Bayesian Causalities, Mappings, and Phylogenies: A Social Science Gateway for
Modeling Ethnographic, Archaeological, Historical Ecological and Biological Variables
CS-DC’15 Panel on Synthesis of Ecological, Biological and Ethnographic Data 9-13
Paul Rodriguez (SDSC), Eric Blau, Stuart Martin, Lukasz Lacinski, Thomas Uram (Argonne),
Feng Ren (Xiamen), Wesley Roberts (Carnegie), Tolga Oztan, Douglas White (UCI)
Abstract
186BayesianCausalities_8-30.docx
Extending the innovative “Def Wy” procedures for modeling evolutionary
network effects (Dow, 2007; Dow and Eff’s, 2009a, 2009b), a Complex Social Science
http://intersci.ss.uci.edu (CoSSci) Gateway was developed to provide complex
analyses of ethnographic, archaeological, historical, ecological, and biological
datasets with easy open access. Analysis begins with dependent variable y with n
observations and X independent and other variables, and imputes missing data for all
variates. Several (n x n) W* matrices measure evolutionary network effects such as
diffusion or phylogenetic ancestries. W* is row-normalized to sum to 1 and combined
to obtain a W, multiplied by X as WX, and allowing X and y multiplication by W:
Wy measures the evolutionary autocorrelation portion of y discounting evolutionary
effects of propinquity and phylogenetics. Tested for exogeneity (error terms
uncorrelated with Wy or independent variables) the two-stage Ordinary Least
Squares (OLS) results include measures of independent variable and deep
evolutionary autocorrelation predictors. We show how these methods apply to a
wide variety of problems in the social sciences to which ecological and biological
variables will apply once contributed.
Introduction. We use DEf Wy procedures to begin to integrate analysis of ethnographic,
archaeological, ecological, historical empires and biological variables into complex system
analyses of human evolutionary (deep structure and processes) and causal analysis of
networks of variables at a given set or series of time periods (surface structure). At this time
our datasets of societal samples and variables relate to specific places and times, and the
focus of these variables has been largely sociocultural but it is imperative – given the role of
biology in evolutionary processes – that our data are collated to variables (and models)
from bioecological sciences pinpointed or encompassing where possible the spatiotemporal
sites of our datasets. This is an ambitious undertaking but one successful example of this is
a result of Botero et al. (2014) who predict the High Moral God variable from multiplicative
relationships between bioecological stability and abundance variables (Map Figures 1 and
2) when added to the Ethnographic Atlas dataset of 1265 societies (White 2015). The idea
that multiplying features of two subcontinents can get predictive results is dubious.1
Figure 1: Ethno Atlas Stability Biovariable
Figure 2: Ethno Atlas Abundance Biovariable
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Specifically, when the lower-values (small yellow datapoints) of the bioecological variables
are multiplied the resultant variable predicts “Moral High God” religions such as
Christianity (Map 1) and Islam (Map 2) which together constitute the majority of “Moral
High God” religions, as shown by White (2015). The deficiency of the Botero et al. model is
that their results are invalidated by endogeny due to failure to apply appropriate tests of
whether their predictors of the deep evolutionary background effects of language
phylogeny and spatial diffusion alongside predictors of specific independent variables are
correlated with the error terms of their regression.
Methods: When we use DEf Wy R software implementation with appropriate diagnostics
(Dow, personal communication), we are able to solve diagnostics for exogeneity by use of
two-stage OLS to separate the evolutionary network effects from independent variable
effects. By producing Wy variables explicitly, as well as imputing missing values, their
method enables further analysis and comparisons, such as using a Bayesian Network to
examine the network of variables. Dow and Eff (2013:218-219) provide a succinct
description of their method:
Dow (2007) recently proposed that Galton’s Problem be formulated as a network
autocorrelation effects regression model, where the autocorrelated dependent variable is
included as an endogenous predictor variable. It is straightforward to extend the usual
regression model to include an additional variable that incorporates the effects of trait
transmission on the distribution of percentage of married females in monogamous
marriages, the dependent variable (y) in our proposed model:
y = α + λWy + Xβ+ ε
(1)
where ε is a n x 1 vector of normally distributed error terms with zero mean, X is a n x k
matrix of exogenous variables, β is a k x 1 vector of regression coefficients, α is the
intercept, and λ is the scalar network autocorrelation effect coefficient. The first
independent variable on the right of the equals sign is the product of the square n x n trait
transmission matrix W and the n x 1 vector of scores on the dependent y variable.
