![[PDF]](/pb-assets/Icons/adobe-pdf-icon-1498649896243.png){kind=icon}
Abstract
Regression coefficients cannot be interpreted as causal if the relationship can be attributed to an alternate mechanism. One may control for the alternate cause through an experiment (e.g., with random assignment to treatment and control) or by measuring a corresponding confounding variable and including it in the model. Unfortunately, there are some circumstances under which it is not possible to measure or control for the potentially confounding variable. Under these circumstances, it is helpful to assess the robustness of a statistical inference to the inclusion of a potentially confounding variable. In this article, an index is derived for quantifying the impact of a confounding variable on the inference of a regression coefficient. The index is developed for the bivariate case and then generalized to the multivariate case, and the distribution of the index is discussed. The index also is compared with existing indexes and procedures. An example is presented for the relationship between socioeconomic background and educational attainment, and a reference distribution for the index is obtained. The potential for the index to inform causal inferences is discussed, as are extensions.
|
Abbott, Andrew
. 1998. “The Causal Devolution.” Sociological Methods & Research 27:148-181. Google Scholar | SAGE Journals | ISI | |
|
Abelson, Robert P. 1997. “On the Surprising Longevity of Flogged Horses.” Psychological Science 8 (1): 12-15. Google Scholar | SAGE Journals | ISI | |
|
Anderson, Sharon
, A. Auquier , W.W. Hauck , D. Cakes , W. Vandaele , H. I. Weisberg , A. S. Bryk , and J. Kleinman . 1980. Statistical Methods for Comparative Studies. New York: Wiley. Google Scholar | |
|
Angrist, Joshua D.
, Guide W. Imbens , and Donald B. Rubin . 1996. “Identification of Causal Effects Using Instrumental Variables [With Discussion].” Journal of the American Statistical Association 91:444-472. Google Scholar | Crossref | ISI | |
|
Aroian, L.A.
, V.S. Taneja , and L. W. Cornwell . 1978. “Mathematical Forms of the Distribution of the Product of Two Normal Variables.” Communications in Statistics: Part A—Theory and Methods 7:165-172. Google Scholar | Crossref | ISI | |
|
Asher, H.
1983. Causal Modeling. Beverly Hills, CA: Sage. Google Scholar | |
|
Bakan, D. 1966. “The Test of Significance in Psychological Research.” Psychological Bulletin 66:423-437. Google Scholar | Crossref | Medline | ISI | |
|
Blalock, H. M., Jr.
1971. Causal Models in the Social Sciences. Chicago: Aldine. Google Scholar | |
|
Blau, P. M.
and O. D. Duncan . 1967. The American Occupational Structure. New York: Wiley. Google Scholar | |
|
Bourdieu, P.
1986. “The Forms of Capital.” Pp. 241-258 in Handbook of Theory and Research for the Sociology of Education, edited by J. G. Richardson . New York: Greenwood. Google Scholar | |
|
Bowles, S.
and H. Gintis . 1976. Schooling in Capitalist America. New York: Basic Books. Google Scholar | |
|
Bross, Irwin D. J. 1966. “Spurious Effects From an Extraneous Variable.” Journal of Chronic Disease 19:637-647. Google Scholar | Crossref | Medline | |
|
Bross, Irwin D. J. 1967. “Pertinency of an Extraneous Variable.” Journal of Chronic Disease 20:487-495. Google Scholar | Crossref | Medline | |
|
Bryk, Anthony S.
, Valerie E. Lee , and Peter B. Holland . 1993. Catholic Schools and the Common Good. Cambridge, MA: Harvard University Press. Google Scholar | |
|
Carver, Ronald
. 1978. “The Case Against Statistical Significance Testing.” Harvard Educational Review 48:378-399. Google Scholar | Crossref | ISI | |
|
Chatterjee, Samprit
and Ali S. Hadi . 1988. Sensitivity Analysis in Linear Regression. New York: Wiley. Google Scholar | |
|
Chow, Siu L. 1988. “Significance Test or Effect Size.” Psychological Bulletin 103(1): 105-110. Google Scholar | Crossref | ISI | |
|
Clogg, Clifford C.
