Empirical likelihood inference and goodness-of-fit test for logistic regression model under two-phase case-control sampling

Figures & data

Table 1. Coverage probabilities (%) of the Wald and ELR confidence intervals with $n_0=n_1=100$.

		γ	$(\gamma ,{\text{β}}_1)$	$(\gamma ,{\text{β}}_1,{\text{β}}_2)$	$(\gamma ,{\text{β}}_1,{\text{β}}_2,{\text{β}}_3)$
Level (%)	Method	The true parameter value is $(0,0,1,2)$
90	ELR	88.20	88.60	88.90	87.90
	Wald	90.85	93.60	94.50	92.75
95	ELR	93.95	93.85	94.20	93.35
	Wald	96.30	97.75	97.55	96.60
99	ELR	98.60	98.25	98.40	98.10
	Wald	99.50	99.55	99.60	99.25
		The true parameter value is $(1,0,1,2)$
90	ELR	89.05	88.95	87.20	88.25
	Wald	92.05	92.25	92.50	92.25
95	ELR	94.10	94.25	93.15	93.65
	Wald	96.05	96.20	96.40	96.05
99	ELR	98.65	98.60	98.35	97.60
	Wald	99.30	99.45	99.35	98.75

Table 2. Coverage probabilities (%) of the Wald and ELR confidence intervals with $n_0=n_1=200$.

		γ	$(\gamma ,{\text{β}}_1)$	$(\gamma ,{\text{β}}_1,{\text{β}}_2)$	$(\gamma ,{\text{β}}_1,{\text{β}}_2,{\text{β}}_3)$
Level (%)	Method	The true parameter value is $(0,0,1,2)$
90	ELR	89.50	89.85	89.85	87.85
	Wald	90.55	91.60	92.40	90.90
95	ELR	94.15	94.25	94.35	93.35
	Wald	95.40	96.25	96.40	94.95
99	ELR	98.70	98.40	98.60	97.85
	Wald	99.25	99.40	99.40	98.60
		The true parameter value is $(1,0,1,2)$
90	ELR	88.10	88.80	89.05	88.95
	Wald	88.90	90.85	90.80	91.25
95	ELR	93.70	94.40	94.35	94.40
	Wald	94.60	95.70	95.70	95.05
99	ELR	98.85	98.85	98.85	98.20
	Wald	99.25	99.40	99.05	98.70

Figure 1. Rejection probabilities of the ELR (dashed line) and Wald (solid line) tests in Scenarios (A1) and (A2) (First row), and (A3) and (A4) (Second row) of Example 3.1.

Figure 2. Rejection probabilities of the ELR (dashed line) and Wald (solid line) tests in Scenarios (B1) and (B2) (First row), and (B3) and (B4) (Second row) of Example 3.2.

Table 3. Rejection probabilities (%) of model checking of Examples 3.3 and 3.4.

$k=$	0	1	2	3	4
Example 3.3	6.1	10.5	39.8	75.4	93.4
Example 3.4	5.1	14.5	39.9	71.2	90.5

Table 4. IH and outcome for Wilms tumour.

IH	Relapsed	Non-relapsed
Favourable	415	3,207
Unfavourable	156	250

Figure 3. Histograms of Age in different combinations of outcome (Row 1, relapses or cases; Row 2, non-relapses or controls) and the stratum variable (IH; Column 1, unfavourable; Column 2, favourable).

Table 5. Results of regression parameter estimates in Wilms tumour study.

		Confidence interval
Variable	Estimate	Wald	ELR
Intercept	−1.3026	(−1.7570, −0.8482)	(−1.8399, −0.7932)
CH	3.1958	(1.5867, 4.8049)	(1.7909, 5.1445)
Age	0.1316	(0.0483, 0.2150)	(0.0500, 0.2173)
Stage-II	0.6399	(0.0080, 1.2718)	(0.0093, 1.2759)
Stage-III	0.3685	(−0.2929, 1.0298)	(−0.2970, 1.0294)
Stage-IV	1.0052	(0.2196, 1.7908)	(0.2272, 1.8053)
IH	−0.9811	(−2.4389, 0.4766)	(−2.9375, 0.5030)