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The 2 x 2 x 2 case in causality, of an effect, a cause and a confounder. A cross-over’s guide to the 2 x 2 x 2 contingency table

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  • Colignatus, Thomas

Abstract

Basic causality is that a cause is present or absent and that the effect follows with a success or not. This happy state of affairs becomes opaque when there is a third variable that can be present or absent and that might be a seeming cause. The 2 x 2 x 2 layout deserves the standard name of the ETC contingency table, with variables Effect, Truth and Confounding and values {S, -S}, {C, -C}, {F, -F}. Assuming the truth we can find the impact of the cause from when the confounder is absent. The 8 cells in the crosstable can be fully parameterized and the conditions for a proper cause can be formulated, with the parameters interpretable as regression coefficients. Requiring conditional independence would be too strong since it neglects some causal processes. The Simpson paradox will not occur if logical consistency is required rather than conditional independence. The paper gives a taxonomy of issues of confounding, a parameterization by risk or safety, and develops the various cases of dependence and (conditional) independence. The paper is supported by software that allows variations. The paper has been written by an econometrician used to structural equations models but visiting epidemiology hoping to use those techniques in experimental economics.

Suggested Citation

  • Colignatus, Thomas, 2007. "The 2 x 2 x 2 case in causality, of an effect, a cause and a confounder. A cross-over’s guide to the 2 x 2 x 2 contingency table," MPRA Paper 3351, University Library of Munich, Germany, revised 14 May 2007.
    • Handle: RePEc:pra:mprapa:3351
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    References listed on IDEAS

    as
    1. Saari,Donald G., 2001. "Decisions and Elections," Cambridge Books, Cambridge University Press, number 9780521808163, January.
    2. Colignatus, Thomas, 2007. "A comparison of nominal regression and logistic regression for contingency tables, including the 2 × 2 × 2 case in causality," MPRA Paper 3615, University Library of Munich, Germany, revised 19 Jun 2007.
    3. Colignatus, Thomas, 2007. "Correlation and regression in contingency tables. A measure of association or correlation in nominal data (contingency tables), using determinants," MPRA Paper 3394, University Library of Munich, Germany, revised 07 Jun 2007.
    4. Saari,Donald G., 2001. "Decisions and Elections," Cambridge Books, Cambridge University Press, number 9780521004046, January.
    5. Colignatus, Thomas, 2007. "A measure of association (correlation) in nominal data (contingency tables), using determinants," MPRA Paper 2662, University Library of Munich, Germany, revised 10 Apr 2007.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Experimental economics; causality; cause and effect; confounding; contingency table; Simpson paradox; conditional independence; risk; safety; epidemiology; correlation; regression; Cornfield’s condition; inference;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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