Methods for Computing the Standard Errors of Branching
Points in an Evolutionary Tree and Their Application to
Molecular Data from Humans and Apes’
Masatoshi Nei, J. Claiborne Stephens, and Naruya Saitou
Center for Demographic

and Population

Genetics,

Universityof Texas at Houston

Statistical methods for computing the standard errors of the branching points ofan evolutionary tree are developed. These methods are for the unweighted pairgroup method-determined(UPGMA) trees reconstructedfrom molecular datasuch as amino acid sequences, nucleotide sequences, restriction-sites
data, and
electrophoretic
distances. They were applied to data for the human, chimpanzee,
gorilla, orangutan,
and gibbon species. Among the four different sets of data
used, DNA sequences for an 895nucleotide
segment of mitochondrial
DNA(Brown et al. 1982) gave the most reliable tree, whereas electrophoretic
data(Bruce and Ayala 1979) gave the least reliable one. The DNA sequence datasuggested that the chimpanzee is the closest and that the gorilla is the next closest
to the human species. The orangutan and gibbon are more distantly related to
man than is the gorilla. This topology of the tree is in agreement with that for
the tree obtained from chromosomal
studies and DNA-hybridization
experiments.
However, the difference between the branching point for the human and thechimpanzee
species and that for the gorilla species and the human-chimpanzee
group is not statistically significant. In addition to this analysis, various factorsthat affect the accuracy of an estimated tree are discussed.

Introduction

In recent years, an increasing number of authors have been using molecular
data to construct evolutionary trees of species. There are several different methods
for constructing evolutionary trees, but in all of them the accuracy of a reconstructed
tree is quite low, unless the lengths (number of mutational changes) of all branches
are sufficiently large (Peacock and Boulter 1975; Tateno et al. 1982; Nei et al.
1983). There are two types of errors involved in a reconstructed tree: (1) topologicalerrors and (2) errors in the estimates of branch lengths. These two types of errors
are intricately related, since the topology and branch lengths are usually estimatedsimultaneously. If we know the pattern of amino acid replacement or nucleotide
substitution in evolution, the standard error (SE) of an estimate of evolutionary
distance between a pair of species can be computed relatively easily (e.g., Kimura
1969; Kimura and Ohta 1972; Nei 1978; Nei and Tajima 1983). Some authors(e.g., Kumazaki et al. 1983) have substituted this error for the SE of a branching
point. However, this does not give a correct value except under certain circumstances,and we need a general method for computing the SE of a branching point.
1. Key words: topological errors, UPGMA trees, molecular phylogeny.
Address for correspondence and reprints: Dr. Masatoshi Nei, Center for Demographic and Population
Genetics, University of Texas at Houston, P.O. Box 20334, Houston, Texas 77225.
Mol. Biol. Evol. 2(1):66-85. 1985.
0 1985 by The University of Chicago. All rights reserved.

0737-4038/85/020 l-0009$02.00

66

Standard

Errors of Branching

Points67

Evaluationof the SE is importantbecause each branching
point suggests that an
important
event of speciationor population
splitting occurred there. In practice,
however, topological errors are often introduced
at the time of tree reconstruction,
and an estimated topology may not represent the actual process of evolutionarychanges of genes or organisms (Tateno et al. 1982; Nei et al. 1983).
In this paper we present methods for computing
the SE values of branching
points for a tree reconstructedby the UPGMA
method (see Sneath and Sokal
1973). The UPGMA is known as a method of phenetic clustering, but Nei (1975)
suggested that this method would give a good evolutionary
tree when the expected
rate of gene substitution
is constant but the actual number of substitutions
is subject
to stochastic errors. Later, using computer simulation,
Tateno et al. (1982) and Nei
et al. (1983) showed that UPGMA is more efficient for getting the correct (true)
tree than are several other alternative
methods when a tree is reconstructedfromgenetic distances the expectationsof which are proportional
to evolutionarytime.
In this paper we consider four different types of molecular
data: ( 1) amino acidreplacement
data, (2) nucleotide substitution
data, (3) restriction-sites
data, and (4)electrophoretic
data. We then apply the methods developed to study the reliabilityof the evolutionary
trees reconstructed
for the human and several ape species.
TheoryAmino

Acid Replacement

Data

Let us consider the evolutionary
tree given in figure 1. In this figure, numerals
1, 2, 3, and 4 represent four different species, whereas numerals
5, 6, and 7 standfor branching
points (or ancestral species). For simplicity, we assume that the rateof amino acid replacement
for a protein is h per year per amino acid site and is thesame for all amino acids. We also assume that the evolutionary
time considered isrelatively short, so that parallel mutations
in different evolutionary
lineages or backmutationsin the same lineage are negligible.
(The effect of violationof these
assumptions
will be discussed later.) Under these assumptions,the expected numberof amino acid replacements
between a pair of species is given by d = 2ht, where t
is the time since divergence between the two species. If the total number of aminoacids compared between the two species is n and the proportionof identical aminoacids between them is i, then d is estimated by
d = -lo&i

1b5

CLUSTERA

CLUSTER

B

2

7
b3
b6

3

4
FIG. 1.-A hypotheticalevolutionary
7 represent branching points (or ancestraltree. I, 2, 3, and 4 represent extant species, whereas 5, 6, andspecies). The hatched boxes represent the magnitude of SE.

68

Nei, Stephens, and Saitou

with sampling

variancel-iV(d) = in(2)(e.g., see Nei [ 19751, p. 14).
It should be noted that if we knew the number of amino acid replacements
forall amino acid sites, the variance of the number of amino acid replacements
persite would have been 2ht/n under the Poisson process. In practice, however, it is
impossible to know these numbers, and we must estimate d by equation (1). Since
equation(1) is based on incomplete
informationon amino acid replacements,
equation (2) gives a variance larger than 2ht/n.
In a tree reconstructedby UPGMA, the branching
point between two species(species 1 and 2 in fig. 1) is given by b = d/2 with variance V(b) = V(d)/4, whereb = b, = b2 in figure 1. Therefore, the SE of this branching
point (5 in fig. 1) isgiven by the square root of V(b).
In a UPGMA tree, the distance between cluster A (species 1 and 2) and cluster
B (species 3 and 4) in figure 1 is given by

d

_

AB -

63 + &4 + 63 + d24

,

4

(3)

where djk is the number of amino acid replacements
between species j and k, anddi2 and d34 are assumed to be smaller than di3, d14, d23, and dz4. The djk value isestimated by the equation djk = -log, ijk, where ijk is the proportionof identicalamino acids between species j and k. From equation(3), the variance of dAB isgiven byV(d,,)= &[V(d13)+ V(dJ+ V(d&+ V(d24) + 2 Cov(d13, d14) + 2 Cov(d13, d23)

+ 2 Cov(d13, d24) + 2 Cov(d14, dz) + 2 Cov(di4,d24) + 2 Cov(d23,and the variance of branching
point 7 in figure 1 is V(bAB) = V(d,,)/4.
The variance V(djk) in equation(4) is given by equation(2), whereas
covariance between di 3 and di4 is

Cov(d13, d,4) = Cov(b,+ bS + bs + b3, bl + bS + b6 + b4) = V(d,,)

&A
(4)
the

.
= 1-116,116n(5)
because b3 and b4 are independent(see fig. 1). Here, di6 is bl i- b5 + b6. In equation(5), i16 cannot be measured from actual data, since the ancestral species 6 does notexist. However, it can be estimated by

116=-(bl+bS+b6)e=e-dAB+d34/2.

