Academia.eduAcademia.edu

Figure from:Interior-Branch and Bootstrap Tests of Phylogenetic Trees

See full PDFdownloadDownload figure
This integral can be evaluated numerically (see Drezner and Wesolowsky 1990). Computer simulations have shown that equation (22) gives a surprisingly good es- timate of the probability of obtaining the true tree by the NJ method as long as 7 > 100 even if the distribution of evolutionary distances deviates from normality to some extent, as in the case of nucleotide sequences. We are now in a position to derive the expression for Pg. For each bootstrap pseudosample we can com- pute d*, D* and D3, where the asterisk indicates that the quantity is computed from a pseudosample. As in the case of the original data set, the correct tree is re- covered from a pseudosample only when Df and DF are both positive. The joint distribution of Df and D} for the pseudosamples obtained from a particular data set can be approximated by a bivariate normal distri- bution with mean vector (D,, Dy) and variance-covari- ance matrix V. Here, D, and D> are computed by equa- tions (18) and (19) from the original set of data, and Vv is the estimate of matrix V in equation (20). Therefore, Pz is equal to the proportion of cases where both D¥ and D¥ are positive and can be computed by

Figure 4 This integral can be evaluated numerically (see Drezner and Wesolowsky 1990). Computer simulations have shown that equation (22) gives a surprisingly good es- timate of the probability of obtaining the true tree by the NJ method as long as 7 > 100 even if the distribution of evolutionary distances deviates from normality to some extent, as in the case of nucleotide sequences. We are now in a position to derive the expression for Pg. For each bootstrap pseudosample we can com- pute d*, D* and D3, where the asterisk indicates that the quantity is computed from a pseudosample. As in the case of the original data set, the correct tree is re- covered from a pseudosample only when Df and DF are both positive. The joint distribution of Df and D} for the pseudosamples obtained from a particular data set can be approximated by a bivariate normal distri- bution with mean vector (D,, Dy) and variance-covari- ance matrix V. Here, D, and D> are computed by equa- tions (18) and (19) from the original set of data, and Vv is the estimate of matrix V in equation (20). Therefore, Pz is equal to the proportion of cases where both D¥ and D¥ are positive and can be computed by

Related Figures (11)

Fic. !.—A, Hypothetical true tree for six sequences. B and C, Incorrect trees for the same sequences. D, Three model trees for six sequences used for generating the relationship between P, and Pz in fig. 68. When bg = 0, trees (i), (ii), and (iii) behave as six-, five-, and four-sequence star trees, respectively. Bootstrap Test
Fic. !.—A, Hypothetical true tree for six sequences. B and C, Incorrect trees for the same sequences. D, Three model trees for six sequences used for generating the relationship between P, and Pz in fig. 68. When bg = 0, trees (i), (ii), and (iii) behave as six-, five-, and four-sequence star trees, respectively. Bootstrap Test
Fic. 2.—A, Hypothetical true tree for four sequences. B and ¢ Incorrect trees for the same sequences.
Fic. 2.—A, Hypothetical true tree for four sequences. B and ¢ Incorrect trees for the same sequences.
Analytical Formulation for the Case of Normally Distributed Evolutionary Distances
Analytical Formulation for the Case of Normally Distributed Evolutionary Distances
Fic. 3.—Comparison of the average values of Pe and Pz for four sequence model trees for the cases of 6; > 0. We used the followin two sets of the expected exterior branch lengths for the model tree i1 fig. 2A: b, = 0.07, b, = 0.075, b3 = 0.065, b, = 0.07 (short brancl trees, solid circles), and b, = 0.3, b: = 0.3, b3 = 0.3, b4 = 0.3 (lon; branch trees, open circles). The expected length of the interior branch b;, was changed from 0 to 0.7 by incrementing it by 0.005 for eac! model tree of short branches and by incrementing it by 0.03 for eac! model tree of long branches. The sample size (number of nucleotides was equal to 100. Each point was obtained by averaging results o 3,000 replications. A, Normally distributed distance data; B, simulate nucleotide sequence data. The relationship between Pc and Pg depends or the expected lengths of the exterior branches of the model tree. The points (Pc, P,) tend to be “upward” (P:<P,) when the model tree has short exterior branches Let us consider the relationship of the averages ( P; and Ps) of 3,000 Pc’s and P,’s for the cases of bs > 0 for the model tree of four sequences in figure 2A when bs gradually increases from 0 (fig. 3). Po and P, are quite close to each other, though the individual values of P- and P, obtained for each data set are not necessarily close (data not shown). Figure 3.4 and B represent the cases of normally distributed distance data and simulated nucleotide sequence data, respectively. Although these two figures are slightly different for small values of P, and P, (small values of bs), the relationships between the two estimates are very similar for the two types of data.
