Electronic Supplementary Material
Cultural Macroevolution on Neighbor Graphs: Vertical and Horizontal Transmission among
Western North American Indian Societies
Mary C. Towner, Mark N. Grote, Jay Venti, Monique Borgerhoff Mulder
Gibbs Sampler
For a given trait, the parameters θm, λm, βm; m=1,…4 (known as “driving values”) are chosen to
be in a region of high likelihood support under model m. We accomplished this by carrying out
preliminary runs of the Gibbs sampler, using the MCMC maximum-likelihood method described
by Geyer (1991, 1996) to approximate the likelihood surface for model m=1,…4. We then chose
θm, λm, βm close to the approximate maximum-likelihood estimates for model m. Although the
realizations x1,… xR and mixture distribution described below can be generated by any family of
distributions having positive support on the entire set of binary arrays, we chose to concentrate
our simulation efforts on the focal models 1–4. For each trait and model, the initial state of the
Gibbs sampler has all observations equal to the same trait value, as suggested by Geyer (1991).
We also implement stochastic symmetry swaps, under which (at randomly chosen scans) the
positive and negative signs of trait values are reversed, to enhance mixing of the sampler.
We use an importance sampling technique (Geyer 1994, 1996) which combines information
from realizations at parameter values θm, λm, βm; m=1,…4, to approximate likelihoods for
models 1-4 for each trait. We sample from models 1-4 in equal proportions; thus the
importance sampling distribution (up to a constant of proportionality) is
hmix(x) = Σm hm(x) eηm
(ESM 1)
where hm(x) = exp{ θm S(x) + λm T(x) + βm U(x)} and eηm = 1 / 4 z(θm, λm, βm) (see Geyer
1996:253-254). We use the “reverse logistic regression” method of Geyer (1994) to estimate
ηm; m=1,…4. Finally, the log-likelihood ratio for the observation x at parameter values θ, λ, β is
approximated as
Towner et al. ESM-1
log (L[θ, λ, β; x] / Lmix) ≈ log (h(x) / ĥmix(x)) – log (R−1 Σr [h(xr) / ĥmix(xr)])
(ESM 2)
where Lmix is the likelihood under the importance sampling distribution (eq. ESM 1), h(x) = exp{
θ S(x) + λ T(x) + β U(x)}, ĥmix(x) is obtained by substituting the reverse-logistic estimates of ηm;
m=1,…4 in equation (ESM 1), R is the number of realizations from the Gibbs sampler and x1,… xR
are the realizations themselves. We evaluate the right-hand side of equation (ESM 2) on a
three-dimensional grid of parameter values (θ, λ, β), and fit models 1-4 to the observation x by
maximizing equation (ESM 2) on the grid (or on an appropriate lower-dimensional subset, such
as the two-dimensional grid having θ=0 for model 2).
Model Comparison
Likelihoods under models 1–4 are maximized with respect to an importance sampling
distribution shared in common, via equation (ESM 2); this facilitates AIC (and, respectively, BIC)
comparisons among models as follows. Maximizing the right-hand side of equation (ESM 2)
over the parameter set for model m produces the stochastic approximation log LR*m ≈ log(L*m /
Lmix). For an alternative model m′, the difference in AIC is approximated as
−2(log LR*m − Km − log LR*m′ + K m′) ≈ −2(log[L*m / Lmix] − Km – log[L*m′ / Lmix] + K m′)
= −2(log L*m − Km) + 2(log L* m′ − K m′)
= AIC m - AIC m′
(ESM 3)
A similar calculation produces the approximate difference in BIC. AIC (and, respectively, BIC)
values for models 1–4 can be ordered from smallest to largest by examining the differences
approximated by equation (ESM 3).
Exact Calculation
Evaluating the autologistic likelihood (eq. 1 of the main text) exactly involves the enumeration
of 2n binary arrays, which requires long computing times even for small samples; therefore we
chose a subsample from our original 172 societies for exact model fitting. We reasoned that the
subsample should contain relatively few language groups and should be geographically limited,
Towner et al. ESM-2
as compared with the original sample. We focused on the Northwest Coast Group identified by
Jorgenson (1980), eliminating societies from this group that were language isolates, were
geographically distant from the bulk of the group, or had missing values in a subset of traits
under consideration; this produced a subsample of n=24. The three chosen traits—digstick,
brideservice, and brideprice—have levels of variation in the subsample similar to those typical
of traits in the original sample. We defined linguistic and spatial neighbors in the subsample in
the same way as for the original sample (see “Neighbor Graphs” in the main text). The average
number of linguistic neighbors in the subsample is 7.1 (range 2–11), and the average number of
spatial neighbors is 4.3 (range 1–8). Our intention here is not to make inferences about cultural
evolution in the subsample, but to investigate the accuracy of results obtained by the MCMC
method by comparing them with exact results.
