584[36] Maximum

General

I361

ESTIMATINGSEOUENCEDIVERGENCE

Likelihood

Methods

Principle

Application of the maximum likelihood (ML) method to the problemof phylogenetic tree reconstruction was first studied for the case of genefrequency data.’ Later, an ML algorithm for constructing unrooted phylogenetic trees from nucleotide sequence data was developed by Felsenstein.*
Recently, Saitou3 proposed a stepwise tree-searching algorithm for the ML
method. This is similar to that of the neighbor-joining method,4 in which
distance matrices are used.
Let us first explain the general principle of the ML method for nucleotide sequence data. Consider the tree in Fig. 1. Let us ignore the root R andconsider the tree as an unrooted tree. We first restrict our attention to a
specific nucleotide site, and assume that nucleotide Ni was observed atsequence (node) i (i = A, B, C, D, or E). On the other hand, nucleotide Nj
at node j (j = X, Y, or Z) is unknown, and it can be one of four nucleotides. Thus, the likelihood (L) for this site becomes

L = 2NV1gYpYc
c hx&A&BI( cNZPYZ&DPzE
(1)
( NX
>1
where gy is the probability
that node Y has nucleotide NY,
Pii E P(Ni&pd,)
is the probability of observing nucleotides Ni and Nj insequences i and j, respectively, and du is the expected number of nucleotide
substitutions between these two sequences. Summation is for four possible
nucleotides, because Nx , NY, and Nz are variables. To obtain P(Ni,Nj,du),we must specify the pattern of nucleotide substitution. If we use therandom substitution model (the one-parameter model),sP(Ni,Nj,do)P(Ni,Nj,du)= $ + (3) exp(-4d,/3)= $ - (4) exp(-4d,/3)(if Ni = 4)
(if w # 3)

(240)

’ L. L. Cavalli-Sforza and A. W. F. Edwards, Am. J. Hum. Genet. 19,233 (1967).
2 J. Felsenstein, J. Mol. Evol. 17, 368 (1981).
3 N. Saitou, J. Mol. Evol. 27,26 1 (1988).
4 N. Saitou and M. Nei, Mol. Biol. Evol. 4,406 (1987).
5 T. H. Jukes and C. R. Cantor, in “Mammalian Protein Metabolism” (H. N. Munro, ed.),p. 2 1. Academic Press, New York, 1969.
METHODSIN ENZYMOLOGY,

VOL.

183

Copyright 0 1990 by Academic PISS, Inc.
All rights of reproduction in any form reserved.

1361

MAXIMUM

FIG.

LIKELIHOODMETHODS585

1. Phylogenetic tree for five nucleotide sequences.

If we assume the two-parameter modeL6 in which transitions andtransversions can occur at different rates, the expressions that correspondto Eqs. (2a) and (2b) are somewhat complicated [see Eqs. (9a)-(9c)]. It isalso true when other kinds of data such as amino acid sequences’ or
restriction site data* are considered. However, the essential nature of thelikelihood function remains the same.
In any case, the likelihood for each nucleotide site defined by Eq. (1) isthen multiplied for all sites and computed for many combinations ofbranch lengths for a given tree topology, and the combination that shows
the highest likelihood is chosen as the ML solution. It has been proved that
there exists a single ML point in the possible parameter range for a given
tree topology when the one-parameter model of nucleotide substitution
was assumed.9
For the case of a rooted tree, it is necessary to assume a constant rate ofnucleotide substitution. For example, the following constraints should be
invoked for the tree in Fig. 1: du = dXB, d, = dzE, dye = dyx + dxA,
and dRy + dye = dRz + dm . Because of this, the number of parameters tobe estimated for n sequences is n - 1 for rooted trees, compared to 2n - 3
parameters (branch lengths) for unrooted trees.
Each nucleotide site is considered separately in Felsenstein’s method.*
When each site is assumed to evolve at the same evolutionary rate, however, a more essential unit of comparison for the ML method is the“nucleotide configuration.”
A nucleotide configuration is the distribution
pattern of nucleotides for a given set of sequences (see Tables I and II for
examples). When the one-parameter model is assumed, the possible number of configurations for n sequences is (4n-1 + 3 X 2”-’ + 2)/6.i” There
6 M. Kimura, J. Mol. Evol. 16, 111 (1980).
7 R. L. Kashyap and S. Subas, J. Theor. Biol. 47,75 (1974).
* P. E. Smouse and W.-H. Li, Evolution 41, 1162 (1987).
9 K. Fukami and Y. Tateno, J. Mol. Evol. 28,460 (1989).
lo N. Saitou and M. Nei, J. Mol. Evol. 24, 189 (1986).

