Observational studies of the association between an explanatory measure and an outcome are complicated by hosts of potential confounding covariates. With large numbers of confounders conventional regression methods suffer from difficulty in identifying suitable regression functions without introducing additional bias. Inference is more easily conducted if we can greatly reduce the number covariates. Propensity theory was developed to reduce covariate dimension and avoid bias issues. However, propensity theory has not been adequately employed when estimating causal effects for polytomous interventions and is not directly applicable in settings such as case-control studies.
We develop a theory of sufficient summaries, and in particular conditional density ratios, for dimension reduction in observational studies that extends and generalizes propensity theory to address selection bias and case-mix imbalance when examining differences in outcomes among multiple populations based on categorical “intervention”, “treatment”, or “trait” measures.
We adapted statistical sufficiency to the estimation of covariate balanced differences in outcomes to extend the dimension reduction theory contained in propensity theory. We continued this adaptation of sufficiency to extend this dimension reduction theory into dimension reduction in regression yielding a more general sufficient dimension reduction summary theory. We examined use of principal components as means to identify minimal linear dimensional forms of sufficient summaries and developed a nonparametric method of sufficient summary estimation. We examined the performance of applying the developed theory in simulated and existing data.
We developed a theory for sufficient summaries, in particular conditional density ratios, for dimension reduction. We show conditional density ratios can greatly reduce covariate dimension and still effectively address case-mix imbalance and selection issues while losing none of the information in the covariates. We directly link this theory to dimension reduction approaches considered in regression theory and to propensity theory and show that these theories can be subsumed into this new dimension reduction theory. We show conditional density ratios play a role analogous to minimal sufficient statistics and possess optimal properties related to dimension and expected loss or variance. The theory also presents a mathematical framework for dimension reduction in the estimation of casual effects in the presence of covariates outside the counter-factual structure in which propensity theory was developed. The theory is applicable in situations where propensities do not exist or are not readily identified, such as case-control studies and for static trait characteristics. We have identified a fully nonparametric method of estimating sufficient summaries and have demonstrated the applicability and usefulness of these sufficient dimension reduction summaries and conditional density ratios using both simulated and existing datasets.
Applications of this dimension reduction theory exist in balancing multiple populations with respect to hosts of covariates to deal with selection and case-mix imbalance more fully than achievable with propensities and in situations where propensities do not exist. The theory is also applicable in regression analyses and in randomized studies to assess the effect of polytomous interventions in the presence of issues such as nonresponse bias and noncompliance. The theory will provide researchers will a valuable set of new methods for facilitating sound statistical inference in complex situations.
- Nelson D, Noorbaloochi S. Dimension reduction summaries for balanced contrasts. Journal of Statistical Planning and Inference. 2009 Feb 1; 139(2):617-628.
- Noorbaloochi S, Nelson D. Conditionally specified models and dimension reduction in the exponential families. Journal of multivariate analysis. 2008 Sep 1; 99(8):1574-1589.
- Nelson DB, Meeden G. Noninformative nonparametric quantile estimation for simple random samples. Journal of Statistical Planning and Inference. 2006 Jan 1; 136(1):53-67.
- Korthauer K, Nelson DB, Noorbaloochi S. Assessing the robustness of combining propensity theory and methods to assess the impact of unmeasured confounders on the estimation of causal effects. Paper presented at: Health Policy Statistics Annual International Conference; 2010 Jan 21; Washington, DC.
- Nelson DB. Propensity Scores and Related Summaries. Paper presented at: Task Force on Design and Analysis in Oral Health Research Scientific Session; 2009 Nov 10; Newark, NJ.
- Nelson D. Confounder reduction in group comparisons. Paper presented at: Mayo Clinic Division of Biomedical Statistics and Informatics Annual Seminar; 2008 Sep 2; Rochester, MN.
- Nelson DB, Noorbaloochi S. Use of propensities with unmeasured confounders. Paper presented at: American Statistical Association Joint Statistical Annual Meeting; 2008 Aug 3; Denver, CO.
- Noorbaloochi S, Nelson DB. Regression graphics for bias reduction in observational studies. Paper presented at: American Statistical Association Joint Statistical Annual Meeting; 2008 Aug 3; Denver, CO.
- Nelson D, Noorbaloochi S. Sufficient Dimension Reduction Summaries. Paper presented at: Health Policy Statistics Annual International Conference; 2008 Jan 17; Philadelphia, PA.
- Nelson DB, Noorbaloochi S. Novel approaches for the analysis of observational studies. Paper presented at: Society of General Internal Medicine Annual Meeting; 2007 Apr 25; Toronto, Canada.
- Nelson DB, Noorbaloochi S. Bias reduction versus dimension reduction. Paper presented at: University of Minnesota Department of Biostatistics Annual Seminar; 2007 Apr 2; Minneapolis, MN.
- Noorbaloochi S, Nelson DB, Grill JP. Dimensionality Reduction of High-Dimensional Tables. Paper presented at: American Statistical Association Joint Statistical Annual Meeting; 2006 Aug 1; Seattle, WA.
- Nelson DB, Noorbaloochi S. Estimation of covariate balanced contrasts of expectations. Paper presented at: Joint Statistical Annual Meeting; 2005 Aug 11; Minneapolis, MN.
- Noorbaloochi S, Nelson DB. Dimension reduction in exponential family. Paper presented at: Joint Statistical Annual Meeting; 2005 Aug 11; Minneapolis, MN.
- Noorbaloochi S, Nelson DB. Regression graphics and dimension reduction in exponential families. Paper presented at: American Statistical Association Joint Statistical Annual Meeting; 2005 Aug 7; Minneapolis, MN.
Research measure, Research method, Risk adjustment
Selection Bias, Statistical Distributions, Bias (Epidemiology), Causality