Hypotheses Testing from Complex Survey Data Using Bootstrap Weights: A Unified Approach

Figures & data

Table 1 Test power for the hypothesis test $H_0:{\theta }_2={\theta }_2^{(0)}$ based on 1 000 Monte Carlo simulations, the true parameter is ${\theta }_2=1.00$, and the nominal significance level is 0.05.

(N, n)	${\theta }_2^{(0)}$	NLR	NQS	LS	M = 500		M = 1000
					BLR	BQS	BLR	BQS
$(2 000,200)$	1.00	0.20	0.20	0.07	0.07	0.07	0.07	0.07
	1.10	0.53	0.53	0.25	0.24	0.24	0.23	0.23
	1.20	0.89	0.89	0.73	0.72	0.72	0.72	0.72
$(15 000,500)$	1.00	0.22	0.23	0.06	0.06	0.06	0.06	0.06
	1.10	0.76	0.76	0.52	0.53	0.53	0.53	0.53
	1.20	1.00	1.00	0.98	0.98	0.98	0.98	0.98

NOTE: NLR, naive likelihood-ratio method; NQS, naive quasi-score method; LS, Lumley and Scott (2014) method; BLR, bootstrap quasi-likelihood-ratio method; BQS: bootstrap quasi-score method. The numbers of bootstrap repetitions are set to be M = 500 and M = 1000.

Table 2 Power of the test procedures for independence based on 1000 Monte Carlo simulation samples, Case I corresponds to the independence scenario, and the nominal significance level is 0.05.

(N, n)	Case	NP	RS	BP_I	BP_II
$(2 000,75)$	I	0.16	0.06	0.07	0.06
	II	0.21	0.10	0.12	0.11
	III	0.71	0.53	0.56	0.55
$(10 000,150)$	I	0.14	0.05	0.07	0.06
	II	0.25	0.13	0.14	0.14
	III	0.94	0.84	0.84	0.84

NOTE: NP, naive Pearson method; NLR: naive likelihood-ratio method; RS, Rao-Scott method using second-order correction; BP_I, bootstrap Pearson method with M = 500; BP_II, bootstrap Pearson method with M = 1000. The numbers of bootstrap repetitions are set to be M = 500 and M = 1000.

Table 3 Estimates (Est) and the p-values (Unit: ${10}^{-2}$) for testing $H_{0,i}:{\theta }_i=0$, where $i=1,\dots ,6$, in (20) for the middle school and high school.

School level	Cov	Est	p-value (Unit: ${10}^{-2}$)
			NLR	NQS	BLR	BQS	Wald
Middle School	After-school Edu	0.03	16.9	16.6	39.4	39.0	35.6
	Father’s Edu	0.42	0.0	0.0	0.0	0.0	0.0
	Gender	0.3	0.3	0.3	0.7	0.6	0.6
	Income	0.14	0.0	0.0	0.2	0.2	0.4
	Mother’s Edu	0.37	0.2	0.2	0.0	0.0	0.0
	Private Edu	0.05	0.0	0.0	0.0	0.0	0.0
High School	After-school Edu	0.03	2.0	2.0	0.3	0.3	0.1
	Father’s Edu	0.19	10.4	10.3	14.1	14.0	13.8
	Gender	0.21	2.5	2.5	17.0	16.8	15.7
	Income	0.1	0.0	0.0	0.1	0.1	0.1
	Mother’s Edu	0.15	20.4	20.4	41.2	41.1	40.3
	Private Edu	0.04	0.0	0.0	3.5	3.0	3.2

NOTE: Cov, covariate; Est, estimate; NLR, naive likelihood-ratio method; NQS, naive quasi-score method; BLR, bootstrap quasi-likelihood-ratio method; BQS: bootstrap quasi-score method; Wald, two-sided Wald test.

Lumley, T., and Scott, A. J. (2014), “Tests for Regression Models Fitted to Survey Data,” Australian & New Zealand Journal of Statistics, 56, 1–14. DOI: 10.1111/anzs.12065.

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