586

ESTIMATINGSEQUENCE

DIVERGENCE

TABLE
I
NUCLEOTIDECONFIGURATIONS
THREE SEQUENCES

Configurationa

FOR

Observed no.

No.

A

B

C

Casea

Caseb

1
2
3
4
5

i
i
i
j
i

i
i
j
i
j

i
j
i
i
k

789
98
59
50
4
l,ooo

40
20
15
5
20
ii%

a A, B, and C are different sequences, and i, j,
and k are nucleotides that are different fromeach other.

TABLE
II
NUCLEOTIDECONFIGURATIONSFOUR SEQUENCES

FOR

Configurations
No.

A

B

C

D

Observedno.

1
2
3
4

i
i
i
i
j
i
i
i
i
i
j
j
j
i
i

i
i
i
j
i
i
j
j
i
j
i
k
i
j
j

i
i
j
i
i
j
i
j
j
i
i
i
k
k
k

i
j
i
i
i
j
j
i
k
k
k
i
i
i
1

196
61
40
77
22
12
5
8
11
21
10
12
3
15
1
500

5
6
7
8
9
10
II
12
13
14
15

(1A, B, C, and D are. sequences of Fig. 4, and i, j,
k, and 1 are different nucleotides.

1361

I361

MAXIMUM

LIKELIHOODMETHODS587

are 5, 15, and 5 1 configurations for three, four, and five sequences, respectively.
If we consider the nucleotide configuration, the general formula for the
likelihood becomesm!
1. 2....

C.
i-1
where c is the number of possible nucleotide configurations, U, the probability of obtaining the ith configuration, mi the observed number of the ith
configuration, and m the sum of mi values, or the total number of nucleotides compared. Because the number of configurations depends on thenumber of sequences compared, we consider the cases of two, three, four,and five sequences separately. The case of unrooted trees is first considered.
L=fpFxm,m,mr

Two Sequences
When we have only two sequences, there are only one unrooted and
one rooted trees. Thus the ML solution for the rooted tree becomes identical to that for the unrooted tree.
Let us consider the one-parameter model of nucleotide substitution.
There are only two configurations for this case: two nucleotides are thesame (configuration 1) or different each other (configuration 2). Therefore,
Eq. (3) becomes

L=uyxP=x2(ml + m2Y
m,!m,!

where U, = l/4 + (3/4) exp(-4d,/3)[see Eq. (2a)], and U2= 3/4 (3/4) exp(-4d,/3)[see Eq. (2b)]. First, we take the logarithm,
log L = m, log[( 1 + 3x)/4] + m, log[(3 - 3x)/4] + constant(5)

where x = exp(-4d,/3).
To obtain the maximum likelihood estimate ofd,, we differentiate log L with respect to x, and equate it to zero. This gives-4m,x+

4m,x

1+3x

-=
3-3x

0

(6)

Solving Eq. (6) (note that x # 0 while dii is finite),k= 1 -4m2/3(m,+m2)(7)d,= -(3/4)log(l - 47~3)(8)Hence,

588

ESTIMATINGSEQUENCE

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1361

where rrti = m&r,+ mz). Equation (8) is identical to the famous formulaof Jukes and Cantor.5
When the two-parameter model6 is assumed, we have to distinguishtransitional and transversional differences. Thus, the number of configurations becomes three. Probabilities Vi (i = 1,2, 3) are given byu, = l/4 + p2/4 + pq/2
u, = l/4 + p2/4 - pq/2
u, = l/2 - p2/2

