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Model Misspecification due to Site Specific Rate Heterogeneity: how is tree inference affected? Stephen Crotty School of Mathematical Sciences, University of Adelaide October, 2013 Stephen Crotty (School of Math. Sci.) Model Misspecification

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Model Misspecification due to Site Specific Rate Heterogeneity: how is tree inference affected?

Stephen Crotty

School of Mathematical Sciences, University of Adelaide

October, 2013

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 1 / 21

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What is Site Specific Rate Heterogeneity (SSRH)?

A B C D 0.1 0.4 0.4 0.4 0.4

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 2 / 21

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What is Site Specific Rate Heterogeneity (SSRH)?

A B C D 0.1 0.4 0.4 0.4 0.4 The model contains 3 site types: Invariable sites

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 2 / 21

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What is Site Specific Rate Heterogeneity (SSRH)?

A B C D 0.1 0.4 0.4 0.4 0.4 The model contains 3 site types: Invariable sites Variable sites

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 2 / 21

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What is Site Specific Rate Heterogeneity (SSRH)?

A B C D 0.1 0.4 0.3 0.3 0.1 0.1 0.4 The model contains 3 site types: Invariable sites Variable sites Switching sites

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 2 / 21

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Why should we care about SSRH?

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

Tasmanian Pygmy Possum Tasmanian Native Hen

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

Tasmanian Pygmy Possum Tasmanian Native Hen

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

Tasmanian Devil

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

What’s up Doc? Devil Facial Tumour Syndrome

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

What’s up Doc? Devil Facial Tumour Syndrome

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Why should we care about SSRH?

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 3 / 21

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Experimental Procedure

1 Data was simulated using the program LineageSpecificSeqgen1

1Source: L. Shavit Grievink, D. Penny, M. D. Hendy, and B. R. Holland.

BMC Evolutionary Biology, 8:317, 2008.

2http://evolution.genetics.washington.edu/phylip/ Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 4 / 21

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Experimental Procedure

1 Data was simulated using the program LineageSpecificSeqgen1 2 The Phylip2 software package was used to perform tree inference

using the maximum parsimony (MP), neighbour joining (NJ) and maximum likelihood (ML) methods.

1Source: L. Shavit Grievink, D. Penny, M. D. Hendy, and B. R. Holland.

BMC Evolutionary Biology, 8:317, 2008.

2http://evolution.genetics.washington.edu/phylip/ Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 4 / 21

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Experimental Procedure

1 Data was simulated using the program LineageSpecificSeqgen1 2 The Phylip2 software package was used to perform tree inference

using the maximum parsimony (MP), neighbour joining (NJ) and maximum likelihood (ML) methods.

3 A theoretical analysis of each method was carried out in an effort to

understand their performance.

1Source: L. Shavit Grievink, D. Penny, M. D. Hendy, and B. R. Holland.

BMC Evolutionary Biology, 8:317, 2008.

2http://evolution.genetics.washington.edu/phylip/ Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 4 / 21

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Simulation Parameters

A B C D 0.1 0.4 0.3 0.3 0.1 0.1 0.4

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 5 / 21

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Simulation Parameters

A B C D 0.1 0.4 0.3 0.3 0.1 0.1 0.4 pinv = 80% pvar = 20% pswitch = 0, 1, 2, . . . , 100%

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 5 / 21

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Simulation Parameters

A B C D 0.1 0.4 0.3 0.3 0.1 0.1 0.4 pinv = 80% pvar = 20% pswitch = 0, 1, 2, . . . , 100% 100000 base pairs Jukes Cantor substitution model 100 replications

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 5 / 21

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Maximum Parsimony

25 50 75 100 25 50 75 100 pswitch Correct Tree Inferred %

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 6 / 21

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Maximum Parsimony

Site pattern analysis predicts the asymptotic failure point of MP to be 26.56%.