Standard Errors of Branching Points

Similarly,

all the other covariances69

can be obtained.
1 - i35

CoWn, &d = -

’

i35n

Cov(d13, &cd = CMh,

1 - is6= &i-r ’

&d

.
1 -

Covbh, &d = 7,

145

145n

1Cov(d23, d24) = 7

126

.

126n

These covariances

can be computedfrom the values of dAB and djk. For example,

i56 = e-dm+(dn+d34)/2.

Therefore, the SE of branching
point 7 in figure 1 can be estimated.
It is now obvious that this procedure can be used for any branching point in a
UPGMA tree irrespective of the number of species involved. In general, the UPGMA
distance between two species clusters A and B is given by
CjkdjkdAB = -

rs

(6)

’

where djk is the inter-cluster distance between the jth species in cluster A and thekth species in cluster B, and r and s are the numbers of species in clusters A andB, respectively. Therefore, the variance of dAB is

2 V(djk) + C CM&, 4nJ
V&d

=

.

b-G2

(7)

There are rs variances and rs(rs - 1) covariancesof intercluster
distances. (COV[djk,djk].)
The
variancesinvolvedaredirectly
obtainablefromdl,] is equal to Cov[dlm,an equation equivalent
to equation (2). The distance contributing
to COV(djk, di,)is given by

d AB -

d(jl) + d(km)2’where djl, and d#m) are the intrachster distancesk and m, respectively. Therefore, we havel-i
Cov(djk, dl,) = in

(8)

between

7

species j and 1 and species

(9)

70

Nei, Stephens, and Saitou

i = e-dAB+(d(jl)+4km))/2.

Note that djr, = 0 for j = 1 and dkmJ = 0 for k = m.
Although the above theory is straightforward,
the actual computation
can betedious, particularly
when the number of species involved is large. However, thevariances of branching points can be computed simultaneouslywith the estimationof UPGMA branch lengths. Such a computer program has been developed and may
be obtained on request.
At this point, some readers might wonder how the above formulationis related
to Chakraborty’s(1977) study of the variance of a branch length. His study isdifferent from ours on two points. First, he assumed a Poisson variance for d, sothat it is smaller than our variance, as mentioned
earlier. Second, he was interested
in the variance of a branch length, rather than in the variance of a branching point.
Since he assumed a Poisson variance for d, his expressions look simple. If we use
the correct variance of d as given in equation (2), however, Chakraborty’s
methodbecomes more complicatedthan ours.
The variance given by equation(7) can be used for testing the statisticalsignificanceof the difference between two branching
points in a tree, but somecautionis necessary.
When the two branching
points to be tested belong to
independent
clusters (e.g., branching
points 5 and 6 in fig. l), the variance of thedifference is simply the sum of the variances of the two branching points. However,when the two branching
points are hierarchicallyrelated (e.g., 5 and 7 in fig. I),
there is a correlation
between them. For example, the variance of the difference(6 = bAB - br2) between points 7 and 5 in figure 1 is given by

V(S) =
Therefore,

WL,d + W&d - 2 COV(dAB,
42)
4

.

(11)

if we use

Vu@)=

V&d

+ Wd

(12)

4

as the variance of 6, it will be an overestimateof the true variance, since COV(dAB,d12) is always positive. Cov(dAB, dr2) can easily be evaluated in the present case.
That is,Cov(dAB, d12) = Cov

43 + 44 + d23+ b
4

3 42

Wd

1

= V(b12) = 4.

(13)

Therefore, we can use equation ( 11) for testing bAB - br2.
The above procedure can easily be extended to cases where both clusters A
and B in figure 1 involve more than two species. A few examples are presented in
the Appendix.
When the numberof species in cluster A is large, however, thecomputation
can be quite tedious. In this case, it is convenient
to note that thedifference between V(6) and Vu(s) is usually small and that Vu(s) can thus be used

Standard Errors of Branching Points

71

for an approximate
test. Since Vu(s) 2 V(bAB - b12), such a test will be aconservative
one. In the Appendix, we also present a formula [Vi_@)] for the lower
bound of the variance of bAB - br2. This formula can be used for a liberal test if
necessary. (See the numerical examples in the next section.)In the above formulation,
we assumed that the rate of amino acid replacement
is the same for all amino acid sites. In practice, this assumption
does not hold formost proteins (Fitch and Margoliash
1967a), and the actual number of amino acidreplacements
seems to follow the negative
binomial
rather than the Poisson
distribution(Uzzell and Corbin 197 1). Nevertheless,
Nei and Chakraborty(1976)
have shown that equation (1) holds approximately
for most replacement
patterns
as long as d is ~1. Of course, the variance of d will be larger than that given byequation (2), but the extent of underestimationof V(d& given by equation(7)
even if it is an underestimate,would not be serious as long as dAB < 1. Furthermore,
the variance of branching points will be useful in making a proper interpretationofestimated trees, as long as we understandit is an underestimate.
Nucleotide

Substitution

Data

In the case of nucleotide
substitution
data, the relationshipbetweenthe
expected number of nucleotide substitutions
per site (2ht) and the proportion(i) ofidentical nucleotides between two DNA sequences that are compared is given by
d = -3/4 lo&[ 1 - “/s( 1 - i)](Jukes and Cantor1969), and the variance(14)of d isi( 1 - i)V(d) = n[ 1 - 4( 1 - i)/312(15)(Kimura and Ohta 1972). The variances of branching points can be obtained in thesame way as is that in the case of amino acid substitution.
However, note that theformula corresponding
to equation ( 10) is given byi = ‘/4 +y4e -4h[dAB-(4jl)+4km))/21.

