Sampling strategies for overlapping eligibility groups in the World Trade Center Health Registry

Abstract

Sub-studies addressing specific research questions may necessitate selecting a random sample from different eligibility groups of The World Trade Center Health Registry (WTCHR) for estimating population parameter mean. In such a situation, Singh (1988) and Osahan (1997) proposed three two-stage sampling strategies in overlapping clusters. These sampling strategies are compared using WTCHR’s overlapping eligibility groups, in terms of their bias and efficiency.

Keywords

Overlapping clusters two-stage sampling eligibility groups probabilities proportional to size bias efficiency

1. Introduction

The terrorist events of September 11, 2001 (9/11) in New York City led to almost 3,000 deaths, including more than 2,200 civilians, 343 firefighters, and 60 police officers. Tens of thousands of lower Manhattan building occupants, residents, and school children were evacuated and had their lives and livelihoods disrupted for months to years afterward. The rescue, recovery, and clean-up effort involved city, state, and federal agency employees as well as contracted workers and volunteers from all 50 states (Farfel et al., 2008).

The World Trade Center Health Registry (WTCHR) is the largest post-disaster registry in U.S. history (Brackbill et al., 2009). Its goals are to document short- and long-term 9/11-related physical and mental health impacts and gaps in care, connect individuals to care, and inform 9/11 health care policy and response planning for future disasters.

The WTCHR is comprised of enrollees from five overlapping eligibility groups (rescue and recovery workers; building occupants and passersby; residents; school students/staff). Sub-studies addressing specific research questions may necessitate selecting a random sample from different WTCHR eligibility groups. However, many enrollees are members of multiple eligibility groups, so if we are selecting a random sample separately from two eligibility groups, there are chances some sampled units belong to either both eligibility groups or another third eligibility group.

This overlapping eligibility complicates the process of construction of estimators. To minimize the estimator bias and variability, a sampled person should belong to only one group or a statistical weight should be applied to a person that accounts for the number of eligibility groups that the person is included in. Using this technique, efficient estimators can be constructed from WTCHR samples even with significant overlap in eligibility groups.

Overlapping clusters are common in many types of epidemiological investigations. Special sampling strategies can be used to accommodate overlapping clusters so that resulting estimates remain statistically valid. Such special sampling strategies have been used in regional epidemiological surveys for infectious diseases like mycobacterium tuberculosis (TB) (Osahan, 1997). These improved strategies were also used to study tuberculosis transmission in correctional facilities (Bellin et al., 1993), in schools (Braden et al., 1995), in shelters (CDC, 1991), in an airplane (Driver et al., 1994), in a church (Dutt et al., 1995) and in a bar (Kline et al., 1995).

These sampling strategies are equally applicable to household clusters in ecological surveys formed around factories burning coal and emitting polycyclic aromatic hydrocarbons (PAH’s) (Menzie et al., 1992). Overlapping clusters are also important in conducting market research when an advertising company may wish to identify the range of TV stations to determine the size of clusters of their viewers. Similarly, many other situations benefit from its application.

The objective of this study is to examine three sampling strategies’ relative efficiencies and compare the variances of unbiased estimators of the following strategies: Probabilities Proportional to Size (PPS), Probabilities Proportional to Size Without Repetitions of second stage units (PPSW), and mean square error of biased estimator of strategy Simple Random Sampling (SRS) as applied to the WTCHR population. The scope of WTCHR dataset to fit in dual sampling methodology is yet to explore (Singh et al., 2014).

Figure 1.

Geographic distribution of Lower Manhattan registrants’ primary residence by Zip Code in New York City on September 11, 2001 (Farfel et al., 2008).

2. World Trade Center Health Registry

The WTCHR was created in July 2002 as a collaborative effort between the Agency for Toxic Substances and Disease Registry (ATSDR) and the New York City Department of Health and Mental Hygiene (NYC DOHMH). Development of eligibility criteria (Farfel et al., 2008) took into account proximity by time and place to the WTC attack, acute exposure to the dust and debris cloud that resulted from the collapse of the towers, and chronic exposure to dust, smoke and fumes in the vicinity of the WTC site through the end of the recovery phase in June 2002. The WTCHR was approved by the institutional review boards of NYC DOHMH and U.S. Centers for Disease Control and Prevention.

3. Initial data collection

Of the 71,437 baseline interviews conducted from September 2003 through November 2004 (2–3 years post-9/11) with people who voluntarily enrolled in the WTCHR, 67,527 (95%) were completed using computer-assisted telephone interviewing (CATI) and the remaining 3,910 were completed using in-person, computer-assisted personal interviewing (CAPI).

4. Overlapping eligibility groups

All enrollees were categorized into four broad overlapping categories known as eligibility groups:

(1) (1)
Rescue and recovery workers and volunteers,
(2)
Building occupants, passersby, and people in transit,
(3)
Residents south of Canal Street,
(4)
School students and staff.

