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Our third example again uses the same data matrix, PLANT, but now the clada used are the partial monothetic sets (PMS) of terminal taxa plus all additive binary codings possible for each block of homologous states (PMS + ABC). The best cladograms are selected by using the minimum step criterion. To run this example you can follow the same steps as given for the first example in the beginning of this chapter, except for the CAFCA parameters dialog where you must click the new settings, i.e., # 3: PMS + ABC.
We now find 16 clada instead of 12 when using PMS or 14 when using SMS (table 3.11). Compared with the first example the following groups are added to the list of clada: {1 2}, {2 5}, {2 3 4}, and {1 3 4 5}. It is clear from table 3.11 with sets of character states that these new clada do not possess at least one unique character state (clada 8, 9, 12, and 15).
Partial Monothetic Sets of terminal taxa in Plant
-----------------
1 | 1
2 | 2
3 | 3
4 | 4
5 | 5
6 | 3 4
7 | 4 5
8 | 2 5
9 | 1 2
10 | 3 4 5
11 | 1 2 3
12 | 2 3 4
13 | 2 3 4 5
14 | 1 2 3 4
15 | 1 3 4 5
16 | 1 2 3 4 5
-------------
Partial Monothetic Sets of character states in Plant
------------------
1 | 2 5 13
2 | 6 8 11
3 |
4 |
5 | 10
6 | 9
7 | 14 16 20
8 |
9 |
10 | 7 12 19
11 | 15
12 |
13 | 3 4 18
14 | 17
15 |
16 | 1
------------
Table 3.11: Clada with corresponding character states obtained with option 4, PMS plus all additive binary codings.
These groups are based on an additive combination of character states within a block of homologous states. For instance, {1 2} is based on columns 11 + 13 [= state 1 plus 3 of character 6], {1 3 4 5} on columns 12 + 13 [= state 2 plus 3 of character 6], {2 5} on columns 8 + 10 [= state 1 plus 3 of character 5], {2 3 4} on 8 + 9 [= state 1 plus 2 of character 5].
In fact all additive codings possible within each block of homologous states are used to add new clada to the list that can be build on the basis of PMS only. (CAFCA poses a limit of 6 states per block, as a default, to derive additive codings, thus implying a maximum of 945 character state trees per character. You can change this default in the CAFCA parameter dialog, up to maximum value of 12. Note, however, that the number of implied cladograms increases exponentially relative to the number of taxa, such that 12 in this context is an extremely high number).
The use of all possible additive codings for each character adds considerable resolving power to the set of clada. It boils down to exploring and comparing all polarity and order decisions possible for each multi-state character (transformation series).
By means of the option in the menu we can generate the strict monothetic sets of character states, instead of only the partial sets, that characterise the clada resulting from the PMS+ABC option listed in table 3.11. These strict sets of character states indicate that some of the clada, as # 9 & 11, 8 & 13 and 15 & 16, besides lacking unique character states, don't even have unique combinations of character states that could be used to characterise them (table 3.12).
Strict Monothetic Sets of character states in Plant
---------------------------------------------------
1| 1 2 5 13 15 17
2| 1 3 4 6 8 11 15 17 18
3| 1 3 4 7 9 12 15 17 18 19
4| 1 3 4 7 9 12 14 16 17 18 19 20
5| 1 3 4 7 10 12 14 16 18 19 20
6| 1 3 4 7 9 12 17 18 19
7| 1 3 4 7 12 14 16 18 19 20
8| 1 3 4 18
9| 1 15 17
10| 1 3 4 7 12 18 19
11| 1 15 17
12| 1 3 4 17 18
13| 1 3 4 18
14| 1 17
15| 1
16| 1
---------------------------------------
Table 3.12: Strict monothetic sets of character states for the clada based on PMS + all additive binary codings for multi-state characters.
In contrast to SMS (second example) your homology hypotheses are not broken down relative to the pattern exhibited by other states of other characters, but these hypotheses are merged (added) in all different permutations possible within each block of homologous states. These extended combinatorial possibilities produced by the increased number of mutually inclusive groups of terminal taxa result in a larger number of cladograms (12) compared to PMS alone.