We enumerated the 224 binary arrays that form the summands of z(θ, λ, β) using the binary
representations of the integers 0, 1, …, 224 – 1. A programming loop through these integers
produces, in turn, each possible binary array of length n=24. Additional programming steps
embedded in the loop calculate the contributions to z(θ, λ, β) from each array, on a grid of (θ, λ,
β) values. Direct maximization of the likelihood L(θ, λ, β; x), as well as calculations leading to AIC
and BIC weights, are straightforward once z(θ, λ, β) has been evaluated. We obtained
maximum-likelihood parameter estimates and AIC weights for digstick, brideservice, and
brideprice in the subsample using the exact scheme and then obtained analogous results in
independent runs of the MCMC method, using the implementation details described in the
main text and above. ESM Table 1 shows that the MCMC method produces estimates and
model weights very close to the exact values. The computing time for a grid of 70,000
parameter points was approximately 5.5 days using the exact method (on a Dell Precision
Workstation 650), as compared with a few hours for the MCMC method (on a Dell laptop). For a
given grid size, each addition of a society to the sample doubles the computing time when the
exact method is used.
Simulation
Towner et al. ESM-3
We designed a simulation study in which models 1–4 were fitted to datasets generated under
controlled levels of horizontal transmission. Charles Nunn graciously modified the simulation
program described in Nunn et al. (2006) to produce a single binary trait for each society and
generated 200 simulated datasets for our use. The program is based on an explicit spatiotemporal evolutionary model unrelated to the autologistic model. Each society in a simulated
dataset has a binary trait value, a position on a square lattice, and a known phylogenetic
lineage. The parameters held constant over all simulations were number of societies (100),
number of discrete generations (800), per-generation probability of extinction (0.1), pergeneration probability of diversification (propagation of a society along with its trait to an
adjacent, unoccupied position on the lattice, 0.9), and per-generation probability of trait
evolution (a random switch to the other trait value, 0.01). The per-generation probability of
horizontal trait transmission (donation of a society’s trait value to an adjacent society already in
existence) was systematically varied over the simulations, with 50 simulations at each of the
values 0.0, 0.001, 0.01, 0.1.
To turn a simulated dataset into a neighbor-graph dataset, we converted the known phylogeny
into a phylogenetic neighbor graph (resulting in a coarsening of information about historical
relationships between societies). In the phylogenetic neighbor graph, tips of the tree are
collected into mutually exclusive sets of closely related societies, such that the average number
of phylogenetic neighbors across the sample is approximately the same as the average number
of spatial neighbors (3.6, derived from the geometry of the 10-by-10 lattice). This calibration
procedure is analogous to the one used for the WNAI sample, except that here the spatial
neighbor graph of the square lattice is fixed, and the phylogenetic neighbor graph is calibrated
to it.
To achieve the calibration, for each simulated dataset we progressively moved a phylogeny
horizon from the tips of the unrooted tree inward to the center (see ESM Figure 2). As the
horizon crosses each internal node, cultures are segregated into clades which branch at a
distance from the center greater than or equal to the distance from the center to the horizon.
The phylogenetic neighbor graph treats each member of a clade as equally related to all other
Towner et al. ESM-4
members (thus the clades are converted to cliques). As the horizon moves inward, the number
of cliques decreases; at the center there would be only one clique. At some internal node, the
average neighbor number of the resulting graph most closely approximates 3.6. This is the
graph we chose for analysis of the simulated datset.
We developed a semi-automated batch processing program to fit autologistic models to the
simulated datasets using the MCMC method described in the main text and above. Some
numerical compromises were necessary to keep overall computing times reasonable: for each
simulated dataset the approximate log-likelihood ratio (eq. ESM 2) is based on R=42,000
realizations, with thinning and burn-in as for the WNAI analysis.