(16)

Equation (14) is based on the assumptionthat nucleotide
substitution
occursat random among the four nucleotides
A, T, C, and G. However, this equation is
known to hold quite well, even under nonrandom
nucleotide
substitution,
as long
as d is smaller than 0.5 (Kimura 1980, 198 1; Takahata and Kimura 198 1; Gojobori
et al. 1982). Therefore, the above method for computing
the variances of branching
points seems to apply as long as d < 0.5. In the case of nonrandom
nucleotide
substitution
with d 2 0.5, a more elaborate method for estimatingd is necessary.
There are several such methods, but most of them are model-dependent,
and thecomputationof the variance of d is complicated.
The only method in which thevariance can be computed
relatively easily is Tajima and Nei’s ( 1984). In thismethod, d is estimated by

d=-clog,[ l-(!$)I,

72

Nei, Stephens, and Saitou

where c is a function of the frequencies of the four nucleotides and nucleotide pairs
between the two DNA sequences compared. In Jukes and Cantor’s formula, c = 3/4.
Since this c can be estimated from data by Tajima and Nei’s method, one can use
the same method of computing the variance of branching points as that used in thecase of c = %, simply replacing 3/4in equations ( 14), ( 15), and (16) by the estimateof c that is obtained.
Restriction-Sites Data
The evolutionary change of nucleotide sequence can also be studied byexamining restriction cleavage site differences among different species or genes.
When restriction enzymes with r nucleotides in the recognition sequence are used,
the maximum likelihood estimate of the number of nucleotide differences per site
is given byd=[-1o&slr(17)’

where S = 2mxu/(mx + my) (Nei and Li 1979; Kaplan and Risko 1981; Nei andTajima 1983). Here, m x and my are the numbers of restriction sites for DNA
sequences X and Y, respectively, and m xy is the number of restriction sites sharedby the two sequences. The variance of d is given by,(,)=(2-s)(1

-‘I

2r2mS

’

(18)

where m = (mx + my)/2 (Nei and Tajima 1983). Equations (17) and (18) are quite
accurate as long as d I 0.25. When d > 0.25, Nei and Tajima’s (1983) equations(2 1) and (23) should be used. However, as d increases, the reliability of the estimateof d gradually declines.
Since we have the formulae for computing the mean and variance of d, the SE
values of branching points can be estimated as in the case of amino acid replacement
data. The equation corresponding to equation (10) iss=e-r[dAB-(4jl)+4km))/21

.

(19)

When restriction enzymes with various values of r are used, the maximum
likelihood estimate of d can still be obtained by the methods of Kaplan and Langley(1979), Gotoh et al. (1979), Kaplan and Risko (198 l), and Nei and Tajima (1983).
The simplest method among these is Nei and Tajima’s (1983) iteration method.
The variance of the estimate of d thus obtained is given by

V(d) = n1

C [l/Vi(di)l

’

(20)

i=l

where n is the number of types of enzymes used, and Vi(d) is the value of equation(18) for the ith type of enzymes whens = e-ridStandard Errors of Branching Points

73

is used. Here, ri is the value of r for the ith type of restriction enzymes. The
expected value of S for the ith type of enzymes can be obtained by replacing r by
ri in equation (19), and the corresponding variance can be obtained by equation(20). (See the numerical example given later in the text.)
Electrophoretic

Distance

In the construction of evolutionary trees for closely related species or populations,electrophoretic data are often used. Although many distance measures are available
for electrophoretic data, Nei’s (1972) measure seems to be most appropriate for thispurpose, since it is linearly related to evolutionary time, at least theoretically. By
using computer simulation, Nei et al. ( 1983) have also shown that this distancemeasure, in combination with UPGMA, is most efficient in recovering the true tree
among the several alternative distance measures examined. Farris ( 198 1) criticized
this measure for not permitting evolutionary path:length interpretation. However,this criticism is apparently based on his failure to distinguish between expected andobserved distances (Nei et al. 1983; Felsenstein 1984).
Let xi and yi be the frequency of the ith allele at a locus in populations X andY, respectively. The gene identities within and between populations are thencomputed by jx = 2 x2, j, = 2 yf , and jx, = C xiyi, where 2 stands for thesummation for all alleles at the locus. The average gene identities over all loci aresimply the averages of these quantities, that is,
i

Jx

=k=ln

5jYk

jXk

5jXYk

Jy = k=l
’

n

and

’

k=l

Jxy = -n’where k refers to the kch locus and n is the number of loci examined. Nei’s 1972)
standard genetic distance is then defined as D = -log, I, where I = Jxy/ $_JxJy. If
the time after divergence between populations X and Y is t years and the two
populations are in equilibrium with respect to the effects of mutation, selection,and genetic drift, the expectation of I is given by I = e-2ut, where a is the rate ofgene substitution per locus per year. Therefore, D = 2at.
The D value between a pair of species or populations and its variance can be
estimated by the methods of Nei and Roychoudhury (1974) and Nei (1978),although these methods are more complicated than those for amino acid replacement
or nucleotide substitution. However, it is not always easy to compute the varianceof branching points, since the allele frequencies of ancestral populations such as
those for branching points 5 and 6 in figure 1 are not known. The only cases inwhich this is not a problem are those in which (1) all populations examined have arelatively low average heterozygosity and are fixed for different alleles at some locior (2) different strains in asexual haploid organisms such as Escherichia co/i (e.g.,
Selander and Levin 1980) are studied. In these cases the variance of D between apair of populations is given by
1-I

V(D) = --jy-(21)(Nei 197 I), and all other procedures for obtaining the variance of branching points
are the same as those for amino acid sequence data.

74

Nei, Stephens, and Saitou

In practice, the applicability of equation (2 1) depends on the value of I relative
to average heterozygosity (H). When I is do.9 for all pairs of populations, and H is~0.2, equation (21) seems to give a quite reliable estimate, as long as many loci are
examined. This is because in this case the genetic identity for a single locus(Ij = jXY/iE)
shows a distribution close to the binomial distribution, with Ij
taking a value of 0 or 1 for a majority of loci. When I is ~0.9 for most population
pairs, however, equation (2 1) may give a serious overestimate of the true variance.
Although it is difficult to develop a general method for computing the exact
variance of a branching point, it is possible to get an upper bound or a maximum
estimate of the variance. This maximum estimate is the average of the variances ofall pairwise distances used for computing a branching point. For example, branching
point 7 in figure 1 is given by half the average (DAB) of D13, Did, D23, and DZ4 in
the case of electrophoretic data. The maximum variance of DABis then given by
Wh3)
V@AB)

=

+

V(Dl4)+V(D23)

4+

V(D24).