(no difference)(transitional difference)(transversional difference)

VW
Pb)
(SC)

where p = exp(- 2p) and q = exp(- 2o). (Yand /3 are the expected numbersof transitions and transversions, respectively. A likelihood function can beobtained using these Vi values, and we equate the differentiations of the log
likelihood with p and q to be zero, as in the case of the one-parametermodel. We solve the two equations,a = (1 - 2??2&92)“2UW

G = [l - (37% + %wmfi(lob)where mi is the observed number of the ith configuration and m = m, +m2 + m3. Thus, the total number of nucleotide substitutions d = a + 28 isestimated by
13= -(l/2)= -(l/2)log(i%))
log[( 1 - 2P - Q)( 1 - 2Q)“2](11)

where P = m,/m and Q = m,/m. Equation ( 11) is equivalent to Kimura’s
estimation formula of evolutionary distance.6
It is hypothesized that a similar correspondence between the ML estimate and the distance formula above can be applied to other types ofnucleotide substitution models (see this volume, [33] for other models). If
this conjecture is true, the ML method is identical with the distancemethod for the case of two sequences.

Three Sequences&rooted

Trees

There is only one unrooted tree for three sequences (Fig. 2a). Let us
assume the one-parameter model of nucleotide substitution. There are fivenucleotide configurations (Table I). Probabilities Vi for a given set of three
distances (dxA , dXB , and d,,) are given by
U, = 4h(i,i,i),
U2 = 12h(i,ij)

(i +A

(1W
(12b)

WI

MAXIMUM

LIKELIHOOD

U, = 12h(ij,i)U, = 12h( j,i,i)
U, = 24h(i,j,k)

METHODS

0 W
0 #.i)(i#j#k)

589
(1W

(12d)
(124

where i, j, and k are any one of the four nucleotides and(13)
where g, is the probability of observing nucleotide x at node X andP,(d&= P(x,i,d,)[see Eqs. (2a) and (2b)]. By substituting these Ui
values into Eq. (3), the likelihood function is obtained.
An analytical solution for the ML value is not easily found, but the ML
value can be obtained numerically by changing three distances (dxA, dXB ,and d,,). In this case, however, the ML estimates of these distances are nolonger the same as those obtained by the distance method,” in which casethe branch length estimates are given by:
Jti = (dm + dAc - d&/2
&x = (dm + 4c - dd/2
&x = (dAc + 4ic - daJI2

VW
(14b)
(14c)

where dAB, dAc, and dec are given using Eq. (8), which is the ML estimateof each distance value. In applying Eq. (8) the proportion of nucleotide
differences (a& between sequences i and j is estimated by
7rm = (m3 + m4 + m#m
XAC = (m2 + m4 + m&m
7r, = (m2 + m3 + m&/m

(154
(15b)
(15c)

It is clear from Eqs. (15a)- ( 15~) that different sets of mi values (1 5 i 5 5)can give the same set of 7rB, nAc, and rrx values. Thus, the estimates ofdd,,,andd
xc by the distance method are not the ML estimates.
Hyiever, it has been shown that the ML estimates of branch lengths arequite similar to those obtained by the distance method.r2
Rooted Trees
There are three rooted trees for three sequences, and we have twoparameters (d, and d2) to be estimated (see Fig. 2b). Let us designate thetree in Fig. 2b as “tree 1,” in which sequences A and B are closer to eachother than to C. As for the other two possible trees, sequences A and C are
closer in tree 2, and B and C are closer in tree 3.
‘I W. M. Fitch and E. Margoliash, Science 155,279 (1967).
I2 N. Saitou, Ph.D. dissertation, University of Texas at Houston, 1986.

590

ESTIMATINGSEQUENCE

a

FIG.

