BACKGROUND/RATIONALE:
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.
OBJECTIVE(S):
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.
METHODS:
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.
FINDINGS/RESULTS:
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.
IMPACT:
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.
PUBLICATIONS:
Journal Articles
- Noorbaloochi S, Nelson D. Conditionally specified models and dimension reduction in the exponential families. Journal of multivariate analysis. 2008; 99(8): 1574-1589.
- Nelson DB, Meeden G. Noninformative nonparametric quantile estimation for simple random samples. Journal of statistical planning and inference. 2006; 136(1): 53-67.
