A Comment on Palmore and Whittington's Index of Similarity

Since we share an interest in age inequality, I approached Palmore and Whittington's articles on racial and age inequalities in this journal (1970; 1971) with some enthusiasm. Their Equality Index (alternatively, Similarity Index), proposed as a standard measure of "overlap" between two groups in the percentage distributions of some desideratum, seemed a simple and useful way to examine the nature and trends of a variety of inequalities. But after close scrutiny, I find myself quite troubled by their exaggerated claims for, and misleading use of, this index. Because Palmore and Whittington clearly anticipate more widespread use of the Similarity Index, misunderstanding as to its value may retard the painstaking analysis and interpretation of inequality that are sorely needed. The Similarity Index employs a single figure to describe, for a single variable, a comparison between the percentage distributions of two subpopulations. It is defined as the sum of the lesser of the two percentages that represent proportions of the subpopulations within each category of the variable of interest. Like most summary measures, the Similarity Index achieves simplification of the data through sacrifice of information available in the full set. Because of this characteristic, it is particularly important that the user of a summary measure distinguish clearly between information revealed and information concealed by his index, and that he interpret his analysis accordingly. For example, it is not uncommon for summary measures to contain "unwanted equivalencies" in which differences of central theoretical or empirical interest are concealed by equal numbers. Furthermore, the investigator, being familiar with the full set of data, may treat a summary measure as revealing more information than it actually does. Despite the advantage of simplicity, the Similarity Index conceals so much information about the original distributions as to be practically useless in the analysis of equality. It does not take into account the absolute levels of the variable under consideration (e.g., the index doesn't tell us whether two groups are equally miserable or equally blessed). It reveals little about the distribution of overlap (or, conversely, of differences) through categories of the variable of interest. (For example, we can't tell whether differences between two groups are uniformly distributed or created primarily by a few categories.) Nor does the index reveal which of two groups being compared suffers relative deprivation (although this deficiency could easily be remedied by appropriate labeling in the tabular presentation). Most troublesome, a change in the Similarity Index between two time periods tells us almost nothing we want to know about the nature of changes in the subpopulations compared. Palmore and Whittington seem largely unaware of the paucity of information contained in their index, and of the attendant dangers of misinterpretations. Their articles contain such misleading statements as: "The similarity indexes show that from one-third to one-half of the aged are poorer, are employed less, work in lower-status occupations, have less education and poorer health than similar proportions of the nonaged" (1971:89). "Even among those [aged] who are working there has been a sharp decline in the number of hours worked as shown by the 18-point drop in the hours worked SI" (1971:88). "The housing El shows a fairly steady improvement from 1950.... Thus, the quality of nonwhite housing has improved steadily, even relative to white housing . . ." (1970:113). "The rising discontent among Negroes cannot be explained by changes in their absolute or relative socioeconomic position because the El shows that Negroes are gaining in most areas, both absolutely and relative to whites" (1970:115). In fact, there is nothing about the SI (or, by its other name, the El) that makes any of these