Fic. 3.—Comparison of the average values of Pe and Pz for four sequence model trees for the cases of 6; > 0. We used the followin two sets of the expected exterior branch lengths for the model tree i1 fig. 2A: b, = 0.07, b, = 0.075, b3 = 0.065, b, = 0.07 (short brancl trees, solid circles), and b, = 0.3, b: = 0.3, b3 = 0.3, b4 = 0.3 (lon; branch trees, open circles). The expected length of the interior branch b;, was changed from 0 to 0.7 by incrementing it by 0.005 for eac! model tree of short branches and by incrementing it by 0.03 for eac! model tree of long branches. The sample size (number of nucleotides was equal to 100. Each point was obtained by averaging results o 3,000 replications. A, Normally distributed distance data; B, simulate nucleotide sequence data. The relationship between Pc and Pg depends or the expected lengths of the exterior branches of the model tree. The points (Pc, P,) tend to be “upward” (P:<P,) when the model tree has short exterior branches Let us consider the relationship of the averages ( P; and Ps) of 3,000 Pc’s and P,’s for the cases of bs > 0 for the model tree of four sequences in figure 2A when bs gradually increases from 0 (fig. 3). Po and P, are quite close to each other, though the individual values of P- and P, obtained for each data set are not necessarily close (data not shown). Figure 3.4 and B represent the cases of normally distributed distance data and simulated nucleotide sequence data, respectively. Although these two figures are slightly different for small values of P, and P, (small values of bs), the relationships between the two estimates are very similar for the two types of data.
Fic. 4.—Frequency distributions of Pg (A) and Pc (B) for normally distributed distance data obtained in 10,000 replicate simulations. The frequencies of Py and Pc for 5; > 0 and bs < 0 are shown separately (corresponding bars are stacked one on top of the other). Here we used the model tree in fig. 24 with b, = b, = b; = b, = 0.3, and b; = 0. The number of nucleotides sampled (”) was equal to 100.
Fic. 4.—Frequency distributions of Pg (A) and Pc (B) for normally distributed distance data obtained in 10,000 replicate simulations. The frequencies of Py and Pc for 5; > 0 and bs < 0 are shown separately (corresponding bars are stacked one on top of the other). Here we used the model tree in fig. 24 with b, = b, = b; = b, = 0.3, and b; = 0. The number of nucleotides sampled (”) was equal to 100.
Fic. 5.—Comparison of Pc and Ps for interior branch bg for the case of six-sequence model trees (fig. LA) in simulation for nucleoti sequence data. 4, Simulations for model trees with the expected exterior branch lengths 5; = b, = hb; = bg = hs = be = 0.07 and b; = by = (short branch trees, solid circles) and b, = 0.31, Bb, = 0.3, b3 = 0.29, by = 0.3, bs = 0.28, bg = 0.3, and b; = by = 0 (long branch trees, op circles). The value of bg was changed from 0 to 0.6 by incrementing it by 0.005 for each model tree of short branches and by incrementing by 0.015 for each model tree of long branches. Pc and Pz, were obtained by averaging all Pc’s and P,’s (given bg > 0) for each bg value. Ea point was obtained by averaging the results for 3,000 replications. The number of nucleotides sampled (7) was equal to 100. B, Simulations f the model trees with the exterior branches b, = b, = b; = by = bs = be = 0.07 and various values of the interior branch lengths (solid circl« b, = by = 0; solid squares: b; = 0 and by = 0.09; solid triangles: b; = 0.08, by = 0.09). The value of bg in all three cases was increased from to 0.25 by incrementing it by 0.005 for each model tree. Each point was obtained by averaging results of 3,000 replications; n = 100. eee ee ne ae EIS ERE! EIDE ABE EAI For the first tree (b7=b,=0; solid circles in fig. 5 B), we obtain an already familiar relationship in figure 5A (solid circles). In the second case (b;=0, by>0; solid squares in fig. 5B), Pg’s increase. In the third case (b7>0 and bo>0; solid triangles in fig. 5B), we observe a re- lationship of P¢ and P, that is virtually identical with So far we have considered the interior-branch anc Ee ee ee eT Ee EY Ce : a ee OT ME Te ee men Cn nee mms Ml cerry
