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On Average Predictive Comparisons and Interactions

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  • Juxin Liu
  • Paul Gustafson

Abstract

In a regression context, consider the difference in expected outcome associated with a particular difference in one of the input variables. If the true regression relationship involves interactions, then this predictive comparison can depend on the values of the other input variables. Therefore, one may wish to consider an average predictive comparison as a target of inference, where the averaging is with respect to the population distribution of the input variables. We consider inferences about such targets, with emphasis on inferential performance when the regression model is misspecified. Particularly, in light of the difficulties in dealing with interaction terms in regression models, we examine inferences about average predictive comparisons when additive models are fitted to relationships truly involving pairwise interaction terms. We identify some circumstances where such inferences are consistent despite the model misspecification, notably when the input variables are independent, or have a multivariate normal distribution. Dans un contexte de régression, considérons la différence de la valeur espérée de la variable d'intérêt, associée à une différence donnée d'une des variables d'entrée. Si la relation de régression véritable implique des termes d'interaction, alors cette comparaison prédictive peut dépendre de la valeur des autres variables d'entrée. Ainsi, il peut être souhaitable de considérer la comparaison prédictive moyenne comme objet d'inférence, où la moyenne est calculée sur la distribution des variables d'entrée dans la population. Nous nous intéressons à ce contexte d'inférence, et plus particulièrement à la performance inférentielle lorsque le modèle de régression est mal spécifié. À la lumière des difficultés existantes de traiter les termes d'interaction dans les modèles de régression, nous examinons l'inférence sur les comparaisons prédictives moyennes lorsque des modèles additifs sont ajustés, alors que les vraies relations impliquent des termes d'interaction deux‐à‐deux. Nous identifions quelques situations où l'inférence est convergente malgré la mauvaise spécification du modèle, notamment lorsque les variables d'entrée sont indépendantes ou possèdent une loi normale multivariée.

Suggested Citation

  • Juxin Liu & Paul Gustafson, 2008. "On Average Predictive Comparisons and Interactions," International Statistical Review, International Statistical Institute, vol. 76(3), pages 419-432, December.
  • Handle: RePEc:bla:istatr:v:76:y:2008:i:3:p:419-432
    DOI: 10.1111/j.1751-5823.2008.00056.x
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    References listed on IDEAS

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    4. Guido W. Imbens, 2004. "Nonparametric Estimation of Average Treatment Effects Under Exogeneity: A Review," The Review of Economics and Statistics, MIT Press, vol. 86(1), pages 4-29, February.
    5. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
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    2. Manuel Gomes & Richard Grieve & Richard Nixon & Edmond S.‐W. Ng & James Carpenter & Simon G. Thompson, 2012. "Methods For Covariate Adjustment In Cost‐Effectiveness Analysis That Use Cluster Randomised Trials," Health Economics, John Wiley & Sons, Ltd., vol. 21(9), pages 1101-1118, September.
    3. Lefebvre Geneviève & Gustafson Paul, 2010. "Impact of Outcome Model Misspecification on Regression and Doubly-Robust Inverse Probability Weighting to Estimate Causal Effect," The International Journal of Biostatistics, De Gruyter, vol. 6(2), pages 1-27, March.
    4. Peng Zhang & Juxin Liu & Jianghu Dong & Jelena L. Holovati & Brenda Letcher & Locksley E. McGann, 2012. "A Bayesian Adjustment for Multiplicative Measurement Errors for a Calibration Problem with Application to a Stem Cell Study," Biometrics, The International Biometric Society, vol. 68(1), pages 268-274, March.
    5. Gomes, M & Grieve, R, 2011. "Estimating the Effects of Friendship Networks on Health Behaviors of Adolescents," Health, Econometrics and Data Group (HEDG) Working Papers 11/14, HEDG, c/o Department of Economics, University of York.

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