The `Out-of-Africa' hypothesispmcc/seminars/SASA/outofafr.pdf · The ‘Out-of-Africa’ hypothesis Peter McCullagh Department of Statistics University of Chicago South Africa, November

The ‘Out-of-Africa’ hypothesis

Peter McCullagh

Department of StatisticsUniversity of Chicago

South Africa, November 2013

Outline

Atkinson paperPhonemic inventories and linguistic diffusion hypothesisWALS dataFitting Atkinson’s cline model

Spatial and variance-component models

Controversies: Feb/Mar 2012

Socotra

Providence Is.

Agalega Is.Farquhar Is.

Mayotte Is.(under French

admin.)

Aldabra Is.

Amirante Is.Zanzibar

Pemba

Carajos

Réunion

TromelinSt. Helena

Ascension

Madeira Is.

Canary Is.

Principe

Cargados

Annobón

São Tomé

Asmara

LibrevilleKampala

Nairobi

Moroni

Brazzaville

Kinshasa

Yaoundé

Khartoum

Addis Ababa

N'Djamena

Bangui

Kigali

Bujumbura

Lilongwe

Djibouti

Banjul

Conakry

Yamoussoukro Accra

Freetown

Monrovia

Abidjan

Abuja

Nouakchott

Dakar

Bissau

Bamako

Ouagadougou

Malabo

Niamey

Luanda

PortoNovo

Tripoli

TunisAlgiers

Rabat

Laayoune

Cairo

Lusaka

Harare

Pretoria

MaseruBloemfontein

MbabaneMaputo

Cape Town

Windhoek Gaborone

Antananarivo

Dodoma Victoria

Mogadishu

Praia

Lom

e

Port Louis

SãoTomé

Juba

A T L A N T I C

O C E A N

LakeTurkanaLake

Albert

LakeTanganyika

LakeNyasa

LakeKariba

LakeChad

LakeVictoria I N D I A N O C E A N

M e d i t e r r a n e a n S e a

Re d S e a

Gulf of Aden

SUDAN

SOUTHSUDAN

NIGERIA

NAMIBIA

LIBYA

CHAD

SOUTHAFRICA

UNITED REPUBLIC OFTANZANIA

MOROCCO

SAO TOME AND PRINCIPE

ZAMBIA

CENTRALAFRICAN REPUBLIC

TUNISIA

UGANDA

CÔTE- D'IVOIRE

LIBERIA

SIERRALEONE

BURKINA FASOGAMBIA

CAMEROON

EQUATORIAL GUINEA

WesternSahara

MAURITIUS

CAPE VERDE

ERITREA

C

ONGO

NIGER

DEMOCRATIC REPUBLIC

OF THECONGO

GABON

MALI

Cabinda(ANGOLA)

MAURITANIA

BOTSWANA

SWAZILAND

LESOTHO

MALAWI

BURUNDI

RWANDA

ZIMBABWE

DJIBOUTI

KENYA

COMOROS

SEYCHELLES

MO

ZA

M

BIQ U E

MA

DA

GA

SCAR

ANGOLA

ALGERIA

SENEGAL

GUINEA-BISSAU GUINEA

EGYPT

ETHIOPIA

(EQUATORIAL GUINEA)

(PORTUGAL)

(SPAIN)

(UK)

(UK)

(YEMEN)

(MAURITIUS)

(FRANCE)

(FRANCE)

GHANA

BE

NIN

T

OG

O

SOMA

LIA

Map No. 4045 Rev. 7 UNITED NATIONSNovember 2011

Department of Field SupportCartographic Section

0

0

500 1000 km

500 mi

AFRICA

The boundaries and names shown and the designations used on this map do not imply official endorsement or acceptance by the United Nations.

Final boundary between the Republic of the Sudan and the Republic of South Sudan has not yet been determined.

