Table 1.

The K = 6 nominal variables ν_k that were balanced as closely as possible by the matching algorithm, where ν₁ consists of L₁ = 176 surgical procedures, and ν₆ is the interaction of 176 surgical procedures with 14 binary covariates, making L₆ = 176 × 2¹¹ categories, or about 2.9 million categories. An × indicates that the row variable contributes to nominal variable ν_k. The algorithm minimized the total imbalance ${\sum }_{\ell =1}^{L_{k^{\prime }}}\mid {\beta }_{k^{\prime }\ell }\mid$ for ν_k′ among all matches that minimized ${\sum }_{\ell =1}^{L_k}\mid {\beta }_{k\ell }\mid$ for ν_k for k < k′. The balance obtained by matching is much better than the best balance obtained in 10,000 simulated randomized experiments with the same marginal totals.

Covariate	Levels	Nested nominal covariate, νk k = 1, …, 6
		1	2	3	4	5	6
Procedure	176	×	×	×	×	×	×
Hospital Group	2		×	×	×	×	×
Male	2			×	×	×	×
ER-admit	2			×	×	×	×
Transfer	2			×	×	×	×
Paraplegia	2				×	×	×
Stroke	2				×	×	×
PPF	2				×	×	×
CC	2					×	×
CHF	2					×	×
Dementia	2					×	×
Renal	2					×	×
Liver	2						×
Past A	2						×
Past MI	2						×
# Categories Lk		176 = 176	176 × 2 = 352	176 × 24 = 2, 816	176 × 27= 22, 528	176 × 211= 360, 448	176 × 214 =2, 883, 584
Imbalance ${\sum }_{\ell =1}^{L_k}\mid {\beta }_{k\ell }\mid$		0	12	52	176	664	1242
% of maximum		0.0%	0.1%	0.4%	1.4%	5.3%	9.9%
Independence χ2		0.0	4.9	43.3	142.3	588.9	1158.7
Balance in 10,000 simulated randomized experiments with the same margins
Simulated χ2 statistics for independence
Mean χ2		174.9	302.9	767.5	1062.0	1946.0	2814.0
Minimum χ2		117.0	226.5	645.7	933.6	1777.0	2645.0
Simulated Total Imbalance ${\sum }_{\ell =1}^{L_k}\mid {\beta }_{k\ell }\mid$
Mean ${\sum }_{\ell =1}^{L_k}\mid {\beta }_{k\ell }\mid$		768	1051	1749	2086	3010	3812
Minimum ${\sum }_{\ell =1}^{L_k}\mid {\beta }_{k\ell }\mid$		540	814	1500	1826	2752	3578