Notwithstanding a large body of literature on log-linear models and odds ratios, no general marginal-free index of the association in a contingency table has gained a wide acceptance. Building on a framework developed by L. A. Goodman, we put into light the direct links between odds ratios, the Altham index, the intrinsic association coefficient, and coefficients in log-multiplicative models including Unidiff and row-column association models. We devise a normalized version of the latter coefficient varying between 0 and 1, which offers a simpler interpretation than existing indices similar to the correlation coefficient. We illustrate with the case of educational and socioeconomic homogamy among 149 European regions how this index can be used either alone in a non- or semiparametric approach or combined with models, and how it can protect against incorrect conclusions based on models which rely on strong assumptions to summarize the strength of association as a single parameter.
AgrestiAlan. 2002. Categorical Data Analysis. 2nd ed. New York: Wiley.
2.
AlthamPatricia M. E.1970. “The Measurement of Association of Rows and Columns for an R x S Contingency Table.” Journal of the Royal Statistical Society. Series B (Methodological)32(1):63–73.
3.
AlthamPatricia. M. E.FerrieJoseph P.. 2007. “Comparing Contingency Tables Tools for Analyzing Data from Two Groups Cross-classified by Two Characteristics.” Historical Methods40(1):3–16.
4.
BeckerMark P.1989. “On the Bivariate Normal Distribution and Association Models for Ordinal Categorical Data.” Statistics & Probability Letters8(5):435–40.
5.
BeckerMark P.CloggClifford C.. 1988. “A Note on Approximating Correlations from Odds Ratios.” Sociological Methods & Research16(3):407–24.
6.
BeckerMark P.CloggClifford C.. 1989. “Analysis of Sets of Two-way Contingency Tables Using Association Models.” Journal of the American Statistical Association84(405):142–51.
7.
BishopYvonne M.FienbergStephen E.HollandPaul W.. [1975] 2007. Discrete Multivariate Analysis: Theory and Practice. New York: Springer.
8.
BlandenJo. 2013. “Cross-country Rankings in Intergenerational Mobility: A Comparison of Approaches from Economics and Sociology.” Journal of Economic Surveys27(1):38–73.
9.
Bouchet-ValatMilan. 2018a. “Educational and Socioeconomic Homogamy, Development Level, and Metropolisation across 149 European Regions.” Revue européenne des sciences sociales/European Journal of Social Sciences56(1): 53–84.
10.
Bouchet-ValatMilan. 2018b. “logmult: Log-multiplicative Models, Including Association Models.” R package. Version 0.7.0. Retrieved May 24, 2019 (http://CRAN.R-project.org/package=logmult).
11.
BourdieuJérômeFerrieJoseph P.KesztenbaumLionel. 2009. “Vive La Différence? Intergenerational Mobility in France and the United States during the Nineteenth and Twentieth Centuries.” Journal of Interdisciplinary History39(4):523–57.
12.
BreenRichardLuijkxRuudMüllerWalterPollakReinhard. 2009. “Nonpersistent Inequality in Educational Attainment: Evidence from Eight European Countries.” American Journal of Sociology114(5):1475–521.
13.
CloggClifford C.1982. “Some Models for the Analysis of Association in Multiway Cross-classifications Having Ordered Categories.” Journal of the American Statistical Association77(380):803–15.
DomańskiHenrykPrzybyszDariusz. 2007. “Educational Homogamy in 22 European Countries.” European Societies9(4):495–26.
18.
EbbinghausBernhard. 2005. “When Less Is More. Selection Problems in Large-N and Small-N Cross-national Comparisons.” International Sociology20(2):133–52.
19.
EriksonRobertGoldthorpeJohn H.. 1992. The Constant Flux: A Study of Class Mobility in Industrial Societies. Oxford, England: Clarendon Press.
20.
FerrieJoseph P.2005. “History Lessons: The End of American Exceptionalism? Mobility in the United States Since 1850.” The Journal of Economic Perspectives19(3):199–215.
21.
GilulaZviHabermanShelby J.. 1986. “Canonical Analysis of Contingency Tables by Maximum Likelihood.” Journal of the American Statistical Association81(395):780–88.
22.
GoodmanLeo A.1969. “How to Ransack Social Mobility Tables and Other Kinds of Cross-classification Tables.” American Journal of Sociology75(1):1–40.
23.
GoodmanLeo A.1981a. “Association Models and Canonical Correlation in the Analysis of Cross-classifications Having Ordered Categories.” Journal of the American Statistical Association76(374):320–34.
24.
GoodmanLeo A.1981b. “Association Models and the Bivariate Normal for Contingency Tables with Ordered Categories.” Biometrika68(2):347–55.
25.
