On the Delta Q-Test of Templeton’Naruya Saitou
Center for Demographic and Population Genetics, The University of Texas
Health Science Center at Houston

Using a modified form (delta Q [AQ]-test) of Pielou’s (1979) Q-test, Templeton(1985) reanalyzed Sibley and Ahlquist’s (1984) DNA-DNA hybridization data forhominoid species. He concluded that the phylogeny (fig. 1A) supported by Sibley andAhlquist is not significantly better than the phylogeny (fig. 1B) favored by Templeton(1983). However, Templeton’s (1985) AQ-test has several statistical problems, and hisconclusion does not seem to be justified.
In brief, Pielou’s Q-test and Templeton’s AQ-test are as follows: Consider thedistance matrix in table 1A, where dij (i > j) denotes the evolutionary distance between
species i and j. In the Q-test, dii is compared with dkl (k > I) with the restrictioni < k, and a random variable X, which takes 1 when dij < dkl, 0 when do > dkl, and0.5 when do = dkl, is considered. The Q-statistic is the sum of x for all possible comparisons of do and dkl. If we assume that phylogeny A in figure 1 is correct and thatdij increases in proportion to evolutionary time without error, we will have the inequalitydz,<d3j<d4j<dsj(j=1, a* 4).InthiscaseQ=9+(2X7)+(3X4)= 35 for distance matrix A of table 1. This is the maximum value Q can take for thefive-species case. In practice, dij may be subject to random errors, and, furthermore,
the phylogeny A of figure 1 may not be true. Pielou’s ( 1979) original test is for examining
whether or not there is a hierarchical structure of species (strata in her original problem),and the null hypothesis is that there is no hierarchical structure, i.e., that all species
diverged at the same time. To test this null hypothesis, Pielou’s Q-test is justified. In
evolutionary studies, however, there is almost always some hierarchical structure. For
this reason, Templeton (1985) introduced the AQ-test. In this test the Q-statistic is
computed for two different phylogenies (e.g., A and B in fig. l), and the difference inQ between the two phylogenies is called AQ. Templeton obtained the probabilitydistribution of AQ and used this distribution for testing the statistical significance ofAQ. However, this distribution is based on the assumption that there is no hierarchicalstructure.
As mentioned earlier, the Q-value for matrix A of table 1 is 35. By contrast, theQ-value for matrix B, which corresponds to phylogeny B of figure 1 becomes 29-3 1,depending on the differences among ddl, d42, and d43 (see table 1B). The distributionof AQ (Templeton’s table 2) indicates that AQ = 6 is required to achieve statisticalsignificance at the 5% level, but other, smaller values are nonsignificant. If we apply
the above test to Sibley and Ahlquist’s (1984) data, AQ becomes 4. Therefore, thistest suggests that phylogeny A is not favored against phylogeny B.
However, we note that AQ depends only on the ranks among d4j)s in the present4) maintained, AQ becomescase. With the inequality d2j < dsj < dbj < dsj (j = 1,
6 only when db3is smaller than both ddl and dd2. Even when the strict rate constancy
is assumed, where d4, = dd2 = d43, AQ = 5 (see table 1). Obviously, d4, , d42, and dJ3
may vary by chance effects even if species 4 is remotely related to species 1, 2, and 3,and AQ = 4 may be obtained with a high probability. It is therefore clear that the AQtest is inadequate for testing topological differences.
llll

1. Key words: delta Q test, DNA-DNA hybridization data, hominoid evolution, molecular phylogeny.
Address for correspondence and reprints: Mr. Naruya Saitou, Center for Demographic and Population Genetics, The University of Texas Health Science Center at Houston, P.O. Box 20334, Houston,Texas 77225.
Mol. Biol. Evol. 3(3):282-2&I. 1986.
0 1986 by The University of Chicago. All rights reserved.
0131-4038/86/0303-3303%02.00

282

Delta Q-Test
Phylogeny AL

Phylogeny B

A1

common chimpanzee

2

pygmy

3

human

4

gorilla5orangutan

1

common chimpanzee

2

pygmy

4

gorilla

3

human

5

orangutan-3Phylogeny C

u283

chimpanzee

chimpanzee

human
4

gorilla1common chimpanzee

2

pygmy

5

orangutan

chimpanzee

1' chimpanzee (made-up)

