I am trying to do a Canonical correspondence analysis (CCA) using the community data and chemical data. I have my family level taxonomic data as community data. In chemical data I have 18 variables: Ni Cr Cu Pb Cd Co Zn Fe As Ammonia_mean Silicate_mean Phosphate_mean Nitrite_mean Nitrate_mean Sulphate_mean pH_mean Salinity_mean D.O_mean

Obviously CCA only shows the unconstrained loadings. Can anybody please explain me how these are chosen. The reason is I see its always takes from the first columns. So if in my chemical data variables are displayed as Ni Cr Cu Pb Cd Co Zn Fe As Ammonia_mean Silicate_mean Phosphate_mean Nitrite_mean Nitrate_mean (in this order)

I get the CCA plot as enter image description here

But if the chemical data variables are displayed as Ammonia_mean Silicate_mean Phosphate_mean Nitrite_mean Nitrate_mean Ni Cr Cu Pb Cd Co Zn Fe As (in this order)

I get the CCA plot as enter image description here

Now the unconstrained loading changes based on which variable I keep in first columns. I should I know which one are more important? Help suggestion to this?

I am also providing my data here.

dput(Chemical.data.mean) structure(list(Ammonia_mean = c(91.2808, 38.337936, 13.69212, 58.419712, 20.994584, 17.343352, 25.558624, 15.517736), Silicate_mean = c(4733.721109, 2799.512484, 1221.605447, 712.6031777, 1934.208625, 865.3038584, 4606.470542, 916.204092), Phosphate_mean = c(256.191072, 258.859729, 325.576154, 280.208985, 301.558241, 293.55227, 242.847787, 309.564212 ), Nitrite_mean = c(92.53356407, 133.6595926, 131.3748132, 89.10639501, 142.79871, 121.0933061, 211.3420908, 166.7888933), Nitrate_mean = c(224480.5, 224092, 121617.5, 119583.5, 121188.5, 158316, 207189.5, 212209.5 ), Sulphate_mean = c(167818, 159793.5, 61225, 116131, 128932.5, 139670, 958423.5, 172161.5), pH_mean = c(7.74, 7.775, 7.915, 7.85, 7.63, 7.58, 7.57, 7.735), Salinity_mean = c(23.805, 23.35, 20.495, 20.37, 19.275, 18.55, 22.65, 22.55), D.O_mean = c(13.835, 15.46, 11.715, 13.45, 12.605, 11.995, 17.56, 18.03), Ni = c(63.76, 53.52, 78.88, 71.6, 87.8, 111.92, 82.6, 64.72), Cr = c(91.04, 88.16, 113.44, 131.88, 113.6, 103.48, 91.16, 89.24), Cu = c(46.96, 30.96, 48.16, 42.56, 34.96, 32.82, 32.77, 30.1), Pb = c(32.56, 16.12, 32.256, 16.82, 15.02, 22.04, 9.98, 11.97), Cd = c(0.164, 0.49, 1.48, 3.02, 2.48, 1.21, 0.2, 0.004), Co = c(940, 696, 1052.8, 1076.8, 983.2, 1216, 863.2, 723.6), Zn = c(1.66, 1.94, 3.69, 4.24, 2.33, 2.56, 2.21, 2), Fe = c(9.94, 14.18, 10.16, 76.16, 9.33, 11.23, 7.27, 7.3), As = c(4.02, 4, 4.36, 3.7, 6.08, 6.58, 4.98, 3.56)), .Names = c("Ammonia_mean", "Silicate_mean", "Phosphate_mean", "Nitrite_mean", "Nitrate_mean", "Sulphate_mean", "pH_mean", "Salinity_mean", "D.O_mean", "Ni", "Cr", "Cu", "Pb", "Cd", "Co", "Zn", "Fe", "As" ), class = "data.frame", row.names = c("S_1", "S_2", "SO_3", "SO_4", "SO_5", "SO_6", "SO_7", "SO_8")) dput(Taxa.family) structure(list(SO_4832_1 = c(260L, 0L, 79L, 0L, 32L, 356L, 0L, 324L, 130L, 30L, 758L, 11L, 0L, 0L, 55L, 0L, 0L, 86L, 666L, 42L, 679L, 18L, 22L, 523L, 0L, 101L, 0L, 42L, 2715L, 37L, 0L, 12L, 94L, 0L, 0L, 198L, 0L, 0L, 104L, 9L, 218L, 61L, 1068L, 1902L, 0L, 0L, 312L, 97L, 21L, 44L, 0L, 73L, 0L, 23L, 240L, 0L, 176L, 119L, 0L, 87L, 390L, 504L, 0L, 758L, 437L, 27L, 0L, 0L, 448L, 199L, 78L, 524L, 94L, 128L, 0L, 91L, 62L, 0L, 39L, 34L, 14L, 33L, 0L, 0L, 0L, 83L, 239L, 326L, 61L, 34L, 13L, 75L, 53L, 0L, 0L, 76L, 124L, 12L, 0L, 41L, 0L, 18L, 0L, 646L, 0L, 21L, 27L, 38L, 0L, 134L, 35L, 489L, 14L, 0L, 8L, 32L, 112L, 0L, 1323L, 0L, 40L, 2516L, 43L, 78L, 23L, 15L, 21L, 565L, 100L, 32L, 31L, 0L, 0L, 0L, 13L, 22L, 14L, 15L, 0L, 14L, 0L, 0L, 0L, 0L, 1473L, 67L, 33L, 0L, 11L, 0L, 0L, 0L, 0L, 33L, 0L, 371L, 0L, 17L, 0L, 0L, 54L, 8L, 35L, 0L, 39L, 20L, 21L, 60L, 147L, 0L, 645L, 209L, 0L, 85L, 13L, 0L, 0L, 57L), SO_4832_2 = c(230L, 0L, 91L, 11L, 46L, 297L, 0L, 260L, 249L, 12L, 986L, 10L, 0L, 9L, 71L, 10L, 10L, 0L, 445L, 26L, 494L, 0L, 27L, 647L, 0L, 117L, 0L, 38L, 2632L, 32L, 0L, 9L, 114L, 0L, 0L, 222L, 0L, 9L, 81L, 0L, 336L, 37L, 676L, 1530L, 0L, 0L, 265L, 139L, 37L, 38L, 0L, 48L, 11L, 0L, 189L, 0L, 379L, 60L, 0L, 150L, 684L, 706L, 0L, 481L, 390L, 39L, 14L, 0L, 339L, 136L, 46L, 252L, 53L, 134L, 0L, 124L, 144L, 31L, 55L, 25L, 10L, 15L, 11L, 22L, 0L, 26L, 190L, 66L, 30L, 27L, 20L, 139L, 40L, 0L, 14L, 40L, 186L, 23L, 0L, 13L, 0L, 18L, 0L, 706L, 9L, 12L, 15L, 27L, 40L, 126L, 35L, 3933L, 21L, 0L, 8L, 36L, 58L, 0L, 502L, 0L, 25L, 1410L, 14L, 42L, 13L, 10L, 15L, 312L, 101L, 23L, 38L, 0L, 46L, 0L, 0L, 21L, 0L, 9L, 0L, 0L, 0L, 0L, 0L, 0L, 2163L, 64L, 96L, 0L, 21L, 0L, 14L, 37L, 0L, 31L, 0L, 294L, 13L, 17L, 0L, 0L, 0L, 0L, 10L, 0L, 62L, 46L, 0L, 0L, 238L, 0L, 363L, 330L, 0L, 61L, 42L, 0L, 0L, 61L), SO_4832_3 = c(70L, 0L, 57L, 0L, 0L, 28L, 1L, 188L, 128L, 25L, 1632L, 15L, 192L, 174L, 196L, 251L, 86L, 55L, 892L, 91L, 7760L, 45L, 9L, 60L, 12L, 10L, 0L, 0L, 308L, 89L, 0L, 0L, 76L, 45L, 0L, 178L, 0L, 0L, 490L, 18L, 55L, 11L, 7552L, 441L, 11L, 0L, 5009L, 770L, 47L, 39L, 83L, 0L, 37L, 0L, 37L, 39L, 109L, 93L, 17L, 61L, 201L, 628L, 55L, 119L, 522L, 8L, 0L, 13L, 160L, 74L, 19L, 557L, 30L, 29L, 77L, 1745L, 0L, 9L, 152L, 290L, 39L, 0L, 0L, 54L, 9L, 171L, 130L, 110L, 23L, 25L, 11L, 32L, 51L, 28L, 0L, 444L, 93L, 14L, 9L, 220L, 9L, 551L, 45L, 196L, 0L, 10L, 13L, 37L, 0L, 70L, 19L, 718L, 47L, 0L, 0L, 13L, 31L, 0L, 727L, 82L, 24L, 4006L, 0L, 168L, 0L, 48L, 0L, 2198L, 321L, 49L, 31L, 0L, 0L, 21L, 0L, 813L, 44L, 10L, 0L, 31L, 27L, 0L, 0L, 0L, 664L, 109L, 0L, 0L, 0L, 13L, 0L, 13L, 11L, 41L, 0L, 317L, 8L, 0L, 0L, 0L, 