To show very simply how weighting can affect an outcome, consider the data set

one   aaaa
two   acac
three cccc
four  caca

which defines two equally parsimonious trees (homoplasy in red):

 _____ one   aaaa
|_____ two   acac
|   __ three cccc
`--|__ four  caca

and
 _____ one   aaaa
|   __ two   acac
|--|__ three cccc
|_____ four  caca

 _____ one   aaaa
|   __ two   acac
|--|__ three cccc
|_____ four  caca

1. a priori and
3. a posteriori

A Priori Weighting

adaptive

• the notion that some characters inherently are worth more than others
• not based on expected frequency of change
• usually based on notions of adaptive importance of characters, such as "limb morphology in tetrapods is adaptively important but eye color is not , or is less so"
  
  • but even if the argument can be made it is hard to arrive at an integer (or other) value to express "importance"

  independence

• compensatory changes in rDNA not independent in stem regions, a change at one site implies a change at another, therefore, down-weight stem regions by half, or increase weight of loop regions by two (c.f., Wheeler and Honeycutt).

  
                        a       a
               c       g a     t g
        cagtcgg tgattcc   actga   c
        ||||||| |||||||   |||||    t
        ttcagcc actaagg   tgact   c
               t       a g     c a
                        c       a
  

  chemistry

• a transversion purine <--> pyrimidine is chemically harder than a transition
  • this is not really how substitutions occur but rather by misincorporation of a base during replication
• a change from leucine to isoleucine is less chemically substantial that leucine to proline
  • but how much so?
• a priori once can get a measure of the tolerance to change of codon positions as follows:

  # tolerated changes amino acid codons pos1 pos2 pos3 Ala GCA Ala GCC Ala GCG Ala GCU 0 0 3 Arg AGA Arg AGG Arg CGA Arg CGC Arg CGG Arg CGU 1 0 3 Asn AAC Asn AAU 0 0 1 Asp GAC Asp GAU 0 0 1 Cys UGC Cys UGU 0 0 1 Gln CAA Gln CAG 0 0 1 Glu GAA Glu GAG 0 0 1 Gly GGA Gly GGC Gly GGG Gly GGU 0 0 3 His CAC His CAU 0 0 1 Ile AUA Ile AUC Ile AUU 0 0 2 Trp UGG 0 0 0 Tyr UAC Tyr UAU 0 0 1

  # tolerated changes amino acid codons pos1 pos2 pos3 Leu CUA Leu CUC Leu CUG Leu CUU Leu UUA Leu UUG 1 0 3 Lys AAA Lys AAG 0 0 1 Met AUG 0 0 0 Phe UUC Phe UUU 0 0 1 Pro CCA Pro CCC Pro CCG Pro CCU 0 0 3 Ser AGC Ser AGU Ser UCA Ser UCC Ser UCG Ser UCU 1 1 3 STOP UAA STOP UAG STOP UGA 0 1 1 Thr ACA Thr ACC Thr ACG Thr ACU 0 0 3 Val GUA Val GUC Val GUG Val GUU 0 0 3

  Which indicates a ratio of tolerance of 0.143:0.095:1.71, and the inverse weighting
  would be 12:18:1.

A posteriori Weighting

For example,

• contra the independence based argument, because loops are more free to change (i.e., no compensatory change required in some other base) then one would expect them to change more frequently so one should increase the weight of stem regions not of loop regions(see for example Hillis and Dixon).

  
                        a       a
               c       g a     t g
        cagtcgg tgattcc   actga   c
        ||||||| |||||||   |||||    t
        ttcagcc actaagg   tgact   c
               t       a g     c a
                        c       a
  

• 3rd positions of codons all things being equal are expected to change more frequently than 1st positions which also are expected to change more frequently than 2nd positions

• transitions (A<-->G, or C<-->T) are expected to occur more frequently than transversions.

Knight and Mindell (1993) stated that "a level of TIs near 50% indicates that this class of change is saturated with multiple changes and is therefore not an indicator of phylogeny but is largely 'noise'" (see also Mindell and Honeycutt, 1990). In some cases (e.g., Mindell et al., 1995) all gapped regions in alignments as well as third position transitions are eliminated. And this can be determined from saturation plots (below) in which the number of transitions and transversions are plotted against total substitutions for all pairwise comparisons of taxa.

More often, though, it is argued that one should simply weight according to the inverse of the frequency.

If so, the frequency has to be determined across some tree. This can be a preliminary tree based on unweighted data (which leaves you open to the criticism of cricularly biasing your results towards that unweigted tree because ofweighting in light of the unweigted tree), or across a random set of trees.

Given the tree here for primate mtDNA (left), one can get a preliminary tree (by a search on the unweighted data) and then calulate the amount of change in 1st, 2nd and 3rd positions, which is 260:146:594 and would imply a weighting scheme for positions of about 2:4:1.

Random trees for the same data suggest a number of Ti's between 576 and 715 and a number of Tv's between 938 and 1113. So, weighting transversion somewhere between 1.5 and 2 appears to be indicated.

How is this implemnented? In PAUP, one can set up a Sankoff matrix to define these weights as follows:

BEGIN ASSUMPTIONS;
USERTYPE tv = 4 a c g t
- 2 1 2
2 - 2 1
1 2 - 2
2 1 2 -;
END;

Williams and Fitch (and Fitch and Ye) took this one step further by suggesting dynamic weighting. In this, each class of trnasformation in each direction is weighted inversely to it frequency. For the primate data, the frequencies of each transformation is:

	To
From	A C G T A - 0.101 0.180 0.060 C 0.080 - 0.015 0.295 G 0.029 0.004 - 0.001 T 0.044 0.183 0.007 -

Because transformations are weighted inversely to their frequency, this would suggest the following Sankoff matrix:

BEGIN ASSUMPTIONS;
USERTYPE tv = 4 a c g t
  -  900  820  940
 920  -   985  705
 970 996   -   999
 956 817  993   - ;
END;

Cautionary note

Frequencies are inherently rate-based arguments and thus if one is going to justify weighting according to this principle, one should weight not according to the inverse of the frequency but accoring to the natural logarithm of the inverse of the frequency.

Thus, a Sankoff matrix that looks like this:

BEGIN ASSUMPTIONS;
USERTYPE tv = 4 a c g t
- 2 1 2
2 - 2 1
1 2 - 2
2 1 2 -;
END;

Really applies to a situation where transitions are more than 7 times as frequent as transversions.

Successive Approximations

The procedure involves

1. finding a preliminary tree based on unweighted data
2. reweighting by the rescaled consistency index
3. searching again
4. repeat steps 2 and 3 until the weights don't don't change

mh;bb;xs w;

hsearch/swap = tbr; reweight /fit = rc;

Goloboff's Implied Weights

                  234  1238
Steps on tree 1 :  2    14
Steps on tree 2 :  1    15

That is, for character 234 the proportional disagreement is:

(2 - 1)/(2*1) = 0.500

(15-14)/(15*14) = 0.005

The argument is that one should prefer tree #2.

Goloboff developed a way to weight characters without requiring a priori reference to a tree in which the implied weight for a character is:

W = K/(K+ (maxsteps - observed steps))