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| 3 | <HEAD> |
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| 4 | <TITLE>restml</TITLE> |
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| 5 | <META NAME="description" CONTENT="restml"> |
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| 6 | <META NAME="keywords" CONTENT="restml"> |
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| 7 | <META NAME="resource-type" CONTENT="document"> |
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| 8 | <META NAME="distribution" CONTENT="global"> |
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| 9 | <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> |
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| 10 | </HEAD> |
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| 11 | <BODY BGCOLOR="#ccffff"> |
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| 12 | <DIV ALIGN=RIGHT> |
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| 13 | version 3.6 |
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| 14 | </DIV> |
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| 15 | <P> |
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| 16 | <DIV ALIGN=CENTER> |
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| 17 | <H1>RESTML -- Restriction sites Maximum Likelihood program</H1> |
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| 18 | </DIV> |
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| 19 | <P> |
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| 20 | © Copyright 1986-2000 by The University of |
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| 21 | Washington. Written by Joseph Felsenstein. Permission is granted to copy |
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| 22 | this document provided that no fee is charged for it and that this copyright |
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| 23 | notice is not removed. |
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| 24 | <P> |
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| 25 | This program implements a maximum likelihood method for restriction sites |
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| 26 | data (not restriction fragment data). This program is one of the slowest |
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| 27 | programs in this package, and can be very tedious to run. It is |
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| 28 | possible to have the program search for the maximum likelihood tree. |
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| 29 | It will be more practical for some users (those that do not have |
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| 30 | fast machines) to use the U (User Tree) |
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| 31 | option, which takes less run time, optimizing branch lengths and computing |
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| 32 | likelihoods for particular tree topologies suggested by the user. The |
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| 33 | model used here is essentially identical to that used by Smouse and Li (1987) |
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| 34 | who give explicit expressions for computing the likelihood for three-species |
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| 35 | trees. It does not place prior probabilities on trees as they do. The present |
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| 36 | program extends their approach to multiple species by |
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| 37 | a technique which, while it does not give explicit expressions for likelihoods, |
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| 38 | does enable their computation and the iterative improvement of branch |
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| 39 | lengths. It also allows for multiple restriction enzymes. The algorithm |
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| 40 | has been described in a paper (Felsenstein, 1992). Another relevant |
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| 41 | paper is that of DeBry and Slade (1985). |
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| 42 | <P> |
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| 43 | The assumptions of the present model are: |
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| 44 | <P> |
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| 45 | <OL> |
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| 46 | <LI>Each restriction site evolves independently. |
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| 47 | <LI>Different lineages evolve independently. |
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| 48 | <LI>Each site undergoes substitution at an expected rate which we specify. |
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| 49 | <LI>Substitutions consist of replacement of a nucleotide by one of the |
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| 50 | other three nucleotides, chosen at random. |
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| 51 | </OL> |
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| 52 | <P> |
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| 53 | Note that if the existing base is, say, an A, the chance of it being |
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| 54 | replaced by a G is 1/3, and so is the chance that it is replaced by |
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| 55 | a T. This means that there can be no difference in the (expected) |
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| 56 | rate of transitions and transversions. Users who are upset at this |
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| 57 | might ponder the fact that a version allowing different rates of |
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| 58 | transitions and transversions would run an estimated 16 times |
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| 59 | slower. If it also allowed for unequal frequencies of the four bases, |
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| 60 | it would run about 300,000 times slower! For the moment, until a better |
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| 61 | method is available, I guess I'll stick with this one! |
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| 62 | <P> |
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| 63 | <H2>INPUT FORMAT AND OPTIONS</H2> |
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| 64 | <P> |
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| 65 | Subject to these assumptions, the program is an approximately |
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| 66 | correct maximum likelihood method. The |
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| 67 | input is fairly standard, with one addition. As usual the first line of the |
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| 68 | file gives the number of species and the number of sites, but there is also |
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| 69 | a third number, which is the number of different restriction enzymes that were |
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| 70 | used to detect the restriction sites. Thus a data set with 10 species and |
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| 71 | 35 different sites, representing digestion with 4 different enzymes, would |
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| 72 | have the first line of the data file look like this: |
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| 73 | <P> |
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| 74 | <PRE> |
