Galtons Problem A Network Autocorrelation Approach to
Cross-Cultural Research
Introduction to and Overview of DEfAn R software package for
cross-cultural research
E. Anthon EffMalcolm M. Dow Wes Routon
Anthropological Sciences Conference, Albuquerque, March 18,
20141The two major problems with cross-cultural data analysis
addressed by DEf are:Missing DataAll of the major cross-cultural
data sets have substantial missing data. Single imputation methods
mean substitution, regression predicted scores, hot deck, etc.
result in coefficient variance estimates of that are downwardly
biased. Data editing procedures, e.g. listwise deletion, generally
result in small samples (loss of power) and also require very
strong assumptions about why data are missing. These assumptions
are very unlikely to hold. Single imputation methods are no longer
recommended.
DEf employs the Multiple Imputation by Chained Equations (mice)
approach to handling missing data.
Non-Independence of Sample Units Sample cases in cross-cultural
and cross-national data are frequently not independent of one
another due to various inter-societal network processes: cultural
trait borrowing, conquest, emulation, inheritance from ancestral
populations, etc. This is the classic Galtons Problem in
anthropology, understood more generally as the problem of cultural
trait transmission. DEf addresses this issue by incorporating
networks of relations into regression models, and employing
instrumental variables procedures to generate consistent and
relatively efficient estimates.
2First problem: Missing DataSociety markin markout money
commland sharefoodNama Hottentot NA NA 1 NA NAKung Bushmen 1 4 1 3
6Thonga 4 4 3 3 6Lozi 3 3 1 3 NAMbundu NA NA 4 NA NASuku 2 2 4 2
2Two solutions: Listwise deletionMultiple imputation3Listwise
deletionSociety markin markout money commland sharefoodNama
Hottentot NA NA 1 NA NAKung Bushmen 1 4 1 3 6Thonga 4 4 3 3 6Lozi 3
3 1 3 NAMbundu NA NA 4 NA NASuku 2 2 4 2 2 Lose three observations.
Lose all of the information in the cells marked in red. Of 186
societies, 156 would have been dropped using listwise deletion. No
longer testing against the full range of human societies. Losing
the big advantage of the SCCS. Probable sample selection
bias.4Multiple imputationSociety markin markout money commland
sharefoodNama Hottentot 3 4 1 2 3Kung Bushmen 1 4 1 3 6Thonga 4 4 3
3 6Lozi 3 3 1 3 6Mbundu 4 5 4 3 1Suku 2 2 4 2 2Society markin
markout money commland sharefoodNama Hottentot 2 3 1 1 2Kung
Bushmen 1 4 1 3 6Thonga 4 4 3 3 6Lozi 3 3 1 3 4Mbundu 3 5 4 5 3Suku
2 2 4 2 2Society markin markout money commland sharefoodNama
Hottentot 3 5 1 2 3Kung Bushmen 1 4 1 3 6Thonga 4 4 3 3 6Lozi 3 3 1
3 5Mbundu 2 6 4 4 2Suku 2 2 4 2 2Replace missing values with
imputed values, drawn from conditional distribution. Create several
(5 to 10) new data sets with imputed values.5Step 1 of the DEf
Approach to Multiple Imputation of Missing Data: finding auxiliary
variables.The mice procedure imputes values for missing
observations on the variables specified in the structural
regression model of interest, using both these variables themselves
plus a set of auxiliary variables. Ideal auxiliary variables are
usually a subset of those with no missing values in the full data
set. Auxiliary variables must be correlated with the variables in
the structural regression model that have missing values, since the
imputation procedure is designed to borrow information from them to
help impute the missing values.
DEf will employ auxiliary variables provided by the user.