39L, 0L, 50L, 44L, 22L, 204L, 21L, 106L, 620L, 0L, 583L, 507L, 0L, 52L, 54L, 0L, 12L, 0L), SO_4832_4 = c(130L, 0L, 126L, 15L, 13L, 175L, 2L, 247L, 51L, 192L, 490L, 9L, 11L, 205L, 398L, 12L, 37L, 30L, 726L, 108L, 2435L, 35L, 24L, 152L, 0L, 16L, 9L, 34L, 875L, 68L, 10L, 0L, 116L, 12L, 0L, 264L, 11L, 0L, 241L, 18L, 169L, 32L, 3004L, 1069L, 0L, 13L, 1181L, 239L, 19L, 62L, 11L, 19L, 24L, 0L, 56L, 0L, 177L, 99L, 18L, 197L, 1118L, 1964L, 29L, 254L, 425L, 11L, 0L, 0L, 215L, 107L, 48L, 529L, 84L, 121L, 130L, 3875L, 0L, 17L, 23L, 136L, 30L, 41L, 24L, 59L, 0L, 135L, 429L, 375L, 45L, 48L, 12L, 240L, 41L, 0L, 40L, 103L, 196L, 19L, 13L, 133L, 0L, 59L, 0L, 726L, 0L, 35L, 65L, 63L, 0L, 160L, 104L, 2493L, 20L, 9L, 0L, 30L, 97L, 10L, 444L, 11L, 23L, 2962L, 11L, 70L, 0L, 23L, 10L, 670L, 317L, 26L, 57L, 13L, 0L, 11L, 53L, 291L, 49L, 10L, 0L, 13L, 8L, 10L, 0L, 0L, 1625L, 122L, 26L, 0L, 10L, 0L, 0L, 22L, 0L, 22L, 0L, 223L, 9L, 13L, 0L, 18L, 43L, 0L, 32L, 52L, 40L, 90L, 41L, 19L, 206L, 0L, 788L, 250L, 0L, 72L, 9L, 61L, 10L, 22L), SO_4832_5 = c(185L, 0L, 84L, 10L, 11L, 304L, 0L, 532L, 64L, 0L, 292L, 0L, 0L, 14L, 10L, 39L, 0L, 68L, 1059L, 59L, 1940L, 10L, 18L, 528L, 33L, 21L, 0L, 41L, 1712L, 41L, 0L, 0L, 105L, 0L, 0L, 135L, 0L, 0L, 218L, 0L, 168L, 64L, 1822L, 2119L, 14L, 10L, 848L, 222L, 15L, 14L, 0L, 79L, 0L, 0L, 151L, 0L, 228L, 155L, 0L, 89L, 379L, 705L, 0L, 420L, 159L, 9L, 20L, 0L, 757L, 479L, 55L, 594L, 92L, 203L, 0L, 189L, 31L, 0L, 31L, 100L, 18L, 17L, 0L, 0L, 0L, 80L, 652L, 414L, 36L, 44L, 19L, 133L, 73L, 0L, 10L, 28L, 190L, 17L, 0L, 109L, 11L, 25L, 18L, 1084L, 0L, 0L, 0L, 0L, 0L, 125L, 29L, 1361L, 11L, 0L, 0L, 26L, 64L, 0L, 620L, 0L, 16L, 1335L, 29L, 88L, 14L, 8L, 14L, 576L, 68L, 21L, 41L, 0L, 0L, 0L, 17L, 39L, 12L, 0L, 0L, 19L, 21L, 0L, 0L, 0L, 1220L, 77L, 0L, 0L, 8L, 0L, 0L, 10L, 0L, 41L, 0L, 393L, 8L, 12L, 0L, 0L, 13L, 0L, 0L, 0L, 53L, 0L, 0L, 0L, 95L, 0L, 57L, 112L, 0L, 38L, 0L, 0L, 0L, 33L), SO_4832_6 = c(134L, 9L, 61L, 9L, 37L, 495L, 0L, 426L, 36L, 0L, 370L, 0L, 0L, 0L, 0L, 0L, 0L, 72L, 480L, 16L, 227L, 0L, 10L, 699L, 0L, 0L, 0L, 14L, 2733L, 24L, 0L, 0L, 73L, 0L, 29L, 291L, 0L, 0L, 72L, 0L, 322L, 154L, 341L, 2206L, 0L, 0L, 101L, 83L, 16L, 22L, 0L, 19L, 0L, 0L, 174L, 0L, 88L, 102L, 0L, 38L, 180L, 499L, 0L, 907L, 93L, 26L, 9L, 0L, 514L, 283L, 102L, 499L, 82L, 23L, 0L, 106L, 34L, 0L, 27L, 17L, 0L, 0L, 0L, 0L, 0L, 23L, 170L, 470L, 53L, 0L, 23L, 50L, 10L, 0L, 0L, 18L, 97L, 20L, 0L, 12L, 0L, 0L, 0L, 316L, 0L, 9L, 0L, 21L, 0L, 26L, 0L, 53L, 24L, 0L, 0L, 37L, 34L, 0L, 883L, 69L, 28L, 919L, 24L, 45L, 18L, 0L, 24L, 136L, 67L, 0L, 48L, 0L, 0L, 0L, 20L, 0L, 12L, 21L, 0L, 67L, 33L, 0L, 10L, 10L, 1165L, 34L, 10L, 31L, 12L, 0L, 0L, 0L, 0L, 0L, 450L, 601L, 3901L, 10L, 156L, 0L, 0L, 0L, 0L, 0L, 15L, 0L, 0L, 0L, 88L, 0L, 18L, 112L, 1155L, 