Phylogeny A'U

1common chimpanzee

2pygmy

3

human

4

gorilla5orangutan

chimpanzee

FIG. l.-Three probable phylogenies (A, B, and C) for five hominoid
phylogeny A in which one chimpanzee species (made-up) is included.

and a derivative (N) of

Of course, one can still argue that phylogeny A is better than phylogeny B when
AQ becomes 6. When we consider the branching orders of humans, chimpanzees, andgorillas, however, there is another possible phylogeny (fig. 1C). This phylogeny cannot
be compared with phylogeny A or B in terms of AQ (Templeton 1985). Therefore,
even if phylogeny A is judged to be superior to phylogeny B with AQ = 6, it is notnecessarily the best tree; the true tree could be phylogeny C.
Another problem of the AQ-test is that the power of the test depends on thenumber of OTUs (operational taxonomic units) compared. Templeton ( 1985) acknowledged this point but considered it as a good support for the AQ-test. Supposethat we include the third species of chimpanzee (made-up) and obtain phylogeny A’of figure 1. In this case, if we switch species 3 and 4, AQ becomes 9- 12, again dependingon the values of d4, ( =d4p), d42, and d43. The smallest AQ (=9) is still not significant
at the 5% level, but the probability of AQ r 9 is now 0.0643. As the number of OTUs
increases from the chimpanzee lineage gradually, even the smallest AQ becomes statistically significant. This peculiar property has occurred because the probability distribution of AQ is obtained under the assumption that no hierarchical structure ofOTUs exists. This assumption is certainly unrealistic in most statistical studies ofphylogenetic trees.

284

SaitouTable 1
Various AQ-Values for Two Distance Matrices
B. Reordered Matrix(Species 3 and 4 Exchanged)

A. Correctly Ordered Matrix
d21

2345d::
dd51

d42
32
d52

d43
d53

Species

123d21
d41

d42

43

d31

d32

d54

5d51

d52

d43
d54

d53

4Species12432(Q’ = 29-31)(Q = 35)
Q’AQ

31 .......
30.5 ......
30 .......
29.5 ......
29 .......

44.5
55.5
6Conditions
and& > &d4, < d42= d43& < & < &d4, > d42= d43and& < &ororor& > && < & = && < & < && > 64, = && < &or& = & = &In my view, Templeton’s (1985) criticism of Sibley and Ahlquist’s (1984) use ofthe t-test is not really justified. Since Sibley and Ahlquist considered only experimental
errors, all the TsoH values are independent. Therefore, I do not think that there isanything wrong with the f-test. The pooling of “Gorilla X Pan, Homo” comparisons
as done by Sibley and Ahlquist (1984) also seems to be valid, since they wanted toshow that the human-chimpanzee clustering obtained by the distance Wagner method
was significant. Templeton (1985) used the t-test for Gorilla-Homo vs. Gorilla-P. trogZodytes comparisons and showed that the former is significantly larger than the latter.
This may suggest that Sibley and Ahlquist’s ( 1984) DNA-DNA hybridization data are
not strictly rate constant. However, this result alone should not, despite Templeton’s(1985) suggestion to the contrary, be taken as the evidence of the existence of inconsistency in Sibley and Ahlquist’s data.
At any rate, if we can apply this t-test to DNA-DNA hybridization data, why is
it necessary to use Templeton’s AQ-test, which has almost no statistical power?
Acknowledgments

I thank Dr. M. Nei for encouraging discussions and detailed suggestions. I also
thank Drs. P. E. Smouse, W.-H. Li, P. Pamilo, J. C. Stephens, and M. Ruvolo for
discussions and comments on the manuscript.
LITERATURECITEDPIELOU,E. C. 1979. Interpretationof paleoecologicalsimilarity

matrices.

Paleobiology

543%

443.

SIBLEY,C. G., and J. E. AHLQUIST.1984. The phylogeny of the hominoid primates. J. Mol.
Evol. 20:2- 15.
TEMPLETON,
A. R. 1983. Phylogenetic inference from restriction endonuclease cleavage site
map with particular reference to the evolution of humans and the apes. Evolution 37:22 l244.
1985. The phylogeny of the hominoid primates: a statistical analysis of the DNA-DNA
hybridization data. Mol. Biol. Evol. 2:420-433.
WALTER M. FITCH, reviewing editor

Received October 4, 1985; revision received October 28, 1985.