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| 75 | 10 35 4 |
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| 76 | </PRE> |
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| 77 | <P> |
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| 78 | The first line of the data file will also contain a letter W following |
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| 79 | these numbers (and separated from them by a space) if the Weights option |
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| 80 | is being used. As with all programs using the weights option, a line or lines |
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| 81 | must then follow, before the data, with the weights for each site. |
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| 82 | <P> |
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| 83 | The site data are in standard form. Each species starts with a species name |
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| 84 | whose maximum length is given by the constant "nmlngth" |
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| 85 | (whose value in the |
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| 86 | program as distributed is 10 characters). The name should, as usual, be padded |
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| 87 | out to that length with blanks if necessary. The sites data then follows, one |
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| 88 | character per site (any blanks will |
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| 89 | be skipped and ignored). Like the DNA and protein sequence data, the |
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| 90 | restriction sites data may be either in the "interleaved" form or the |
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| 91 | "sequential" form. Note that if you are analyzing restriction sites |
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| 92 | data with the programs DOLLOP or MIX or other discrete character |
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| 93 | programs, at the moment those programs do not use the "aligned" or |
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| 94 | "interleaved" data format. Therefore you may want to avoid that format |
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| 95 | when you have restriction sites data that you will want to feed into |
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| 96 | those programs. |
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| 97 | <P> |
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| 98 | The presence of a site is indicated by a "+" and the absence by a "-". I have |
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| 99 | also allowed the use of "1" and "0" as synonyms for "+" and "-", for |
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| 100 | compatibility with MIX and DOLLOP which do not allow "+" and "-". If the |
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| 101 | presence of |
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| 102 | the site is unknown (for example, if the DNA containing it has been deleted so |
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| 103 | that one |
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| 104 | does not know whether it would have contained the site) then the state "?" can |
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| 105 | be used to indicate that the state of this site is unknown. |
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| 106 | <P> |
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| 107 | User-defined trees may follow the |
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| 108 | data in the usual way. The trees must be unrooted, which means that at their |
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| 109 | base they must have a trifurcation. |
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| 110 | <P> |
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| 111 | The options are selected by a menu, which looks like this: |
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| 112 | <P> |
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| 113 | <TABLE><TR><TD BGCOLOR=white> |
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| 114 | <PRE> |
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| 115 | |
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| 116 | Restriction site Maximum Likelihood method, version 3.6 |
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| 117 | |
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| 118 | Settings for this run: |
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| 119 | U Search for best tree? Yes |
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| 120 | A Are all sites detected? No |
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| 121 | S Speedier but rougher analysis? Yes |
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| 122 | G Global rearrangements? No |
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| 123 | J Randomize input order of sequences? No. Use input order |
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| 124 | L Site length? 6 |
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| 125 | O Outgroup root? No, use as outgroup species 1 |
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| 126 | M Analyze multiple data sets? No |
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| 127 | I Input sequences interleaved? Yes |
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| 128 | 0 Terminal type (IBM PC, ANSI, none)? (none) |
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| 129 | 1 Print out the data at start of run No |
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| 130 | 2 Print indications of progress of run Yes |
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| 131 | 3 Print out tree Yes |
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| 132 | 4 Write out trees onto tree file? Yes |
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| 133 | |
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| 134 | Y to accept these or type the letter for one to change |
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| 135 | </PRE> |
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| 136 | </TD></TR></TABLE> |
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| 137 | <P> |
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| 138 | The U, J, O, M, and 0 options are the usual ones, described in the main |
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| 139 | documentation file. The user trees for option U are read from a file whose |
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| 140 | default name is <TT>intree</TT>. The I option selects between Interleaved and |
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| 141 | Sequential input data formats, and is described in the documentation file for |
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| 142 | the molecular sequences programs. |
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| 143 | <P> |
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| 144 | The G |
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| 145 | (global search) option causes, after the last species is added to the tree, |
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| 146 | each possible group to be removed and re-added. This improves the result, |
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| 147 | since the position of every species is reconsidered. It approximately triples |
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| 148 | the run-time of the program. |