[361

DIVERGENCEb

2. (a) Unrooted tree and (b) rooted tree for three sequences.

Assuming the one-parametercomemodel, expressions for Ui for tree 1 be-

U,=(A+BaZ)/16

(164
WW

U2= (3A - Ba*)/ 16
U3 = (C + Da*)/16
u, = u,(164(164
U, = 2(C - Da*)/ 16
We)
where A= 1 +3b*, B=6(1 +b)b2, C=3(1 -b*, D=6(1 -b)b*, anda = exp(- 4d, /3) and b = exp(- 4d2/3).3 [Equation (16e) corresponds to
Eq. (4e) of Saitou,3 which was incorrect.] From these probabilities, one canevaluate the likelihood, L(l), of tree 1 under specific di and d2 values.
Computation of L(2) and L(3) for trees 2 and 3, respectively, is done in asimilar manner.
Saitou3 derived the conditions for obtaining tree 1 as the ML estimate,or for obtaining the relationship L(1) > y2) and L(1) > L(3), i.e., m2 >m3 and m2 > m,, where mi is the observed number of the ith nucleotide
configuration (see Table I). When this condition is satisfied, d- becomes
the smallest among dAB, d,,, and dBc. Thus, tree 1 is chosen if we use
UPGMA (unweighted pair-group method),”in which the pair of sequences with the smallest distance is first clustered. This means that the
topology of the UPGMA tree is always identical with that of the maximum
likelihood tree, though the estimates of branch lengths (d, and d2) may bedifferent to some extent.
Figure 3 shows two examples of likelihood surfaces for artificial data
(Cases a and b of Table 1).3 In Case a, the above condition (m2 > m3 andm2 > m4) is satisfied and tree 1 has the ML value at dl = 0.0259,whereas
the trifurcating tree (d, = 0) gives the ML value for the other two trees (Fig.

I3 R. Sokal and P. H. A. Swath, “Principles
Francisco, California, 1963.

of NumericalTaxonomy.”

Freeman, San

MAXIMUM

1361

LIKELIHOODa591

METHODSb-755-150 I-

M:
1

ML

-760

-151

/\

L

1

1r
”
s -152
”
8
A

2

-770

-15:

3

-775

.02

-15r
.04

dl

FIG. 3. Two examples of likelihood surfaces for three possible trees. (a) and (b) correspondto Cases a and b in Table I. See text for details.

3a). The likelihood surfaces represent the ML values for given d, values (anML value was computed by varying d2 for a given d, value).
The above condition is also satisfied in Case b, and tree 1 gives thehighest ML value. However, tree 2 also gives an ML value higher than thatfor the trifurcating tree (Fig. 3b). In this case the following relationship is
satisfied:
4tc -C (dm + Ad/2

(17)

Thus, if sequences A and C are first clustered (tree 2), a positive estimate ofdl is obtained. This situation is different from that for Case a, in which a
negative estimate of d, is obtained if tree 2 is considered. It seems that apositive estimate of d, corresponds to the ML value higher than that for thetrifurcating tree.3 For tree 3, however, d, becomes negative, and the loglikelihood value decreases as the value of dl increases (Fig. 3b).

592

ESTIMATINGSEQUENCE

DIVERGENCE

A

X
0.1
x

0.4

FIG.

1361

D

0.3

0.05

Y

0.2

C
B
4. Unrooted tree for four sequences. Numbers are lengths for each branch.

