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Table 1 Results of the simulation study

From: Spurious interaction as a result of categorization

   Normal Uniform Chi-square
60th percentile
ρ = 0.2
\( {\widehat{\overset{\sim }{\beta}}}_1 \) 1.74 (0.11) 0.54 (0.04) 2.99 (0.25)
\( {\widehat{\overset{\sim }{\beta}}}_2 \) 1.73 (0.11) 0.54 (0.04) 3.00 (0.26)
\( {\widehat{\overset{\sim }{\beta}}}_3 \) −0.06 (0.20) − 0.03 (0.06) 0.63 (0.40)
60th percentile
ρ = 0.5
\( {\widehat{\overset{\sim }{\beta}}}_1 \) 1.89 (0.14) 0.59 (0.04) 2.81 (0.28)
\( {\widehat{\overset{\sim }{\beta}}}_2 \) 1.88 (0.14) 0.59 (0.04) 2.83 (0.29)
\( {\widehat{\overset{\sim }{\beta}}}_3 \) −0.16 (0.21) − 0.08 (0.06) 1.38 (0.43)
60th percentile
ρ = 0.7
\( {\widehat{\overset{\sim }{\beta}}}_1 \) 1.95 (0.15) 0.62 (0.04) 2.60 (0.31)
\( {\widehat{\overset{\sim }{\beta}}}_2 \) 1.94 (0.15) 0.61 (0.04) 2.63 (0.32)
\( {\widehat{\overset{\sim }{\beta}}}_3 \) −0.24 (0.22) − 0.12 (0.07) 1.82 (0.48)
80th percentile
ρ = 0.2
\( {\widehat{\overset{\sim }{\beta}}}_1 \) 1.96 (0.14) 0.57 (0.04) 4.22 (0.27)
\( {\widehat{\overset{\sim }{\beta}}}_2 \) 1.96 (0.14) 0.57 (0.04) 4.23 (0.28)
\( {\widehat{\overset{\sim }{\beta}}}_3 \) −0.23 (0.26) − 0.11 (0.07) 0.25 (0.55)
80th percentile
ρ = 0.5
\( {\widehat{\overset{\sim }{\beta}}}_1 \) 2.22 (0.15) 0.67 (0.04) 4.35 (0.31)
\( {\widehat{\overset{\sim }{\beta}}}_2 \) 2.22 (0.16) 0.67 (0.05) 4.35 (0.32)
\( {\widehat{\overset{\sim }{\beta}}}_3 \) −0.58 (0.25) − 0.27 (0.07) 0.50 (0.54)
80th percentile
ρ = 0.7
\( {\widehat{\overset{\sim }{\beta}}}_1 \) 2.35 (0.16) 0.74 (0.05) 4.35 (0.35)
\( {\widehat{\overset{\sim }{\beta}}}_2 \) 2.36 (0.18) 0.74 (0.05) 4.34 (0.35)
\( {\widehat{\overset{\sim }{\beta}}}_3 \) −0.81 (0.26) − 0.40 (0.08) 0.58 (0.56)
  1. The true regression coefficients βi = 1,  i = 1,  2. The table gives estimated regression coefficients with corresponding empirical standard errors in parentheses