Robinson's Ecological Correlations and the Behavior of Individuals: methodological corrections.

Sixty years ago the late William S. Robinson (1913–96) published his Ecological Correlations and the Behaviors of Individuals. 1 The paper became an all-time classic and it is one of the most influential methodological papers in social sciences. To underscore its impact on epidemiology , this classic was reprinted in this journal, 2 along with an original re-analysis of Robinson's data and conclusion from a multilevel and historical perspective , 3 with discussions. 4–7 In this research letter, we identify, and correct, an error in Robinson's original analysis. Robinson used data from a US Census Bureau 1933 publication, Table 10 (p. 1229). 8 We coded these data from the original paper records and have made them available at Robinson opened his paper with a scatter diagram (his Figure 1) in which the 'percent illiterate' is plotted against the 'percent negro', using observations of the US Census Bureau's nine geographical divisions in 1930. We note that Robinson used 'negro' in the original paper, while in our own results we will use the term 'Black'. The illiteracy–race relationship was positive and the 'ecological' Pearson's correlation was 0.946. This figure is a weighted correlation coefficient, using the number of individuals in each division as weights. The data were taken from the row margins of a 2 (race) by 2 (illiteracy) by 9 (division) table (his Table 2). Robinson then aggregated this table into a 2 (race) by 2 (illiteracy) table (his Table 1) and used the counts in the interior cells to calculate an individual correlation of 0.203. Here, we have the well-known illustration of a potential ecological fallacy as the ecological (division-level) correlation is substantially different than the individual correlation. Somewhat less known, Robinson also demonstrated that the weighted ecological correlation equals the weighted difference between the overall individual correlation and the average of all within-division individual correlations (Equation 1, p. 340). 2 This relation between ecological and individual correlation holds as long as the weighted ecological correlation is calculated from the total margins of the underlying individual data tables. This is important to note because when Robinson analysed the data at state level, this mathematical relationship did not hold any more. There are two reasons for this remarkable fact. First, Robinson unwarily used state-level data, which were not the result of the underlying individual data table. Secondly, for reasons not known to us, Robinson used unweighted ecological correlations. Robinson's Figure 1, …