Table 1.

Bias using maximum likelihood (ML), biweight (BI, B = 4), and Huber (HU, H = 1) estimation of the latent trait when there is back random responding in the data at a severity of 10%. (Nonconvergence rates higher than 30% displayed in parentheses)

			$\theta$
	K	J	−2	−1.5	−1	−0.5	0	0.5	1	1.5	2
ML	3	10	(79.5%)	(43.3%)	0.18	0.08	−0.05	−0.18	−0.28	(67.5%)	(89.5%)
		30	0.42	0.37	0.28	0.17	0.06	−0.01	−0.09	−0.15	−0.21
		50	0.23	0.12	0.02	−0.04	−0.08	−0.14	−0.24	−0.30	−0.31
		70	0.31	0.23	0.15	0.07	−0.01	−0.08	−0.16	−0.19	−0.25
	5	10	(66.8%)	0.15	−0.03	−0.17	−0.27	−0.30	−0.24	(35.5%)	(80.8%)
		30	(33.7%)	0.18	0.12	0.03	−0.02	−0.07	−0.10	−0.17	−0.22
		50	(48.8%)	0.16	0.10	0.05	0.00	−0.05	−0.11	−0.17	−0.31
		70	0.24	0.17	0.13	0.09	0.04	−0.02	−0.09	−0.15	(33.2%)
	7	10	(78.7%)	0.29	0.20	0.17	0.11	0.01	−0.06	−0.20	(44.0%)
		30	(89.8%)	0.15	0.05	0.02	−0.01	−0.06	−0.09	−0.15	(70.3%)
		50	(80.3%)	0.11	0.07	0.04	−0.01	−0.04	−0.09	−0.15	(84.0%)
		70	(91.0%)	0.14	0.10	0.07	0.03	−0.02	−0.06	−0.13	(88.0%)
HU	3	10	−0.08	−0.05	0.13	0.10	−0.03	−0.12	−0.08	0.04	0.05
		30	−0.10	0.02	0.07	0.05	0.02	−0.01	−0.02	−0.03	0.07
		50	−0.08	−0.04	−0.08	−0.02	0.01	0.02	−0.07	−0.11	0.06
		70	−0.01	0.01	−0.02	0.01	0.04	0.01	−0.02	−0.04	0.02
	5	10	0.01	−0.04	−0.05	−0.10	−0.13	−0.12	−0.04	0.09	0.21
		30	−0.06	0.04	0.03	−0.04	−0.02	−0.03	0.02	0.05	0.07
		50	−0.04	0.03	0.02	0.01	0.03	0.00	−0.03	−0.01	0.02
		70	0.00	0.03	0.01	0.03	0.05	0.00	−0.04	−0.04	0.02
	7	10	−0.15	0.02	0.09	0.10	0.09	0.02	−0.04	−0.08	−0.06
		30	−0.04	0.01	0.03	0.02	0.00	−0.03	−0.03	−0.01	0.02
		50	−0.02	0.01	0.03	0.01	−0.01	−0.03	−0.02	−0.03	0.00
		70	−0.04	0.04	0.05	0.02	0.02	−0.01	−0.03	−0.04	−0.02
BI	3	10	−0.33	−0.12	0.12	0.09	−0.02	−0.12	−0.08	0.09	0.11
		30	−0.13	−0.03	0.00	0.02	0.00	−0.03	0.00	0.00	0.09
		50	−0.13	−0.09	−0.10	−0.02	0.03	0.05	−0.04	−0.06	0.07
		70	−0.05	−0.04	−0.06	0.01	0.03	0.02	0.01	0.00	0.05
	5	10	−0.09	−0.05	−0.04	−0.08	−0.10	−0.09	−0.02	0.07	(35.0%)
		30	−0.11	0.00	0.00	−0.05	−0.03	0.00	0.04	0.07	0.08
		50	−0.07	−0.01	0.00	0.00	0.02	0.01	−0.01	0.02	0.08
		70	−0.04	−0.01	−0.02	0.02	0.03	0.01	−0.02	−0.01	0.08
	7	10	−0.29	−0.06	0.06	0.08	0.07	0.01	−0.03	−0.05	−0.03
		30	−0.08	−0.01	0.02	0.01	0.00	−0.01	−0.01	0.02	0.05
		50	−0.05	−0.01	0.01	0.01	−0.02	−0.01	0.00	−0.01	0.03
		70	−0.07	0.00	0.02	0.01	0.01	0.00	−0.02	−0.01	0.01