A note on median regression for complex surveys

Summary

There is a great need for statistical methods for analyzing skewed responses in complex sample surveys. Quantile regression is a logical option in addressing this problem but is often accompanied by incorrect variance estimation. We show how the variance can be estimated correctly by including the survey design in the variance estimation process. In a simulation study, we illustrate that the variance of the median regression estimator has a very small relative bias with appropriate coverage probability. The motivation for our work stems from the National Health and Nutrition Examination Survey where we demonstrate the impact of our results on iodine deficiency in females compared with males adjusting for other covariates.

1. Introduction

Complex sample surveys are used to collect data on the economic, social, and health status of individuals in a finite population. These data are generally used to make policy decisions and to conduct research. When conducting large national surveys, various design features may be incorporated in the sample survey design. These design features include stratification, clustering, and unequal probability sampling. Due to the complex nature of survey data, two issues are of paramount concern. The first is the representativeness of the sample and its impact on the parameter estimates. The second is the estimation of population variances. Our primary focus is on the latter. Specifically, we address the issue of variance estimation of median regression for complex survey data.

Ordinary least squares regression models are widely used in the analysis of complex survey data. In many complex sample surveys, the outcome of interest may be skewed with no suitable transformation. Under these conditions, median regression is a viable alternative (Koenker, 2005). One application of median regression to complex survey data is to bootstrap the primary sampling units within each stratum since the primary sampling units are independent. Unfortunately, this approximation may be incorrect because the number of strata and/or primary sampling units may be too small to apply the bootstrap successfully. This may help to explain why there are very few examples of median or quantile regression being applied to complex survey data (Geraci, 2013; Chen and others, 2010). Additionally, perhaps the most important reason, is that the quantile regression estimating equation is a discontinuous function of the regression parameters. Hence, the usual sandwich estimate of variance will not be consistent (Binder, 1983). For the same reason, other well-known estimators such as Taylor series linearization (Woodruff, 1971; Lohr, 2009) and the jackknife (Shao and Tu, 2012) are not consistent for estimates obtained via the quantile regression estimating equation.

Further, resampling methods such as the bootstrap (Efron and Tibshirani, 1994; He and Hu, 2002) and balanced repeated replication (McCarthy, 1966, 1969; Lohr, 2009) tend to overestimate variance. In practice, the primary sampling units are sampled without replacement to avoid selecting the same primary sampling unit more than once. However, it is common practice to treat the primary sampling units as if they were sampled with replacement in order to simplify variance estimation calculations. A consequence of this approach is that the variance may be overestimated. Resampling-based variance estimates have been shown to be useful in certain specialized problems (Canty and Davison, 1999). However, while resampling methods may be useful in some problems, there is not enough evidence of their usefulness as a general-purpose technique for the analysis of complex sample surveys. More importantly, it is generally unclear how to extend resampling methods to complex sample surveys (Presnell and Booth, 1994). Resampling methods such as the bootstrap cannot be applied appropriately to complex survey data unless we know the individual probabilities that make up the final sampling weights. Unfortunately, it is often the case that only the composite or final sampling weights are known; partly done to protect the privacy of the survey participants. The objective of this note is to offer an approximation to the estimation of the asymptotic covariance matrix for median regression estimators for complex survey data.

2. Asymptotic normality

Consider a stratified multistage sampling design in which the primary sampling units are selected with replacement. Let denote a sequence of finite populations, with strata in . We assume that the strata sample sizes are fixed in each population and that the number of strata is tending to infinity. The linear median (i.e., ) regression model can be written as

(2.1)

where is the response variable; is the population index; is the stratum index; is the cluster index within stratum and are individuals within cluster of stratum . The total number of observations in the sample is . The unknown error terms are dependent random variables each with median zero (i.e. ). The functional regression parameter is a vector of unknowns at the quantile and is an explanatory covariate vector. For simplicity, we omit the population index in our discussion that follows.

The regression quantiles can be estimated as

(2.2)

where are the complex survey sampling weights and is a check function. A minimizer of (2.2) is a solution of the following estimating equation:

(2.3)

The estimating equation (2.3) for the quantile regression estimator is not differentiable with respect to at the quantile. To remove this barrier, we must establish a stochastic equicontinuity condition on the sample average moment function,

Specifically, the stochastic equicontinuity condition (Andrews, 1994) states,

converges in probability to zero. The stochastic equicontinuity condition has been applied to independent data but in complex sample surveys with stratification and clustering, it has not been established. Since and the expected value of the first-order condition is given by,

It follows that

converges in probability to zero. Therefore, the Taylor series expansion of around is,

where

such that . The regularity conditions necessary to find the limiting distribution of the -estimator are as follows:

• (R1)The distribution functions are absolutely continuous, with continuous densities uniformly bounded away from 0 and at the points .
• (R2) ( positive definite).
• (R3) .
• (R4) .
• (R5) where is the fraction of the population in stratum .