Figure 1 is a map of lower Manhattan showing the WTC site (Farfel et al., 2008). The largest group of WTCHR enrollees included people present in lower Manhattan near the WTC site on the morning of 9/11 ( $n=$ 43,487), including 10,393 occupants of damaged or destroyed buildings, 19,900 occupants of other nearby buildings, and 13,194 passersby or people in transit. The Registry’s 30,665 rescue and recovery workers included police, firefighters, emergency medical services workers, construction or engineering personnel, and sanitation workers ( $n=$ 14,747) as well as volunteers affiliated with organizations and unaffiliated volunteers ( $n=$ 7,389). The Registry also included 14,665 lower Manhattan residents, 2,075 students (pre-K to 12 ${}^{\rm th}$ grade) and 571 staff from schools in the vicinity of the WTC site.

The number of persons eligible for the Registry was estimated to be approximately 409,000, of whom 71,437 (17.4%) enrolled (Murphy et al., 2007). Outreach and multilingual media campaigns encouraged enrollment through a toll-free number or Web site (classified as “self-identified”). Lists of persons potentially exposed were provided by entities such as employers and governmental agencies (classified as “list-identified”). Final enrollment was 70% self-identified and 30% list-identified (Brackbill et al., 2009). The percentage of list-identified enrollees also varied across eligibility groups, ranging from 14% among students to 37% among workers.
5. Methods

SAS ${}^{\@setsize{\scriptsize}{8pt}{\viipt}{\@viipt}\textregistered}$ (version 9.4) was used to empirically compare overlapping eligibility groups, repetitions of units within eligibility groups (i.e., enrollees belonging to more than one eligibility group), and sampling efficiency of proposed estimators. Sample sizes were rounded to the nearest integer. To avoid implying more precision than is appropriate, estimates/variances of Zi were rounded to two decimal places and final variances/mean square errors/relative efficiencies to three decimal places. Microsoft PowerPoint software was used to create a Venn diagram (not proportional to size) to depict overlap in eligibility criteria among enrollees.

6. Sampling strategies in overlapping clusters

Cluster sampling is sometimes used in surveys to avoid constructing a list of sampling units (sampling frame) which can be challenging to develop. In contrast, generating such a frame for clusters is economical and is often readily available. Developing a list of households for a State in the USA may be challenging, but lists are available for smaller units like cities, counties and zip codes. Similarly, it may not be possible to list all of the customers of a chain of stores in an area. However, it would be possible to have such a list from each store and treat them as clusters. In such situations, cluster sampling is beneficial. Clusters are formed either before selecting the sample (CBS) or after selecting the sample (CAS). In the two practical situations discussed above, the CBS system of cluster formation is applicable and clusters may be overlapping or non-overlapping. For non-overlapping clusters, there are a number of usual available methodologies in the literature for cluster sampling that can be applied (Cochran, 1977).

For overlapping clusters, however, the literature is limited (Sethi, 1965; Goel & Singh, 1977; Agarwal & Singh, 1982; Amdekar, 1985; Singh, 1988; Osahan, 1997). Singh (1988) developed two sampling strategies in overlapping clusters in the CBS system for estimating the population mean, assuming known population size. An extension of this method was undertaken by Tracy and Osahan (1994).

The first strategy SRS of Singh (1988):

a) a)
Clusters were selected with equal probabilities, and
b)
Second stage units were taken with equal probabilities.

In the second strategy PPS:

a) a)
Clusters were selected with probabilities proportional to size of cluster, and
b)
Second stage units were taken with equal probabilities.

An improved sampling strategy PPSW was proposed by Osahan (1997) in which:

a) a)
Clusters were selected with probabilities proportional to size of cluster, and
b)
For second stage units, a single big sub-sample was selected and thus there was no chance of repetition of second stage units.

The above approach was expected to provide a better estimator in comparison to Singh’s (1988). Also here, the comparison was made relatively more simple and direct, whereas in Singh’s approach the support of evidence given by Hansen and Hurwitz (1943) was required. The proposed strategy was discussed in the context of a tuberculosis outbreak but equally applicable to other situations where sampling was required from a population consisting of overlapping clusters.

The two sampling strategies of Singh (1988) and third of Osahan (1997) are two-stage sampling strategies that can be efficiently applied to a population comprised of overlapping clusters/first-stage units. The goal of the present investigation is to empirically compare the efficiency of these three sampling strategies in overlapping eligibility groups in the WTCHR population.
7. Proposed sampling strategies for comparison

Let $\Omega$ be a population under investigation of $N$ distinct and identifiable units, $y$ be the characteristic of study and the parameter of interest be its population mean

$\displaystyle\bar{Y}=\frac{1}{N}\sum\limits_{q=1}^{N}Y_{q}$

Assume that these $N$ units may be expressed in the form of $K$ overlapping clusters with $N_{i}(i=1,2,\ldots,K)$ units associated with the i-th cluster and

$\displaystyle\sum\limits_{i=I}^{K}N_{i}=M\geqslant N,$

equality holds for non-overlapping clusters, a situation where no unit is common to more than one cluster. But in fact, a unit may be associated with more than one cluster and let Fj be the frequency of j-th unit occurring in $K$ clusters.

