Sample empirical likelihood approach under complex survey design with scrambled responses
Section 5. Real application

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In this section, we applied the proposed approach to 2015-2016 National Health and Nutrition Examination Survey (NHANES) to evaluate its practical performance. NHANES provides timely health- and nutrition-related information for the noninstitutionalized civilian resident population of the United States. It uses a complex, multistage probability design based on in-person survey to collect information. (see https://wwwn.cdc.gov/nchs/nhanes/continuousnhanes/overview.aspx?BeginYear=2015 for more information). The sample size for the 2015-2016 NHANES is about 9,000. We treated the original NHANES sample as a finite population and selected one sample by using a simple random sampling design with sample sizes $\left(\text{}n\text{}\right)$ as 30, 40, 50, 100, and 200, respectively. Suppose our parameters of interest include population means of systolic blood pressure, diastolic blood pressure, HDL cholesterol, and total cholesterol. We created scramble responses for these parameters by using $p\mathrm{<mo>=</mo>}$ 0.6, $a\mathrm{<mo>=</mo>}$ 1.5, and $b^2\mathrm{<mo>=</mo>}\text{0}\text{.2}/\text{1}\text{.5}.$ In addition, body mass index (BMI) was selected as a covariate in the estimation process, since BMI is correlated with those study variables.

We compared the performances of two approaches, HJ and SEL, in terms of point estimates and interval estimates (Table 5.1). Point estimates obtained by using both methods were similar, and they were close to finite population parameters (120.47, 66.17, 54.43, and 180.25 for systolic blood pressure, diastolic blood pressure, HDL cholesterol, and total cholesterol), especially for larger sample sizes (Table 5.1). For systolic blood pressure, diastolic blood pressure, and total cholesterol, intervals produced by SEL shifted slightly to the right compared with the results produced by HJ for small sample sizes. However, when sample sizes increased, the results from the two approaches were similar. For HDL cholesterol, the results are comparable. The results from this application verified the validity of the proposed SEL approach.

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