. Author manuscript; available in PMC: 2023 Nov 20.

Published in final edited form as: Stat Med. 2022 Sep 20;41(26):5335–5348. doi: 10.1002/sim.9572

TABLE 2.

Performances of different methods when data is simulated based on Model 3 (random slope model with ear-level covariates)

Parameter		Bothears A^a	Bothears B^b	Worse-ear, misspecified^c	Average-ear, misspecified^d	Worseear A^e	Worseear B^f	Averageear^g
β₁ (truth = 0.2) (Model 3)	Relative bias	1.10%	0.52%	−28.00%	−28.70%	−1.89 %	1.01%	0.37%
	Empirical SE	0.0985	0.148	0.173	0.168	0.153	0.155	0.149
	Estimated SE	0.101	0.150	0.166	0.163	0.150	0.150	0.146
	CR of 95% CI	95.6%	94.6%	92.7%	92.5%	94.8%	94.7%	93.8%
β₂ (truth = 0.1) (Model 3)	Relative bias	−0.33%	−0.19%	−34.90%	−35.00%	−2.66%	−0.18%	−0.19%
	Empirical SE	0.00880	0.0141	0.0162	0.0158	0.0147	0.0147	0.0142
	Estimated SE	0.00883	0.0139	0.0159	0.0156	0.0145	0.0145	0.0141
	CR of 95% CI	95.1%	95.4%	43.0%	41.5%	94.5%	95.2%	95.4%

Note: Sample size N = 1000, with four audiologists at each of the ten sites.

Both-ears method taking into account participants, sites, and audiologists-formed clusters and ear-level covariates, based on mixed-effects model $E (Y_{i j} ∣ X_{i}, W_{i}, Z_{i j}, b_{s_{i}}^{(s i t e)}, b_{a_{i 0}}^{(a u d 0))}, b_{a_{i 1}}^{(a u d 1)}, b_{1 i}^{(i n d)}, b_{2 i}^{(i n d)}) = β_{0} + (β_{1} + b_{2 i}^{(i n d)}) X_{i} + β_{2} W_{i} + β_{3} Z_{i j} + b_{s_{i}}^{(s i t e)} + b_{a_{i 0}}^{(a u d 0)} + b_{a_{i 1}}^{(a u d 1)} + b_{1 i}^{(i n d)}$ .

Both-ears method taking into account ear-level covariates but ignoring sites and audiologists, based on mixed-effects model $E (Y_{i j} ∣ X_{i}, W_{i}, Z_{i j}, b_{1 i}^{(i n d)}, b_{2 i}^{(i n d)}) = β_{0} + (β_{1} + b_{2 i}^{(i n d)}) X_{i} + β_{2} W_{i} + β_{3} Z_{i j} + b_{1 i}^{(i n d)}$ .

Worse-ear method ignoring sites, audiologists and ear-level covariates, based on linear regression model $E (Y_{i}^{w} ∣ X_{i}, W_{i}) = β_{0} + β_{1} X_{i} + β_{2} W_{i}, Y_{i}^{w} = \max (Y_{i 1}, Y_{i 2})$ .

Average-ear method ignoring sites, audiologists and ear-level covariates, based on linear regression model $E (Y_{i}^{a} ∣ X_{i}, W_{i}) = β_{0} + β_{1} X_{i} + β_{2} W_{i}, Y_{i}^{a} = (Y_{i 1} + Y_{i 2}) / 2$ .

Worse-ear method taking into account ear-level covariates but ignoring sites and audiologists, based on linear regression model $E (Y_{i}^{w} ∣ X_{i}, W_{i}, Z_{i}^{w}) = β_{0} + β_{1} X_{i} + β_{2} W_{i} + β_{3} Z_{i}^{w}, Y_{i}^{w} = \max (Y_{i 1}, Y_{i 2}), Z_{i}^{w} = \max (Z_{i 1}, Z_{i 2})$ .

Worse-ear method ignoring sites and audiologists, using averaged ear-level covariates, based on linear regression model $E (Y_{i}^{w} ∣ X_{i}, W_{i}, Z_{i}^{a}) = β_{0} + β_{1} X_{i} + β_{2} W_{i} + β_{3} Z_{i}^{a}, Y_{i}^{w} = \max (Y_{i 1}, Y_{i 2}), Z_{i}^{a} = (Z_{i 1} + Z_{i 2}) / 2$ .

Average-ear method taking into account ear-level covariates but ignoring sites and audiologists, based on linear regression model $E (Y_{i}^{a} ∣ X_{i}, W_{i}, Z_{i}^{a}) = β_{0} + β_{1} X_{i} + β_{2} W_{i} + β_{3} Z_{i}^{a}, Y_{i}^{a} = (Y_{i 1} + Y_{i 2}) / 2, Z_{i}^{a} = (Z_{i 1} + Z_{i 2}) / 2$ .