Results of the simulation study are summarized in ESM Figures 3 and 4. ESM Figure 3 is a
scatter plot of approximate maximum-likelihood estimates of the spatial (θ; horizontal axis) and
phylogenetic (λ; vertical axis) association parameters from model 4, for each simulated dataset.
Plotting characters are shaded according to the level of horizontal transmission in effect.
Simulations with lower horizontal transmission rates tend to have larger estimates of λ and
smaller estimates of θ. The converse is true for simulations with higher horizontal transmission
rates. Calibration of the spatial and phylogenetic neighbor graphs produces θ and λ estimates
roughly on the same scale. The bar graphs of ESM Figure 4 depict model weights averaged over
50 simulated datasets, for each level of horizontal transmission (ht). From top to bottom, it is
evident that as levels of horizontal transmission increase, average support for model 3 (spatial
neighbors only) increases while support for model 2 (phylogenetic neighbors only) decreases.
We understand the relatively high support for model 4 (both spatial and phylogenetic
neighbors) in the ht=0 simulations to be a consequence of an assumption built into the model
of Nunn et al. (2006): parent societies propagate only into adjacent, unoccupied positions of
the lattice. Thus spatial association carries information about trait similarity in the simulation
model even when there is no horizontal transmission.
Towner et al. ESM-5
ESM TABLE 1. Comparison of parameter estimates and model weights obtained using the
exact and MCMC methods for the subsample of n=24.
θ, λ, and β are maximum likelihood estimates under model 4, including both the spatial and
linguistic neighbor graphs. M4, M3, and M2 are AIC weights for the respective models (the AIC
weight for model 1 can be obtained by subtraction: see “Model Comparison” in the main text).
Trait
Method
θ
λ
β
M4
M3
M2
exact
0.26
−0.02
−0.07
0.22
0.58
0.12
MCMC
0.26
−0.02
−0.08
0.22
0.58
0.12
exact
0.38
−0.04
−0.08
0.28
0.68
0.03
MCMC
0.38
−0.04
−0.08
0.28
0.68
0.03
exact
0.31
−0.11
0.05
0.29
0.43
0.08
MCMC
0.31
−0.11
0.05
0.29
0.43
0.08
digstick
brideservice
brideprice
Towner et al. ESM-6
ESM TABLE 2. Forty-four cultural traits within six domains.
WNAI traits were selected on these criteria: breadth of trait type across the domains, few
missing cases, and low skew (such that the trait exhibited enough variation for statistical
analyses to be meaningful). All WNAI traits are already coded categorically; we combined
categories as appropriate in order to create meaningful binary traits. For example, our binary
variable agriculture is based on V. 187 in the WNAI, which has seven outcomes: absent (n=81)
and six other categories describing the nature (food, nonfood) and extent (incipient, % of diet)
of horticulture and agriculture. We collapsed the latter into one category, for which the answer
to the question “Is there agricultural or horticultural production (including nonfoods)?” would
be “yes.”
Description (modified from Jorgensen 1980)
Binary
Variable
yes
(1)
no
(−1)
n
WNAI
Is there agricultural or horticultural production (including
nonfoods)?
agriculture
90
81
171
V187
At least 1-10% of diet contributed by local agriculture?
agrodiet
37
135
172
V193
At least 26-50% of diet contributed by aquatic animals?
aquaticdiet
79
93
172
V199
At least 26-50% of diet contributed by non-aquatic hunting?
huntdiet
92
80
172
V204
At least 26-50% of diet contributed by local gathering?
gatherdiet
115
57
172
V211
At least semisedentary settlements occupied throughout the year?
fixedsettle
90
81
171
V284
At least 1-5 persons per square mile?
popdens
53
118
171
V288
At least 11-25% incidence of polygyny?
polygyny
54
118
172
V294
Is there a marked tendency toward exogamous marriages?
exogamy
69
99
168
V301
Are there unequal gift exchanges, which tend to approach brideprice, at marriage?
brideprice
48
122
170
V302
Is there continued exchange of goods and services between
relatives of the bride and groom after marriage?
affinalexch
56
107
163
V303
Does the man perform services for his bride's family before or after
marriage?
brideservice
56
108
164
V304
Domain: Subsistence and Settlement
Domain: Marriage and Residence
Towner et al. ESM-7
Is dominant postnuptial residence with husband's kin (patrilocal,
virilocal, avunculocal)?