We are now ready to state our theorem.

Theorem 2.1

Assume the model (2.1) and that the stratum sample sizes are fixed. Under the following regularity conditions R1–R5, as the number of strata tends to infinity the regression quantile (median) converges in distribution to a normal vector with mean zero and covariance matrix given by ,

where

and

with

The sampling weights correspond to the sampling vector , denotes the design matrix with rows, . We assume that is independent of and . The density is estimated using a kernel-based method, where we use a variant of the kernel estimator proposed by Powell (1991) of the form

(2.4)

where is the Gaussian kernel, , and is a kernel bandwidth parameter. The first and third quantiles of the response are denoted and , respectively. The bandwidth can be computed in a number of ways (Bofinger, 1975; Hall and Sheather, 1988; Chamberlain, 1994). In this article, we use the very elegant formula of Chamberlain, 1994,

where is the standard normal distribution, is the total number of observations in the sample, and satisfies for the construction of confidence intervals. Therefore, by incorporating the survey design in the matrix , the covariance matrix of is .

3. Monte-Carlo study

To illustrate the appropriateness of our variance estimator, we present a complex survey design where resampling methods cannot be applied appropriately; specifically, a two-stage cluster sample. We use the median regression model where . To ensure skewness and heteroscedasticity, we sampled data from the exponential distribution. The median model used for generating our population is

(3.5)

where and such that are the primary sampling units (PSUs), secondary sampling units (SSUs), and unit level indices, respectively. The covariate is at the PSU-level, is at the SSU-level and at the unit-level. We assumed the number of PSUs, SSUs, and units follows a Poisson distribution with mean 1000, 100, and 60. The parameter values are and . The response where is the inverse cumulative distribution function of the exponential density such that . This population had 5 887 353 observations and 978 PSUs. We sampled 1000 two-stage cluster sample replicates without replacement, via simple random sampling at each stage, from the population with the number of observations ranging from 5703 to 6264. In the first stage, we sampled 50 PSUs, followed by 2 SSUs within PSUs at the second stage. At each stage, the sampling weights were computed and used to compute the final or composite sampling weights which was used for the simulation study.

The results of this simulation study are summarized in Table 1. The simulation standard deviation, average standard error, relative bias, and 95% coverage probabilities are reported. Our small simulation study showed that the bootstrap (He and Hu, 2002) performed poorly. That is, the bootstrap estimator will have a larger relative bias and lower coverage probability when compared with the proposed method. On the other hand, the proposed variance estimator had small relative bias and showed the correct coverage probability. Hence, the proposed estimator should be preferred over the bootstrap estimator for complex sample survey. Finally, we conclude that resampling methods including the bootstrap cannot be applied appropriately to complex sample survey without knowing the individual sampling weights that make up the final sampling weight. The novelty of this article is that it shows how to obtain a consistent estimator of the standard error of a median regression estimator in complex survey data, which has not been previously proposed in the literature.

Table 1.

Simulation study of 1000 replicates for median regression model comparing the proposed variance estimator and the bootstrap variance estimator for a two-stage cluster sampling design

			Performance measures
Method	True parameter value	Simulation standard deviation	Average standard error	Ratio	Relative bias of standard error (%)	Coverage probability
Proposed		2.267	2.282	1.007	0.677	0.945
bootstrap			2.049	0.904	–9.642	0.916
Proposed		0.796	0.783	0.984	–1.626	0.949
bootstrap			0.760	0.955	–4.466	0.928
Proposed		0.395	0.407	1.031	3.056	0.950
bootstrap			0.374	0.948	–5.207	0.930
Proposed		0.190	0.199	1.050	5.044	0.949
bootstrap			0.182	0.960	–4.015	0.936

Standard error is estimated using the method of He and Hu (2002).

Average of simulation standard errors divided by the simulation standard deviation.

4. Application: predictors of urinary iodine concentration in NHANES

The National Health and Nutrition Examination Survey (NHANES) (CDC, 2008) is a program of studies designed to assess the health and nutritional status of adults and children in the United States. NHANES uses a stratified, multistage survey to provide a representative sample of the noninstitutionalized US population. It consists of an initial in-person interview at the household, followed by a physical examination in a mobile examination center and follow-up questionnaires. During the NHANES physical examinations, spot urine specimens were collected from participants, and aliquots of these specimens were generated and stored cold or frozen until shipped. Our analysis is restricted to the 2007–2008 cycle of NHANES laboratory data involving urinary iodine (UI) concentration. Severe iodine deficiency of UI can lead to increased risks of many cancers, including thyroid, breast, endometrial, and ovarian cancer (Feldt-Rasmussen, 2001; Stadel, 1976). The objective of the analysis is to identify potentially important characteristics of individuals that are associated with urinary iodine concentration; in particular, it is of interest to determine whether females are at a higher risk of iodine deficiency than males.

