0.34 1 0.56
1,190 1.88 1 0.17
1,191 1.98 1 0.16
1,192 3.53 1 0.06
1,193 3.83 1 0.05
1,194 2.31 1 0.13
1,195 0.94 1 0.33
1,196 0.84 1 0.36
1,197 0.7 1 0.4
1,198 2.42 1 0.12
1,199 0.14 1 0.71
1,200 0.59 1 0.44
1,201 0.85 1 0.36
1,202 1.93 1 0.17
1,203 6.19 1 0.01
1,204 9.99 1 0
1,205 3.33 1 0.07
1,206 0.42 1 0.52
1,207 1.74 1 0.19
1,208 0.72 1 0.4
1,209 0.28 1 0.59
1,210 0.21 1 0.65
1,211 2.05 1 0.15
1,212 0.02 1 0.89
1,213 8.81 1 0
1,214 0.02 1 0.89
1,215 0.78 1 0.38
1,216 10.49 1 0
1,217 0.11 1 0.74
1,218 0.06 1 0.8
1,219 0.03 1 0.86
1,220 7.47 1 0.01
1,221 3.05 1 0.08
1,222 0.92 1 0.34
1,223 2.18 1 0.14
1,224 0.02 1 0.89
1,225 1.74 1 0.19
1,226 1.27 1 0.26
1,227 0.02 1 0.9
1,228 0 1 0.98
1,229 0.02 1 0.89
1,230 0.53 1 0.47
1,231 2.15 1 0.14
1,232 1.33 1 0.25
1,233 0.04 1 0.83
1,234 0.23 1 0.63
1,235 2.75 1 0.1
1,236 0.51 1 0.47
1,237 0 1 0.99
1,238 5.97 1 0.01
1,239 2.15 1 0.14
1,240 1.6 1 0.21
1,241 0.09 1 0.76
1,242 1.15 1 0.28
1,243 1.43 1 0.23
1,244 0.14 1 0.71
1,245 4.58 1 0.03
1,246 0 1 0.95
1,247 0.53 1 0.47
1,248 0.99 1 0.32
1,249 1.33 1 0.25
1,250 0.53 1 0.47
1,251 0.59 1 0.44
1,252 0.01 1 0.92
1,253 0 1 0.95
1,254 0.32 1 0.57
1,255 0.28 1 0.6
1,256 2.04 1 0.15
1,257 0.11 1 0.74
1,258 0.17 1 0.68
1,259 3.2 1 0.07
1,260 0.12 1 0.73
1,261 4.07 1 0.04
1,262 0.31 1 0.58
1,263 0.12 1 0.73
1,264 0.67 1 0.41
1,265 0.1 1 0.75
1,266 0.07 1 0.79
1,267 0.01 1 0.93
1,268 1.36 1 0.24
1,269 0.07 1 0.79
1,270 1.06 1 0.3
1,271 0.06 1 0.81
1,272 0.05 1 0.82
1,273 0.66 1 0.42
1,274 0.36 1 0.55
1,275 3.05 1 0.08
1,276 2.18 1 0.14
1,277 1.17 1 0.28
1,278 1.62 1 0.2
1,279 0.77 1 0.38
1,280 0.01 1 0.92
1,281 0.01 1 0.92
1,282 1.65 1 0.2
1,283 0.18 1 0.67
1,284 0.11 1 0.74
1,285 0.25 1 0.62
1,286 1.06 1 0.3
1,287 0.13 1 0.71
1,288 0.46 1 0.5
1,289 4.14 1 0.04
1,290 0.68 1 0.41
1,291 0.1 1 0.75
1,292 0.66 1 0.42
1,293 0.94 1 0.33
1,294 0 1 0.98
1,295 2.63 1 0.1
1,296 0.09 1 0.77
1,297 0.11 1 0.74
1,298 1.32 1 0.25
1,299 0.22 1 0.64
1,300 0.81 1 0.37
1,301 0.55 1 0.46
1,302 0.68 1 0.41
1,303 0.26 1 0.61
1,304 0.35 1 0.55
1,305 0.56 1 0.45
1,306 1.23 1 0.27
1,307 0.18 1 0.67