(22)

This variance is obviously larger than the true variance given by an equationequivalent to equation (4) or equation (7). (Note that the divisor of the right handside of equation (22) is rs = 4, whereas that of equation (4) is (rs)2 = 16.) However,as long as we understand that equation (22) gives an upper bound of the truevariance, we can use it for getting a rough idea of the reliability of branching points.
The accuracy of V(bAB) = V(DAB)/4 as an estimate of the variance of a branching
point depends on tree structure. If most intracluster branches are short, the accuracy
is quite high. However, if there are many long branches within each cluster, thevariance may be seriously overestimated. In general, this method should be used
only when equation (22) gives a smaller value than does equation (21). This is
expected to occur when the I values are large, for example, >0.9.
Evolutionary Tree For the Human and Ape Species

In the past 10 yrs the genetic relationship of the human and ape species has
been studied intensively by using various types of molecular data. In the following,
we apply our methods to four different types of data.
Amino Acid Sequence Data
Although amino acid sequencing of proteins was started more than 20 yr ago,the sequence data for the human and ape species are still limited. The only datathat can be used for our purpose are those for hemoglobins c1 and p, myoglobin,
and fibrinopeptides A and B for the human, chimpanzee, gorilla, and orangutan
species (Dayhoff 1972, 1973, 1978; Goodman et al. 1983). There are also data for
two partial sequences (13 amino acids each) of the duplicate hemoglobin y chainsfor these organisms (Huisman et al. 1973). The rate of amino acid substitution for
fibrinopeptides is known to be significantly higher than that for hemoglobins.
However, since the sequence differences among these four species are very small,
we can pool the sequences together to compute the proportion of different aminoacids (p = 1 - i) for each pair of these species, using a total of 496 amino acids forcomparison. From these proportions, the number of amino acid replacements persite (d) and its SE can be computed by equations (1) and (2). The results obtained
are presented in table 1. In the present case, d = -lo&( 1 - p) is virtually identicalStandard

Table 1
Proportions of Different Amino Acids and Estimatesfor Four Primate Species
Human

.....
Human
...
Chimpanzee
Gorilla . .
..
..
Orangutan

0.00200.00610.0204+ 0.0020+ 0.0035+- 0.0064

Errors of Branching

Points75

of Amino Acid Replacements

Chimpanzee

GorillaOrangutan

0.00200.00600.008 10.02020.02220.02220.0081
0.0224+ 0.0040
f 0.0068

0.0224

f 0.0068

NOTE.-Numbers
above the diagonal are the proportion (p = 1 - i) of different amino acids for hemoglobins a, p,‘y, G~; myoglobin; and fibrinopeptides
A and B for the species pairings indicated. Numbers below the diagonal arenumbers of amino acid replacements per site (*SE) for the species pairings indicated. The number of amino acids used
for hemoglobins a, p, 3. and Go; myoglobin; and fibrinopeptides A and B are 141, 146, 13, 13, 153, and 30, respectively,
for a total of 496. The data for hemoglobin 9 and “7 refer to a region (13 amino acids) sequenced by Huisman et al.
(1973). The amino acid sequence for the remaining region is not known for all four of the species examined. The dataon sequence differences for the orangutan hemoglobin-aand -p chains are those of Goodman et al. (1983). All otherdata were obtained from Dayhoff (1972, 1973, 1978).

with p, since p is very small. The evolutionary tree reconstructed from the d values
by using UPGMA is given in figure 2. The branching point (a) between the human(H) and chimpanzee (C) species is bHc = 0.0020/2 = 0.0010 (from table I), and the
SE is 0.0020/2 = 0.00 10, which is identical with b HC, Since bHc is Small. Similarly,
the branching point (b) between the gorilla (G) and the human-chimpanzee lines is
btHcG = (0.0061 + 0.0081)/4 = 0.0036.
The variance [V(b (H&I of b(HCG is V@(HC)C)/~~ where
V(d(Hc&=V6h-d + V&c)+

4

2 COV(dm,&G)

(23)

,

from equation (7). We have V(d& = (0.0035)2 and V(d& = (0.0040)2 (from theSE values in table 1). To compute Cov(d HG, &G), we must know the expecteddistance between the gorilla species (G) and the branching point (a) between thehuman and chimpanzee species. It is given by d = 2btHCG - bHC = 0.0062.
Therefore, the corresponding i value is e -“.oo62= 0.9938, from equation (10). Thus,
COV(dHG, d& = V(d) becomes 1.258 X 10W5,from equation (9). Hence, the SE ofb(HcG = [v(d(HC)C)/411’2= 0.0018. The SE of branching point c can be obtained in
.OOl

HUMAN

CHIMPANZEEGOR I LLA

ORANGUTAN
FIG. 2.-Evolutionary
tree for four hominoid
species, which was reconstructedfrom amino acidsequence data. The number given for each branch represents the branch length or the number of aminoacid replacements
per site. The hatched box represents 1 SE on each side of the mean branch length.

76Nei, Stephens, and Saitou

Table 2
Numbers of Nucleotide Substitutions

Chimpanzee . . .
Gorilla . . .
.
Orangutan.
Gibbon . . .

per Site (d) for Five Primate Species

Human

Chimpanzee

GorillaOrangutan

0.0939 -+ 0.0107
0.1106 -+ 0.0118
0.1797 f. 0.0156
0.2072 -t 0.0 170

0.1145 f 0.0120
0.1940 + 0.0163
0.2175 f 0.0175

0.1882 + 0.0160
0.2175 -e 0.0175

0.2160 f 0.0174

NOTE-Data from Brown et al. 1982. In the computation
nucleotide deletion in the segment of the orangutan mtDNA.

of d, 895 nucleotides were used because there was one

the same way, and it becomes 0.0032. These results are presented graphically infigure 2. The difference between branching points a and b can be tested by using
the normal deviate (t). In the present case, t becomes (0.0036 - O.OOlO)/[O.OOl82+ 0.00 102/2] ‘I2 = 1.34 by using equation (A3), so that the difference is notstatistically significant. If we had used the approximate variance, as calculable byequation ( 12), t would have been 1.26, which is not much different from the correct
value. On the other hand, the difference between branching points b and c issignificant at the 5% level, since we have t = 0.0074/0.0035 = 2.10 by using
equation (A5). The same conclusion is obtained by using equation (12), which givest = 2.00. A similar t value (t = 2.15) is also obtained by using the lower boundvariance, V&), in equation (A7).
Nucleotide Sequence Data
Brown et al. (1982) sequenced a segment (896 nucleotides) of mitochondrial
DNA (mtDNA) for the human, chimpanzee, gorilla, orangutan, and gibbon species.
From these sequence data, we can estimate d and its SE by using equations (14)and (15), respectively. The results obtained are presented in table 2, and the
UPGMA tree obtained from these distance data is given in figure 3. If we exclude
the gibbon, the topology of this tree is identical with that of the tree derived fromamino acid sequence data. The branching point (a) between the human andchimpanzee species is bnc = 0.0939/2 = 0.0470 (from table 2). The SE of this bnc.047. ooga
+
.037
.014

d

.047

b

c

.056

.093

.107

HUMAN

CHIMPANZEEGORILLA

ORANGUTANGIBBON

FIG. 3.-Evolutionary
tree for five hominoid species, which was reconstructed from nucleotidesequence data on a segment of mitochondrial DNA. The number given for each branch represents thebranch length or the number of nucleotide substitutions per site. The hatched box represents 1 SE oneach side of the mean branch length.

