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Simpson-type paradoxes, dependence, and ageing

  • Marco Scarsini (a1) and Fabio Spizzichino (a2)

We will state a general version of Simpson's paradox, which corresponds to the loss of some dependence properties under marginalization. We will then provide conditions under which the paradox is avoided. Finally we will relate these Simpson-type paradoxes to some well-known paradoxes concerning the loss of ageing properties when the level of information changes.

Corresponding author
Postal address: Dipartimento di Scienze, Università D'Annunzio, Viale Pindaro 42, I-65127 Pescara, Italy. Email address:
∗∗ Postal address: Dipartimento di Matematica, Università ‘La Sapienza’, Piazzale Aldo Moro 5, I-00185 Roma, Italy. Email address:
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Arjas, E. (1981a). A stochastic process approach to multivariate reliability systems: Notions based on conditional stochastic order. Math. Operat. Res. 6, 263276.
Arjas, E. (1981b). The failure and hazard processes in multivariate reliability theory. Math. Operat. Res. 6, 551562.
Arjas, E., and Norros, I. (1984). Life lengths and association: a dynamic approach. Math. Operat. Res. 9, 151158.
Arjas, E., and Norros, I. (1991). Stochastic order and martingale dynamics in multivariate life length models: a review. In Stochastic Orders and Decision under Risk, ed. Mosler, K. and Scarsini, M. IMS Lecture Notes/Monograph Series, Hayward, California, pp. 724.
Barlow, R. E. (1985). A Bayesian explanation of an apparent failure rate paradox. IEEE Trans. Rel. 34, 107108.
Barlow, R. E., and Pereira, C. A. de B. (1990). Conditional independence and probabilistic influence diagrams, in Topics in Statistical Dependence, ed. Block, H. W., Sampson, A. R. and Savits, T. H. IMS Lecture Notes/Monograph Series, Hayward, California, pp. 1933.
Barlow, R. E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing, Holt, Rinehart, and Winstion, New York, NY.
Blyth, C. R. (1972). On Simpson's paradox and the sure thing principle. J. Amer. Statist. Assoc. 67, 364366.
Blyth, C. R. (1973). Simpson's paradox and mutually favorable events. J. Amer. Statist. Assoc. 68, 746.
Cartwright, N. (1979). Causal laws and effective strategies. Noûs 13, 419437.
Cohen, J. E. (1986). An uncertainty principle in demograpy and the unisex issue. Amer. Statistician 40, 3239.
Eells, E. (1987). Cartwright and Otte on Simpson's paradox. Philosophy of Science 54, 233243.
Good, I. J., and Mittal, Y. (1987). The amalgamation and geometry of two-by-two contingency tables. Ann. Statist. 15, 694711.
Gudder, S. P. (1988). Quantum Probability, Academic Press, Boston, MA.
Hardcastle, V. G. (1991). Partitions, probabilistic causal laws, and Simpson's paradox. Synthése 86, 209228.
Jensen, U. (1996). Stochastic models of reliability and maintenance: an overview. In Reliability and Maintenance of Complex Systems (ed. Özekici, S.). Springer, Berlin, pp. 336.
Jogdeo, K. (1978). On a probability bound of Marshall and Olkin. Ann. Statist. 6, 232234.
Karlin, S., and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467498.
Kimeldorf, G., and Sampson, A. R. (1989). A framework for positive dependence. Ann. Inst. Statist. Math. 41, 3145.
Langberg, N. A., Leòn, R. V., and Proschan, F. (1980). Characterization of nonparamteric classes of life distributions. Ann. Prob. 8, 11631170.
Lindley, D. V., and Novik, M. R. (1981). On the role of exchangeability in inference. Ann. Statist. 9, 4558.
Mittal, Y. (1991). Homogeneity of subpopulations and Simpson's paradox. J. Amer. Statist. Assoc. 86, 167172.
Norros, I. (1985). Systems weakened by failures. Stochast. Proc. Appl. 20, 181196.
Otte, R. (1985). Probabilistic causality and Simpson's paradox. Philosophy of Science 52, 110125.
Ross, S. M. (1984). A model in which component failure rates depend on the working set. Naval Res. Logist. Quart. 31, 297300.
Saari, D. G. (1995). A chaotic exploration of aggregation paradoxes. SIAM Rev. 37, 3752.
Samuels, M. L. (1993). Simpson's paradox and related phenomena. J. Amer. Statist. Assoc. 88, 8188.
Shaked, M., and Shanthikumar, J. G. (1988). Multivariate conditional hazard rates and the MIFRA and MIFR properties. J. Appl. Prob. 25, 150168.
Shaked, M., and Shanthikumar, J. G. (1990). Multivariate stochastic orderings and positive dependence in reliability theory. Math. Operat. Res. 15, 545552.
Shaked, M., and Shanthikumar, J. G. (1991). Dynamic multivariate ageing. Stoch. Proc. Appl. 38, 8597.
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications, Academic Press, Boston, MA.
Simpson, E. H. (1951). The interpretation of interaction in contingency tables. J. Roy. Statist. Soc. B 13, 238241.
Vaupel, J. V., and Yashin, A. I. (1985). Some surprising effects of selection on population dynamics. Amer. Statistician 39, 176185.
Yule, G. U. (1903). Notes on the theory of association of attributes in statistics. Biometrika 2, 121134.
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Journal of Applied Probability
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