17L, 0L, 0L, 0L, 35L), SO_4832_7 = c(147L, 0L, 92L, 19L, 18L, 409L, 0L, 492L, 44L, 15L, 545L, 0L, 0L, 13L, 52L, 0L, 15L, 59L, 731L, 35L, 992L, 0L, 14L, 606L, 0L, 14L, 0L, 50L, 2284L, 16L, 0L, 0L, 134L, 0L, 0L, 143L, 0L, 0L, 134L, 0L, 165L, 51L, 1040L, 2994L, 11L, 12L, 440L, 121L, 15L, 10L, 0L, 41L, 0L, 8L, 159L, 0L, 280L, 121L, 0L, 103L, 1023L, 1183L, 0L, 551L, 273L, 60L, 10L, 0L, 823L, 421L, 90L, 492L, 117L, 212L, 0L, 116L, 21L, 19L, 39L, 57L, 8L, 29L, 0L, 0L, 0L, 37L, 411L, 473L, 72L, 18L, 29L, 159L, 37L, 0L, 13L, 18L, 181L, 15L, 31L, 35L, 0L, 16L, 0L, 626L, 11L, 12L, 13L, 29L, 0L, 219L, 14L, 614L, 0L, 0L, 0L, 35L, 64L, 0L, 483L, 0L, 0L, 949L, 98L, 60L, 22L, 0L, 21L, 250L, 109L, 9L, 28L, 0L, 0L, 0L, 16L, 0L, 10L, 0L, 0L, 13L, 9L, 0L, 0L, 13L, 1647L, 38L, 25L, 0L, 13L, 0L, 0L, 0L, 0L, 12L, 0L, 317L, 14L, 18L, 0L, 0L, 0L, 0L, 16L, 0L, 28L, 0L, 0L, 0L, 181L, 0L, 61L, 227L, 0L, 42L, 8L, 0L, 0L, 32L), SO_4832_8 = c(42L, 0L, 125L, 18L, 13L, 83L, 0L, 169L, 323L, 259L, 2687L, 37L, 10L, 186L, 325L, 11L, 36L, 0L, 357L, 102L, 1867L, 0L, 16L, 137L, 0L, 18L, 0L, 16L, 556L, 36L, 0L, 0L, 56L, 12L, 0L, 221L, 9L, 0L, 246L, 0L, 124L, 19L, 2932L, 899L, 0L, 17L, 1155L, 218L, 55L, 16L, 0L, 8L, 0L, 31L, 28L, 0L, 140L, 125L, 0L, 171L, 1104L, 2530L, 20L, 80L, 380L, 0L, 0L, 0L, 151L, 91L, 23L, 380L, 38L, 106L, 156L, 5953L, 0L, 23L, 145L, 165L, 12L, 14L, 20L, 72L, 0L, 81L, 452L, 308L, 31L, 42L, 0L, 326L, 23L, 0L, 0L, 97L, 158L, 20L, 0L, 85L, 0L, 57L, 0L, 845L, 0L, 28L, 8L, 31L, 0L, 171L, 79L, 1630L, 18L, 24L, 0L, 0L, 0L, 0L, 197L, 0L, 0L, 793L, 0L, 45L, 0L, 0L, 0L, 243L, 212L, 35L, 56L, 9L, 0L, 0L, 72L, 407L, 17L, 0L, 12L, 0L, 12L, 0L, 9L, 10L, 1814L, 75L, 77L, 0L, 22L, 0L, 18L, 0L, 0L, 10L, 0L, 161L, 15L, 8L, 0L, 9L, 13L, 0L, 37L, 85L, 41L, 73L, 38L, 0L, 837L, 10L, 100L, 1016L, 0L, 210L, 44L, 16L, 0L, 13L)), .Names = c("S_1", "S_2", "SO_3", "SO_4", "SO_5", "SO_6", "SO_7", "SO_8"), class = "data.frame", row.names = c("Acidobacteriaceae", "Acanthopleuribacteraceae", "Holophagaceae", "Bryobacteraceae", "Solibacteraceae", "Calditrichaceae", "Deferribacteraceae", "Rhodothermaceae", "Bacteroidaceae", "Porphyromonadaceae", "Prevotellaceae", "Rikenellaceae", "Marinifilaceae", "Marinilabiliaceae", "Prolixibacteraceae", "Catalimonadaceae", "Cyclobacteriaceae", "Cytophagaceae", "Flammeovirgaceae", "Cryomorphaceae", "Flavobacteriaceae", "Sphingobacteriaceae", "Ignavibacteriaceae", "Gemmatimonadaceae", "Longimicrobiaceae", "Fusobacteriaceae", "Leptotrichiaceae", "Nitrospinaceae", "Nitrospiraceae", "Caulobacteraceae", "Kordiimonadaceae", "Micropepsaceae", "Parvularculaceae", "Aurantimonadaceae", "Bradyrhizobiaceae", "Hyphomicrobiaceae", "Methylobacteriaceae", "Methylocystaceae", "Phyllobacteriaceae", "Rhizobiaceae", "Rhodobiaceae", "Xanthobacteraceae", "Rhodobacteraceae", "Rhodospirillaceae", "Rickettsiaceae", "Sneathiellaceae", "Erythrobacteraceae", "Sphingomonadaceae", "Alcaligenaceae", "Comamonadaceae", "Oxalobacteraceae", "Hydrogenophilaceae", "Methylophilaceae", "Neisseriaceae", "Nitrosomonadaceae", "Rhodocyclaceae", "Bacteriovoracaceae", "Bdellovibrionaceae", "Pseudobacteriovoracaceae", "Desulfarculaceae", "Desulfobacteraceae", "Desulfobulbaceae", "Desulfovibrionaceae", "Desulfurellaceae", "Desulfuromonadaceae", "Geobacteraceae", "Anaeromyxobacteraceae", "Myxococcaceae", "Kofleriaceae", "Nannocystaceae", "Polyangiaceae", "Sandaracinaceae", "Syntrophaceae", "Syntrophobacteraceae", "Campylobacteraceae", "Helicobacteraceae", "Acidiferrobacteraceae", "Aeromonadaceae", "Succinivibrionaceae", "Alteromonadaceae", "Colwelliaceae", "Pseudoalteromonadaceae", "Psychromonadaceae", "Shewanellaceae", "Arenicellaceae", "Cellvibrionaceae", "Halieaceae", "Microbulbiferaceae", "Porticoccaceae", "Spongiibacteraceae", "Chromatiaceae", "Ectothiorhodospiraceae", "Granulosicoccaceae", "Halothiobacillaceae", "Thioalkalispiraceae", "Enterobacteriaceae", "Coxiellaceae", "Legionellaceae", "Methylococcaceae", "Alcanivoracaceae", "Hahellaceae", "Halomonadaceae", "Kangiellaceae", "Oceanospirillaceae", "Oleiphilaceae", "Pasteurellaceae", "Moraxellaceae", "Pseudomonadaceae", "Salinisphaeraceae", "Piscirickettsiaceae", "Thiotrichaceae", "Vibrionaceae", "Xanthomonadaceae", "Mariprofundaceae", "Chlamydiaceae", "Parachlamydiaceae", "Simkaniaceae", "Oligosphaeraceae", "Phycisphaeraceae", "Tepidisphaeraceae", "Gemmataceae", "Planctomycetaceae", "Opitutaceae", "Puniceicoccaceae", "Chthoniobacteraceae", "Rubritaleaceae", "Verrucomicrobia subdivision 3", "Verrucomicrobiaceae", "Spirochaetaceae", "Bifidobacteriaceae", "Mycobacteriaceae", "Nocardiaceae", "Frankiaceae", "Kineosporiaceae", "Cellulomonadaceae", "Demequinaceae", "Microbacteriaceae", "Micrococcaceae", "Promicromonosporaceae", "Micromonosporaceae", "Nocardioidaceae", "Propionibacteriaceae", "Pseudonocardiaceae", "Streptomycetaceae", "Anaerolineaceae", "Caldilineaceae", "Dehalococcoidaceae", "Ktedonobacteraceae", "Sphaerobacteraceae", "Cyanobacteriaceae", "Microcoleaceae", "Hyellaceae", "Synechococcaceae", "Trueperaceae", "Alicyclobacillaceae", "Bacillaceae", "Paenibacillaceae", "Staphylococcaceae", "Thermoactinomycetaceae", "Enterococcaceae", "Lactobacillaceae", "Leuconostocaceae", "Streptococcaceae", "Christensenellaceae", "Clostridiaceae", "Clostridiales Family XII. Incertae Sedis", "Defluviitaleaceae", "Eubacteriaceae", "Lachnospiraceae", "Peptococcaceae", "Peptostreptococcaceae", "Ruminococcaceae", "Symbiobacteriaceae", "Erysipelotrichaceae", "Selenomonadaceae", "Sporomusaceae", "Acholeplasmataceae", "Halobacteriaceae"))