1,308 0.66 1 0.42
1,309 0.49 1 0.48
1,310 0.44 1 0.51
1,311 1.9 1 0.17
1,312 1.06 1 0.3
1,313 0.06 1 0.81
1,314 0.08 1 0.78
1,315 1.04 1 0.31
1,316 0.18 1 0.67
1,317 0.63 1 0.43
1,318 1.69 1 0.19
1,319 0.45 1 0.5
1,320 4.17 1 0.04
1,321 0.15 1 0.7
1,322 0.81 1 0.37
1,323 0.01 1 0.9
1,324 0.08 1 0.78
1,325 0.45 1 0.5
1,326 0.04 1 0.85
1,327 1.21 1 0.27
1,328 1.21 1 0.27
1,329 0.55 1 0.46
1,330 0.08 1 0.78
1,331 0.01 1 0.94
1,332 0.04 1 0.84
1,333 0.01 1 0.94
1,334 0.4 1 0.53
1,335 0.22 1 0.64
1,336 0.57 1 0.45
1,337 1.97 1 0.16
1,338 0.58 1 0.45
1,339 0.08 1 0.77
1,340 1.72 1 0.19
1,341 1.45 1 0.23
1,342 0.28 1 0.6
1,343 0.52 1 0.47
1,344 0.33 1 0.56
1,345 5.97 1 0.01
1,346 1.13 1 0.29
1,347 0.08 1 0.78
1,348 0.26 1 0.61
1,349 1.57 1 0.21
1,350 0.8 1 0.37
1,351 2.49 1 0.11
1,352 0.22 1 0.64
1,353 1.03 1 0.31
1,354 1.98 1 0.16
1,355 1.06 1 0.3
1,356 0.31 1 0.58
1,357 0.02 1 0.88
1,358 3.14 1 0.08
1,359 3.84 1 0.05
1,360 0.27 1 0.6
1,361 1.57 1 0.21
1,362 0 1 0.99
1,363 1.03 1 0.31
1,364 1.68 1 0.2
1,365 0.07 1 0.78
1,366 0.04 1 0.85
1,367 0.81 1 0.37
1,368 0.1 1 0.75
1,369 1.05 1 0.31
1,370 1.04 1 0.31
1,371 0.02 1 0.88
1,372 2.72 1 0.1
1,373 0.04 1 0.85
1,374 1.28 1 0.26
1,375 0 1 0.99
1,376 0.29 1 0.59
1,377 1.85 1 0.17
1,378 5.51 1 0.02
1,379 0.03 1 0.87
1,380 0.06 1 0.8
1,381 0.81 1 0.37
1,382 1.4 1 0.24
1,383 0.22 1 0.64
1,384 1.69 1 0.19
1,385 0.01 1 0.9
1,386 0.04 1 0.85
1,387 0.55 1 0.46
1,388 0.5 1 0.48
1,389 0.18 1 0.67
1,390 1.04 1 0.31
1,391 0.39 1 0.53
1,392 0.04 1 0.85
1,393 2.04 1 0.15
1,394 7.4 1 0.01
1,395 0.13 1 0.72
1,396 1.88 1 0.17
1,397 0.07 1 0.79
1,398 0.55 1 0.46
1,399 0.13 1 0.72
1,400 0.15 1 0.7
1,401 0.56 1 0.45
1,402 0 1 0.97
1,403 0.02 1 0.88
1,404 0.05 1 0.82
1,405 0.51 1 0.48
1,406 0.9 1 0.34
1,407 0.69 1 0.41
1,408 0 1 0.97
1,409 2.86 1 0.09
1,410 0.89 1 0.35
1,411 0.31 1 0.58
1,412 0.1 1 0.75
1,413 0.39 1 0.53
1,414 3.72 1 0.05
1,415 0.23 1 0.63
1,416 0.99 1 0.32
1,417 1.25 1 0.26
1,418 3.75 1 0.05
1,419 0.03 1 0.87
1,420 0.01 1 0.94
1,421 0.06 1 0.81
1,422 0.2 1 0.66
1,423 7.22 1 0.01
1,424 0.32 1 0.57
1,425 2.89 1 0.09