, Eva Petkova , and Edward S. Shihadeh . 1992. “Statistical Methods for Analyzing Collapsibility in Regression Models.” Journal of Educational Statistics 17(1):51-74. Google Scholar | SAGE Journals | |
|
Cochran, W. G. 1953. “Matching in Analytical Studies.” American Journal of Public Health 43:684-691. Google Scholar | Crossref | ISI | |
|
Cochran, W. G. 1965. “The Planning of Observational Studies of Human Populations.” Journal of the Royal Statistical Society Series A, 234-266. Google Scholar | Crossref | ISI | |
|
Cochran, W. G.
1970. “Performance of a Preliminary Test of Comparability in Observational Studies.” Office of Naval Research Technical Report No. 29, Harvard University. Google Scholar | |
|
Cohen, Jacob
. 1990. “Things I Have Learned (So Far).” American Psychologist 45:1304-1312. Google Scholar | Crossref | ISI | |
|
Cohen, Jacob
. 1994. “The Earth Is Round (p<.05).” American Psychologist 49:997-1003. Google Scholar | Crossref | ISI | |
|
Cohen, J.
and P. Cohen . 1983. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences. Hillsdale, NJ: Lawrence Erlbaum. Google Scholar | |
|
Cohen, M.R.
and E. Nagel . 1934. An Introduction to Logic and the Scientific Method. New York: Harcourt Brace. Google Scholar | |
|
Cook, Thomas
and Donald T. Campbell . 1979. Quasi-Experimentation: Design and Analysis Issues for Field Settings. Boston: Houghton Mifflin. Google Scholar | |
|
Cordray, D. S. 1986. “Quasi-Experimental Analysis: A Mixture of Methods and Judgment.” New Directions for Program Evaluation 31:9-27. Google Scholar | Crossref | |
|
Cornfield, J.
, W. Haenszel , A. M. Lilienfeid , M. B. Shimkin , and E.L. Wynder . 1959. “Smoking and Lung Cancer: Recent Evidence and a Discussion of Some Questions.” Journal of the National Cancer Institute 22:173-203. Google Scholar | Medline | |
|
Cortina, Jose M.
and William P. Dunlap . 1997. “On the Logic and Purpose of Significance Testing.” Psychological Methods 2:161-172. Google Scholar | Crossref | ISI | |
|
Davis, W. A.
1988. “Probabilistic Theories of Causation.” Pp. 133-160 in Probability and Causality, edited by James H. Fetter . Dordrecht: D. Reidel. Google Scholar | |
|
Dempster, Arthur P. 1988. “Employment Discrimination and Statistical Science.” Statistical Science 3:149-195. Google Scholar | Crossref | |
|
Dowe, Phil
. 1992. “Wesley Salmon's Process Theory of Causality and the Conserved Quantity Theory.” Philosophy of Science 59:195-216. Google Scholar | Crossref | ISI | |
|
Draper, N. R.
and H. Smith . 1981. Applied Regression Analysis. New York: Wiley. Google Scholar | |
|
Duncan, Otis D.
, David L. Featherman , and Beverly Duncan . 1972. Socioeconomic Background and Achievement. New York: Seminar Press. Google Scholar | |
|
Einhorn, Hillel J.
and Robin M. Hogarth . 1986. “Judging Probable Cause.” Psychological Review 99 (1): 3-19. Google Scholar | |
|
Featherman, David L.
and Robert M. Hauser . 1976. “Sexual Inequalities and Socioeconomic Achievement in the U.S. 1962-1973.” American Sociological Review 41:462-483. Google Scholar | Crossref | ISI | |
|
Freedman, David
. 1997. “From Association to Causation via Regression.” Advances in Applied Mathematics 18:59-110. Google Scholar | Crossref | ISI | |
|
Frick, Robert W. 1999. “Defending the Statistical Status Quo.” Theory and Psychology 9:183-189. Google Scholar | SAGE Journals | ISI | |
|
Gage, N. L.
1978. The Scientific Basis of the Art of Teaching. New York: Teachers College Press. Google Scholar | |
|
Giere, R.
