Skip to main content

Table 3 Performance of different weighted measures of dependence reported as \(100 \int |\hat {C}({\mathbf {u}})- C({\mathbf {u}})|\mathrm {d}{\mathbf {u}} / \int |\hat {C}_{\boldsymbol {\lambda }}({\mathbf {u}})-C({\mathbf {u}})|\mathrm {d}{\mathbf {u}}\) or by a ratio of the kind \(100\,\text {MSE}(\hat {\rho })/\text {MSE}(\hat {\rho }_{\boldsymbol {\lambda }})\)

From: Rank correlation under categorical confounding

  ρ=0.1 ρ=0.5 ρ=0.9
  n=10 20 50 n=10 20 50 n=10 20 50
\(\hat {C}_{\boldsymbol {\lambda }}\) 100 100 100 100 100 100 100 100 100
\(\hat {C}_{{\boldsymbol {\mu }}}\) 93 93 93 94 95 95 99 99 99
\(\hat {\rho }_{\boldsymbol {\lambda }}\) 93 94 98 78 87 95 35 48 69
\(\hat {\rho }_{{\boldsymbol {\mu }}}\) 60 64 67 53 63 72 33 46 68
\(\hat {\tau }_{\boldsymbol {\lambda }}\) 79 86 95 76 86 95 66 77 91
\(\hat {\tau }_{{\boldsymbol {\mu }}}\) 52 59 65 54 64 72 61 73 88
\(\widetilde {\tau }_{{\boldsymbol {\mu }}}\) 45 56 63 46 59 70 39 50 72
  1. In a practical situation, the confounding would make it impossible to calculate \(\hat {C}\), \(\hat {\rho }\) and \(\hat {\tau }\) on the whole dataset, but they are used here as unattainable ideal benchmarks. Five samples of size n are simulated from a Clayton distribution with Spearman’s correlation ρ. Each scenario is repeated 10,000 times