OLS regression requires that all of the independent variables be uncorrelated with the
errors, ε, otherwise all of the estimated regression coefficients will be biased. Two-stage
least squares (2SLS) estimation procedures are commonly used to deal with endogenous
predictor variables. The first step in the 2SLS estimation of equation (1) is to generate an
estimate of Wy that is independent of the ε term. This can be done by regressing Wy on
one or more “instrumental” variables, which are independent variables that predict Wy
but are uncorrelated with ε. Kelejian and Prucha (1998) suggest WX as a suitable set of
instrumental variables,2 so that the following equation is estimated at stage 1 using OLS:
Wy = a + WXc + e
(2)
Which allows computation of ŷ, an exogenous predictor for Wy:
ŷ = â+WXĉ
(3)
The vector of predicted scores is then entered into equation (1) and the following
stage 2 equation is estimated, again using OLS regression:
y=α+λŷ +Xβ+ε
(4)
The estimates from this second stage model permit valid inferences about the effect
of trait diffusion net of the functional associations (assessed by the estimated λ and its
associated significance level), and the functional associations net of diffusion (the
estimated β coefficients and their significance levels).
It is straightforward to extend the 2SLS approach to handle multiple W matrices
simultaneously. However, network matrices representing commonly posited diffusion
processes can be highly correlated with each other in any particular study, thus the use of
multiple matrices may result in problems of multicollinearity and disentangling their
2
separate contributions may be difficult. Dow and Eff (2009a) suggest one approach to
handling this problem: combine multiple matrices into a single W matrix.
Results: Table 1 gives a result of DEf Wy applied to a “Stages of Religious Evolution” (SRE)
variable v2013 (Sanderson and Roberts 2008), using the 186 society Standard CrossCultural Sample (Murdock and White 1969) where the number of variables contributed by
many other researchers has grown to 2109.
Table 1: CoSSci DEf01 Wy Result for an SCCS model with a Box-Cox power coefficient
for the Dependent Variable v2013, Evolution of God (Sanderson and Roberts 2008),
where lambda (λ) is the Box-Cox coefficient of predictability. More recent analyses
lead to inclusion of more independent variables with exogeneity in Diagnostics and
higher R2=0.72. The analysis employs a Box-Cox dependent variable with coefficient
y(λ)=(yλ-1)/λ=9.656 that calculates an optimal power threshold for the model (R2=0.5478,
modified to R2=0.72 with the additional variables.
Group significance tests may be used to select among these candidate variables those that
pass the Holm-Bonferroni (H-B) test using the OLS derived p-values. We then further
analyzed the network of variables using Bayesian networks learning procedures with
bootstrapping to produce the analysis shown in Figure 3, which includes the SRE v2013
dependent and independent variables and a set of candidate model variables. In this step,
all variables are analyzed for potential inter-relations using Bayesian networks with
structure learning methods as implemented in the R library(bnlearn). Structure learning
finds likely edges by combining a search for likely conditional dependencies and global
network scores (Scutari and Denis, 2014). Choosing edges are made more robust by taking
random bootstrap samples of the data, learning a network for each sample, and then taking
statistics over the samples of networks. HPC (High Performance Computing) resources at
San Diego Supercomputer Center was used to enable a very large number bootstrap
samples.
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Figure 3: Edge selection over bootstrapped samples of Bayes network shows separate graph
clusters after thresholding. Variables are: v2013 EvoGod (the dependent variable of the DEf01 Wy
model) | Superjurisdictional HierarchyXWriting=SuperjhWriting | v149 Writing | v155 Money |
Wy=Evolutionary effects of DEf01 Wy diffusion and language phylogeny | v232 Intensity of
Cultivation | v207 Dependence on Agriculture | AnimXbwealth | AnimXbwealthLoRain2003. Some of
the latter variables are connected to the EvoGod DEf01 Table 1 Wy variable but not intrinsically. This
graph includes dependent and independent variables in Table 1, and additional set of “unrestricted
variables” (to be explored). Additional clusters show a set of // climate variables bio.1 …bio.11 |
v2005 Water | 2003 Rain | No_rain_Dry // and two pairs of variables: // DepOnHuntGath | v203
Dependence on Gathering // 2125 Wage labor | FxCmntyWages //. Wy is correlated with the EvoGod
variable as shown in the Figure, which is also a feature of Table 1.