1981. “Causal Systems and Statistical Hypotheses.” Pp. 251-270 in Applications of Inductive Logic, edited by L. J. Cohen and M. Hesse . Oxford, UK: Clarendon. Google Scholar | |
|
Gigerenzer, G.
1993. “The Superego, the Ego, and the Id in Statistical Reasoning.” Pp. 311-339 in A Handbook for the Data Analysis in the Behavioral Sciences: Methodological Issues, edited by G. Keren and C. Lewis . Hillsdale, NJ: Lawrence Erlbaum. Google Scholar | |
|
Godfrey, L. G.
1988. Misspecification Tests in Econometrics. Cambridge, UK: Cambridge University Press. Google Scholar | |
|
Granger, C.W.J. 1969. “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods.” Econometrica 37:424-438. Google Scholar | Crossref | ISI | |
|
Grayson, D.A. 1998. “The Frequentist Facade and the Flight From Evidential Inference.” British Journal of Psychology 89:325-345. Google Scholar | Crossref | ISI | |
|
Hanushek, E.
1977. Statistical Methods for Social Scientists. New York: Academic Press. Google Scholar | |
|
Harlow, L. L.
, S. A. Mulaik , and J. H. Steiger . 1997. What If There Were No Significance Tests. Mahwah, NJ: Lawrence Erlbaum. Google Scholar | |
|
Harris, Richard J. 1997. “Significance Tests Have Their Place.” Psychological Science 8 (1): 8-11. Google Scholar | SAGE Journals | ISI | |
|
Hausman, J. A. 1978. “Specification Tests in Econometrics.” Econometrica 46:1251-1271. Google Scholar | Crossref | ISI | |
|
Heckman, J.
and R. Robb . 1985. “Alternative Methods for Evaluating the Impact of Interventions.” Pp. 156-245 in Longitudinal Analysis of Labor Market Data, edited by J. Heckman and B. Singer . New York: Cambridge University Press. Google Scholar | |
|
Holland, Paul W. 1986. “Statistics and Causal Inference.” Journal of the American Statistical Association 81:945-970. Google Scholar | Crossref | ISI | |
|
Holland, Paul W.
1988. “Causal Inference, Path Analysis, and Recursive Structural Equations Models (With Discussion).” Pp. 449-493 in Sociological Methodology, edited by C. C. Clogg . Washington, DC: American Sociological Association. Google Scholar | |
|
Hunter, John E. 1997. “Needed: A Ban on the Significance Test.” American Psychological Society 8 (1): 3-7. Google Scholar | |
|
Jencks, C. 1985. “How Much Do High School Students Learn?” Sociology of Education 58:128-135. Google Scholar | Crossref | ISI | |
|
Kitcher, P.
1989. “Explanatory Unification and the Causal Structure of the World.” Pp. 410-505 in Minnesota Studies in the Philosophy of Science, vol. 13, Scientific Explanation, edited by P. Kitcher and W. Salmon . Minneapolis: University of Minnesota Press. Google Scholar | |
|
Lareau, A.
1989. Home Advantage: Social Class and Parental Intervention in Elementary Education. New York: Falmer. Google Scholar | |
|
Lewis, S. 1973. “Causation.” Journal of Philosophy 70:553-567. Google Scholar | |
|
Lin, D. Y.
, B. M. Psaty , and R. A. Kronmal . 1998. “Assessing the Sensitivity of Regression Results to Unmeasured Confounders in Observational Studies.” Biometrics 54:948-963. Google Scholar | Crossref | Medline | ISI | |
|
MacIver, R. M.
1964. Social Causation. New York: Harper Torchbooks. Google Scholar | |
|
Mackie, J.