GoodmanLeo A.1985. “The Analysis of Cross-classified Data Having Ordered and/or Unordered Categories: Association Models, Correlation Models, and Asymmetry Models for Contingency Tables with or Without Missing Entries.” The Annals of Statistics13(1):10–69.
26.
GoodmanLeo A.1986. “Some Useful Extensions of the Usual Correspondence Analysis Approach and the Usual Log-linear Models Approach in the Analysis of Contingency Tables.” International Statistical Review54(3):243–70.
27.
GoodmanLeo A.1991. “Measures, Models, and Graphical Displays in the Analysis of Cross-classified Data.” Journal of the American Statistical Association86(416):1085–11.
28.
GoodmanLeo A.1996. “A Single General Method for the Analysis of Cross-classified Data: Reconciliation and Synthesis of Some Methods of Pearson, Yule, and Fisher, and also Some Methods of Correspondence Analysis and Association Analysis.” Journal of the American Statistical Association91(433):408–28.
29.
GoodmanLeo A.HoutMichael. 1998. “Statistical Methods and Graphical Displays for Analyzing How the Association Between Two Qualitative Variables Differs Among Countries, among Groups, or Over Time: A Modified Regression-type Approach.” Sociological Methodology28(1):175–230.
30.
GoodmanLeo A.HoutMichael. 2001. “Statistical Methods and Graphical Displays for Analyzing How the Association Between Two Qualitative Variables Differs Among Countries, Among Groups, or Over Time. Part II: Some Exploratory Techniques, Simple Models, and Simple Examples.” Sociological Methodology31(1):189–221.
31.
HoutMichaelBrooksClemManzaJeff. 1995. “The Democratic Class Struggle in the United States, 1948–1992.” American Sociological Review60(6):805–28.
32.
KatrňákTomášFučíkPetrLuijkxRuud. 2012. “The Relationship between Educational Homogamy and Educational Mobility in 29 European Countries.” International Sociology27(4):551–73.
33.
KatrňákTomášKreidlMartinFónadováLaura. 2006. “Trends in Educational Assortative Mating in Central Europe: The Czech Republic, Slovakia, Poland, and Hungary, 1988–2000.” European Sociological Review22(3):309–22.
34.
LiebetrauAlbert M.1983. Measures of Association. Thousand Oaks, CA: Sage.
35.
LongJasonFerrieJoseph. 2013. “Intergenerational Occupational Mobility in Great Britain and the United States Since 1850.” The American Economic Review103(4):1109–37.
ParkHyunjoonSmitsJeroen. 2005. “Educational Assortative Mating in South Korea: Trends 1930–1998.” Research in Social Stratification and Mobility23:103–27.
38.
RaymoJames M.XieYu. 2000. “Temporal and Regional Variation in the Strength of Educational Homogamy.” American Sociological Review65(5):773–81.
39.
RudasTamás. 1998. Odds Ratios in the Analysis of Contingency Tables. Thousand Oaks, CA: Sage.
40.
SkinnerChrisValletLouis-André. 2010. “Fitting Log-linear Models to Contingency Tables from Surveys with Complex Sampling Designs: An Investigation of the Clogg-Eliason Approach.” Sociological Methods & Research39(1):83–108.
41.
SmitsJeroen. 2003. “Social Closure among the Higher Educated: Trends in Educational Homogamy in 55 Countries.” Social Science Research32(2):251–77.
42.
SmitsJeroenUlteeWoutLammersJan. 1998. “Educational Homogamy in 65 Countries: An Explanation of Differences in Openness Using Country-level Explanatory Variables.” American Sociological Review63(2):264–85.
43.
SmitsJeroenUlteeWoutLammersJan. 1999. “Occupational Homogamy in Eight Countries of the European Union, 1975–89.” Acta Sociologica42(1):55–68.
44.
SmitsJeroenUlteeWoutLammersJan. 2000. “More or Less Educational Homogamy? A Test of Different Versions of Modernization Theory Using Cross-temporal Evidence for 60 Countries.” American Sociological Review65(5):781–88.
45.
van der HeijdenPeter G. M.MooijaartAb. 1995. “Some New Log-bilinear Models for the Analysis of Asymmetry in a Square Contingency Table.” Sociological Methods & Research24:7–29.
46.
WongRaymond Sin-Kwok. 2010. Association Models. Thousand Oaks, CA: Sage.
47.
XieYu. 1992. “The Log-multiplicative Layer Effect Model for Comparing Mobility Tables.” American Sociological Review57(3):380–95.
48.
ZhouXiang. 2015. “Shrinkage Estimation of Log-odds Ratios for Comparing Mobility Tables.” Sociological Methodology45(1):320–56.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
