Laplace approximation, penalized quasi-likelihood, and adaptive Gauss–Hermite quadrature for generalized linear mixed models: towards meta-analysis of binary outcome with sparse data

Ju, Ke; Lin, Lifeng; Chu, Haitao; Cheng, Liang-Liang; Xu, Chang

doi:10.1186/s12874-020-01035-6

Table 2 Convergence rate for each estimation procedure in each scenario (based on 20,000 iterations)

From: Laplace approximation, penalized quasi-likelihood, and adaptive Gauss–Hermite quadrature for generalized linear mixed models: towards meta-analysis of binary outcome with sparse data

Procedure	Convergence rate (0–100%)
Procedure	OR = 1					OR = 2					OR = 3					OR = 4					OR = 5
	τ=0.2	τ=0.4	τ==0.6	τ=0.8	τ=1.0	τ=0.2	τ=0.4	τ==0.6	τ=0.8	τ=1.0	τ=0.2	τ=0.4	τ==0.6	τ=0.8	τ=1.0	τ=0.2	τ=0.4	τ==0.6	τ=0.8	τ=1.0	τ=0.2	τ=0.4	τ==0.6	τ=0.8	τ=1.0
Laplace Approximation (LA)
Model 1	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100
Model 2	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100
Model 3	100	100	100	100	100	100	100	100	100	100	100	100	100	99.99	100	100	100	100	99.99	100	100	100	100	100	100
Model 4	100	100	100	100	100	100	100	100	100	100	100	100	100	99.99	100	100	99.99	100	99.99	100	100	100	100	100	100
Model 5	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	100	99.99	100	100	100	100	100	100	100	100
Penalized quasi-likelihood estimation (PQL)
Model 1	98.46	98.47	98.25	97.48	96.31	99.07	99.01	98.94	98.51	97.01	99.08	99.14	99.01	98.52	96.30	99.27	99.38	99.17	98.58	93.22	99.50	99.44	99.39	98.84	96.84
Model 2	16.37	19.01	17.23	16.16	13.51	17.03	16.97	16.48	15.20	13.76	17.45	17.07	17.08	15.66	14.04	18.85	18.41	17.64	15.84	14.28	19.65	19.62	18.51	16.25	14.56
Model 3	99.99	99.98	99.99	99.96	99.95	99.99	100	99.99	99.99	99.93	99.92	99.92	99.91	99.86	99.86	99.89	99.93	99.93	99.84	99.42	100	99.99	100	99.98	99.95
Model 4	15.59	18.50	16.13	13.06	9.91	16.24	15.82	15.23	12.55	9.49	16.97	16.48	15.01	12.49	8.43	17.94	17.58	15.48	12.21	6.32	18.63	18.05	15.80	11.55	7.82
Model 5	19.67	22.22	19.52	16.57	14.08	20.57	20.15	19.24	16.77	13.72	21.31	20.73	19.22	16.74	13.14	22.73	22.04	20.25	16.31	11.41	23.96	23.16	20.37	15.94	12.50
Adaptive Gauss-Hermite quadrature (AGHQ)
Model 1	98.87	99.14	99.01	98.98	98.12	99.97	99.98	99.67	99.97	99.97	99.43	99.47	99.24	99.16	98.64	99.67	99.53	99.50	98.96	98.87	99.99	99.99	99.99	99.98	99.98
Model 2	86.59	89.45	88.34	87.69	81.55	99.50	99.47	99.41	99.31	99.22	96.26	96.15	95.13	94.70	92.77	96.40	96.60	95.87	94.44	92.44	99.63	99.64	99.70	99.60	99.49
Model 3	99.07	99.32	99.01	98.69	97.62	99.97	99.96	99.95	99.93	99.98	99.49	99.47	99.10	98.59	97.66	99.57	99.58	99.34	98.72	98.02	99.98	99.96	99.96	99.96	99.98
Model 4	87.09	89.15	88.06	87.44	82.10	98.72	98.64	98.35	97.85	97.26	96.07	95.97	95.45	94.24	92.58	96.74	96.51	96.49	94.65	92.79	99.29	99.24	98.85	97.93	97.31
Model 5	90.96	92.11	91.35	89.90	86.51	99.73	99.73	99.71	99.65	99.69	96.58	96.64	96.12	94.96	93.77	97.20	97.12	96.08	94.50	91.17	99.79	99.74	99.72	99.70	99.60

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