Quentin Atkinson paper: Science 2011Linguistic diffusion hypothesis:

Single point of origin (where?)diversity process accumulates like genetic mutationlanguage spreads by migration/diffusion

At migration/fragmentation:the ancestral population maintains linguistic inventorythe migrating group loses inventory

Implications of the diversity/inventory hypothesis:older languages have greater vowel inventoryolder languages have greater consonant inventoryolder languages have greater tone inventoryolder languages have greater phoneme inventory

Greater phoneme inventory↔ closer to geographic sourceLess phoneme inventory↔ more distant from source

⇒ Phoneme inventory gradient

Atkinson’s WALS dataFrom World Atlas of Language Structures

1. Lname Abkhaz, Aikan??, B? c©t? c©, . . ., Zuni

2. Fam Arawakan, Indo-European, Sino-Tibetan,...

3. Lat as a decimal number: −12.67

4. Long e.g. −60.67 (meaning 60◦ 40′W)

5. Nvd Normalized vowel diversity: WALS feature No. 2

6. Ncd Normalized consonant diversity: WALS feature No. 1

7. Ntd Normalized tone diversity: WALS feature No. 13

8. Tnpd Total normalized phoneme diversity

9. Popn Estimated speaker population: integer1–873 014 298

10. Dbo Distance in km. from Atkinson’s best-fit origin

504 languages; 109 families Niger-Congo 62; Austronesian (42); Indo-European (42);

Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)





4. Long e.g. −60.67 (meaning 60◦ 40′W)








Afro-Asiatic (??)

Worldwide language distribution

**

*

*

*

*

*

*

*

*

*

Malagasy

*

*

Restrictions on migration routes: choke points

**

*

*

*

Restrictions on migration routes: choke points

Africa

Europe

Asia

Austral

N.Amer

S.Amer

Cairo

Istanbul Bering

PnomPenh

Panama

A sample of Atkinson’s WALS dataLname WALS Fam Lat Long Tnpd ISO PopnAbkhaz abk Nw Caucasian 43.08 41 -1.186 abk 105952Acoma aco Keresan 34.92 -107.58 0.220 kjq 3391Andoke adk Andoke -0.67 -72 0.127 ano 619Aleut E aea Eskimo-Aleut 54.75 -164 -0.348 ale 490Arabic (Egyptian) aeg Afro-Asiatic 30 31 -0.377 arz 46321000Aghem agh Niger-Congo 6.67 10 0.965 agq 26727Ahtna aht Na-Dene 62 -145 -0.936 aht 80Aikan?? aik Arawakan -12.67 -60.67 0.061 tba 90Ainu ain Ainu 43 143 0.061 ain 15Akan akn Niger-Congo 6.5 -1.25 0.127 aka 8300000Akawaio akw Cariban 6 -59.5 -0.311 ake 5000Alamblak ala Sepik -4.67 143.33 0.247 amp 1527Albanian alb Indo-European 41 20 0.247 aln, als 5823075Alladian ald Niger-Congo 5.17 -4.33 0.127 ald 23000Alawa alw Australian -15.17 134.25 -0.348 alh 17Amele ame Trans-New Guinea -5.25 145.58 0.181 aey 5300Amharic amh Afro-Asiatic 10 38 0.526 amh 17417913Amo amo Niger-Congo 10.33 8.67 0.686 amo 12263Arrernte amp Australian -24 136 -0.348 aer 2175Amuesha amu Arawakan -10.5 -75.42 -0.348 ame 9831Amuzgo amz Oto-Manguean 16.83 -98 0.845 amu 23000Araona ana Tacanan -12.33 -67.75 -0.069 aro 81Angas anc Afro-Asiatic 9.5 9.5 0.566 anc 400000//Ani ani Khoisan -18.92 21.92 -0.218 hnh 1000Angaatiha ant Trans-New Guinea -7.22 146.25 0.339 agm 2100

Yulu yul Nilo-Saharan 8.5 25.25 1.682 yul 7000Yurok yur Algic 41.33 -124 -0.098 yur 12Yupik yus Eskimo-Aleut 65 -173 -0.348 ess 1350Zande zan Niger-Congo 4 26 0.685 zne 1142000Zoque zqc Mixe-Zoque 17 -93.25 -0.657 zoc 10000Zulu zul Niger-Congo -30 30 0.339 zul 9563422Zuni zun Zuni 35.08 -108.83 -0.377 zun 9651