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| 149 | <P> |
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| 150 | The two options specific to this program are the A, and L options. The L |
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| 151 | (Length) option allows the user to specify the length in bases of the |
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| 152 | restriction sites. Allowed values are 1 to 8 (the |
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| 153 | constant "maxcutter" in file <TT>phylip.h</TT> |
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| 154 | controls the maximum allowed value). At the |
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| 155 | moment the program assumes that all sites have the same length (for example, |
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| 156 | that all enzymes are 6-base-cutters). The default value for this parameter is |
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| 157 | 6, which will be used if the L option is not invoked. A desirable future |
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| 158 | development for the package would be allowing the L parameter to be different |
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| 159 | for every site. It would also be desirable to allow for ambiguities in the |
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| 160 | recognition site, since some enzymes recognize 2 or 4 sequences. Both of these |
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| 161 | would require fairly complicated programming or else slower execution times. |
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| 162 | <P> |
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| 163 | The A (All) option specifies that all sites are detected, even those for which |
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| 164 | all of the species have the recognition sequence absent (character state |
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| 165 | "-"). The default condition is that it is assumed that such sites will not |
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| 166 | occur in the data. The likelihood computed when the A option is not used is |
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| 167 | the probability of the pattern of sites given that tree and conditional on the |
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| 168 | pattern not being all absences. This will be realistic for most data, except |
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| 169 | for cases in which the data are extracted from sites data for a larger number |
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| 170 | of species, in which case some of the site positions could have all absences in |
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| 171 | the subset of species. In such cases an effective way of analyzing the data |
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| 172 | would be to omit those sites and not use the A option, as such positions, even |
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| 173 | if not absolutely excluded, are nevertheless less likely than random to have |
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| 174 | been incorporated in the data set. |
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| 175 | <P> |
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| 176 | The W (Weights) option, which is invoked in the input file rather than in |
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| 177 | the menu, allows the user to select a subset of sites to |
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| 178 | be analyzed. It is invoked in the usual way, except that only weights |
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| 179 | 0 and 1 are allowed. If the W option is not used, all sites will be |
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| 180 | analyzed. If the Weights option is used, there must be a W in the first |
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| 181 | line of the input file. |
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| 182 | <P> |
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| 183 | <H2>OUTPUT FORMAT</H2> |
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| 184 | <P> |
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| 185 | The output starts by giving the number of species, and the number of sites. If |
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| 186 | the default condition is used instead of the A option the program states that |
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| 187 | it is assuming that sites absent in all species have been omitted. The value |
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| 188 | of the site length (6 bases, for example) is also given. |
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| 189 | <P> |
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| 190 | If option 2 (print out data) has been selected, |
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| 191 | there then follow the restriction site sequences, printed in |
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| 192 | groups of ten sites. The trees found are printed as an unrooted |
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| 193 | tree topology (possibly rooted by outgroup if so requested). The |
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| 194 | internal nodes are numbered arbitrarily for the sake of |
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| 195 | identification. The number of trees evaluated so far and the log |
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| 196 | likelihood of the tree are also given. |
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| 197 | <P> |
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| 198 | A table is printed |
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| 199 | showing the length of each tree segment, as well as (very) rough confidence |
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| 200 | limits on the length. As with DNAML, if a confidence limit is |
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| 201 | negative, this indicates that rearrangement of the tree in that region |
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| 202 | is not excluded, while if both limits are positive, rearrangement is |
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| 203 | still not necessarily excluded because the variance calculation on which |
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| 204 | the confidence limits are based results in an underestimate, which makes |
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| 205 | the confidence limits too narrow. |
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| 206 | <P> |
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| 207 | In addition to the confidence limits, |
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| 208 | the program performs a crude Likelihood Ratio Test (LRT) for each |
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| 209 | branch of the tree. The program computes the ratio of likelihoods with and |
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| 210 | without this branch length forced to zero length. This done by comparing the |
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| 211 | likelihoods changing only that branch length. A truly correct LRT would |
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| 212 | force that branch length to zero and also allow the other branch lengths to |