Standard Errors of Branching Points

77= 0.0107/2 = 0.0054. Similarly, the branching point (b) between the gorilla andthe human-chimpanzee groups is b (nc)G = (0.1106 + 0.1145)/4 = 0.0563.
The variance of bfHCF can be obtained by using equation (23). In the present
case, V(dHo) = (0.01 18)2 and V(d,& = (0.0120)2 (from table 2). On the other
hand, the expected distance between the gorilla and branching point a is given
by d = 2bcHCG - bHC = 0.0656, so that the corresponding i value is 1/4+ (3/4)
X exp[-(4/3) X 0.06561 = 0.9372, from equation (16). Thus, Cov(dHG, dcG) = V(d)becomes 7.833 X 10v5, from equation (15). Therefore, the SE of bcHCK = 0.0052.
Similarly, the SE values of branching points c and d in figure 3 become 0.007 1 and0.0074, respectively. The differences between branching points a and b and thosebetween branching points c and d in figure 3 are not statistically significant.
However, the difference between branching points b and c is significant, t being 4.5.
In this case, equation (12) gives t = 4.2, whereas equation (A7) gives t = 4.6.
Restriction-Sites Data
A phylogenetic tree for these five primate species can also be constructed from
Ferris et al.‘s (198 1) restriction-sites data. Estimates of the numbers of nucleotide
substitutions are obtained from the data in table 3 by using Nei and Tajima’s (1983)
equations (25) and (28). This table includes data on the number of restriction sites
and the number of shared restriction sites for eighteen six-base enzymes and one
four-base enzyme. Ferris et al. (198 1) used two six-base enzymes with four
recognition sequences (AvaI and HincII). Two of the four recognition sequences ofeach of these two enzymes were identical with those of two other six-base enzymes.
For example, HincII recognizes the four sequences GTTGAC, GTCAAC, GTTAAC,
and GTCGAC, but the sequences GTTAAC and GTCGAC are also recognized by
HpaI and SaZI, respectively. In the case of AvaI and HincII, we have therefore
considered only those restriction sites that were not recognized by the other enzymes.
We note that two mutational changes are required to transform one of the tworemaining recognition sequences (e.g., GTTGAC and GTCAAC in HincII) to theother, so that the shared restriction sites for a pair of species (unique to these
enzymes) must almost always have an identical sequence (e.g., GTTGAC or
GTCAAC in HincII). Therefore, each of these two enzymes can be regarded asconsisting of two different enzymes with r = 6. This makes the total number ofenzymes with r = 6 equal to 20.

Table 3
Numbers of Restriction Sites (m) and Shared Restriction Sites (mxv) for the Mitochondrial
DNAs from Five Primate Species

Human .
.
Chimpanzee..
Gorilla .
..
Orangutan . . .
Gibbon . . . . .

HumanChimpanzee

GorillaOrangutan

Gibbon42 (6)
19 (6)
22 (4)
20 (4)
17 (4)

42 (6)
25 (6)
15 (4)
15 (4)

42 (6)
16 (4)
18 (4)

36 (10)14 (5)

47 (7)

Nom-Thefigures in front of parentheses are the sums of m or mxy for 18 6-base enzymes, whereas those inparentheses are the values of m or m xy for one 4-base enzyme used. The values on and off the diagonal are m’s andmxy’s, respectively. Data from Ferris et al. (198 1) were used.

78

Nei, Stephens, and Saitou

Table 4
Numbers of Nucleotide Substitutions

Chimpanzee . . .
Gorilla .
. ..
Orangutan . . . .
Gibbon . . . . . . .
NOTE.-Data

per Site for Five Primate Species

Human

Chimpanzee

GorillaOrangutan

0.1142 f 0.0209
0.1071 + 0.0198
0.1194 + 0.0219
0.1556 f 0.0262

0.0880 + 0.0171
0.1613 f 0.0283
0.1735 k 0.0289

0.1521 f 0.0268
0.1474 + 0.0250

0.1729 f 0.029 1

from Ferris et al. (198 1).

The estimates of the number of nucleotide substitutions obtained are presented
in table 4. The estimates of substitutions between the human and chimpanzee
species and between the human and gorilla species are virtually the same as thosederived from the nucleotide sequence data, but the estimates for the other pairs oforganisms are smaller than those derived from the sequence data, the difference
between the two sets of data increasing with increasing d. This difference could be
due to either inaccuracy of the estimates obtained from restriction-sites data or thedifference in the rate of nucleotide substitution between the sequenced region andthe entire region of mtDNA. We note that the SE values of d’s derived fromrestriction-sites data are considerably larger than those derived from sequence data.
Figure 4 shows the UPGMA tree reconstructed from the d values in table 4.
This tree has a topology different from that in figure 3. That is, among the human,
chimpanzee, and gorilla species, the latter two cluster first in this tree, whereas in
the tree in figure 3, the human and chimpanzee species make the first cluster. Allother parts of the topology are the same for the two trees.
The branching point (a) between the chimpanzee and the gorilla and its SE
can be obtained directly from table 4. That is, bco = 0.0440 + 0.0085. The
branching point (b) between the human and the chimpanzee-gorilla group is bn(co)= (0.1142 + 0.1071)/4 = 0.0553. The variance of this branching point can be
.044

.044

CHIMPANZEEGOR I LLA

HUMAN

ORANGUTAN

I,082

GIBBON

FIG. 4.-Evolutionary
tree for five hominoid species, which was reconstructed from restriction sites
data for mitochondrial DNA. The number given for each branch represents the branch length (thenumber of nucleotide substitutions per site). The hatched box represents 1 SE on each side of the meanbranch length.

Standard Errors of Branching Points

79

obtained by an equation similar to equation (23). From table 4, V(duc) = (0.0209)*
and V(duo) = (0.0199)*. We also have Cov(duc, dno) = V(d), where d = 2bH(cGJ- bco = 0.0667. The value of V(d) can be computed by [ 2, V,(d)-‘]-‘, where VXd)
is the variance of d for r-base enzymes given by (2 - S)( 1 - S)/[2r2mS] (Nei andTajima 1983). Here, S = eYrd, and m is the mean number of restriction sites. Nei
and Tajima (1983) defined m as the mean for the two species to be compared, but
in the present case, m should be defined as the mean for all species being investigated,since we are computing the expected variance of d for the entire set of data. This
mean can be obtained from the data in table 3, and it becomes 42 for the six-baseenzymes and 7 for the four-base enzyme. Therefore, we have V,(d) = 0.00022, andV,(d) = 0.0017. Thus, V(d) = 0.00019, and V(du(co)) = 0.00030. Hence, the SE ofbH(eGJ= 0.0087. The SE values of all other branching points can be obtained in thesame way. The results obtained are presented graphically in figure 4. It can be seenthat the SE values of the branching points are considerably larger than those of thetree obtained from nucleotide sequencing (fig. 3). The difference between branching
points a and b is again statistically nonsignificant. In the present case, even the
difference between branching points b and c is not significant.
Electrophoretic