1974. The Cement of the Universe. Oxford, UK: Oxford University Press. Google Scholar | |
|
Mauro, Robert
. 1990. “Understanding L.O.V.E. (Left Out Variables Error): A Method of Estimating the Effects of Omitted Variables.” Psychological Bulletin 108:314-329. Google Scholar | Crossref | ISI | |
|
McDonald, R. P. 1997. “Haldane's Lungs: A Case Study in Path Analysis.” Multivariate Behavioral Research 32:1-38. Google Scholar | Crossref | Medline | ISI | |
|
McKinlay, Sonja M. 1975. “The Design and Analysis of the Observational Study—A Review.” Journal of the American Statistical Association 70:859-864. Google Scholar | ISI | |
|
Meehl, Paul E. 1978. “Theoretical Risks and Tabular Asterisks: Sir Karl, Sir Ronald and the Slow Progress of Soft Psychology.” Journal of Consulting and Clinical Psychology 46:806-834. Google Scholar | Crossref | ISI | |
|
Meeker, W. Q.
, L. W. Cornwell , and L. A. Aroian . 1980. “The Product of Two Normally Distributed Random Variables.” Pp. 1-256 in Selected Tables in Mathematical Statistics, edited by W. J. Kenney and R. E. Odeh . Providence, RI: American Mathematical Society. Google Scholar | |
|
Meeker, William Q.
and Luis A. Escobar . 1994. “An Algorithm to Compute the CDF of the Product of Two Normal Random Variables.” Communications in Statistics—Simulations 23:271-280. Google Scholar | Crossref | ISI | |
|
Mill, J. S.
[1843] 1973. “A System of Logic: Ratiocinative and Inductive.” In The Collected Works of John Stuart Mill, vols. 7, 8, edited by J. M. Robson . Toronto: University of Toronto Press. Google Scholar | |
|
Montgomery, D. C.
and E. A. Peck , 1982. Introduction to Linear Regression Analysis. New York: Wiley. Google Scholar | |
|
Morrison, D. E.
and R. E. Henkel . 1970. The Significance Test Controversy. Chicago: Aldine. Google Scholar | |
|
Oakes, M.
1986. Statistical Inference: A Commentary for the Social and Behavioral Sciences. New York: Wiley. Google Scholar | |
|
Pedhazur, E.
1982. Multiple Regression in Behavioral Research: Explanation and Prediction. 2d ed. San Francisco: Holt, Rinehart & Winston. Google Scholar | |
|
Pollard, Paul
. 1993. “How Significant Is `Significance?' ” Pp. 449-460 in A Handbook for the Data Analysis in the Behavioral Sciences, Methodological Issues, edited by G. Keren and C. Lewis . Hillsdale, NJ: Lawrence Erlbaum. Google Scholar | |
|
Pratt, John W.
and Robert Schlaifer . 1988. “On the Interpretation and Observation of Laws.” Journal of Econometrics 39:23-52. Google Scholar | Crossref | ISI | |
|
Ramsey, J.
1974. “Classical Model Selection Through Specification Error Tests.” Pp. 13-47 in Frontiers in Econometrics, edited by P. Zarembka . New York: Academic Press. Google Scholar | |
|
Reichenbach, H.
1956. The Direction of Time. Berkeley: University of California Press. Google Scholar | |
|
Rice, J. A.
1988. Mathematical Statistics and Data Analysis. Pacific Grove, CA: Wadsworth & Brooks. Google Scholar | |
|
Rosenbaum, Paul R. 1986. “Dropping Out of High School in the United States: An Observational Study.” Journal of Educational Statistics 11:207-224. Google Scholar | SAGE Journals | |
|
Rosenbaum, Paul R.
and Donald B. Rubin . 1983. “The Central Role of the Propensity Score in Observational Studies for Causal Effects.” Biometrika 70 (1): 41-55. Google Scholar | Crossref | ISI | |
|
Rozenboom, William W. 1960. “The Fallacy of the Null Hypothesis Significance Test.” Psychological Bulletin 57:416-428. Google Scholar | Crossref | Medline | ISI | |
|
Rubin, Donald B. 1974. “Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies.” Journal of Educational Psychology 66:688-701. Google Scholar | Crossref | ISI | |
|
Ryan, T. P.