Four Sequences
Unrooted Trees
There are three unrooted trees for four sequences, and we have toconsider 15 nucleotide configurations (Table II). The likelihood function(not shown) can be obtained in a manner similar to the case of three
sequences.
Let us consider a numerical example. This is a result based on asimulated data set (see Table II), assuming the tree of Fig. 4.3 The tree
in which sequences A and B are clustered (i.e., the true tree -a in Fig. 4)
had the highest ML value (log likelihood is - 1004.2) with dJy = 0.026.
The estimates of pther branch lengths were dm = 0.110, dxB = 0.36 1,
dye = 0.188, and dm = 0.303. On the other hand, the same tree was obtained when we applied the neighbox-joining
method for a di$ance
matrix. -Branch length estimates were dxy = 0.033, dm = 0.10 1, dxB =0.366, d,, = 0.179, and d,, = 0.3 16, which were close to the estimatesobtained by the ML method. The corresponding log-likelihood value forthese estimates was - 1004.5, slightly lower than that of the ML estimate.
The maximum likelihood value for the other two unrooted trees was
obtained for the case with no interior branch (a quadrifurcating tree). Thisis analogous to the trifurcation of the rooted tree for three sequences.
However, other trees may have ML values with positive estimates of theinterior branch.
Rooted Trees
The number of possible rooted trees for four sequences is 15, and 3 ofthem (trees 3a, 3b, and 3c) are shown in Fig. 5. Ui of Eq. (3) is computed ina manner similar to the case of three sequences, and the likelihood function can be obtained using these Vi values.3
Although the ML method and UPGMA may no longer give the same
tree on every occasion, it has been shown that the tree estimated by the ML

1361

MAXIMUM
Level

I

Level

II

LIKELIHOOD

ABCD

Tree

1

ABCD

Tree
Level

ABCD

Tree

2a

2b

111

ABCD

ACBD

BCAD

dYl-y

y

WJ

Tree
FIG.

593

METHODS

3a

Tree

ib

Tree

3c

5. Three levels of the topological relationships for rooted trees for four sequences.

method is similar to that obtained by UPGMA.3 In this connection, it
should be noted that UPGMA gives least-squares estimates of branch
lengths for the tree obtained. l4 That is, UPGMA minimizes the quantity,
Z(d, - 2Lt,)2, where I is the rate of nucleotide substitution and tij is thetime since divergence between sequences i and j.
Let us explain Saitou’s algorithm3 using Fig. 5. This algorithm is reminiscent of the neighbor-joining
method. First the ML value for tree 1(Level I) is computed. There is no information on the clustering of sequences with tree 1, whereas there are trifurcating and bifurcating points intrees at the next step (Level II). The ML values for all 10 trees at Level II
are computed and are compared with each other. We choose the tree withthe highest ML value among the 10. Suppose it is tree 2a. We thencompute the ML values for trees 3a, 3b, and 3c, which are produced when
the trifurcating point of tree 2a is resolved. In any case, the tree with thehighest ML value is chosen as the final tree. In this way, we may be able tofind the tree with the highest ML value. When the number of sequences is
I4 R. Chakraborty, Can. .I. Genet. Cytol. 19,217 (1977).

594

ESTIMATINGSEQUENCE

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1361

large, this algorithm requires much less computationaltime comparedwith tree-by-tree examination.
The two trees (2a and 2b) at Level II are related to tree 3a at Level III,and they may be used as null trees to test the significance of d, and d2 oftree 3a. If d, is not significantly greater than zero, we assume that tree 2a isthe correct one and proceed to test the significance of d2 with tree 1 as thenull tree. In general, there are n - 1 levels for trees with n sequences, andwe can successively test the significance of all interior branches of a tree.
Five Sequences
There are 15 unrooted trees and 105 rooted trees for five sequences,and the number of possible nucleotide configurations under the one-parameter model of nucleotide substitution is now 5 1. Construction of thelikelihood function is straightforward
as before. We consider only
unrooted trees.
Algorithm for Finding ML Tree
Let us explain Saitou’s algorithm3 for the ML method for unrooted
trees, with a numerical example of five nucleotide sequences of higher
primates (human, common chimpanzee, pygmy chimpanzee, gorilla, andorangutan).i5 As in the case of rooted trees, we first compute the ML
estimate for the Level I tree of Fig. 6. The initial value (d,) for the length of
the branch between node X and sequence i may be computed from the
distance matrix, applying the neighbor-joining method as follows:’
d~i=~d~-~~dj~

