A model for the analysis of paired data

There is a certain opposition between the ideal of equal opportunity and that of family responsibility. Responsibility involves autonomy, which will produce divergence among families, which, in turn, will mean divergent conditions for the children; that is, unequal opportunities. (Charles Horton Cooley, cited in Blau & Duncan, 1967, p. vii) This quote, from a turn-of-the-century social philosopher, reflects the longstanding importance social scientists have placed on the family as an agent in the generation of social and economic well-being. Perhaps the most accepted fact in the social sciences is that what goes on in the family is important for understanding the opportunities and constraints children face as they grow into adults, Indeed, a substantial body of literature has evolved in an attempt to measure the impact of family background, broadly conceived, on attainment and well-being. Unfortunately, while the family is given theoretical primacy, the methodological tools most commonly applied to study the effects of families are inadequate. The most common and damaging deficiency is that data from one level of analysis, individuals, are used to examine relationships at another level of analysis, families. This is a common problem, and one that has long been recognized. For example, several articles appeared in the 1982 special methodology issue of Journal of Marriage and the Family that argued for the use of measurement and statistical procedures appropriate to the consideration of relationships between and within dyads (Gottman, 1982; Hill & Scanzoni, 1982; Huston & Robins, 1982; Thompson & Walker, 1982; Thomson & Williams, 1982). Based on work by Acock and Schumm (1992) and Hauser (1984, 1988), we expand on the approach taken by Thomson and Williams (1982) by outlining a procedure by which data collected from individuals can be used to examine statistical relationships between concepts on more than one level. The basic requirement of the procedure is that paired data be collected from two or more members of the same group (for our purposes, a family). This is a common feature of many data bases, given the fact that households are often the smallest sampling unit in a survey. The procedure proposed to analyze such data is based on a latent-variable structural equation model and allows for consideration of both within- and between-family variation in patterns of effects. The model assumes that measurements on individuals can serve as indicators of family-level concepts. The pedagogical example used focuses on mental ability and grades in high school, The analysis uses data on siblings collected in a large, national study of educational attainment. While the example is sometimes contrived, it is sufficiently complex to illustrate the wide variety of issues that can be addressed with the model proposed. THE PROBLEM Consider the sources of variation between children in different families that may be attributed to family background. These include, but are not limited to, genetic heritage, parental financial and human resources, parental standards and forms of parent-child interaction (barring substantial temporal change in family form due to factors like divorce and remarriage), and a common social environment involving neighborhoods, schools, and time-specific historical conditions (wars, business cycles, local labor market conditions). Even this limited list indicates the diversity of sources of variation in child well-being that can be attributed to families. Indeed, many of these factors may not represent what some researchers would ordinarily consider to be direct influences of the family, but are, rather, more indirect effects operating through accidents of history or the decisions families make concerning residential mobility and so on. It remains true, though, that each of these sources of variation are tied to membership in a family. To avoid confusion with prior research, we use the term family environment to reference broader sources of variation between families, while the term family background is used in a more limited sense to indicate more standard measures of parental characteristics and family composition. …