Subtleties of Distributions

How are we to investigate causes with complex effects? The effect of divorce and remarriage on children's academic performance is complex, and yet more and more of a pressing reality in our fragmenting society. What is the academic penalty suffered by children as a result of divorce or the death of a parent? Should society be encouraging single or widowed mothers to remarry (e.g., by tax incentives) in hopes of establishing a new stable family, or should we be encouraging them to stay single, to avoid yet another emotional trauma in the child's life?

The National Education Longitudinal Study (NELS) contains data that is intended to throw light on this issue. One substantively important academic indicator is reading performance. The 1988 sample of 24,599 children contains reading measures for children in all three family conditions. Here are their summary statistics:

Reading Measure, 8th Grade 1988 Mean Standard Deviation
Intact Families
Single-parent Families
Reconstituted Families

What are we to make of this? The intact families do best - as we would predict. Children in single-parent families perform better than in reconstituted families. The policy-makers appear to have their answer: tell the parent not to remarry! But what about dispersions?

Now look at the plot. It challenges the accepted wisdom about the affect of family status on academic performance. Are reconstituted families just like intact families? No, the distribution of children in intact families is to the right, but it is not a huge effect. Are reconstituted families automatically better than single-parent families? No, their means essentially coincide - in fact, there is no advice to policy-makers here. The standard deviations, however, are even more perplexing. The tails of the single-parent distribution are slightly wider than the reconstituted, so that the cumulative curves cross. These data are ambiguous.

William H. Jeynes
Department of Education, University of Chicago

Subtleties of distributions. Jeynes W.H. … Rasch Measurement Transactions, 1996, 10:3 p. 515.

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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