It is worth noting that the Wy variables from two-stage OLS provides a kind of latent
variable for the Bayesian network structure learning. This helps alleviate a potential
concern with hidden causes among variables in Bayesian networks. One of the deficiencies
of comparing DEf01 Wy results with Bayesian structure learning is that some variables
need to be discretized for Bayesian learning of conditional probability tables. The Bnlearn
package contains discretization functions to best transform variables to maintain
correlation between independent and dependent variables.
Discussion - Deep (DEf01 Wy) and Surface (Bnlearn) Causality: DEf01 Wy regression
often renders, in two stages, a deep model of causality where past history is shown to have
affected, as in Table 1: diffusion, and linguistic ancestry, characteristic of deep evolutionary
effects on human societies contrasted with surface causalities as in Figure 3. This approach
is potentially applicable to all analyses where indicators of historical or evolutionary
processes are significant.
Endogeneity and Exogeneity: Generic and Specific
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Endogeneity “includes omitted variables, omitted selection, simultaneity, commonmethod variance, and measurement errors – renders estimates causally uninterpretable”
(Antonakis et al. 2010:1086). This includes regression results in which the independent
variables are correlated with residual errors. DEf01 Wy results are potentially replete with
measures of significant endogeneity that destroy attempts to obtain results that represent
potential causality, as noted in Antonakis et al. (2010) for, e.g., the Ramsey RESET test
(p18), Hausman test (p19), Breusch-Pagan Heteroskedascity (p. 28), shown as Diagnostics
in Table 1 but that also include the Shapiro-Wilks, and Breusch–Godfrey autocorrelation LM
tests for distance, language and ecology.
Exogeneity, where dependent and independent variables are uncorrelated with
regression errors, is intrinsic to validity in Bayesian network learning, although omitted
variables, omitted selection, simultaneity, common-method variance, and measurement
errors may still produce endogeneity. If properly specified on all these dimensions,
library(bnlearn) results, recently enhanced in numerous publications for implementation
(Højsgaard, Edwards, Lauritzen 2012, Scutari and Nagarajan 2013, and Nagarajan, Scutari
and Lèbre 2013), provide determinant results that we call surface causality, as above:
causality that omits autocorrelation. These are distinct from the effects of common histories
of the units in a sample, or what has been called Galton’s problem in the social sciences.
The only other alternative approaches to causal estimates in observational studies that
provide results comparable to experiments are those of Cook, Stadish and Wong (2008),
Stadish and Cook (2009), and Thistlethwaite and Campbell (1960), but these approaches
(see Antonakis et al. 2010), like controlled experimental studies that don’t measure deep
histories as does DEf01 Wy, measure surface rather than deep causalities.
Summary
The ongoing goal of this project has been articulated as possibilities of collaboration
between bioecological scientists such as Botero et al. (2014), and exemplified by James
Brown’s (1995) innovation in developing an early framework in macroecology that has
been continued by many others, including Lomolino, Brown, Whittaker, and Riddle (2010),
and Harcourt (2012), a member of our panel, and other members of our panel.
Acknowledgements: Douglas R. White thanks the Santa Fe Institute for hosting multiple
one to two week Causality Working Groups engaging Tolga Oztan, Peter Turchin, Amber
Johnson and many others on this topic in 2010-2014 and to Jü rgen Jost and the MPI for
Mathematics in the Sciences for hosting of our working group in June 2011. We thank
Anthon Eff for his immense work in building the R code prior to and as used in CoSSci.
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This was a feature of a 2014 publication of the PNAS predicting moral god religion using the
data of the Ethnographic Atlas (Murdock 1967). The direction of this article is to explore the use of
Bayesian learning graphs, in an early stage of analysis, using the more extensive dataset of the
Standard Cross-Cultural Sample (Murdock and White 1967). Our variables SuperjhWriting and
AnimXbwealth substitute for the PNAS variables; in doing so they do not multiply but rather
distinguish a dominant ecological region of pastoralism, associated with Islam as a major Moral God
religion from the large states and empires present in both the Islamic and the Christian regions of
West Eurasia and North Africa.
2 The Wy variable is thus intended to be endogenous by definition.
1
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