Fic. 5.—Comparison of Pc and Ps for interior branch bg for the case of six-sequence model trees (fig. LA) in simulation for nucleoti sequence data. 4, Simulations for model trees with the expected exterior branch lengths 5; = b, = hb; = bg = hs = be = 0.07 and b; = by = (short branch trees, solid circles) and b, = 0.31, Bb, = 0.3, b3 = 0.29, by = 0.3, bs = 0.28, bg = 0.3, and b; = by = 0 (long branch trees, op circles). The value of bg was changed from 0 to 0.6 by incrementing it by 0.005 for each model tree of short branches and by incrementing by 0.015 for each model tree of long branches. Pc and Pz, were obtained by averaging all Pc’s and P,’s (given bg > 0) for each bg value. Ea point was obtained by averaging the results for 3,000 replications. The number of nucleotides sampled (7) was equal to 100. B, Simulations f the model trees with the exterior branches b, = b, = b; = by = bs = be = 0.07 and various values of the interior branch lengths (solid circl« b, = by = 0; solid squares: b; = 0 and by = 0.09; solid triangles: b; = 0.08, by = 0.09). The value of bg in all three cases was increased from to 0.25 by incrementing it by 0.005 for each model tree. Each point was obtained by averaging results of 3,000 replications; n = 100. eee ee ne ae EIS ERE! EIDE ABE EAI For the first tree (b7=b,=0; solid circles in fig. 5 B), we obtain an already familiar relationship in figure 5A (solid circles). In the second case (b;=0, by>0; solid squares in fig. 5B), Pg’s increase. In the third case (b7>0 and bo>0; solid triangles in fig. 5B), we observe a re- lationship of P¢ and P, that is virtually identical with So far we have considered the interior-branch anc Ee ee ee eT Ee EY Ce : a ee OT ME Te ee men Cn nee mms Ml cerry
FiG. 6.—Frequency distributions of the individual values of Ps(A) and P.(B) for the branch bg of the model tree in fig. 1A for nucleotide sequence data. The frequencies of P¢ and Pg corresponding to bg > 0 and by < 0 are shown separately (corresponding bars are stacked one on top of the other). The expected branch lengths of the tree were b, = 0.31, b. = 0.3, b; = 0.29, bg = 0.3, bs = 0.28, bs = 0.3, and b, = by = by = 0. The histogram was obtained for data from 30,000 replicate simulations; n = 100.
FiG. 6.—Frequency distributions of the individual values of Ps(A) and P.(B) for the branch bg of the model tree in fig. 1A for nucleotide sequence data. The frequencies of P¢ and Pg corresponding to bg > 0 and by < 0 are shown separately (corresponding bars are stacked one on top of the other). The expected branch lengths of the tree were b, = 0.31, b. = 0.3, b; = 0.29, bg = 0.3, bs = 0.28, bs = 0.3, and b, = by = by = 0. The histogram was obtained for data from 30,000 replicate simulations; n = 100.
FiG. 7.—A, Distribution of Z = hs/s(hs) obtained for the model tree in fig. 24 by considering only data sets that generated the NJ tree identical with tree A in fig. 2. The histogram was obtained for data from 100,000 replications with n = 100. The distribution represented by a smooth curve is the gamma distribution (equation [25]) with parameters a = 3.10 and b = 2.97. The branch lengths of the model tree were b, = b, = b; = by = 0.3 and b; = 0. “Estimates” of evolutionary distances were computed using a multivariate normal distribution. B, Various conditional distributions of Z for four-species model trees. Solid line, b; = 6, = 63 = b4 = 0.3 and bs = 0. Dashed line, b,; = 53 = 0.6, b2 = bs = 0.02, and b; = 0. Dotted line, b; = b2 = 0.6, b3 = bg = 0.02, and bs = 0. Here the solid line represents the worst-case dis- tribution of Z. The number of replications was 100,000; n = 100. C, Conditional distributions of Z = bs/s(b) for six-sequence model trees (see fig. LA). The distributions of Z for a four-sequence-like tree (solid line, by = by = 0.15), a five-sequence-like tree (dashed line, by = 0.15 and by = 0), and a six-sequence-like tree (dotted line, b; = by = 0). The expected lengths of exterior branches for six-sequence model trees were b, = 0.31, by = 0.3, b3 = 0.29, by = 0.3, bs = 0.28, bg = 0.3, and the interior branch length bg = 0. The number of replications was 1,000,000; n = 100.