Alternatively, DEf will identify suitable auxiliary variables as
follows:
1. identify all categorical, ordinal, interval variables with no
missing values in the complete data set.2. identify variables that
one wants to impute, and, one at a time, treating each as a
dependent variable: i) regress (using binary/ordinal logit,
multinomial, OLS) the dependent variable on the covariate that
provides the highest correlation, and save the residualii) add to
the regression model the covariate that correlates highest with the
residual, and calculate the new residualiii) repeat the above steps
8 times (or more)iv) calculate the relative importance of
predictors, drop variables that fall below a given threshold, and
recalculate the residualv) repeat steps ii iv. 6Step 2: Create m
complete data setsThe mice procedure is repeated m times to create
m copies of the data set, each containing different sets of imputed
values.
Since each data set is now complete, each can be analyzed using
any of the usual statistical models that require complete data.
m = 10 - 100 is currently suggested, depending on sample size
and amounts of missing data. 7Step 3: Analyzing the data and
pooling the results: Rubins Rules
8Analyzing the data and pooling the results, cont.
9Analyzing the data and pooling the results, cont.
10Analyzing the data and pooling the results, cont.
11Analyzing the data and pooling the results, cont.Rubins
pooling procedures can be done with any statistic generated by the
statistical method employed to analyze the m imputed data sets.
12Galtons ProblemIncorporating inter-societal networks into
network autocorrelation effects regression models13Galtons
problemObservations not independent.Common descent (language
phylogeny)Cultural borrowing (geographic distance)
In regression context, Galtons problem will cause biased
coefficients and biased standard errors.
14Galtons problem example:alcohol wivesEcuador 1 0Iran 0
2Ireland 1 0Morocco 0 3Spain 1 0Yemen 0 4Pearson correlation=
-0.9332565, p-value=0.0065 An observed correlation between a pair
of cultural traits across cultures could be due to the borrowing of
the traits, as a package, from a common source (horizontal
transmission), or could be due to their transmission, as a package,
from a common ancestor (vertical transmission), or could be due to
a true functional relationship. Hypothesis: Drinking alcohol
dampens the libido of religious specialists. Adapted from Victor de
Munck and Andrey Korotayev. 2000. Cultural Units in Cross-Cultural
Research. Ethnology 39(4): 335-34815What processes might be
inducing non-independence?Spatial Diffusion: societies in close
proximity have more opportunity to emulate, conform to, adopt,
borrow, etc. neighbors behaviors, beliefs, customs, rituals
(horizontal diffusion.)
Language similarity: Similarity due to populations splitting off
from same ancestral population. (vertical diffusion.)
Religion: Marriage practices spread world-wide by the
colonization of large swaths of the world by European Christian
nations.
Equivalence: units similarly situated in a network and not
necessarily proximate. E.g., economic similarity, core/periphery in
world system, colonial status, ecological setting,
16
Assessing non-independence: Toblers First Law of Geography
Everything is related to everything else, but near things are
more closely related than distant things.
This law suggests that the scores on variable y for the ith
society should be similar to the scores of those societies with
which it has the closest relationships. Call these societies is
neighborhood set.
If so, yi should be similar to the weighted average of the set
of y scores for is neighborhood set, where the weights indicate
relative closeness. If the N scores on y are significantly
correlated with the N weighted average scores, conclude the y
variable is auto-(self)-correlated.
17Weighting sample units.First , need to construct an NxN
connectivity matrix C of pair-wise relatedness scores among sample
units, and then row-normalize C to unity to get the required
weights matrix W. That is, wij = cij jcij.
(If a variable y is premultiplied by W, i.e. Wy, the product
will be an Nx1 vector of weighted averages that are on the same
scale as y.)Raw Connectivity Matrix CWeights Matrix
WyWy011100001/31/31/30006710010001/2001/20005710010001/2001/200087C
=0110100 W
=01/31/301/30085.300010110001/301/31/333.3000010100001/201/212000011000001/21/201218
Incorporating autocorrelated variables into multiple
regression
Most cross-cultural researchers are usually interested in
testing whether hypothesized predictor variables are acting on a
dependent variable, as well as what processes are inducing
autocorrelation in it.
The Network Autocorrelation Regression Effects Models in DEf do
just that.