(18)
J<k
The maximum log-likelihood value for the Level I tree was -638 forthe example data. Ten different trees (Level II trees) are then considered,and the ML value for each tree is computed. These trees have one trifurcation and one bifurcation. The clustering of common and pygmy chimpanzees gave the highest log-likelihood value of - 59 1. From each tree at Level
II, three trees are produced if the trifurcation is resolved (Level III). The
ML estimates of six branch lengths for the case of Level II are used as theinitial values for the computation of the ML estimation for a tree at Level
III (see Fig. 6). In this case, the seven-dimensional likelihood surface isnumerically examined for three possible trees. The log-likelihood values
were - 586 for tree 1 (chimpanzee and gorilla clustered), - 584 for tree 2
j-l

I5 J. Hixson and W. M. Brown, Mol. Biol. Evol. 3, 1 (1986).

f361

MAXIMUM
Level

LIKELIHOODMETHODS595

HI

c-638)

241

0
769
157
*
C

G

291
130
P

Level II(-59 I )FIG. 6. Three steps to obtain the maximum likelihood tree for the data of Hixson and
Brown.16 H, Human; C, common chimpanzee; P, pygmy chimpanzee; G, gorilla; 0, orangutan. Estimated branch lengths for each tree should be divided by 10,000. Figures in parentheses are the maximum log-likelihood values.

(human and chimpanzeeclustered),and - 583 for tree 3 (human andgorilla clustered).Thus, tree 3 was found to be the ML tree under thisalgorithm. This new algorithm can be extendedto any number of sequences,asin the casefor rootedtrees.
Comparison

with Other Methods

When we applied other tree-makingmethodsto the samedata set,thechoice of the tree dependedon the method used.3UFGMA chosetree 1;
the neighbor-joiningmethod found tree 3 as the besttree,and most of the
branchlengthsestimatedby this method werequite similar to thoseof the
ML method. On the other hand, the distanceWagnermethod16chosetree
2 when the proportion of different nucleotideswasusedasdistance.
‘6J. Fan-is, Am. Nut. 106,645(1972).

596

ESTIMATINGRESULTSSEQUENCE

TABLE
III
OF FOUR METHODSFOR PRIMATE

MethodFitch - Margoliash
Minimum evolution
Maximum parsimony
Maximum likelihood

1361

DIVERGENCESEQUENCE

DATA”

Tree 1

Tree2

Tree3

Tree4

0

+0.60
+0.35

+0.47
0

+25.33f6.59+8-33.86

+1.10
+ 1

0

0

-3.97

-2.98

0

o Values for the Fitch - Margoliash and minimum evolution
methods are percent standard deviation and sum of
branch lengths X 1000, respectively. Values for the maximum parsimony method are the required number of nucleotide substitutions, and those for the maximum likelihood method are the log likelihood. Values of the best tree
are set to zero, and the other values represent differencesfrom the best tree. From Saitou and Imanishi.”

Phylogenetictrees are constructedstep by step for methods such as
UPGMA, the neighbor-joiningmethod, and the distanceWagnermethod,and finally a singletree is obtained.This type of method can be called a“stepwise”clusteringmethod.” In contrast,all possibletrees(or a limited
number of plausible trees) are compared under a certain criterion for
Fitch - Margoliash method,I’ the maximum parsimony method,i* theminimum evolution method,” and the standardML method.2Resultsof
thesemethodsappliedfor the samedatasetasabovearepresentedin Table
III.” Trees l-3 in Table III correspondto thoseof Fig. 6, whereaspygmy
chimpanzeeand human as well as gorilla and orangutanare clusteredin
tree 4.
The program DNAML of PHYLIP Version 3.1, developedby J. Felsenstein,was usedfor the ML method. The transition/transversionratio
wassetto be 5.0, and observedfrequenciesof nucleotideswereused.Trees
l-4 were examined using the “user tree” option. Becausea different
model of nucleotidesubstitution was usedin this case,the log-likelihood
valuesfor trees 1- 3 are different from thoseobtainedusing Saitou’salgorithm (Fig. 6). However,the order of the maximum log-likelihood values
among the three treesis the sameas that obtainedabove,and tree 3 waschosen.The ML value for tree4 wasconsiderablylower than thosefor theother threetrees.This is apparentlybecausethe highly probableclusteringof common and pygmy chimpanzeeswasnot realizedin this tree.
As for the other methods,trees 1 and 3 werechosenby the method of
I7 N. Saitou and T. Imanishi, Mol. Biol. Evoi. 6,5 14 (1989).
I8 W. M. Fitch, Am. Natur. 111,223 (1977).