patrilocal
102
70
172
V308
Are raids ever motivated by desire for women (wife-stealing)?
raidwomen
93
69
162
V355
Are houses owned by descent units (lineages or demes) rather than
builder or occupant family?
ownhouse
80
89
169
V273
Are there any conventions regarding the inheritance of houses
upon an owner's death?
inherithouse
97
72
169
V281
Is the dominant family household or coresidential unit lineal or
extended?
linealfamily
106
63
169
V307
Are there descent units beyond the ego-oriented kindred of
bilateral kinsmen?
descentunits
92
80
172
V312
Are there one or more special terms for cousins that distinguish
them from siblings? (non-Hawaiian pattern?)
cousinterms
76
80
156
V334
Are settlements compact, e.g., nucleated villages or concentrated
camps?
compactsettle
106
65
171
V285
Does the typical community in the focal area have 50 or more
people in it?
popsize
93
76
169
V286
Does focal community have political leadership beyond a single
leader and informal council of elders?
politicalleaders
105
66
171
V335
Does local society have any territorial organization larger than the
residential kin group?
politicalorg
98
73
171
V337
Do residential kin groups, villages, or tribes ever form alliances with
other groups?
allies
99
68
167
V342
Are there any restricted sodalities (i.e., organizations with restricted
memberships)?
sodalities
54
118
172
V345
Is the incidence of offensive raids moderate or frequent (i.e., more
than 1 per year)?
raiding
94
63
157
V361
Are there few or many slaves (i.e., more than "absent or very
rare")?
slaves
63
109
172
V436
Domain: Kinship and Family
Domain: Political Organization and Social Stratification
Towner et al. ESM-8
Domain: Material Culture
Are digging sticks straight-handled?
digstick
125
43
168
V150
Are there stone food mortars?
mortar
103
69
172
V157
Are milling stones used?
millstone
92
80
172
V159
Is meat smoked or fire-dried?
drymeat
109
60
169
V160
Is salt (sodium chloride) added to food?
salt
119
53
172
V162
Are houses covered with hide and/or thatch?
hidethatch
109
63
172
V166
Are houses covered by bark and/or woven or sewn mats?
barkmat
124
48
172
V167
Are houses covered with stone, adobe, wattle, and/or sod?
stoneearth
120
52
172
V168
Are there hard- (separate-) soled moccasins?
hardsole
93
79
172
V182
Are there any devices for weaving (e.g., one-bar and/or two-bar
frames)?
weavedevice
109
63
172
V186
Is there a feast and/or public ceremony at the most important
naming event?
namefeast
65
90
155
V387
Do girls' puberty rites of female initiations near puberty include any
running?
girlsrun
61
97
158
V392
Is a flexed burial position (ever) used for corpses?
flexedburial
56
106
162
V402
Is there any sacrifice at death (killing or freeing of dogs,
domesticated animals, slaves and/or captives)?
sacrifice
66
77
143
V406
Are there group rites for one or more novice (e.g., spirit
quests/confirmations in sodality initiations)?
grouprite
79
92
171
V412
Is there spirit impersonation with mask or disguise?
spiritmasks
79
89
168
V416
Is there possessional shamanism (including trances)?