Data

The phylogenetic relationship of the human and ape species was also studied
by Bruce and Ayala (1979) by using electrophoresis. They examined the electrophoretie variation of 23 protein loci for man, two species of chimpanzee (Pan troglodytes
and P. paniscus), the gorilla, two subspecies of orangutan (Pongo pygmaeus abelii
from Sumatra and P. p. pygmaeus from Borneo), two species of gibbon (Hylobates
lar and H. concolor), and the siamang. Bruce and Ayala did not use their own dataon the human species. Instead, they used data obtained by other workers. It istherefore difficult to reconstruct their computation of genetic distances between thehuman and other primate species. In the following, we have eliminated from ouranalysis all data concerning the human species. Data concerning the siamang havealso been eliminated, since they were derived from only one individual. Although
they studied 23 protein loci, Bruce and Ayala found a few species in which genefrequency data could not be obtained for a few loci. We therefore eliminated the
Borneo orangutan and used the 20 loci that were shared by the remaining six
species. Estimates of genetic distances and their SE values were computed by using
Nei’s (1978) method. The results obtained are presented in table 5. In this table,
Table 5
Genetic Distances and Average Heterozygosities

for Six Species of Apes%w
Chimpanzee.
Pygmy chimpanzee
GorillaOrangutan
Lar gibbon
Concolor gibbon
NOTE.-Figures

Chimpanzee

0011
L
0.077
0.176
0.124
0.220
0.255

0.107
0
0.164
0.084
0.219
0.254

(0.075)(0.174)(0.124)(0.218)(0.249)above the diagonal are genetic distances,

are the SE’s of genetic distances.
(1979) are used.

Chimpanzee(0.161)(0.087)(0.216)(0.247)

GorillaOrangutan

Lar Gibbon

Concolor
Gibbon

0.475
0.419
0.054
0.192 (0.191)
0.184 (0.182)
0.223 (0.217)

0.269
0.140
0.546
0L 0550.210 (0.210)
0.215 (0.210)

0.668
0.661
0.510
0.634
0.052
0.082 (0.084)

0.806
0.799
0.666
0.633
0.133
0

those on the diagonal are average heterozygosities,

The SE’s of genetic distances in parentheses

are those obtained by equationand those below the diagonal(21). Data from Bruce and Ayala

80

Nei, Stephens, and Saitou

the SE values obtained by equation (21) are given in parentheses.
It is clear thatthese SE values are very close to those obtained by Nei’s more accurate method.
This is because I is ~0.9 for all pairs of species, and the average heterozygosity
islow for all species (table 5). In computing
the SE values of branching
points,we can therefore use the same method as that for amino acid sequence data.
The UPGMA tree reconstructedfrom the genetic distances listed in table 5 isgiven in figure 5. The topology of this tree is different from that of all previous
trees, the orangutan
being closer to the chimpanzee
than to the gorilla. In thepresent case, the SE values of branching
points a, e, and b become 0.037, 0.042,
and 0.050, respectively.
The SE values of the other branching
points are showngraphicallyin figure 5. It is seen that the phylogenetic
tree constructedfromelectrophoretic
data is even less reliable than that obtained from restriction
sites
data. Although the orangutan
clusters with the chimpanzee
before it joins with thegorilla, the SE of the branching
point (c) between the gorilla and the chimpanzeeorangutangroup is so large that the distance between b and c is not statisticallysignificant. The branching
point between the gibbon and the other apes also has alarge SE.
This low reliability of electrophoretic
data is partly due to the small numberof loci used. In a computer simulation,
Nei et al. (1983) have shown that when thenumber of loci used is ~30, the topology of a reconstructed
tree is subject to a large
stochastic error. The accuracy of a reconstructed tree also depends on the detectabilityof protein differences by electrophoresis.
The higher the detectability,
the higher the
reliability. It should be noted that in Bruce and Ayala’s experiment,
this detectability
was not particularlyhigh. Previously,King and Wilson (1975) had studied thegenetic distance
betweenthe humanand chimpanzee
species and obtained
D = 0.62, which is nearly two times higher than the estimate (0.39) obtained byBruce and Ayala.
Discussion
In the present paper,
substitutions
is proportional

we have assumed that the expected numberof geneto evolutionary
time, although the actual number may.05

CHIMPANZEE

PYGMY

CHIMP

ORANGUTANGORILLA
.07
.27

e
.07

LAR

CONC,

GIBBONGIBBON

FIG. 5.-Evolutionary
tree for six hominoid species, which was reconstructed from electrophoretic
data. The number given for each branch represents the branch length (genetic distance). The hatched box
represents 1 SE on each side of the mean branch length.

Standard

Errors of Branching

Points81

deviate from the expected number because of stochastic errors. This assumptionisnecessary for estimating
the topology and branch lengths of a UPGMA
tree. In
practice, of course, it would not always hold, and if it does not, a reconstructedtree
may deviate substantiallyfrom the true tree. Deviations of reconstructed
trees fromthe true trees also occur because of stochastic
errors in the process of genesubstitution,
unless the number of amino acids or nucleotides
examined
is verylarge (Tateno et al. 1982; Nei et al. 1983). Therefore, caution should be exercisedin the interpretationof a tree reconstructedfrom molecular data.
Some tree-making
methods, such as Fitch and Margoliash’s (1967b) and Farris’s(1972) parsimony
methods,
are designed to take care of unequal
rates of genesubstitution
in different branches. In these methods, however, even stochastic errors
are considered to be a reflection of unequal rates of gene substitutions,and this
introduces another source of error into the process of tree making. Indeed, in thepresence of large stochastic errors, Fitch and Margoliash’s and Farris’s methods are
not necessarily
better than UPGMA
in recovering
the true tree, even if thesubstitution rate varies with branch to some extent (Tateno et al. 1982). Furthermore,in parsimony methods, it is not easy to compute the SE values of branching points
since there is no way to compute the expectationand variance of gene substitutions.
We have computed the SE values of branching
points for four different typesof molecular
data and found that the SE values relative to the means or thecoefficients of variation are smallest for nucleotide
sequence data and largest forelectrophoretic
data. The smaller coefficients of variation for nucleotide
sequence
data than for amino acid sequences are mainly due to two factors: (1) the numberof nucleotides
examinedis larger than the number of amino acids examined and(2) the d values for nucleotide
sequence data are larger than those for amino acidreplacement
data. The coefficients of variation for nucleotide sequence data are also
substantiallysmaller than those for restriction sites data. This is largely because thenumber of nucleotides assayed is larger in the former than in the latter. In general,
the average number of nucleotides assayed by restriction enzymes having r nucleotides
per DNA sequence is given by rrii. In the present case, iii = 42 for r = 6, and m = 7
for r = 4. Therefore, it becomes 6 X 42 + 4 X 7 = 280. This is approximately
onethird of the number of nucleotides
assayed (895) in nucleotide sequencing.
As is clear from figure 5, electiophoretic
data show the largest coefficients ofvariation of d (=D) among the four types of data used here. This is partly because
the number of loci examined (20) is small in the present case. However, even if one
uses 80 loci, the coefficients of variation would be only approximately
half of thosegiven in figure 5 and still considerably
higher than those for nucleotide
sequencesof mtDNA. Therefore, electrophoretic
data seem to be less informative
than mtDNA
data, unless a very large numberof loci are examined.
However, electrophoretic
data have one advantage
over mtDNA
data, namely, that they give an average
evolutionarychange of genes for many loci, whereas mtDNA represents a small
piece of DNA, which is maternallyinheritedwithout gene recombination.
Theevolutionary
tree reconstructedfrom a single piece of DNA without recombination
is subject to larger stochastic errors than that reconstructedfrom many independently
evolving genes.
In this paper, we reconstructed
the evolutionary
tree of the human and several
ape species using four different types of data. The topology of the tree reconstructed
is not the same for all types of data. The most reliable tree among the four thatwere reconstructed
seems to be that obtained from nucleotide sequence data. In this82