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| 213 | adjust to that. The result would be a likelihood ratio closer to 1. Therefore |
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| 214 | the present LRT will err on the side of being too significant. |
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| 215 | <P> |
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| 216 | One should also |
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| 217 | realize that if you are looking not at a previously-chosen branch but at all |
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| 218 | branches, that you are seeing the results of multiple tests. With 20 tests, |
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| 219 | one is expected to reach significance at the P = .05 level purely by |
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| 220 | chance. You should therefore use a much more conservative significance level, |
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| 221 | such as .05 divided by the number of tests. The significance of these tests |
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| 222 | is shown by printing asterisks next to |
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| 223 | the confidence interval on each branch length. It is important to keep |
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| 224 | in mind that both the confidence limits and the tests |
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| 225 | are very rough and approximate, and probably indicate more significance than |
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| 226 | they should. Nevertheless, maximum likelihood is one of the few methods that |
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| 227 | can give you any indication of its own error; most other methods simply fail to |
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| 228 | warn the user that there is any error! (In fact, whole philosophical schools |
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| 229 | of taxonomists exist whose main point seems to be that there isn't any |
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| 230 | error, that the "most parsimonious" tree is the best tree by definition and |
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| 231 | that's that). |
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| 232 | <P> |
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| 233 | The log likelihood printed out with the final tree can be used to perform |
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| 234 | various likelihood ratio tests. Remember that testing one tree topology |
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| 235 | against another is not a simple matter, because two different tree topologies |
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| 236 | are not hypotheses that are |
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| 237 | nested one within the other. If the trees differ by only one branch |
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| 238 | swap, it seems to be conservative to test the difference between their |
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| 239 | likelihoods with one degree of freedom, but other than that little is known and |
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| 240 | more work on this is needed. |
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| 241 | <P> |
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| 242 | If the U (User Tree) option is used and more than one tree is supplied, |
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| 243 | and the program is not told to assume autocorrelation between the |
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| 244 | rates at different sites, the |
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| 245 | program also performs a statistical test of each of these trees against the |
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| 246 | one with highest likelihood. If there are two user trees, the test |
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| 247 | done is one which is due to Kishino and Hasegawa (1989), a version |
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| 248 | of a test originally introduced by Templeton (1983). In this |
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| 249 | implementation it uses the mean and variance of |
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| 250 | log-likelihood differences between trees, taken across sites. If the two |
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| 251 | trees' means are more than 1.96 standard deviations different |
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| 252 | then the trees are |
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| 253 | declared significantly different. This use of the empirical variance of |
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| 254 | log-likelihood differences is more robust and nonparametric than the |
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| 255 | classical likelihood ratio test, and may to some extent compensate for the |
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| 256 | any lack of realism in the model underlying this program. |
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| 257 | <P> |
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| 258 | If there are more than two trees, the test done is an extension of |
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| 259 | the KHT test, due to Shimodaira and Hasegawa (1999). They pointed out |
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| 260 | that a correction for the number of trees was necessary, and they |
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| 261 | introduced a resampling method to make this correction. In the version |
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| 262 | used here the variances and covariances of the sum of log likelihoods across |
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| 263 | sites are computed for all pairs of trees. To test whether the |
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| 264 | difference between each tree and the best one is larger than could have |
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| 265 | been expected if they all had the same expected log-likelihood, |
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| 266 | log-likelihoods for all trees are sampled with these covariances and equal |
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| 267 | means (Shimodaira and Hasegawa's "least favorable hypothesis"), |
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| 268 | and a P value is computed from the fraction of times the difference between |
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| 269 | the tree's value and the highest log-likelihood exceeds that actually |
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| 270 | observed. Note that this sampling needs random numbers, and so the |
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| 271 | program will prompt the user for a random number seed if one has not |
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| 272 | already been supplied. With the two-tree KHT test no random numbers |
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| 273 | are used. |
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| 274 | <P> |
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| 275 | In either the KHT or the SH test the program |
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| 276 | prints out a table of the log-likelihoods of each tree, the differences of |