Statistical modelling: Atkinson style

Response: Total normalized phoneme inventory YiGeographic coords: xi = (x1, x2)i of language iNo of speakers: si

Language origin coordinates: τ = (τ0, τ1)

E(Yi) = β0 + β1‖xi − τ‖+ γ log(si)

cov(Yi ,Yj) = σ2δij

Parameters: β0, β1, γ, τ, σ2

distance ‖xi − τ‖ versus log distance

RSS as a function of τ or F -ratio

F (τ) =(RSS(τ)− RSS(τ̂))/2

RSS(τ̂)/(n − 5)

Phonemic inventory versus distance from τ̂

0 5000 10000 15000 20000 25000 30000

−1

.0−

0.5

0.0

0.5

1.0

1.5

S1$Dbo

S1

$T

np

dAfricaEuropeAsiaAustralasiaN.AmerS.Amer

Phonemic inventory versus distance (by region)

500 1000 1500 2000 2500 3000 3500 4000

−0

.50

.00

.51

.01

.5

6000 6500 7000 7500 8000 8500

−0

.4−

0.2

0.0

0.2

0.4

0.6

6000 8000 10000 12000 14000

−1

.0−

0.5

0.0

0.5

1.0

1.5

15000 20000 25000

−1

.0−

0.5

0.0

0.5

1.0

14000 16000 18000 20000 22000

−1

.0−

0.5

0.0

0.5

1.0

1.5

23000 24000 25000 26000 27000 28000 29000

−1

.0−

0.5

0.0

0.5

Consistency of pattern?

Test for consistency of pattern across continents:

Null: E(Y ) = dist + log(pop)Alt: E(Y ) = Reg + Reg.dist + log(pop)

F =Reduction in RSS/(Reduction in df)

Resid SS/Resid df

F-ratio = 8.6 on (10, 491) df. (should be < 2)

No clear evidence of a consistent cline

RSS (F -ratio) as a function of origin τ

1.6

0.1

1.3

0.1

0

1.9

2.3

0.6

5.7

0.4

0.7

2.83.7

9.2

11.7

21.6

11.3

3.4

9.711.7

8.7

5.7

5.3

4.9

57.1

RSS (F -ratio) as a function of origin τ

1.6

0.1

1.3

0.1

0

1.9

2.3

0.6

5.7

0.4

0.7

2.83.7

9.211.7

21.6

11.3

3.49.7

11.7

8.7

5.7

5.3

4.9

27.6

30.2

29.930

28.9

28.3

26.9

33.2

31

54.5

58.7

56.7

57.1

Atkinson’s posterior density for τ

Confidence regions, HPD regions

Fisher’s F -ratio and the excess variance criterion

F (τ) =(RSS(τ)− RSS(τ̂))/2

RSS(τ̂)/(n − 5)

Distributed as F2,n−5 approx; 95% quantile F < 3.01

BIC ≡ LR:

n log RSS(τ)− n log(RSS(τ̂) = n log(1 + 2F/(n − 5)) ' 2F

distributed as 2F2,n−5 ∼ χ22 approx

95% frequentist region LR(τ) < 6Atkinson’s region as illustrated: BIC < 4

whence 4 BIC units?coverage prob: 86%

Further modelling considerations


cov(Yi ,Yj) = σ2δi,j

I Diversity implies β1 ≤ 0: F -ratio does not distinguishIn fact: β̂1 < 0 for Africa, Europe, Asiaβ̂1 > 0 for America, Oceania (incl Malagasy)

I Covariates versus relationshipscovariate i 7→ si (number of speakers)(i , j) 7→ R1(i , j) same linguistic family(i , j) 7→ ‖xi − xj‖ geographic or linguistic distance

I IndependencePossible correlation within linguistic familiesPossible correlation due to geographic proximity