Selection criteria for cladograms of: PlantB
Column numbers refer to numbers of cladograms
---------------------------------------------
Row 1 : Total number of homoplasous events
Row 2 : Total number of single origins (Support)
Row 3 : Corrected Extra Length (x1000; CEL: Turner + Zandee)
Row 4 : Total number of state changes (S: Steps)
Row 5 : Redundancy Quotient (x1000; RQ: Zandee + Geesink)
Row 6 : Rescaled Redundancy Quotient (x1000; RQc)
Row 7 : Consistency Index (x1000; CI), with autapomorphy correction
Row 8 : Rescaled Consistency Index (x1000; RC: Farris)
Row 9 : Average Unit Character Consistency (x1000; AUCC: Sang)
Row 10: Homoplasy Distribution Ratio (x1000; HDR: Sang)
Row 11: Compatible Character State Index (x1000; CCSI: Zandee)
1 2 3 4 5 6 7 8 9 10 11 12
------------------------------------------------------------------------
1 | 3 2 6 3 5 2 2 0 4 0 0 6
2 | 13 14 8 13 10 14 14 16 9 16 16 6
3 | 4033 4033 7117 4033 6100 3017 4033 1017 5100 0 1017 8183
4 | 19 19 22 19 21 18 19 16 20 15 16 23
5 | 466 475 435 460 449 492 460 488 459 510 472 435
6 | 95 110 43 84 66 138 84 131 83 170 105 42
7 | 667 667 533 667 571 727 667 889 615 1000 889 500
8 | 502 502 248 502 325 606 502 852 409 1000 852 178
9 | 860 860 810 860 817 875 860 975 892 1000 975 863
10 | 335 335 401 335 358 250 335 600 567 1000 600 605
11 | 545 545 455 545 455 682 545 682 636 818 682 500
No-Order Limit for Steps, Extra Steps, RQ, and CI:
S ES RQ CI
-------------------
26 11 410 421
Table 3.13: Selection criteria for cladograms in PMS+ABC example.
Using minimum steps as a selection criterion (table 3.13), only one cladogram sticks out as the best: nr 10. Cladogram # 10 is identical to the cladogram selected in our previous runs (table 3.6).
1 2 3 4 5 6 7 8 9 10 11 12
------------------------------------------------------------------------
1 | 4 3 4 4 5 2 2 0 1 0 0 1
2 | 11 11 11 11 9 14 14 16 14 16 16 14
3 | 4100 3050 3050 4100 4100 3050 3050 0 0 0 0 0
4 | 19 18 18 19 19 18 18 15 15 15 15 15
5 | 436 451 446 436 423 469 467 468 474 481 481 479
6 | 6 34 25 7 0 65 62 63 75 86 86 83
7 | 667 727 727 667 667 727 727 1000 1000 1000 1000 1000
8 | 0 208 208 0 0 208 208 1000 1000 1000 1000 1000
9 | 860 875 875 860 860 875 875 1000 1000 1000 1000 1000
10 | 335 250 250 335 335 250 250 1000 1000 1000 1000 1000
11 | 545 545 455 545 455 682 545 682 636 818 682 500
No-Order Limit for Steps, Extra Steps, RQ, and CI:
S ES RQ CI
-------------------
19 4 432 667
Table 3.14: Cladogram evaluation data for 12 cladograms resulting from a primary analysis using option 3 (PMS + ABC) and zero's in multi-state characters as non-ancestral state.
If, however, we run this analysis again but now with the zero's in multi-state characters 4, 5, and 10 as not indicating putative ancestral states (see CAFCA parameter dialog), we arrive at the following evaluation data for the 12 cladograms involved (table 3.14). Not forcing the zero's in characters 4, 5, and 10 to be present on the root reduces the number of steps in almost all cladograms. Cladogram # 8, 9, 11 and # 12 now also count 15 steps, just like # 10 already did. As the number of steps on the bush decreases as well (26 vs 19) some of the cladograms found (# 1, 4, and 5) appear to be as bad as a bush as regards the number of steps. Note the lower limit of the Redundancy Quotient (0.432) which, due to the reduction of the number of steps on a bush, is higher than in the other runs (0.410). A last item from table 3.14 that is well worth noting is that for MPT's with ci=1 the AUCC can not discriminate any further (its range is [ci,1]). Nor can CEL. However, the compatible character states index (CCSI, row 11) does discriminate among these MPT's and selects # 10 as the best. In this particular case CCSI is an even better performer than RQ (RQ does not discriminate among # 10 and # 11).
Plant: Cladogram - 8 Plant: Cladogram - 11
/--- 4 Dus /--- 4 Dus
/---7 /----7
/------10 \--- 5 Eus /---10 \--- 5 Eus
| | /---14 |
| \------- 3 Cus | | \-------- 3 Cus
| | |
| /--- 1 Aus | \------------ 1 Aus
\----------9 |
\--- 2 Bus \---------------- 2 Bus
Figure 3.1: Alternative most parsimonious cladograms resulting from PMS+ABC and non-ancestral zero's in multi-state characters.