shamantrance
105
60
165
V418
Domain: Rituals, Beliefs, and Attitudes
Towner et al. ESM-9
ESM TABLE 3. WNAI sample (n=172 populations) with tribe names and language neighbor
groups (as classified in this study)
WNAI ID
WNAI Tribe Name
Language Neighbor Group
36
Yurok
Algic
39
Wiyot
Algic
147
No. Tonto W Apache
Apachean
148
So. Tonto W Apache
Apachean
149
San Carlos W Apache
Apachean
150
Cibecue W Apache
Apachean
151
White Mtn. W Apache
Apachean
152
Wrm Sprngs Chir Apache
Apachean
153
Huachuca Chir Apache
Apachean
154
Mescalero Apache
Apachean
155
Lipan Apache
Apachean
156
Jicarilla Apache
Apachean
157
Western Navaho
Apachean
158
Eastern Navaho
Apachean
93
Alkatcho Carrier
CentralBritishColumbia
94
Lower Carrier
CentralBritishColumbia
95
Chilcotin
CentralBritishColumbia
27
Lower Chinook
Chinookan
110
Wishram
Chinookan
29
Alsea
CoastOregonPenutian
30
Siuslaw
CoastOregonPenutian
31
Coos
CoastOregonPenutian
11
Bella Coola Salish
CoastSalish
14
Klahuse Salish
CoastSalish
15
Pentlatch Salish
CoastSalish
16
Squamish Salish
CoastSalish
17
Cowichan Salish
CoastSalish
18
West Sanetch Salish
CoastSalish
19
Upper Stalo Salish
CoastSalish
20
Lower Fraser Salish
CoastSalish
21
Lummi Salish
CoastSalish
22
Klallam Salish
CoastSalish
Towner et al. ESM-10
23
Twana Salish
CoastSalish
24
Quinault Salish
CoastSalish
25
Puyallup Salish
CoastSalish
28
Tillamook Salish
CoastSalish
3
N Masset Haida
Haida
4
S Skidegate Haida
Haida
96
Shuswap
InteriorSalish
97
Upper Lillooet
InteriorSalish
98
Upper Thompson
InteriorSalish
99
Southern Okanagon
InteriorSalish
100
Sanpoil
InteriorSalish
101
Columbia
InteriorSalish
102
Wenatchi
InteriorSalish
103
Coeur dAlene
InteriorSalish
104
Kalispel
InteriorSalish
105
Flathead
InteriorSalish
161
Acoma
Keres
162
Sia Keres
Keres
163
Santa Ana Keres
Keres
164
Santo Domingo Keres
Keres
165
Cochiti
Keres
166
San Juan Tewa
KiowaTanoan
167
San Ildefonso Tewa
KiowaTanoan
168
Santa Clara Tewa
KiowaTanoan
169
Nambe Tewa
KiowaTanoan
170
Taos
KiowaTanoan
171
Isleta
KiowaTanoan
172
Jemez
KiowaTanoan
106
Kutenai
Kutenai.isolate
54
Valley Maidu
Maiduan
55
Foothill Maidu
Maiduan
56
Mountain Maidu
Maiduan
57
Foothill Nisenan
Maiduan
58
Mountain Nisenan
Maiduan
59
Southern Nisenan
Maiduan
Towner et al. ESM-11
71
San Joaquin Mono
Numic
72
Kings River Mono
Numic
79
Kawaiisu
Numic
114
Wada-Dokado N Paiute
Numic
115
Kidu-Dokado N Paiute
Numic
116
Kuyui-Dokado N Paiute
Numic
117
Owens Valley N Paiute
Numic
118
Panamint Shoshone
Numic
120
Reese River Shoshone
Numic
121
Spring Valley Shoshone
Numic
122
Ruby Valley Shoshone
Numic
123
Battle Mtn Shoshone
Numic
124
Gosiute Shoshone
Numic
125
Bohogue Shoshone
Numic
126
Agaiduka Shoshone
Numic
127
Hukundika Shoshone
Numic
128
Wind River Shoshone
Numic
129
Uintah Ute
Numic
130
Uncompaghre Ute
Numic
131
Wimonuch Ute
Numic
132
Shivwits S Paiute
Numic
133
Kaibab SPaiute
Numic
134
San Juan S Paiute
Numic
135
Chemehuevi S Paiute
Numic
32
Tututni Athapaskan
PacificCoastAthabaskan
33
Chetco Athapaskan
PacificCoastAthabaskan
34
Galice Creek Athapas
PacificCoastAthabaskan
35
Tolowa Athapaskan
PacificCoastAthabaskan
38
Hupa Athapaskan
PacificCoastAthabaskan
40
Sinkyone Athapaskan
PacificCoastAthabaskan
41
Mattole Athapaskan
PacificCoastAthabaskan
42
Nongatl Athapaskan
PacificCoastAthabaskan
43
Kato Athapaskan
PacificCoastAthabaskan
51
East Achomawi
Palaihnihan
52
West Achomawi
Palaihnihan
Towner et al. ESM-12
53
Atsugewei
Palaihnihan
63
Northern Pomo
Pomoan
64
Eastern Pomo
Pomoan
65
Southern Pomo
Pomoan
107
Nez Perce
Sahaptian
108