19Most commonly used network autocorrelation regression model
is:Network Autocorrelation Effects model:y = + Wy + X +
Where: W is a row-normalized NxN weighting matrix with wij >
0 if i and j are related, 0 otherwise, and wii = 0 for all i; is
the network autocorrelation coefficient; y is an Nx1 vector; Wy is
an Nx1 vector where each element i is a weighted average of y
values for is neighborhood set; X is an Nxk matrix of exogenous
variables; is an kx1 vector of coefficients; is an Nx1 vector of
error terms.
Also called the Network Lag model, by analogy to time series,
since W acts similarly to the lag operator in time series models,
except that W lags the y variable in other kinds of social and
physical spaces.
This is the model currently implemented in DEf20Estimating the
network autocorrelation effects regression model y = + Wy + X +
MLE: Maximum Likelihood Estimation. This is usually the method of
choice. But the log-likelihood function contains the term ln|A|,
where A= (I W). Since A is asymmetric and usually not sparse,
finding the eigenvalues is computationally burdensome for large N.
And, for more than two endogenous Wy variables, the likelihood
function is intractable. OLS: Ordinary Least Squares. Basic
assumption of OLS is that all r.h.s. variables be independent of
(uncorrelated with) the error term . If not, all coefficient
estimates ( and ) are biased and inconsistent. Here, y is by
definition a function of , so Wy is also a function of . That is,
Cov(Wy, ) 0. Wy is thus an endogenous regressor.
IV: Instrumental Variables (IV). Provides a way to obtain
consistent parameter estimates for models with endogenous
variables. 2SLS is an IV estimation procedure. Can deal with large
samples and multiple endogenous variables. DEf uses IV estimation
procedures.21An intuitive view of the IV regression approachOLS
model:y = + Wy + Z Wy y
Z is an instrument for Wy if
Cov(Z,) = 0 (Z is valid) and Cov(Z,Wy) 0 (Z is relevant).
So, need to find an additional variable(s) Z that is correlated
with Wy but uncorrelated with to serve as an instrument for Wy.
22An intuitive view of the 2SLS IV estimation procedure Consider
again the network effects model y = + Wy + X +
Suppose we use WX, the lagged values of X, as an instrument for
Wy.
Step 1. Using OLS, estimate Wy = a + WXc + Save the predicted
scores w = + WX
Step 2. Again using OLS, estimate y = + w + X +
(Note: the reported standard errors from step 2 are incorrect.
Not an issue for the 1-step procedures used in all the usual
software packages.) 232SLS Estimation of the network
autocorrelation effects regression model with IVs: general case y =
+ X +
24Where to get appropriate instruments?Usually, its hard to find
additional variables that meet the conditions required. Variables
that affect the endogenous variable(s) are often also likely to
affect the dependent variable.Kelejian and Prucha (1998) show that
the set of {WX, W2X, W3X,} variables are optimal as instruments for
Wy, where W2, W3,. are the 2-step and 3-step connections between
sample units. In practice, the WX variables or some subset of them
will usually be sufficient.
25Evaluating the quality of the instrumental variables Quality
of 2SLS estimators depends on the quality of the IVs. Require
thatCov(Z,) = 0. IVs must be valid. IV estimation is vulnerable on
this point. Tests are available only if there are more instruments
than endogenous variables (overidentification.)IVs also need to be
relevant. i.e., they should predict endogenous variables
independently of other exogenous variables. Shea (1997) proposed a
partial R2 measure of instrument relevance for multiple endogenous
variable models.Marginal associations between endogenous
variable(s) and Z is known as the weak instruments problem. Some
diagnostics are available. No perfect collinearity between all
exogenous variables.
26Overidentification testsIf there is more than 1 instrumental
variable available for Wy, can test the null hypothesis that at
least one of them is correlated with the errors.
Sargan (1958) is the best known test:Ts = NR2u ~ 2 (with df =
#IVs - #endogenous variables) where R2u is the R2 of OLS regression
of 2SLS residuals on the IVs.
Basmann (1960) provides an alternate, though similar, test.