FiG. 7.—A, Distribution of Z = hs/s(hs) obtained for the model tree in fig. 24 by considering only data sets that generated the NJ tree identical with tree A in fig. 2. The histogram was obtained for data from 100,000 replications with n = 100. The distribution represented by a smooth curve is the gamma distribution (equation [25]) with parameters a = 3.10 and b = 2.97. The branch lengths of the model tree were b, = b, = b; = by = 0.3 and b; = 0. “Estimates” of evolutionary distances were computed using a multivariate normal distribution. B, Various conditional distributions of Z for four-species model trees. Solid line, b; = 6, = 63 = b4 = 0.3 and bs = 0. Dashed line, b,; = 53 = 0.6, b2 = bs = 0.02, and b; = 0. Dotted line, b; = b2 = 0.6, b3 = bg = 0.02, and bs = 0. Here the solid line represents the worst-case dis- tribution of Z. The number of replications was 100,000; n = 100. C, Conditional distributions of Z = bs/s(b) for six-sequence model trees (see fig. LA). The distributions of Z for a four-sequence-like tree (solid line, by = by = 0.15), a five-sequence-like tree (dashed line, by = 0.15 and by = 0), and a six-sequence-like tree (dotted line, b; = by = 0). The expected lengths of exterior branches for six-sequence model trees were b, = 0.31, by = 0.3, b3 = 0.29, by = 0.3, bs = 0.28, bg = 0.3, and the interior branch length bg = 0. The number of replications was 1,000,000; n = 100.
Fic. 8.—Frequency distributions of the P¢ and Ps values for an estimated tree of which the topology is identical with that of the model tree. A, Distributions of Pc; and P's for normally distributed distance data obtained from 30,000 replicate simulations. We used the model tree shown in fig. 24 with b, = b) = b; = b, = 0.3, and bs = 0. B, Distributions of P¢ and P's for branch bg of the model tree in fig. 14. These results were obtained from simulated sequence data with 1,000,000 replications. The expected branch lengths of the tree were 5; = 0.31, 2 = 0.3, bs = 0.29, by = 0.3, bs = 0.28, bg = 0.3, and b, = bg = by = 0, and n = 100. signal” occurs when the partition under consideration does not exist in the true tree (incorrect partition). In the case of star topologies with no phylogenetic
Fic. 8.—Frequency distributions of the P¢ and Ps values for an estimated tree of which the topology is identical with that of the model tree. A, Distributions of Pc; and P's for normally distributed distance data obtained from 30,000 replicate simulations. We used the model tree shown in fig. 24 with b, = b) = b; = b, = 0.3, and bs = 0. B, Distributions of P¢ and P's for branch bg of the model tree in fig. 14. These results were obtained from simulated sequence data with 1,000,000 replications. The expected branch lengths of the tree were 5; = 0.31, 2 = 0.3, bs = 0.29, by = 0.3, bs = 0.28, bg = 0.3, and b, = bg = by = 0, and n = 100. signal” occurs when the partition under consideration does not exist in the true tree (incorrect partition). In the case of star topologies with no phylogenetic
Fic. 9.—Relationships between p and the expected value of Z for four-sequence model trees with short (solid circles) and long (open circles) exterior branches. The actual values of the expected exterior branch lengths are the same as in fig. 3. The values of p and Z increase as the length of interior branch (5s) of the model tree in fig. 24 increases. ferences in the relationships of P- and P, between these two types of model trees in figure 5. In the case of a star tree (bs = 0) equations (Ala- c) reduce to
Fic. 9.—Relationships between p and the expected value of Z for four-sequence model trees with short (solid circles) and long (open circles) exterior branches. The actual values of the expected exterior branch lengths are the same as in fig. 3. The values of p and Z increase as the length of interior branch (5s) of the model tree in fig. 24 increases. ferences in the relationships of P- and P, between these two types of model trees in figure 5. In the case of a star tree (bs = 0) equations (Ala- c) reduce to