25 50 75 100 25 50 75 100 pswitch Correct Tree Inferred %

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 6 / 21

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Neighbour Joining

25 50 75 100 25 50 75 100 pswitch Correct Tree Inferred % MP NJ

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 7 / 21

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Neighbour Joining - why the recovery?

The neighbour joining algorithm r = number of taxa. Dij = JC distance between taxa i and j. Qij = (r − 2)Dij − r

k=1 Dik − r k=1 Djk

Q is the matrix used by the NJ algorithm: the pair of taxa with the smallest Qij are joined together and the process is repeated.

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 8 / 21

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The Q matrix for a 4-taxa tree

QAB = (4 − 2)DAB −

  • k∈{B,C,D}

DAk −

  • k∈{A,C,D}

DBk = −(DAC + DAD + DBC + DBD) Similarly, QAD = −(DAB + DAC + DBD + DCD) and, QAC = −(DAB + DAD + DBC + DCD)

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 9 / 21

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Digression - what tree might we infer?

AB|CD

A B C D

min(QAB, QAD, QAC ) = QAB = ⇒

✔

AD|BC

A D B C

min(QAB, QAD, QAC ) = QAD = ⇒

✘

AC|BD

A C B D

min(QAB, QAD, QAC ) = QAC = ⇒

✘

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 10 / 21

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Digression - what tree might we infer?

AB|CD

A B C D

QAB < QAD(QAB, QAD, = ⇒

✔

AD|BC

A D B C

QAD < QAB(QAB, QAD, = ⇒

✘

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 10 / 21

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The Q matrix for a 4-taxa tree

The correct tree (AB|CD) will be inferred given the condition: QAB < QAD = ⇒ < QAD − QAB = ⇒ < DAD + DBC − DAB − DCD We now define C = DAD + DBC − DAB − DCD so that the correct tree will be inferred when C > 0.

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 11 / 21

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Deriving the expected value of C

T = the tree topology Pij = the proportion of differing sites between taxa i and j E[Pij] = f (pswitch, T) E[Dij] = − 3

4ln(1 − 4 3E[Pij])

E[C] = E[DAD] + E[DBC] − E[DAB] − E[DCD]

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 12 / 21

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Expected value of C

−0.01 0.00 0.01 25 50 75 100 pswitch NJ critical quantity

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 13 / 21

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Neighbour Joining

25 50 75 100 25 50 75 100 pswitch Correct Tree Inferred %

E[C] > 0 E[C] < 0 E[C] > 0

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 14 / 21

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Maximum Likelihood

25 50 75 100 25 50 75 100 pswitch Correct Tree Inferred % MP NJ ML

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 15 / 21

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Why is this important?

Traditional methods of phylogenetic inference may be compromised by SSRH.

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 16 / 21

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Why is this important?

Traditional methods of phylogenetic inference may be compromised by SSRH. Diagnostic tools need to be developed to help identify the presence and extent of SSRH in sequence data.

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 16 / 21

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Why is this important?

Traditional methods of phylogenetic inference may be compromised by SSRH. Diagnostic tools need to be developed to help identify the presence and extent of SSRH in sequence data. Data driven model checking will be the focus of my PhD going forward.

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 16 / 21

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Acknowledgements

I would like to thank my supervisory team for their input and guidance:

  • Prof. Nigel Bean - University of Adelaide

Dr Lars Jermiin - CSIRO Dr Barbara Holland - University of Tasmania Dr Jono Tuke - University of Adelaide

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 17 / 21

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That’s all folks!

Questions?

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 18 / 21

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Why is the AC|BD tree never inferred? I’m glad you asked!

QAB − QAC = DAB + DCD − DAC − DBD

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 19 / 21

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Why is the AC|BD tree never inferred? I’m glad you asked!

QAB − QAC = DAB + DCD − DAC − DBD = + − − A B C D A B C D A B C D A B C D

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 19 / 21

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Why is the AC|BD tree never inferred? I’m glad you asked!

