The authors reply to Zax's critique of the double-equation method for ecological regression and of the specific extension to it proposed by Grofman and Migalski. Although Zax does correct two minor errors in Grofman and Migalski's statement of the double-equation approach, neither of those errors affected the final calculations reported in their article. Furthermore, nothing Zax reports affects their fundamental conclusion that double-equation methods can be superior to single-equation techniques if there is substantial error in the measurement of the independent variable. In particular, by analyzing an election for which, from exit polls, the ``true'' parameters of Hispanic and non-Hispanic levels of political cohesion are known, the authors show that double-equation ecological regression estimates derived from registration data are highly accurate in reproducing the true individual-level behavioral parameters (group means).
Abrajano, Marissa, Jonathan Nagler, and Michael Alvarez .2005. ``A Natural Experiment of Race-Based and Issue Voting: The 2001 City of Los Angeles Elections. '' Political Research Quarterly58 (2): 203-18.
2.
Barreto, Matt A., Mario Villarreal, and Nathan Woods.2005. ``Metropolitan Latino Political Behavior: Turnout and Candidate Preference in Los Angeles.''Journal of Urban Affairs27:71-91.
3.
Bullock, Charles S., III and Ronald Keith Gaddie.2005. ``An Assessment of Voting Rights Progress in Georgia.'' Available at http://www.aei.org/docLib/20060210_Georgia.pdf.
4.
--. 2006. ``An Assessment of Voting Rights Progress in Louisiana.'' Available at http://www.aei.org/docLib/20060308_Louisiana.pdf .
5.
Cho, Wendy Tam and Brian J. Gaines .2004. ``The Limits of Ecological Inference: The Case of Split-Ticket Voting.''American Journal of Political Science48 (1): 152-71.
6.
Cho, Wendy Tam, George C. Judge, and Bruce Cain.2002. ``Some Empirical Evidence on the Impact of Measurement Errors in Making Ecological Inferences.'' Unpublished manuscript.
7.
Freedman, David, S.P. Klein, J. Sacks, C.A. Smyth and C.G. Everett.1991. ``Ecological Regression and Voting Rights .'' Evaluation Review15(6): 673-711.
8.
Gelman, Andrew, Stephen Ansolabehere, Phillip N. Price, David K. Park, and Lorraine C. Minnite .2001. ``Models, Assumptions, and Model Checking in Ecological Regressions.''Journal of the Royal Statistical Society164:101-18.
9.
Goodman, Leo.1953. ``Ecological Regression and the Behavior of Individuals .'' American Sociological Review18 (6): 663-64.
10.
--. 1959. ``Some Alternatives to Ecological Correlation. '' American Journal of Sociology64:610-25.
11.
Greiner, D. James.2007. ``Ecological Inference in Voting Rights Disputes: Where Are We Now, and Where Do We Want to Be?''Jurimetrics47 (2): 115-67.
12.
Grofman, Bernard.1991. ``Statistics Without Substance: A Critique of Freedman et al. and Clark and Morrison.''Evaluation Review15 (6): 746-69.
13.
--. 2000. ``A Primer of Racial Bloc Voting.'' Pp. 44-67 in The Real Y2K Problem: Census 2000 Data and Redistricting Technology, edited by Nathaniel Persily.New York : Bren-nan Center for Law and Justice, New York University .
14.
Grofman, Bernard, Lisa Handley, and Richard Niemi.1992. Minority Representation and the Quest for Voting Equality. New York: Cambridge University Press.
15.
Grofman, Bernard and Michael Migalski .1988. ``Estimating the Extent of Racially Polarized Voting in Multicandidate Elections.'' Sociological Methods & Research16 (4): 427-54.
16.
Grofman, Bernard, Michael Migalski, and Nicholas Noviello .1985. ``The `Totality of Circumstances' Test in Section 2 of the 1982 Extension of the Voting Rights Act: A Social Science Perspective.'' Law and Policy7 (2), 209-23.
17.
Herron, Michael C. and Kenneth W. Shotts .2003a. ``Cross Contamination and EI-R.''Political Analysis11:77-85.
18.
--. 2003b. ``Using Ecological Inference Point Estimates as Dependent Variables in Second-Stage Linear Regressions.'' Political Analysis11:44-64.
19.
King, Gary.1997. A Solution to the Ecological Inference Problem. Princeton, NJ: Princeton University Press.
20.
Kousser, Morgan J.1973. ``Ecological Regression and Analysis of Past Politics.''Journal of Interdisciplinary History4:237-62.
21.
--. 2001. ``Ecological Inference From Goodman to King. '' Historical Methods34:100-26.
Liu, Baodong.2001. ``The Positive Effect of Black Density on White Crossover Voting: Reconsidering Social Interaction Theory.'' Social Science Quarterly82:600-13.
24.
--. 2007. ``EI Extended Model and the Fear of Ecological Fallacy .'' Sociological Methods & Research36:3-25.
25.
Loewen, James and Bernard Grofman .1989. ``Comment on Recent Developments in Methods Used in Voting Rights Litigation.'' Urban Lawyer21 (3): 589-604.
26.
Owen, Guillermo and Bernard Grofman .1997. ``Estimating the Likelihood of Fallacious Ecological Inference: Linear Ecological Regression in the Presence of Context Effects.'' Political Geography16 (8): 657-90.
27.
Phelan, Thomas.2001. ``Comparing Individual-Level Returns With Aggregates: A Historical Appraisal of the King Solution.'' Historical Methods34:127-34.
28.
Sonenshein, Raphael and Susan Pinkus.2002. ``The Dynamics of Latino Political Incorporation: The 2001 Los Angeles Mayoral Election as Seen in Los Angeles Times Exit Polls .'' PS: Political Science & Politics35:67-74.
29.
Voss, Stephen.2004. ``Using Ecological Inference for Contextual Research.'' Pp. 69-96 in Ecological Inference: New Methodological Strategies, edited by Gary King, Ori Rosen, and Martin Tanner.New York: Cambridge University Press.
30.
Word, David L. and R. Colby Perkins .1996. ``Building a Spanish Surname List for the 1990's-A New Approach to an Old Problem.'' Technical Working Paper No. 13. Washington: U.S. Census Bureau.
31.
Zax, Jeffrey S.2002. ``Comment on `Estimating the Extent of Racially Polarized Voting in Multicandidate Contests' by Bernard Grofman and Michael Migalski.''Sociological Methods & Research31:75-86.
32.
--. 2005. ``The Statistical Properties and Empirical Performance of Double Regression.'' Political Analysis13 (1): 57-76.
