Neymans causal model with stochastic potential outcomes: implications for the completely randomized design

  • Abstract
  • Keywords
  • References
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  • Abstract

    In Neymans causal model (NCM), each subject participating in a two-arm randomized trial has a pair of potential outcomes one outcome would be observed under treatment and another under control. In the stochastic version of NCM the two potential outcomes are viewed as possibly non-degenerate random variables with finite expectations and variances. The subject-level treatment effect is the expected outcome under treatment minus that under control, and the average treatment effect is the arithmetic mean of the subject-level effects. In the present paper properties of the ordinary difference of means estimator and its associated variance estimator are examined in the completely randomized design with stochastic potential outcomes. Estimation theory is developed under randomization distribution without commitment to any particular probability model for enrollment, because in real trials subjects are not enrolled by a sampling mechanism with known selection probabilities. It is shown that in this theoretical framework, the difference of means estimator is asymptotically normal and consistent for the average treatment effect in the study cohort, while its associated variance estimator is conservative, producing confidence intervals with at least nominal asymptotic coverage. The proofs are not trivial because in the randomization framework sample means under treatment and control are correlated random variables.

    Keywords: Causality; Clinical Trials; Internal Validity; Neymans Causal Model; Randomization-Based Inference; Stochastic Potential Outcomes.

  • References

    1. Neyman, J, Sur les applications de la theorie des probabilites aux experiences agricoles: Essai des principes, Roczniki Nauk Rolniczych. 10 (1923) 151, in Polish.
    2. Dabrowska, D.M., Speed T.P., On the application of probability theory to agricultural experiments: essay on principles, Statistical Science 5 (1990) 465-472. (English translation of the manuscript originally published by J Neyman in 1923 in Roczniki Nauk Rolniczych Tom X, 1-51).
    3. Rubin, D., Comment: Neyman (1923) and causal inference in experiments and observational studies. Statistical Science 5 (1990) 472-480.
    4. Neyman, J., Iwaszkiewicz K., Kolodziejczyk S., Statistical problems in agricultural experimentation, Journal of the Royal Statistical Society 2 (1935) 107-108.
    5. Scosyrev E., Interval estimation of treatment effects in randomized trials: when do confidence intervals have nominal coverage? International Statistical Review 80 (2012) 439-451.
    6. Freedman, D.A., On regression adjustments to experimental data, Advances in Applied Mathematics 40 (2008)180-193.
    7. Casella, G., Berger, R.L., Statistical Inference, Pacific Grove, Duxbury, 2000.
    8. Schochet P.Z., Estimators for clustered education RCTs using the Neyman model for causal inference, Journal of Educational and Behavioral Statistics 38 (2013) 219-238.
    9. Lin W., Agnostic notes on regression adjustments to experimental data: reexamining Freedmans critique, Annals of Applied Statistics 7 (2013) 295-318.
    10. Miratrix LW, Sekhon JS, Yu B., Adjusting treatment effect estimates by post-stratification in randomized experiments, Journal of the Royal Statistical Society 75 (2013) 369-396.
    11. Robins JM., Confidence intervals for causal parameters, Statistics in Medicine 7 (1988) 773-785.
    12. Scosyrev E., Identification of causal effects using instrumental variables in randomized trials with stochastic compliance, Biometrical Journal 55 (2013) 97-113.
    13. Scosyrev E., Estimation of population mean under unequal probability sampling with unknown selection probabilities, American Journal of Theoretical and Applied Statistics 3 (2014) 65-72.
    14. Cochran WG, Sampling Techniques. 3rd ed, Wiley & Sons, New York, 1977, p39-40.
    15. Parzen, E., Modern Probability Theory and Its Applications, Wiley & Sons, New York, 1960, p431.
    16. Johnson, N.L., Kotz S., Balakrishnan N., Continuous Univariate Distributions, Volume 1, Wiley & Sons, New York, 1994, p163.




Article ID: 2895
DOI: 10.14419/ijasp.v2i2.2895

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