QAB − QAC = DAB + DCD − DAC − DBD = + − − A B C D A B C D A B C D A B C D

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 19 / 21

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Why is the AC|BD tree never inferred? I’m glad you asked!

QAB − QAC = DAB + DCD − DAC − DBD = + − − A B C D A B C D A B C D A B C D

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 19 / 21

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Why is the AC|BD tree never inferred? I’m glad you asked!

QAB − QAC = DAB + DCD − DAC − DBD = + − − A B C D A B C D A B C D A B C D = − − ≤

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 19 / 21

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How was the MP crash point derived? I’m glad you asked!

Site Correct Incorrect Pattern Tree Tree xxxx xxxy 1 1 xxyx 1 1 xyxx 1 1 yxxx 1 1 xxyy 1 2 xyyx 2 1 xyxy 2 2 xxyz 2 2 xyzx 2 2 xyxz 2 2 yxxz 2 2 yxzx 2 2 yzxx 2 2 wxyz 3 3

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 20 / 21

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How was the MP crash point derived? I’m glad you asked!

Site Correct Incorrect Pattern Tree Tree xxxx xxxy 1 1 xxyx 1 1 xyxx 1 1 yxxx 1 1 xxyy 1 2 xyyx 2 1 xyxy 2 2 xxyz 2 2 xyzx 2 2 xyxz 2 2 yxxz 2 2 yxzx 2 2 yzxx 2 2 wxyz 3 3

A B C D Consider site pattern xxyy:

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 20 / 21

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How was the MP crash point derived? I’m glad you asked!

Site Correct Incorrect Pattern Tree Tree xxxx xxxy 1 1 xxyx 1 1 xyxx 1 1 yxxx 1 1 xxyy 1 2 xyyx 2 1 xyxy 2 2 xxyz 2 2 xyzx 2 2 xyxz 2 2 yxxz 2 2 yxzx 2 2 yzxx 2 2 wxyz 3 3

A B C D x x y y Consider site pattern xxyy:

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 20 / 21

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How was the MP crash point derived? I’m glad you asked!

Site Correct Incorrect Pattern Tree Tree xxxx xxxy 1 1 xxyx 1 1 xyxx 1 1 yxxx 1 1 xxyy 1 2 xyyx 2 1 xyxy 2 2 xxyz 2 2 xyzx 2 2 xyxz 2 2 yxxz 2 2 yxzx 2 2 yzxx 2 2 wxyz 3 3

A D B C x y x y xxyy = ⇒ Consider site pattern xxyy:

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 20 / 21

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How was the MP crash point derived? I’m glad you asked!

Site Correct Incorrect Pattern Tree Tree xxxx xxxy 1 1 xxyx 1 1 xyxx 1 1 yxxx 1 1 xxyy 1 2 xyyx 2 1 xyxy 2 2 xxyz 2 2 xyzx 2 2 xyxz 2 2 yxxz 2 2 yxzx 2 2 yzxx 2 2 wxyz 3 3

P(xxyy) = f (T) P(xyyx) = g(pswitch, T)

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 20 / 21

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How was the MP crash point derived? I’m glad you asked!

Site Correct Incorrect Pattern Tree Tree xxxx xxxy 1 1 xxyx 1 1 xyxx 1 1 yxxx 1 1 xxyy 1 2 xyyx 2 1 xyxy 2 2 xxyz 2 2 xyzx 2 2 xyxz 2 2 yxxz 2 2 yxzx 2 2 yzxx 2 2 wxyz 3 3

P(xxyy) = f (T) P(xyyx) = g(pswitch, T) The failure point of MP is given by finding pswitch such that: P(xxyy) = P(xyyx)

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 20 / 21

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How was the MP crash point derived? I’m glad you asked!

0.010 0.015 0.020 25 50 75 100 pswitch Proportion of Sites

P(xxyy) P(xyyx)

Stephen Crotty (School of Math. Sci.) Model Misspecification due to SSRH October, 2013 21 / 21