Our complex survey consists of data on 6802 persons. There are a total of 32 primary sampling units and 16 strata, with 2 primary sampling units per stratum. The average cluster size is 213 with the smallest being 51 and the largest 314. The response variable of interest, urinary iodine concentration measured in L, is extremely right-skewed with a median of , mean of and standard deviation of 9460. The minimum and maximum iodine concentrations are 2.1 and 762 010. The individual characteristics of interest were gender, body mass index (BMI), age at screening, race, total grain intake, dairy consumption, dietary supplements, fish, and salt intake. We used a reference coding scheme for all categorical variables. The continuous variables age, BMI, and total grain intake were centered and scaled accordingly: Age - 30, (BMI - 25)/5, and (Total grain - 310)/10.

In Table 2, we compare the results of the proposed method with He and Hu (2002), namely, the Markov chain marginal bootstrap (MCMB). The proposed method incorporates the survey design in the variance estimation process. That is, stratification, clustering, and sampling weights. However, the Markov chain marginal bootstrap estimator only uses the final sampling weights as reported by NHANES. The regression coefficients for the proposed and MCMB method were estimated using median regression.

Table 2.

Point estimates and standard errors for the proposed estimator and markov chain marginal bootstrap (MCMB) applied to the NHANES urinary iodine concentration data consisting of 6802 individuals

		Proposed		MCMB
Covariate	Est.	SE	t	SE	t
Intercept	147.346	10.721	13.74****	8.997	16.38****
Age (years)	–0.229	0.151	–1.52	0.136	–1.69
BMI (kg/m)	4.352	1.335	3.26**	1.891	2.30*
Total grain (g/day)	–0.125	0.052	–2.40*	0.091	–1.37
Gender
Female	–29.355	2.620	–11.20****	5.391	–5.45****
Male
Dairy in-take
Never/rare
Not often	18.849	8.314	2.27*	6.528	2.89**
Often	58.791	8.744	6.72****	6.465	9.09****
Fish in-take
Yes
No	13.408	2.855	4.70****	5.430	2.47*
Race
White
Black	–18.246	5.003	–3.65***	4.746	–3.84****
Hispanic	9.943	5.585	1.78	5.006	1.99*
Other	–0.346	4.303	–0.08	11.379	–0.03
Salt in-take
Never/rarely
Occasionally	7.771	0.612	12.70****	6.635	1.17
Very often	7.152	5.742	1.25	6.511	1.10
Supplements
Yes
No	–13.952	5.006	–2.79**	5.361	–2.60**

, , , .

Overall, the proposed and MCMB models appear to yield similar results in terms of the covariates associated with iodine concentration. However, the proposed model has two additional predictors of iodine concentration that were not identified in the MCMB model, namely, total grain and salt in-take. One explanation for the difference in the result as noted earlier is that while the proposed method, takes into account stratification, clustering, and the sampling weights into the variance estimation procedure the MCMB variance estimator does not. It treats each observation as independent, sampling each observation with equal probability. For complex survey, the probability of selection is not likely to be the same. Thus, the assumption of independent and identically distributed observations is violated. We argue that the correct application of the bootstrap to complex surveys should incorporate the individual sampling weights rather than the final sampling weights. Unfortunately, NHANES does not report “individual” sampling weights.

In summary, results from the proposed model indicate that BMI, total grain, gender, race, supplements, and fish, dairy, and salt intake are significantly associated with urinary iodine concentration. When taken together, this set of predictors may be useful for identifying individuals who are at higher risk for iodine deficiency, and hence may potentially have increased risks of many cancers (e.g., thyroid, breast, endometrial, and ovarian cancer), and who would benefit from interventions to modify lifestyle risk behaviors.

5. Conclusion

In this article, we propose estimating the variance for the median regression estimator for complex survey data. In many complex surveys, which may include stratification and multiple stages of unequal probability sampling, the individual sampling weights are not known, which makes it challenging to use the resampling method to estimate variance. We show how to estimate the variance for median regression by incorporating the survey design and using the final sampling weights. We conclude that we should be cautious when applying resampling methods, especially the bootstrap, to complex sample surveys as this may lead to incorrect inference and ultimately wrong conclusions.