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| 277 | each from the highest one, the variance of that quantity as determined by |
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| 278 | the log-likelihood differences at individual sites, and a conclusion as to |
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| 279 | whether that tree is or is not significantly worse than the best one. |
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| 280 | <P> |
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| 281 | The branch lengths printed out are scaled in terms of expected numbers of |
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| 282 | base substitutions, not counting replacements of a base by itself. Of course, |
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| 283 | when a branch is twice as long this does not mean that there will be twice as |
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| 284 | much net change expected along it, since some of the changes occur in the same |
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| 285 | site and overlie or even reverse each other. Confidence limits on the branch |
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| 286 | lengths are also given. Of course a negative value of the branch length is |
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| 287 | meaningless, and a confidence limit overlapping zero simply means that the |
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| 288 | branch length is not necessarily significantly different from zero. Because of |
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| 289 | limitations of the numerical algorithm, branch length estimates of zero will |
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| 290 | often print out as small numbers such as 0.00001. If you see a branch length |
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| 291 | that small, it is really estimated to be of zero length. |
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| 292 | <P> |
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| 293 | Another possible source of confusion is the existence of negative values for |
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| 294 | the log likelihood. This is not really a problem; the log likelihood is not a |
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| 295 | probability but the logarithm of a probability, and since probabilities never |
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| 296 | exceed 1.0 this logarithm will typically be negative. The log likelihood is |
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| 297 | maximized by being made more positive: -30.23 is worse than -29.14. The log |
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| 298 | likelihood will not always be negative since a combinatorial constant has been |
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| 299 | left out of the expression for the likelihood. This does not affect the tree |
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| 300 | found or the likelihood ratios (or log likelihood differences) between trees. |
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| 301 | <P> |
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| 302 | <H2>THE ALGORITHM</H2> |
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| 303 | <P> |
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| 304 | The program uses a Newton-Raphson algorithm to update one branch length at a |
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| 305 | time. This is faster than the EM algorithm which was described in my |
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| 306 | paper on restriction sites maximum likelihood (Felsenstein, 1992). The |
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| 307 | likelihood that is being |
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| 308 | maximized is the same one used by Smouse and Li (1987) extended for multiple |
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| 309 | species. |
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| 310 | moving down on the likelihood surface. You may have to "tune" the value of |
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| 311 | extrapol to suit your data. |
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| 312 | <P> |
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| 313 | <H2>PROGRAM CONSTANTS</H2> |
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| 314 | <P> |
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| 315 | The constants include "maxcutter" (set in <TT>phylip.h</TT>), |
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| 316 | the maximum length of an enzyme |
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| 317 | recognition site. The memory used by the program will be approximately |
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| 318 | proportional to this value, which is 8 in the distribution copy. |
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| 319 | The program also uses constants |
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| 320 | "iterations" and "smoothings", and decreasing "epsilon". Reducing |
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| 321 | "iterations" and "smoothings" or increasing "epsilon" |
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| 322 | will result in faster execution but a worse result. These values will |
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| 323 | not usually have to be changed. |
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| 324 | <P> |
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| 325 | The program spends most of its time doing real arithmetic. The algorithm, with |
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| 326 | separate and independent computations |
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| 327 | occurring at each site, lends itself readily to parallel processing. |
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| 328 | <P> |
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| 329 | A feature of the algorithm is that it saves time by recognizing sites at which |
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| 330 | the pattern of presence/absence is the same, and does that computation only |
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| 331 | once. Thus if we have only four species but a large number of sites, there are |
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| 332 | only about (ignoring ambiguous bases) 16 different patterns of presence/absence |
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| 333 | (2 x 2 x 2 x 2) that can occur. The program automatically counts |
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| 334 | occurrences of each and does the computation for each pattern only once, so |
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| 335 | that it only needs to do as much computation as would be needed with at most |
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| 336 | 16 sites, even though the number of sites is actually much larger. Thus |
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| 337 | the program will run very effectively with few species and many sites. |
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| 338 | <P> |
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| 339 | <H2>PAST AND FUTURE OF THE PROGRAM</H2> |
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| 340 | <P> |
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| 341 | This program was developed by modifying DNAML version 3.1 and also adding |
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| 342 | some of the modifications that were added to DNAML version 3.2, with which |
