- dePinna's data
- Felsenstein's data
- Meacham's data
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- Strauch's data
- Anderson's data
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All characters in this matrix are two-state characters, so as for the Pairwise Compatibility Theorem to hold, it does not matter whether we ignore the ancestor given in the matrix, or use it to polarise the characters.
1 1 2
char # 5 0 5 0
Viridis 12111 11111 11121 11111
Alba 12111 21111 21121 11121
Pallida 12211 11111 11121 11121
Coccinea 12111 11111 12122 11212
Brunnea 12111 11111 22122 12211
Caerula 12112 11122 12211 21111
Rubra 12112 12211 11111 21111
Nigra 21121 21111 12211 11111
Ancestor 11111 11111 11111 11111
Table 3.21: Data matrix from Meacham (1981, table 1)
Analysing this matrix with CAFCA under option 1 (PMS), characters unordered, and with taxon # 9 as outgroup to polarise the characters, yields the following results. There are 33 partial monothetic sets of taxa, including all terminal taxa, although they may lack a unique character state (e.g., taxon 1, 2, and 9). The sets of taxa correspond with the sets of character states given in table 3.22. All empty sets are removed from this list.Partial Monothetic Sets of character states for Meacham '84 table 1. ---------------- Set # Character State # 3 : 6 4 : 40 5 : 34 6 : 18 20 7: 14 16 8 : 2 3 8 10 : 10 32 11 : 12 12 : 22 13 : 26 14 : 30 36 15 : 38 16 : 27 17 : 23 18 : 24 19 : 28 20 : 9 31 21 : 11 22 : 21 23 : 25 24 : 29 35 25 : 37 26 : 1 4 7 27 : 5 28 : 13 15 29 : 17 19 30 : 33 31 : 39
Table 3.22: Partially monothetic sets based on Meacham's table 1.
I expect this set of states to correspond with the set of compatible characters if all characters concerned are included with (at least) one of their two states (the complementary state is compatible by definition), and there is just one cladogram involved. In case more cladograms do result from the analysis, the list above should summarise at least half the states (for this data matrix) from all maximum cliques of characters.CAFCA generates 30 cladograms on the basis of the 33 partially monothetic sets of terminal taxa. Only one cladogram is the most parsimonious one, with 25 steps (table 3.23).
I expect the characters fully congruent with this cladogram to comprise the set of compatible characters if group compatibility analysis (with PMS) might be considered identical with character compatibility analysis.
The list with character states compatible with the cladogram is given alongside in table 3.23. Empty clada are not listed.
All characters in the data matrix are (directed) two state characters. The states within a character are always compatible, unless one or more taxa are coded to be polymorphic. As the groups in the cladogram do not conflict as to membership of their constituent taxa, so don't the characters that correspond uniquely with these groups. Meacham (1981) presents one maximum clique of 16 mutually compatible characters; char # 1, 3, 4, 7, 8, 9, 10, 17, 20, 15, 18, 5, 14, 16, 19, and 2. The lists are indeed identical.
Cladogram - 5
/-- 4 Coccinea
/--14
| \-- 5 Brunnea
|
/--19 /-- 2 Alba
| |--15
/--26 | \-- 3 Pallida
/--32 | |
| | | \----- 1 Viridis
| | |
| | | /----- 6 Caerula
| | \--10
| | \----- 7 Rubra
| |
| \----------- 8 Nigra
|
\-------------- 9 Ancestor
---------------------------
Cladon : Character : State
---------------------------
3 | 3 | 2
4 | 20 | 2
5 | 17 | 2
6 | 9 | 2
| 10 | 2
7 | 7 | 2
| 8 | 2
8 | 1 | 2
| 4 | 2
10 | 5 | 2
| 16 | 2
14 | 15 | 2
| 18 | 2
15 | 19 | 2
19 | 14 | 2
26 | 2 | 2
---------------------------
Table 3.23: Most parsimonious cladogram as found by CAFCA based on sets in table 3.22, and its list of compatible character states.
