Phylogenetic autocorrelation, also known as Galton's problem after Sir Francis Galton who described it, is the problem of drawing inferences from cross-cultural data, due to the statistical phenomenon now called autocorrelation. The problem is now recognized as a general one that applies to all nonexperimental studies and to some experimental designs as well. It is most simply described as the problem of external dependencies in making statistical estimates when the elements sampled are not statistically independent. Asking two people in the same household whether they watch TV, for example, does not give you statistically independent answers. The sample size, n, for independent observations in this case is one, not two. Once proper adjustments are made that deal with external dependencies, then the axioms of probability theory concerning statistical independence will apply. These axioms are important for deriving measures of variance, for example, or tests of statistical significance.
Origin In 1888, Galton was present when Sir Edward Tylor presented a paper at the Royal Anthropological Institute. Tylor had compiled information on institutions of marriage and descent for 350 cultures and examined the associations between these institutions and measures of societal complexity. Tylor interpreted his results as indications of a general evolutionary sequence, in which institutions change focus from the maternal line to the paternal line as societies become increasingly complex. Galton disagreed, pointing out that similarity between cultures could be due to borrowing, could be due to common descent, or could be due to evolutionary development; he maintained that without controlling for borrowing and common descent one cannot make valid inferences regarding evolutionary development. Galton's critique has become the eponymous Galton's Problem, as named by Raoul Naroll, who proposed the first statistical solutions. By the early 20th century unilineal evolutionism was abandoned and along with it the drawing of direct inferences from correlations to evolutionary sequences. Galton's criticisms proved equally valid, however, for inferring functional relations from correlations. The problem of autocorrelation remained.
Solutions Statistician William S. Gosset in 1914 developed methods of eliminating spurious correlation due to how position in time or space affects similarities. Today's election polls have a similar problem: the closer the poll to the election, the less individuals make up their mind independently, and the greater the unreliability of the polling results, especially the margin of error or confidence limits. The effective n of independent cases from their sample drops as the election nears. Statistical significance falls with lower effective sample size. The problem pops up in sample surveys when sociologists want to reduce the travel time to do their interviews, and hence they divide their population into local clusters and sample the clusters randomly, then sample again within the clusters. If they interview n people in clusters of size m the effective sample size (efs) would have a lower limit of 1 + (n − 1) / m if everyone in each cluster were identical. When there are only partial similarities within clusters, the m in this formula has to be lowered accordingly. A formula of this sort is 1 + d (n − 1) where d is the intraclass correlation for the statistic in question. In general, estimation of the appropriate efs depends on the statistic estimated, as for example, mean, chi-square, correlation, regression coefficient, and their variances. For cross-cultural studies, Murdock and White estimated the size of patches of similarities in their sample of 186 societies. The four variables they tested – language, economy, political integration, and descent – had patches of similarities that varied from size three to size ten. A very crude rule of thumb might be to divide the square root of the similarity-patch sizes into n, so that the effective sample sizes are 58 and 107 for these patches, respectively. Again, statistical significance falls with lower effective sample size. In modern analysis spatial lags have been modelled in order to estimate the degree of globalization on modern societies. Spatial dependency or auto-correlation is a fundamental concept in geography. Methods developed by geographers that measure and control for spatial autocorrelation do far more than reduce the effective n for tests of significance of a correlation. One example is the complicated hypothesis that "the presence of gambling in a society is directly proportional to the presence of a commercial money and to the presence of considerable socioeconomic differences and is inversely related to whether or not the society is a nomadic herding society."
Tests of this hypothesis in a sample of 60 societies failed to reject the null hypothesis. Autocorrelation analysis, however, showed a significant effect of socioeconomic differences. How prevalent is autocorrelation among the variables studied in cross-cultural research? A test by Anthon Eff on 1700 variables in the cumulative database for the Standard Cross-Cultural Sample, published in World Cultures, measured Moran's I for spatial autocorrelation (distance), linguistic autocorrelation (common descent), and autocorrelation in cultural complexity (mainline evolution). "The results suggest that ... it would be prudent to test for spatial and phylogenetic autoccorrelation when conducting regression analyses with the Standard Cross-Cultural Sample." The use of autocorrelation tests in exploratory data analysis is illustrated, showing how all variables in a given study can be evaluated for nonindependence of cases in terms of distance, language, and cultural complexity. The methods for estimating these autocorrelation effects are then explained and illustrated for ordinary least squares regression using again the Moran I significance measure of autocorrelation. When autocorrelation is present, it can often be removed to get unbiased estimates of regression coefficients and their variances by constructing a respecified dependent variable that is "lagged" by weightings on the dependent variable on other locations, where the weights are degree of relationship. This lagged dependent variable is endogenous, and estimation requires either two-stage least squares or maximum likelihood methods.
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