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 experimental design 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 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 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.