There is another statistic on which we can base the same conclusion, and that is the consistency index for characters. Characters that are fully compatible with a cladogram will have a CI equal to one. The list with CI's for characters for all 30 cladograms is given below. Cladogram # 5 has 16 characters with a CI=1. They are the same as in Meacham's list of compatible characters. The other cladograms, which show more steps, have fewer characters that are fully compatible with them.
Consistency Indices for Characters of Wagner80B
-----------------------------------------------
Columnnumbers refer to index numbers of characters
Rownumbers refer to index numbers of cladograms
Row 0 refers to mean c.i. over all cladograms
1 2 3 4 5 6 7 8 9 10 11 12
------------------------------------------------------------------------
0 | 1.00 .55 1.00 1.00 .63 .50 1.00 1.00 1.00 1.00 .50 .48
1 | 1.00 .50 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .33
2 | 1.00 .50 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .33
3 | 1.00 .50 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .33
4 | 1.00 .50 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .33
5 | 1.00 1.00 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .33
6 | 1.00 .50 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .25
7 | 1.00 .50 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .25
8 | 1.00 .50 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00 .50 .33
9 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
10 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
11 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
12 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 1.00
13 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
14 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
15 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 1.00
16 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 1.00
17 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
18 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
19 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
20 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 1.00
21 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .33
22 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
23 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .33
24 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
25 | 1.00 1.00 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
26 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .50
27 | 1.00 1.00 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .33
28 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .33
29 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .33
30 | 1.00 .50 1.00 1.00 .50 .50 1.00 1.00 1.00 1.00 .50 .33
13 14 15 16 17 18 19 20
------------------------------------------------
0 | .77 .63 .90 .63 1.00 .90 .95 1.00
1 | .50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2 | .50 .50 1.00 1.00 1.00 1.00 1.00 1.00
3 | .50 .50 1.00 1.00 1.00 1.00 1.00 1.00
4 | .50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
5 | .50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
6 | .50 .50 .50 1.00 1.00 .50 1.00 1.00
7 | .50 .50 .50 1.00 1.00 .50 1.00 1.00
8 | .50 .50 1.00 1.00 1.00 1.00 .50 1.00
9 | 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00
10 | 1.00 .50 1.00 .50 1.00 1.00 1.00 1.00
11 | 1.00 .50 1.00 .50 1.00 1.00 1.00 1.00
12 | 1.00 .33 1.00 .50 1.00 1.00 1.00 1.00
13 | 1.00 .50 1.00 .50 1.00 1.00 1.00 1.00
14 | 1.00 .33 1.00 .50 1.00 1.00 1.00 1.00
15 | 1.00 .33 1.00 .50 1.00 1.00 1.00 1.00
16 | 1.00 .50 1.00 .50 1.00 1.00 1.00 1.00
17 | 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00
18 | 1.00 1.00 1.00 .50 1.00 1.00 1.00 1.00
19 | 1.00 .50 1.00 .50 1.00 1.00 .50 1.00
20 | 1.00 .33 1.00 .50 1.00 1.00 .50 1.00
21 | 1.00 .50 .50 .50 1.00 .50 1.00 1.00
22 | 1.00 .33 .50 .50 1.00 .50 1.00 1.00
23 | 1.00 .50 .50 .50 1.00 .50 1.00 1.00
24 | 1.00 .33 .50 .50 1.00 .50 1.00 1.00
25 | .50 .50 1.00 .50 1.00 1.00 1.00 1.00
26 | .50 .50 1.00 .50 1.00 1.00 1.00 1.00
27 | .50 1.00 1.00 .50 1.00 1.00 1.00 1.00
28 | .50 1.00 1.00 .50 1.00 1.00 1.00 1.00
29 | .50 1.00 1.00 .50 1.00 1.00 1.00 1.00
30 | .50 1.00 1.00 .50 1.00 1.00 1.00 1.00
Table 3.24: Consistency indices for characters in 30 cladograms generated by CAFCA, using PMS, from Meacham's table 1.