Kirby and Bollen (2009) discuss additional variants of Sargan
and Basmann in the context of SEM.27Weak InstrumentsBound et al
(1995) show that when the instruments are only weakly correlated
with the endogenous variables IV estimates are biased in the same
direction as OLS estimates, and may be more biased than OLS. In
addition, weak IV regression estimates may not be consistent.
Staiger and Stock (1997) suggest that the partial F-statistic
from the increase in the regression R2 after adding the auxiliary
instruments to the exogenous variables in the first stage
regression should be greater than 10. Stock and Yogo (2005) provide
tables that give some guidance as to how much greater than 10 the
F-statistic may have to be.
28Example: Monogamy in the Pre-industrial World Multiple
proposed determinants of the long-term historical shift in marriage
preference from polygynous to monogamous unions are tested using
data from the Standard Cross-Cultural Sample. 29Determinants of
Monogamy (adapted from Dow and Eff 2013)Theoretical
perspectivePrimary Sources Determinants (expected sign)Agent level
perspectivesMales provide essential resourcesOrions 1969;
Borgerhoff-Mulder et al 1990; Marlowe 2000; Low 2003; Alexander et
al 1979male resource inequality (-), female economic contribution
(-), beneficial natural environment (-)Female intra-sexual
aggressionGowaty 1996; Reichard 2003Endemic violence (-)Male
intra-sexual aggressionEmlen & Oring 1977; Hawkes et al 1995;
Marlowe 2000; Borgerhoff-Mulder 1990; Quinlan and Quinlan 2007; van
Schaik and Dunbar 1990; Wrangham et al 1999; Ember & Ember
1992.endemic violence (-), social control (-)Extrinsic RiskQuinland
and Quinlan 2007; Del Guidice 2009; Low 1988, 1990, 2003,
2007pathogen stress (-)Group-level processesCollective action in
small-scale societiesOlson 1971; Alexander et al 1979; Price 1999;
Betzig 1986(Inverse of)societal scale (-)Socially Imposed Monogamy
(SIM)Alexander et al 1979; Betzig 1986societal scale (+)Cultural
Trait TransmissionDivale and Seda 2001; Dow and Eff 2009; Herlihy
2005; Price 1999Distance (+), language (+), modernization (+)30W
matrices employedGeographical Distance: the WD matrix is described
in Dow and Eff (2009), where cij = (1/dij)2 Use only the nearest 20
societies.Language similarity: the WL matrix is described in Eff
(2008), where cij = e-score(ij)
If the Ws are collinear, can combine them into a single matrix:
WDL = DWD + LWL where 0 D, L 1 and D + L =1
Then, run all combinations of WDL and select as best the matrix
that maximizes R2ivAlso obtain information on the weights that
yield the best combined W.
312SLS estimation of network autocorrelation regression model
using composite distance/language W matrix. Dependent variable is a
Box-Cox transform of the percentage of married females in
monogamous marriages [monofem ( 1)/) ]
32Summary: DEf is a new statistical package designed for
cross-cultural and cross-national data sets.Given the ubiquity of
missing data in such data sets, DEf includes a suite of programs
for multiple imputation of missing dataGiven that sample units in
comparative data sets are non-independent due to various processes
of cultural trait diffusion, DEf includes a suite of programs to
implement network autocorrelation effects models. Available as R
workspace and on XSEDE CoSSci/DEf Science Gateway.33
Separate analyses of m multiply imputed samples generates m
estimates of any statistic of interest. In the general case, for
any statistic an analysis of m data sets yields and estimates of
the statistic and its variance for the jth data set (j = 1, 2,.,m).
The multiple imputation point estimate of each parameter is simply
the mean of the m estimates:
To calculate the variance of this estimate, both the m estimated
variances of each and the variance in the across the m estimations
must be combined. First, the mean of the m estimated variances for
each parameter is obtained as a simple average:
This quantity is known as the within-imputation variance.
Next, the variance in the m estimated values is calculated
as:
This quantity is known as the between-imputation variance. These
two variances are then combined to get the total variance in the
combined estimate of Q:
Rubin (1987: 79) shows that the following quantity is
approximately distributed as a t-distribution where the degrees of
freedom, df, is given by