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| 343 | it shares many of its data structures and much of its strategy. Version |
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| 344 | 3.6 changed from EM iterations of branch lengths, which involved arbitrary |
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| 345 | extrapolation factors, to the Newton-Raphson algorithm, which improved the |
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| 346 | speed of the program (though only from "very slow" to "slow"). |
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| 347 | <P> |
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| 348 | There are a number of obvious directions in which the program needs to be |
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| 349 | modified in the future. Extension to allow for different rates of transition |
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| 350 | and transversion is straightforward, but would slow down the program |
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| 351 | considerably, as I have mentioned above. I have not included in the program |
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| 352 | any provision for saving and printing out |
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| 353 | multiple trees tied for highest likelihood, in part because an exact tie is |
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| 354 | unlikely. |
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| 355 | <P> |
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| 356 | <HR> |
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| 357 | <P> |
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| 358 | <H3>TEST DATA SET</H3> |
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| 359 | <P> |
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| 360 | <TABLE><TR><TD BGCOLOR=white> |
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| 361 | <PRE> |
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| 362 | 5 13 2 |
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| 363 | Alpha ++-+-++--+++- |
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| 364 | Beta ++++--+--+++- |
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| 365 | Gamma -+--+-++-+-++ |
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| 366 | Delta ++-+----++--- |
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| 367 | Epsilon ++++----++--- |
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| 368 | </PRE> |
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| 369 | </TD></TR></TABLE> |
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| 370 | <P> |
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| 371 | <HR> |
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| 372 | <P> |
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| 373 | <H3>CONTENTS OF OUTPUT FILE (if all numerical options are on)</H3> |
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| 374 | <P> |
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| 375 | <TABLE><TR><TD BGCOLOR=white> |
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| 376 | <PRE> |
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| 377 | |
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| 378 | Restriction site Maximum Likelihood method, version 3.6 |
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| 379 | |
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| 380 | 5 Species, 13 Sites, 2 Enzymes |
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| 381 | |
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| 382 | Recognition sequences all 6 bases long |
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| 383 | |
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| 384 | Sites absent from all species are assumed to have been omitted |
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| 385 | |
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| 386 | |
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| 387 | Name Sites |
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| 388 | ---- ----- |
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| 389 | |
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| 390 | Alpha ++-+-++--+ ++- |
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| 391 | Beta ++++--+--+ ++- |
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| 392 | Gamma -+--+-++-+ -++ |
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| 393 | Delta ++-+----++ --- |
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| 394 | Epsilon ++++----++ --- |
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| 395 | |
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| 396 | |
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| 397 | |
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| 398 | |
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| 399 | |
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| 400 | +----Gamma |
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| 401 | | |
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| 402 | | +Epsilon |
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| 403 | | +--3 |
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| 404 | 1--2 +Delta |
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| 405 | | | |
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| 406 | | +Beta |
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| 407 | | |
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| 408 | +Alpha |
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| 409 | |
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| 410 | |
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| 411 | remember: this is an unrooted tree! |
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| 412 | |
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| 413 | Ln Likelihood = -40.34358 |
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| 414 | |
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| 415 | |
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| 416 | Between And Length Approx. Confidence Limits |
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| 417 | ------- --- ------ ------- ---------- ------ |
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| 418 | 1 Gamma 0.10813 ( 0.01154, 0.21901) ** |
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| 419 | 1 2 0.01156 ( zero, 0.04578) |
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| 420 | 2 3 0.05885 ( zero, 0.12697) ** |
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| 421 | 3 Epsilon 0.00100 ( zero, 0.00617) |
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| 422 | 3 Delta 0.01460 ( zero, 0.05036) |
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| 423 | 2 Beta 0.00100 ( zero, infinity) |
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| 424 | 1 Alpha 0.01310 ( zero, 0.04806) |
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| 425 | |
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| 426 | * = significantly positive, P < 0.05 |
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| 427 | ** = significantly positive, P < 0.01 |
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| 428 | |
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| 429 | |
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| 430 | </PRE> |
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| 431 | </TD></TR></TABLE> |
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| 432 | </BODY> |
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| 433 | </HTML> |
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