The cladogram in table 3.23 is not completely resolved; cladon 19 is a trichotomy. In chapter 4 we will see how to resolve such nodes by means of a secondary analysis. An heuristic search with PAUP results in two most parsimonious cladograms with 25 steps. One is identical with cladogram #5 given above. The other resolves the trichotomy and has also 16 fully compatible characters.Table 3.25 lists both the number of steps for the cladograms found by CAFCA as well as the number of character states compatible with these cladograms. It appears that cladograms of the same length may show a different number of compatible states.Nevertheless there is a tendency for shorter cladograms to have more compatible states. Whether this tendency is a general feature remains to be seen when we take a look at more complex multi-state data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ----------------------------------------------------------------------------------------- 26 27 27 26 25 30 30 28 26 27 27 27 27 28 27 26 26 26 28 28 30 30 30 30 27 28 27 28 28 28 15 14 14 15 16 12 12 13 14 13 13 14 13 13 14 14 14 14 12 13 11 11 11 11 13 12 14 13 13 13
Table 3.25: Number of steps (row 2) and number of compatible states (row 3) for the 30 cladograms found by CAFCA for option PMS and undirected characters.
Cladogram #6 counts 30 steps. Its compatible character states are given in table 3.26. Empty clada are not listed. There are indeed fewer compatibilities left for this longer cladogram, 12 to be precise, instead of 16 for the shortest cladogram (you may also count the characters with CI=1 for cladogram # 6 in table 3.24). Most larger groupings in the cladogram, like cladon 24, 30, and 32, do not have a character state that is fully compatible with it.
Cladogram - 6
/-- 6 Caerula
/--10
/--16 \-- 7 Rubra
| |
| \----- 8 Nigra
/--24
/--30 | /----- 2 Alba
/--32 | |--15
| | | | \----- 3 Pallida
| | | |
| | | \-------- 1 Viridis
| | |
| | \----------- 4 Coccinea
| |
| \-------------- 5 Brunnea
|
\----------------- 9 Ancestor
Cladogram-6 : COMPATIBILITIES
-------------------------
Cladon | Character | State
-------------------------
3 | 3 | 2
4 | 20 | 2
5 | 17 | 2
6 | 9 | 2
| 10 | 2
7 | 7 | 2
| 8 | 2
8 | 1 | 2
| 4 | 2
10 | 5 | 2
| 16 | 2
15 | 19 | 2
---------------------------
Table 3.26: One of the not-most-parsimonious cladograms and its compatibilities as found by CAFCA from Meacham's table 1.
I could tentatively suggest from this example that cladograms longer than the most parsimonious one, i.e., with more conflicts among character state distributions, have a set of compatible characters that is smaller than the maximum clique related to the shortest cladogram. However, we will also meet examples where the opposite holds, i.e., shorter cladograms have a smaller set of compatible characters. In the introduction to these examples I have already discussed the probable cause of this phenomenon.What happens when we do not direct the characters by pinpointing an ancestor? CAFCA then finds 11 cladograms, all with 25 steps. Not one of these 11 cladograms is identical to the one given above, although # 11 comes close (table 3.27; it has taxon #9 'Ancestor' included in the in-group).
Cladogram - 11
/-- 4 Coccinea
/--15
| \-- 5 Brunnea
|
/--20 /-- 2 Alba
| |--16
/--21 | \-- 3 Pallida
--28 | |
| | | \----- 1 Viridis
| | |
| | | /----- 6 Caerula
| | \--11
| | \----- 7 Rubra
| |
| \----------- 9 Ancestor
|
\-------------- 8 Nigra
Cladogram-11 : COMPATIBILITIES
--------------------------
Cladon : Character : Character State
--------------------------
2 | 3 | 2
3 | 20 | 2
4 | 17 | 2
5 | 9 | 2
| 10 | 2
6 | 7 | 2
| 8 | 2
7 | 1 | 2
| 4 | 2
11 | 5 | 2
| 16 | 2
15 | 15 | 2
| 18 | 2
16 | 19 | 2
20 | 14 | 2
21 | 2 | 2
28 | 1 | 1
| 4 | 1
--------------------------
Table 3.27: Cladogram and its compatible character states, found from Meacham's table 1 when characters are treated as undirected.
Instead of 16 there are now 18 states listed, but two characters, # 1 and 4, are represented twice, due to the complementary clada 8 and 28. Thus the number of compatible characters is still 16, and the set is still identical with Meacham's (1981) maximum clique. As Meacham (1984) explains very clearly, directing characters can only increase conflicts, never remove them. It appears that in our first analyses, with taxon # 9 'Ancestor' as outgroup, no extra conflicts were introduced for cladogram # 5 by directing the characters. In general, however, adding an all-zero outgroup to a (binary) data matrix may very well turn compatible characters into incompatible ones by introducing the otherwise lacking fourth combination of character states (e.g., 0,0; or 1,1 in a multi-state matrix).When we remove taxon # 9 'Ancestor' all together from the data matrix, CAFCA generates 30 cladograms from 31 partially monothetic sets, the same as in our first analysis but now all 25 steps long. Cladogram # 5 (table 3.28) is identical to the already known topology, except for the absence of taxon # 9.
Cladogram - 5
/-- 4 Coccinea
/--13
| \-- 5 Brunnea
|
/--18 /-- 2 Alba
| |--14
/--25 | \-- 3 Pallida
| | |
| | \----- 1 Viridis
| |
| | /----- 6 Caerula
| \--9
| \----- 7 Rubra
|
\----------- 8 Nigra
Cladogram-5 : COMPATIBILITIES
-------------------------
Cladon |Character| State
-------------------------
2 | 3 | 2
3 | 20 | 2
4 | 17 |
5 | 9 | 2
| 10 | 2
6 | 7 | 2
| 8 | 2
7 | 1 | 2
| 2 | 1
| 4 | 2
8 | 5 | 2
| 16 | 2
13 | 15 | 2
| 18 | 2
14 | 19 | 2
18 | 14 | 2
25 | 1 | 1
| 2 | 2
| 4 | 1
-------------------------
Table 3.28: Cladogram found by CAFCA from Meacham's table 1, while omitting the ancestral taxon.
Cladogram # 5 has 19 compatible character states, but some characters, like 1, 2, and 4, are represented twice, due to the fact that the complementary groups are both represented in the cladogram (clada 7 and 25). Thus the number of compatible characters is still 16, and it is the same set as Meacham's. In comparison, it is now made clear that introducing taxon # 9 'Ancestor' as an outgroup in our first analysis, and thus directing the characters, did raise the number of conflicts, or incompatibilities, among the characters for all cladograms other than # 5. In cladogram # 6, for instance, the number of conflicts raises with 5 as the number of steps increases from 25 to 30, and the number of characters compatible with the cladogram (and thus mutually compatible) decreases from 16 to 12.Cladogram # 27 from this analysis has the highest RQ, but also the same 16 compatible characters. In this cladogram the group 'Caerula + Rubra' is broken down (figure 3.5). This cladogram also demonstrates that, if the most parsimonious cladogram, # 5, is not completely resolved, a longer cladogram, # 27, may show the highest RQ. However, after secondary analysis cladogram # 5 is completely dichotomous and still counts 25 steps but its RQc is now 0.123 in contrast to the secondary resolution of cladogram # 27 with a RQc = 0.116
Cladogram - 27
/-- 4 Coccinea
/--13
| \-- 5 Brunnea
|
/--18 /-- 2 Alba
/--22 |--14
/--25 | | \-- 3 Pallida
| | | |
| | | \----- 1 Viridis
| | |
| | \-------- 7 Rubra
| |
| \----------- 6 Caerula
|
\-------------- 8 Nigra
Figure 3.5: Cladogram with highest RQ, from analysis of Meacham's table 1 with ancestral taxon omitted.
The last option that CAFCA offers is running the analysis with all characters ordered and directed (i.e., transforming from state 1 to 2). This should not make any difference with the analysis with all characters directed through the introduction of an outgroup (having state 1 for all characters), because all characters have only two-states. Two-state characters can always be considered ordered (Meacham, 1984). Indeed, it does not make any difference. Only 1 cladogram is found from 18 partially monothetic sets, two of them empty (= with no supporting character state). It is the same as # 5 given above, with 25 steps, and 16 compatible characters.