And I am doing CCA as


Thnaks. SM

  • $\begingroup$ Obviously CCA should be able to deal with all 18 variables. My feeling is that some of these variables are linearly dependent on the others, and get excluded by the CCA algorithm. Are your variables continuous? Start by checking the rank of your covariate matrix. Also try to provide a reproducible example. $\endgroup$ – Knarpie Aug 10 '17 at 12:16
  • $\begingroup$ Dear Knarpie, Thank you. I am adding my data in the question again. This is the microbiome and chemical data that I want to do CCA on. But problem is if they are linierly dependent then also why column possition will matter? If I change certain chemicals from one column to other the selected unconstrained loading changes. I am getting a bit confused here. $\endgroup$ – user3042163 Aug 10 '17 at 13:41
  • $\begingroup$ What's CCA? Canonical correspondence analysis? Please de-abbreviate it in the question somewhere. $\endgroup$ – ttnphns Aug 11 '17 at 8:59

First of all the cause of the problem is not multicollinearity, as I first thought, but low sample size. Since you have only 8 samples, you can only do a CCA with 7 variables, for lack of degrees of freedom.

Your CCA algorithm clearly acknowledges this, and only uses the first 7 variables it is provided with, and disregards the rest. That explains why the ordering of the variables matters.

Solutions are to get more samples, or to only include chemical variables you suspect to be most strongly related to microbiome composition. A more advanced (and in my opinion better) solution is presented in https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3590923/, which comes with automatic model selection.

  • $\begingroup$ Thank you very much for explaining this. I was missing that point. Thanks :) $\endgroup$ – user3042163 Aug 10 '17 at 16:06
  • $\begingroup$ You're welcome. Can you accept the answer then @user3042163 :-) ? $\endgroup$ – Knarpie Aug 11 '17 at 8:18
  • $\begingroup$ How do I accept? Sorry I am new to this site and couldn't find the option $\endgroup$ – user3042163 Aug 11 '17 at 19:24
  • $\begingroup$ @user3042163 See stats.stackexchange.com/help/someone-answers. It's the checkmark you need. $\endgroup$ – Knarpie Aug 16 '17 at 9:24

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