Umatilla
Sahaptian
109
Klikitat
Sahaptian
111
Tenino
Sahaptian
112
Klamath
Sahaptian
113
Modoc
Sahaptian
44
East Shasta
Shastan
45
West Shasta
Shastan
81
Gabrielino
Takic
82
Luiseno
Takic
83
Cupeno
Takic
84
Serrano
Takic
85
Desert Cahuilla
Takic
86
Pass Cahuilla
Takic
87
Mountain Cahuilla
Takic
145
Pima
Tepiman
146
Papago
Tepiman
1
N Tlingit
Tlingit
2
S Tlingit
Tlingit
5
Tsimshian
Tsimshianic
6
Gitksan Tsimshian
Tsimshianic
68
Northern Miwok
Utian
69
Central Miwok
Utian
70
Southern Miwok
Utian
7
Haisla Kwakiutl
Wakashan
8
Haihais Kwakiutl
Wakashan
9
Bella Bella Kwakiutl
Wakashan
10
Fort Rupert Kwakiutl
Wakashan
12
Clayoquot Nootka
Wakashan
13
Makah Nootkan
Wakashan
47
Trinity River Wintu
Wintuan
Towner et al. ESM-13
48
McCloud River Wintu
Wintuan
49
Sacramento R. Wintu
Wintuan
50
Nomlaki Wintun
Wintuan
67
Patwin Wintun
Wintuan
73
Chuckchansi Yokuts
Yokutsan
74
Kings River Yokuts
Yokutsan
75
Kaweah Yokuts
Yokutsan
76
Lake Yokuts
Yokutsan
77
Yauelmani Yokuts
Yokutsan
60
Coast Yuki
Yuki-Wappo
61
Yuki
Yuki-Wappo
66
Wappo Yukian
Yuki-Wappo
88
Mountain Diegueno
Yuman-Cochimi
89
Western Diegueno
Yuman-Cochimi
90
Desert Diegueno
Yuman-Cochimi
91
Kaliwa
Yuman-Cochimi
92
Akwa-ala
Yuman-Cochimi
136
Havasupai
Yuman-Cochimi
137
Walapai
Yuman-Cochimi
138
Northeast Yavapai
Yuman-Cochimi
139
Southeast Yavapai
Yuman-Cochimi
140
Mohave
Yuman-Cochimi
141
Yuma
Yuman-Cochimi
142
Kamia
Yuman-Cochimi
143
Cocopa
Yuman-Cochimi
144
Maricopa
Yuman-Cochimi
26
Quileute Chimakuan
sample isolate
37
Karok
sample isolate
46
Chimariko
sample isolate
62
Yana
sample isolate
78
Tubatulabal
sample isolate
80
Salinan
sample isolate
119
Washo
sample isolate
159
Hopi
sample isolate
160
Zuni
sample isolate
Towner et al. ESM-14
ESM Figure 1. Spatial neighbor graph for the WNAI sample with nodes placed at
latitude/longitude coordinates.
The Pacific coastline can be discerned by following a rough diagonal from upper left to lower
right. The structural properties of the graph below are the same as the graph in Figure 1a of the
main paper: edges connect pairs of societies that we define as spatial neighbors (i.e., societies
less than 175 km apart).
Towner et al. ESM-15
ESM Figure 2. Cartoon depiction showing conversion of a simulated phylogeny to a
phylogenetic neighbor graph.
On the left, concentric circles indicating phylogeny horizons are superimposed on an unrooted
phylogeny. The circles are centered at the point equidistant from all tips (the simulated
phylogenies are ultrametric). The outer horizon yields neighbor groups (A,B,C,D,E) (F,G,H,I)
(J,K,L) (M,N) and average neighbor number 2.9. The inner horizon yields groups (A,B,C,D,E)
(F,G,H,I) (J,K,L,M,N) and average neighbor number 3.7—closest to the target value 3.6. The
geometric figures on the right show the phylogenetic neighbor graph produced by the inner
horizon. The phylogenies actually used in the simulation study have 100 tips.
Towner et al. ESM-16
ESM Figure 3. Scatter plot of estimated θ and λ for 200 simulated data sets.
θ and λ are respectively the spatial and phylogenetic association parameters of model 4.
Towner et al. ESM-17
ESM Figure 4. Average model weights for 50 simulated datasets generated at each of four
horizontal transmission rates (ht).
Average weights are shown schematically as shaded areas of the horizontal bars. Models 1–4
and the method for calculating weights are described in the main paper. Details of the
production and analysis of simulated datasets are given in the “Simulation” section above.
Towner et al. ESM-18