Nei, Stephens,

and Saitou

tree, the gibbon and the orangutan
separate from the human, chimpanzee,
andgorilla species significantly
earlier than the latter three species diverge. Among thelatter three species, the gorilla diverges from the human earlier than the chimpanzee
does, but the difference
between the two branching
points is not statisticallysignificant.
Therefore,
we cannot rule out the possibility
that the three species
diverged at nearly the same time (Sarich and Wilson 1967). The results derived
from protein data are essentially the same as those derived from nucleotide sequence
data. It should also be noted that this topology is in agreementwith that of thetrees reconstructedby both chromosomal
studies (Yunis and Prakash 1982) andDNA hybridization(Sibley and Ahlquist
1984). However, the topology of theparsimony tree obtained by Brown et al. (1982) is different from ours, even though
the same set of data was used. In a statisticalanalysis of the parsimonytree
reconstructedby Ferris et al. (198 I), Templeton(1983) concluded
that the topology
in figure 4 is significantly
betterthanthat
in figure
3. However, his conclusion
is not justified, since the parsimony
method he used introduces many statistical biases when it is applied to restriction
sites data (Nei andTajima 1984).
Acknowledgements
This study was supported
by research grants
Health and the National Science Foundation.

from

the National

Institutes

of

APPENDIX

Since it is not easy to develop a general algorithm to compute the variance ofthe difference between two branching
points that are hierarchicallyrelated, weherewith illustrate the computation
in a considerationof three typical cases. In
general, we consider the case in which the two species clusters A and B join atbranchingpoint b Aa and, within cluster A, the two subclusters Al and A2 join atbranching point bAI A*. We then evaluate the variance of the difference between thetwo branching points (6 = bAB - bAIAZ). For mathematicalconvenience,
we consider
the variance [V(&)] of 6, = 26 = dAB - dA 1A2. Obviously,
V(S) = V&-$4. In
general,v&) = WLB) + WA,,& - 2 COV(dAB,6 d
We already
COmpUtatiOnknow how to computeof Cov(dAB, dA,AZ) only.

V(dAB) and(Al)

V(dAIA2). So, we consider

Case I: Al and A2 each have one species, and B has s species. When
this is identical with the case represented in figure 1. In this case, we have,dAIA2 = sthes = 2,

Ward + Wad=-V(dAw)

2s

4

1

’

point between Al and A2, and baAl and baA2 refer to thewhere a is the branching
branch lengths between a and Al and between a and A2, respectively. Therefore,V&I) = V&d

V@Ad+ --j-*(A3)

Standard

Errors of Branching

Points83

Case 2: Al, A2, and B contain two, one, and s species, respectively.

i=l=s[V(ba,&

+

V&J

+

Wwd

+

V&,4

+

WaAl(2))+

Wa.d/(6~)= W(dm,d + 2Wap)l/6,(A4)

where c1 is the branching
point between Al and A2, and p is that between
species in A 1, that is, A 1( 1) and A l(2). Hence, V(&J is given byv(b) = Vk-bd +V(b,& can be obtainedby equation

2[V(&,A2) - Wap)l3( 10) or its equivalent

the two.

(A%

formula.

Case 3: Al and A2 both contain two species, and B has s species.
COV(dAB,dAIAZ) = COV[i (&l(l) + &+1(Z)+ d iA2( 1) + diA2(2))/(4S),(& 1(l)A2( 1)i=lV@Am) + Wap) + Way)
+ dA1(,)A2(2)+ &,(2)A2(1)+ dA,(2)A2(2))/41
=
8
4

9

646)

where y is the branching point between two species in A2, that is, A2( 1) and A2(2).
Therefore, we can compute V&J and V(S).
The above procedure can be extended to any type of branching pattern, but it
is quite complicatedif many species are involved. In this case, one may use theapproximate
f0mm.h COV(dAB dAIA2) = V(dA,,&/4.
This iS exact for CaSe 1 but
gives a slight overestimateof the true covariancefor the other cases. It can bederived by assuming that the variances of b,,, b,, etc., are equal to the variance ofbaA 1 or baA2. This provides a lower bound of the variance of 6, that is,,v

L

LITERATURE(6)=[V(dAd + V(‘hd21
4

.

W)

CITED

BROWN, W. M., E. M. PRAGER, A. WANG, and A. C. WILSON. 1982. MitochondrialDNA
sequences of primates: tempo and mode of evolution. J. Mol. Evol. l&225-239.
BRUCE, E. J., and F. J. AYALA. 1979. Phylogenetic
relationships
between man and the apes:
electrophoretic
evidence. Evolution 33: 1040-1056.
CHAKRABORTY,
R. 1977. Estimation of time of divergence from phylogenetic studies. Can J.
Genet. Cytol. 19:2 17-223.
DAYHOFF, M. 0. 1972. Atlas of protein sequence and structure.
Vol. 5. National Biomedical
Research Foundation,
Silver Spring, Md.
. 1973. Atlas of protein sequence and structure. Vol. 5. Suppl. 1. National Biomedical
Research Foundation,
Silver Spring, Md.
-.
1978. Atlas of protein sequence and structure. Vol. 5. Suppl. 3. National Biomedical
Research Foundation,
Silver Spring, Md.

