How variables (constrained loadings) are selected in a biplot CCA

Question

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

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

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

 microbiome_and_chemical.cca<-cca(t(Taxa.family),Chemical.data.mean)

Thnaks. SM

Answer 1

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.