1,426 4.2 1 0.04
1,427 2.89 1 0.09
1,428 2.49 1 0.11
1,429 1.41 1 0.24
1,430 0.19 1 0.67
1,431 0.3 1 0.58
1,432 6.54 1 0.01
1,433 0 1 0.97
1,434 5.76 1 0.02
1,435 2.6 1 0.11
1,436 0.13 1 0.72
1,437 1.06 1 0.3
1,438 1.23 1 0.27
1,439 0.1 1 0.75
1,440 0.13 1 0.72
1,441 0 1 0.96
1,442 6.19 1 0.01
1,443 1.43 1 0.23
1,444 2.66 1 0.1
1,445 1.95 1 0.16
1,446 1 1 0.32
1,447 0.11 1 0.73
1,448 0.28 1 0.6
1,449 0.79 1 0.37
1,450 0.01 1 0.92
1,451 3.07 1 0.08
1,452 1.09 1 0.3
1,453 0.09 1 0.76
1,454 0.34 1 0.56
1,455 0.38 1 0.54
1,456 2.5 1 0.11
1,457 0.05 1 0.82
1,458 0.03 1 0.87
1,459 1.61 1 0.2
1,460 0.63 1 0.43
1,461 0.41 1 0.52
1,462 1.28 1 0.26
1,463 0.16 1 0.69
1,464 0.21 1 0.64
1,465 0.15 1 0.7
1,466 1.23 1 0.27
1,467 0.38 1 0.54
1,468 0 1 0.99
1,469 1.09 1 0.3
1,470 0.11 1 0.73
1,471 2.06 1 0.15
1,472 0.28 1 0.6
1,473 0.71 1 0.4
1,474 0.14 1 0.71
1,475 4.08 1 0.04
1,476 5.51 1 0.02
1,477 5.88 1 0.02
1,478 0.25 1 0.62
1,479 0.11 1 0.75
1,480 0.07 1 0.79
1,481 0.95 1 0.33
1,482 0.3 1 0.58
1,483 1.67 1 0.2
1,484 0.23 1 0.63
1,485 2.96 1 0.09
1,486 0.12 1 0.73
1,487 0.26 1 0.61
1,488 0.96 1 0.33
1,489 0.22 1 0.64
1,490 0.62 1 0.43
1,491 0 1 0.97
1,492 0.61 1 0.44
1,493 0.12 1 0.73
1,494 0.62 1 0.43
1,495 0.12 1 0.73
1,496 1.11 1 0.29
1,497 8.72 1 0
1,498 0.27 1 0.61
1,499 0 1 0.96
1,500 5.4 1 0.02
1,501 2.26 1 0.13
1,502 0.63 1 0.43
1,503 0.91 1 0.34
1,504 2.26 1 0.13
1,505 0.41 1 0.52
1,506 0 1 0.99
1,507 0.57 1 0.45
1,508 0.18 1 0.67
1,509 0.02 1 0.88
1,510 0.15 1 0.7
1,511 5.44 1 0.02
1,512 0.31 1 0.58
1,513 0.06 1 0.81
1,514 0 1 0.96
1,515 0.02 1 0.89
1,516 0.17 1 0.68
1,517 1.06 1 0.3
1,518 1.41 1 0.23
1,519 0.23 1 0.63
1,520 1.93 1 0.16
1,521 0.23 1 0.63
1,522 0 1 0.99
1,523 0.51 1 0.47
1,524 0.08 1 0.78
1,525 10.96 1 0
1,526 4.69 1 0.03
1,527 8.06 1 0
1,528 0.08 1 0.78
1,529 0.03 1 0.87
1,530 0.98 1 0.32
1,531 0.57 1 0.45
1,532 1.91 1 0.17
1,533 0.34 1 0.56
1,534 0.02 1 0.9
1,535 0 1 1
1,536 0.01 1 0.91
1,537 0.13 1 0.72
1,538 2.13 1 0.14
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1,1061
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