I Covariates: E(Yi) = β0 + β1xi + · · ·Relationships: cov(Yi ,Yj) = σ2

0δi,j + σ21R1(i , j) + · · ·








0δi,j + σ21R1(i , j) + · · ·








0δi,j + σ21R1(i , j) + · · ·








0δi,j + σ21R1(i , j) + · · ·

A linear Gaussian variance-components model


cov(Yi ,Yj) = σ2δi,j + σ21Fij + σ2

2 exp(−dij/ρ)

τ = (9.5,−1.25); ρ = 1000 km fixed

Model comparison via likelihood-ratio statistics

Covariance modelδ F V F + V + F .V

LR ∗ 2 0.00 88.80 127.17 143.69 147.70β̂1 × 104 −0.391 −0.313 −0.351 −0.331 −0.334

SE 0.036 0.057 0.077 0.079 0.074

Conclusions:Evidence for both family and geographic correlatonsDbo coefficient β̂1 not much affectedOrdinary least-squares SE is misleading

Confidence regions for source (exponential model)

0.2

0

0.2

0.1

0.3

0.5

0.4

0.4

0.8

0.3

0.4

0.71.1

22.3

3.1

1.80.8

1.41.7

1.20.8

0.8

0.7

3.8

4

44

3.9

3.9

3.8

3.7

4.1

3.9

6.2

7.5

7.8

8.2

8.4

8.37.3

7.3

7.3

7.3

7.3

7.3

7.38.395%: all of Africa

99%: Afr, Eur, Levant

Confidence regions for source (Bessel0 model)

0.2

0

0.2

0.1

0.2

0.3

0.3

0.2

0.5

0.1

0

0.50.8

1.51.6

2

1.20.3

0.81.1

0.70.5

0.5

0.5

2.4

2.3

2.42.3

2.4

2.3

2.3

2.3

2.5

2.3

3.1

3.3

3.5

3.6

3.6

3.63.5

3.5

3.5

3.5

3.5

3.5

3.53.695%: Afr, Eur, Levant

99%: World

Summary

I Distinction between covariates and relationshipsI Rel1: family as a block factor Fij

I Rel2: geographic separation Dij

I Spatial covariance models: Matérn’s Bessel familyI Effect of correlation on OLS coefficientsI Effect of correlation on SE of OLS coefficientsI Selection of languages for inclusion in analysisI Choice of total phoneme inventory as responseI Alternative forms of data using specific phonenes

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Published commentary: Science Feb/Mar 2012

Four commentaries: Munich, Shanghai, Stanford, Rochester/MIT

Summary of criticisms

I WALS data: vowels and tones exaggerated: (M, Sh, St, R)(1 tone ≡ 2.2 vowels ≡ 5.7 consonants)

I Inconsistency of pattern: cline in Africa but not elsewhere (St)

I Inconsistency of pattern for v/c/t: (Sh, St)

I BIC-4 allowance: (M)

I Neighbour influence (St)

I Global geography (R)














































References

Atkinson, Q.D. (2011) Science Mag 332, 346.Cysouw, M., Dediu, D. and Moran, S. (2012) Technical comment.Science 335, 657-b.Wang, C.-C, Ding, Q-L, Tao, H. and Li, H. (2012) Technical comment.Science 335, 657-c.van Tuyl, R. and Pereltsvaig, A. (2012) Technical comment. Science335, 657-d.Atkinson, Q.D. (2012) Response. Science Mag 335, 657-eJaeger, T. Pontillo, D. and Graff, P. (2012) Technical comment.Science 335, 1042-ad.Atkinson, Q.D. (2012) Response. Science Mag 335, 1042-bhttp://www.en.uni-muenchen.de/news/newsarchiv/2012/2012_cysow.htmlhttp://www.stat.uchicago.edu/~pmcc/prelims/2011/McC and Clifford, D. (2006) Evidence for conformal invariance of cropyields. Proc. Roy. Soc. Lond. A.Clifford, D. and McC (2006). The regress function. R Newsletter 6,6-10.

Documents

The `Out-of-Africa' hypothesispmcc/seminars/SASA/outofafr.pdf · The ‘Out-of-Africa’ hypothesis Peter McCullagh Department of Statistics University of Chicago South Africa, November