1361

MAXIMUM

LIKELIHOODMETHODS

597

Fitch - Margoliash and the minimum evolution method, respectively. Interestingly, the rank of these trees in terms of the minimum evolution
method was the same as that of the ML method (see Table III). Finally, themaximum parsimony method found both trees 2 and 3 as equally parsimonious. These inconsistent results indicate that the nucleotide sequence data
used are not sufficient to determine the branching order among human,
chimpanzee, and gorilla. This seems to be consistent with the simulation
result of Saitou and Neil0
Discussion
The original formulation of the ML method’ included the probabilityof tree topology, which is given by assuming a Yule process. FelsensteinZ
took a different approach, in which the ML value for each tree is computed
and the tree with the highest ML value is chosen. However, the likelihood
function to be used varies from tree to tree, so the ML values for different
trees are conditional and cannot be compared in the standard way.3*L9 It
has been claimedZo that this problem can be avoided by applying information theory. Yet there still remains a paradox.
Let us consider Fig. 3b. In this case both trees 1 and 2 had higher ML
values than that for the trifurcating tree where d, is necessarily zero.
Because this trifurcating tree is a submodel of either tree 1 or 2, we may use
the likelihood ratio test for the significance of these two trees. As Felsenstein2i indicated, however, it is possible that both trees 1 and 2 are significantly better than the trifurcating tree. Then which tree should we choose?
This paradox seems to apply in trees l-3 of Fig. 6, since the log-likelihood
values for all these trees are considerably larger than that of Level II tree.
Therefore, we should be cautious in applying the ML method for the
problem of phylogenetic tree estimation.
Nevertheless, the ML method may still be useful for practical purposes.
For example, it has been shown by computer simulations3*22 that the ML
method can find the correct tree with an appreciable proportion even whenthe maximum parsimony method is positively misleading under certain
conditions for four sequences. 23 Saitou and Imanishi” compared the relative efficiency of the five tree-making methods (maximum parsimony,
maximum likelihood, Fitch - Margoliash, minimum evolution, and neighI9 M. Nei, “Molecular Evolutionary Genetics.” Columbia Univ. Press, New York, 1987.
2oT. Kishino and M. Hasegawa, J. Mol. Evol. 29, 170 (1989).
*I J. Felsenstein, in “Statistical Analysis of DNA Sequence Data” (B. S. Weir, ed.), p. 133.
Dekker, New York, 1983.
22M. Hasegawa and T. Yano, Bull. Biomed. Sot. Jpn. 5, 1 (1984).
23J. Felsenstein, Cyst. Zool. 27,401 (1978).

598

ESTIMATINGSEQUENCE

DIVERGENCE.

WI

bor-joining methods). They used a computer simulation under the model
tree for six sequences, and showed that the neighbor-joining method, theminimum evolution method, and the ML method performed better thanFitch - Margoliash and maximum parsimony methods.
From a practical point of view, however, a distance matrix method,such as the neighbor-joining
method, seems to be the first choice fordetermining tree topology. The maximum likelihood method is time consuming, and should better be used after a certain number of prospective
trees are chosen by some distance methods. For a review of distance matrix
methods for phylogenetic tree construction, readers may refer to Neil9 andSaitou.24

24 N. Saitou, in “Handbook of Statistics, Volume 8: Statistical Methods for Biological andMedical Sciences” (C. R. Rao and R. Chakraborty, eds.). North-Holland, New York, inpress.