84Nei, Stephens,

and Saitou

FARRIS, J. S. 1972. Estimating phylogenetic trees from distance matrices. Am. Nat. 106:645668.
-.
198 1. Distance data in phylogenetic analysis. Pp. 3-23 in V. FUNK and D. BROOK,
eds. Advances in cladistics. New York Botanical Garden, Bronx, New York.
FELSENSTEIN,J. 1984. Distance methods for inferring phylogenies: a justification. Evolution
38: 16-24.
FERRIS, S. D., A. C. WILSON, and W. M. BROWN. 1981. Evolutionary tree for apes andhumans based on cleavage maps of mitochondrial DNA. Proc. Natl. Acad. Sci. USA
78:2432-2436.
FITCH, W. M., and E. MARGOLIASH.1967a. A method for estimating the number of invariant
amino acid coding positions in a gene using cytochrome c as a model case. Biochem.
Genet. 1:65-7 1.
-.
1967b. Construction of phylogenetic trees. Science 155:279-284.
GOJOBORI, T., K. ISHII, and M. NEI. 1982. Estimation of average number of nucleotide
substitutions when the rate of substitution varies with nucleotide. J. Mol. Evol. l&4 14423.
GOODMAN, M., G. BRAUNITZER,A. STANGL, and B. SCHRANK. 1983. Evidence on humanorigins from haemoglobins of African apes. Nature 303:546-548.
GOTOH, O., J.-I. HAYASHI, H. YONEKAWA,and Y. TAGASHIRA. 1979. An improved method
for estimating sequence divergence between related DNAs from changes in restriction
endonuclease cleavage sites. J. Mol. Evol. 14:301-3 10.
HUISMAN, T. H. J., W. A. SCHROEDER,M. E. KEELING, N. GENGOZIAN, A. MILLER, A. R.
BRODIE, J. R. SHELTON, J. B. SHELTON, and G. APELL. 1973. Search for nonallelicstructural genes for y-chains of fetal hemoglobin in some primates. Biochem. Genet.
10:309-3 18.
JUKES, T. H., and C. R. CANTOR. 1969. Evolution of protein molecules. Pp. 21-132 in
H. N. MUNRO, ed. Mammalian protein metabolism. Academic Press, New York.
KAPLAN, N., and C. H. LANGLEY. 1979. A new estimate of sequence divergence of
mitochondrial DNA using restriction endonuclease mapping. J. Mol. Evol. 13:295-304.
KAPLAN, N., and K. RISKO. 198 1. An improved method for estimating sequence divergenceof DNA using restriction endonuclease mappings. J. Mol. Evol. 17: 156-162.
KIMURA, M. 1969. The rate of molecular evolution considered from the standpoint of
population genetics. Proc. Natl. Acad. Sci. USA 63: 118 1- 1188.
-.
1980. A simple method for estimating evolutionary rates of base substitutions
through comparative studies of nucleotide sequences. J. Mol. Evol. 16: 11 l-l 20.
-.
198 1. Estimation of evolutionary distances between homologous nucleotide sequences.
Proc. Natl. Acad. Sci. USA 78:454-458.
KIMURA, M., and T. OHTA. 1972. On the stochastic model for estimation of mutational
distance between homologous proteins. J. Mol. Evol. 2:87-90.
KING, M. C., and A. C. WILSON. 1975. Evolution at two levels in humans and chimpanzees.
Science 188: 107- 116.
KUMAZAKI, T., H. HORI, and S. OSAWA. 1983. Phylogeny of protozoa deduced from 5s
rRNA sequences. J. Mol. Evol. 19:411-419.
NEI, M. 197 1. Interspecific gene differences and evolutionary time estimated from electrophoretic data on protein identity. Am. Nat. 105:385-398.
-.
1972. Genetic distance between populations. Am. Nat. 106:283-292.
-.
1975. Molecular population genetics and evolution. North-Holland, New York.
-.
1978. Estimation of average heterozygosity and genetic distance from a small numberof individuals. Genetics 89:583-590.
NEI, M., and R. CHAKRABORTY. 1976. Empirical relationship between the number ofnucleotide substitutions and interspecific identity of amino acid sequences in some
proteins. J. Mol. Evol. 7:3 13-323.
NEI, M., and W.-H. LI. 1979. Mathematical model for studying genetic variation in terms ofrestriction endonucleases. Proc. Natl. Acad. Sci. USA 76:5269-5273.

StandardErrorsof BranchingPoints 85
NEI, M., and A. K. ROYCHOUDHURY.1974. Sampling

variances of heterozygosity and genetic
distance. Genetics 76:379-390.
NEI, M., and F. TAJIMA. 1983. Maximum likelihood estimation of the number of nucleotide
substitutions from restriction sites data. Genetics 105:207-2 17.
. 1984. Evolutionary change of restriction cleavage sites and phylogeny inference forhuman and apes. Unpublished manuscript.
NEI, M., F. TAJIMA, and Y. TATENO. 1983. Accuracy of estimated phylogenetic trees frommolecular data. II. Gene frequency data. J. Mol. Evol. 19: 153-170.
PEACOCK,D., and D. BOULTER. 1975. Use of amino acid sequence data in phylogeny andevaluation of methods using computer simulation. J. Mol. Biol. 95:5 13-527.
SARICH,V. M., and A. C. WILSON. 1967. Immunological time scale for homonoid evolution.
Science 158: 1200- 1204.
SELANDER,R. K., and B. R. LEVIN. 1980. Genetic diversity and structure in Escherichia coli
populations. Science 210:545-547.
SIBLEY, C. G., and J. E. AHLQUIST. 1984. The phylogeny of the hominoid primates, as
indicated by DNA-DNA hybridization. J. Mol. Evol. 20:2-16.
SNEATH, P. H. A., and R. R. SOKAL. 1973. Numerical taxonomy. Freeman, San Francisco.
TAJIMA, F., and M. NEI. 1984. Estimation of evolutionary distance between nucleotide
sequences. Mol. Biol. Evol. 1:269-285.
TAKAHATA, N., and M. KIMURA. 1981. A model of evolutionary base substitutions and its
application with special reference to rapid change of pseudogenes. Genetics 98:641-657.
TATENO, Y., M. NEI, and F. TAJIMA. 1982. Accuracy of estimated phylogenetic trees frommolecular data. I. Distantly related species. J. Mol. Evol. l&387-404.
TEMPLETON,A. R. 1983. Phylogenetic inference from restriction endonuclease cleavage site
maps with particular reference to the evolution of humans and apes. Evolution 37:22 l244.
UZZELL, T., and K. W. CORBIN. 197 1. Fitting discrete probability distributions to evolutionary
events. Science 172: 1089- 1096.
YUNIS, J. J., and 0. PRAKASH, 1982. The origin of man: a chromosomal pictorial legacy.
Science 215:1525-1530.
ROBERT K. SELANDER, reviewing

editorReceived

receivedMarch

29, 1984; revision

June 26, 1984.