1997. Modern Regression Methods. New York: Wiley. Google Scholar | |
|
Salmon, Wesley
. 1984. Scientific Explanation and the Causal Structure of the World. Princeton, NJ: Princeton University Press. Google Scholar | |
|
Salmon, Wesley
. 1994. “Causality Without Counterfactuals.” Philosophy of Science 61:297-312. Google Scholar | Crossref | ISI | |
|
Salmon, Wesley
. 1998. Causality and Explanation. New York: Oxford University Press. Google Scholar | |
|
Scheines, R.
, P. Spirites , C. Glymour , C. Meek , and T. Richardson . 1998. “The TETRAD Project: Constraint Based Aids to Causal Model Specification.” Multivariate Behavioral Research 33:65-117. Google Scholar | Crossref | Medline | ISI | |
|
Schlesselman, J. J. 1978. “Assessing Effects of Confounding Variables.” American Journal of Epidemiology 108:3-8. Google Scholar | Medline | ISI | |
|
Schmidt, Frank L. 1996. “Statistical Significance Testing and Cumulative Knowledge in Psychology: Implications for Training of Researchers.” Psychological Methods 1:115-129. Google Scholar | Crossref | ISI | |
|
Sewell, W. H.
, R. M. Hauser , and W. C. Wolf . 1980. “Sex, Schooling, and Occupational Status.” American Journal of Sociology 86:551-583. Google Scholar | Crossref | ISI | |
|
Snedecor, G. W.
and W. G. Cochran . 1989. Statistical Methods. 6th ed. Ames: Iowa State University Press. Google Scholar | |
|
Sobel, Michael E.
1995. “Causal Inference in the Social and Behavioral Sciences.” Pp. 1-38 in Handbook of Statistical Modeling for the Social and Behavioral Sciences, edited by Gerhard Arminger , Clifford C. Clogg , and Michael E. Sobel . New York: Plenum. Google Scholar | |
|
Sobel, Michael E. 1996. “An Introduction to Causal Inference.” Sociological Methods & Research 24:353-379. Google Scholar | SAGE Journals | ISI | |
|
Sobel, Michael E. 1998. “Causal Inference in Statistical Models of the Process of Socioeconomic Achievement: A Case Study.” Sociological Methods & Research 27:318-348. Google Scholar | SAGE Journals | ISI | |
|
Sober, E.
1988. “The Principle of the Common Cause.” Pp. 211-228 in Probability and Causality, edited by James H. Fetter . Dordrecht: D. Reidel. Google Scholar | |
|
Sohn, David
. 1998. “Statistical Significance and Replicability.” Theory & Psychology 8:291-311. Google Scholar | ISI | |
|
Steyer, Rolf
and Thomas Schmitt . 1994. “The Theory of Confounding and Its Application in Causal Modeling With Latent Variables.” Pp. 36-67 in Latent Variables Analysis: Applications for Developmental Research, edited by Alex von Eye and Clifford Clogg . Thousand Oaks, CA: Sage. Google Scholar | |
|
Suppes, P.
1970. A Probabilistic Theory of Causality. Amsterdam: North-Holland. Google Scholar | |
|
Thompson, Bruce
. 1999. “If Statistical Significance Tests Are Broken/Misused, What Practices Should Supplement or Replace Them?” Theory & Psychology 9:165-181. Google Scholar | SAGE Journals | ISI | |
|
Tukey, J. W.
1977. Exploratory Data Analysis. Reading, MA: Addison-Wesley. Google Scholar | |
|
Tukey, J. W. 1991. “The Philosophy of Multiple Comparisons.” Statistical Science 6:100-116. Google Scholar | Crossref | |
|
Wainer, Howard
. 1999. “One Cheer for Null Hypothesis Significance Testing.” Psychological Methods 4:212-213. Google Scholar | Crossref | ISI | |
|
Weisberg, Sanford
. 1985. Applied Linear Regression. New York: Wiley. Google Scholar | |
|
Wilkinson, L.
and Task Force on Statistical Inference . 1999. “Statistical Methods in Psychology Journals: Guidelines and Explanations.” American Psychologist 54:594-604. Google Scholar | Crossref | ISI | |
|
Zellner, A.
1984. Basic Issues in Econometrics. Chicago: University of Chicago Press. Google Scholar |
