Table III.

Class-specific ridge penalization (only β₁₂ penalized) (N =5000).

			β 11		β 12
β * 11	β † 12	λ ‡	Bias§	MSE¶	Bias	MSE
0.69	0	0	0.001	0.033	0.015	0.010
		1000	0.001	0.034	0.001	0.000
1.79	0	0	0.011	0.036	−0.009	0.013
		1000	0.011	0.036	0.000	0.000
1.79	0.69	0	−0.010	0.038	0.011	0.008
		1	−0.013	0.035	−0.001	0.007
1.79	1.79	0∥	0.049	0.038	0.005	0.009
		1	0.031	0.033	−0.028	0.010

^*Simulated log odds ratio for association between x₁ and membership in class 1, relative to class 3.

^†Simulated log odds ratio for association between x₁ and membership in class 2, relative to class 3.

^‡Candidate value for tuning parameter. Results are shown for λ=0 and the candidate λ value which minimized R, the cross-validated log-likelihood loss for the largest number of simulated data sets.

^§${\beta }_{1j}-\left[\right(1∕\mathit{nsi}\left){\Sigma }_{\mathit{si}=1}^{\mathit{nsi}}{\widehat{\beta }}_{1j,\mathit{si},\lambda }\right]$, where si are simulation iterations.

^¶Mean square error =${\Sigma }_{\mathit{si}=1}^{\mathit{nsi}}{\left(\right({\widehat{\beta }}_{1j,\mathit{si},\lambda }-{\stackrel{‒}{\beta }}_{1j,\lambda })}^2∕(n-1))+{\text{bias}}_{\lambda }^2$.

^∥In the case of β₁₁=β₁₂=1.79, R was minimized with λ=0 in all 100 simulated data sets.