PCA Rerandomization

Mahalanobis distance of covariate means between treatment and control groups is often adopted as a balance criterion when implementing rerandomization strategy. However, this may not work well for high-dimensional cases because it balances all orthogonalized covariates equally. We propose using principal component analysis (PCA) to identify proper subspaces in which should be calculated. Not only can PCA effectively reduce the dimensionality covariates, but also provides computational simplicity by focusing on top orthogonal components. The scheme has desirable theoretical properties balancing thereby improving estimation average effects. This conclusion supported numerical studies both simulated real examples. La de entre les moyennes des covariables groupes traités et non est souvent utilisée comme critère d'équilibre lors la mise en œuvre d'une stratégie re-randomisation. Cela dit, ce peut ne pas fonctionner correctement pour cas à grande dimension car il équilibre toutes orthogonalisées manière égale. Les auteurs travail proposent recourir l'analyse composantes principales (ACP) afin d'identifier sous-espaces appropriés dans lesquels devrait être calculée. L'ACP seulement réduire efficacement dimensionnalité dimension, mais elle offre également une simplicité calcul se concentrant sur orthogonales plus importantes. Ce schéma re-randomisation basé l'ACP possède avantages théoriques intéressants équilibrer et, par conséquent, améliorer l'estimation effets moyens du traitement. appuient leur études numériques utilisant fois simulations exemples concrets. Randomized experiments have long been regarded gold standard measure effect an intervention, randomization potential bias estimates distributions average. pure (complete) implemented practice, yields unbalanced allocations, so that rerandomized before experiment actually conducted. Although discussed earlier (Fisher, 1926; Cox, 2009; Worrall, 2010), its formal framework was established until publication Morgan & Rubin (2012). Using treatment–control experiments, shown improve precision estimated Following (2012), effort made extend or modify such schemes. For example, (2015) proposed strategy with different tiers anticipated importance respect outcome variable. extension 2K factorial design developed Branson, Dasgupta (2016) based example educational data. Zhou al. (2018) considered sequentially enrolled units. Li, Ding (2018, 2020) investigated asymptotic estimator settings designs, respectively. Li (2020) further combination regression adjustment. Wang, Wang Liu (2021) studied statistical stratification rerandomization. Zhang Yin incorporated response information into ethical concerns clinical trials. Yang, Qu survey theories. All aforementioned methods use measure, due several appealing characteristics. First, invariant any affine transformation original covariates. Second, (2012) showed preserve unbiasedness equal-sized groups, reduces equal percent sampling variance each covariate. Apart from rerandomization, widely applied matching observational (Rubin, 1973a,b, 1979, 1980; Rosenbaum Rubin, 1985; Stuart, 2010). Despite advantages above, full-rank data (Branson Shao, 2021), difficult equally large number magnitudes variances. In related work, hierarchically prespecified outcome. Branson Shao pointed out might specify relative priori. They including ridge term distance, puts more emphasis components space after (PCA). relies complicated Monte Carlo integration constraint optimization determine value ridging parameter. Rather than Johansson Schultzberg rank-based their heuristic metric designed longitudinal where pre-experimental outcomes are available estimate Moreover, yet under metric. PCA, we calculate associated subspace then perform Our viewed lower dimensional alternative distance. Because selected components, imposes shrinkage them given same acceptance probability. orthogonality simplifies covariance matrix diagonal thus improves efficiency calculating establish theory distribution modified reduction mean differences compared complete randomization. Practically, despite our method easy implement delivers performance without cumbersome parameter specification increased computation required Section 2, review (Morgan 2012). present details 3. 4 reports results show other 5 concludes discussion. Given k ak, derived defer corresponding technical Appendix. average, additionally leads unbiased τ. Theorem 1.Given constant ak>0, According definition Mk, allocations W 1n−W threshold ak>0. Assuming Equation (1) holds ◂◽.▸x‾T−◂◽.▸x‾C τ^ follows 2.1 Corollary 2.2 1 extended unobserved implied addition removing conditional bias, tends make difference concentrated. literature 2012, 2015; adopt normal approximation ◂,▸(◂◽.▸x‾T−◂◽.▸x‾C)|◂∼▸X∼