Define

$\displaystyle Z_{ij}=M∼{}Y_{ij}/N∼{}F_{j};i=1,2,\ldots,K\text{ and }j=1,2,% \ldots,N_{i},$

where $Y_{ij}$ is the value of $y$ for the j-th unit in the i-th cluster.

From (2.3) and (2.7) of Singh (1988), the Mean Square Error (MSE, being a biased estimator) and variance expression for two strategies are given as

$\displaystyle\textit{MSE}(\bar{z}_{\textit{SRS}})=\frac{\sigma_{bz}^{2}}{k}+% \frac{1}{kK}\sum\limits_{i=1}^{K}\left(\frac{1}{n_{i}}-\frac{1}{N_{i}}\right)S% _{iz}^{2}+(\bar{Z}_{K}-\bar{Z})^{2}$ (1)

and

$\displaystyle V(\bar{z}_{\textit{PPS}})=\frac{\sigma_{b^{\prime}z}^{2}}{k}+% \frac{1}{k}\sum\limits_{i=1}^{K}\frac{N_{i}}{M}\left(\frac{1}{n_{i}}-\frac{1}{% N_{i}}\right)S_{iz}^{2}$ (2)

where

$\displaystyle\sigma_{bz}^{2}=\sum\limits_{i=1}^{K}(\bar{Z}_{i}-\bar{Z}_{K})^{2% }/K,$ $\displaystyle\bar{Z}_{K}=\sum\limits_{i=1}^{K}\bar{Z}_{i}/K,$ $\displaystyle S_{iz}^{2}=\sum\limits_{j=1}^{N_{i}}(Z_{ij}-\bar{Z}_{i})^{2}/(N_% {i}-1),$ $\displaystyle\sigma_{b^{\prime}z}^{2}=\sum\limits_{i=1}^{K}N_{i}(\bar{Z}_{i}-% \bar{Z})^{2}/M$

and

$\displaystyle\bar{Z}=\sum\limits_{i=1}^{K}N_{i}\bar{Z}_{i}/M=\bar{Y}.$

In the first and second strategies, clusters are taken with simple random sampling with replacement (SRSWR) and probability proportional to size with replacement (PPSWR), respectively. The second stage units are selected with simple random sampling without replacement (SRSWOR) in both the strategies.

Osahan (1997) proposed sampling strategy in two stage-sampling consists of the following steps:

a) a)

Let $k$ clusters out of $K$ clusters of population be selected by PPSWR with initial probabilities

$\displaystyle P_{i}=N_{i}/M,i=1,2,\ldots,K.$

If i-th cluster is selected $\alpha_{i}$ times in the sample at the first stage, then at the second stage a random sample of size $\alpha_{i}∼{}n_{i}$ units is selected by SRSWOR from the i-th selected cluster of size $N_{i}$ , assuming

$\displaystyle\alpha_{i}n_{i}\leqslant N_{i}(i=1,2,\ldots,K).$

This strategy leads to a fully without replacement sample, i.e., there is no chance of repetition in the ultimate sample of units.

The variance of the estimator is given as

$\displaystyle V(\bar{z}_{\textit{PPSW}})=\frac{\sigma_{b^{\prime}z}^{2}}{k}+% \frac{1}{k}\sum\limits_{i=1}^{K}\frac{N_{i}}{M}\left(\frac{1}{n_{i}}-\frac{1}{% N_{i}}\right)S_{iz}^{2}-\frac{k-1}{k}\sum\limits_{i=1}^{K}\frac{N_{i}}{M^{2}}S% _{iz}^{2}$ (3)

In Singh’s (1988) latter strategy, $k$ clusters out of $K$ were taken in exactly the same manner as Osahan’s (1997) strategy; the difference was in the selection of second stage units. In Singh’s strategy, $n_{i}$ second stage units were selected from Ni units with SRSWOR, but only after replacing the second stage sample that may have been drawn previously in case the i-th first stage unit got selected more than once. Thus, there was a chance that a few units might be repeated in the final ultimate sample of contacts. In Osahan’s (1997) proposed strategy, we selected a single big sub-sample. Thus there was no chance of repetition of second stage units, and it was expected to provide a better estimator in comparison to Singh’s (1988).

Table 1

Eligibility Group Characteristics, World Trade Center Health Registry, 2003–2004, $N=$ 71,228, $M=$ 88,817

	Overlapping eligibility groups
Parameter	Building occupants, passersby, people in transit	Rescue and recovery workers	Residents
Ni	43487	30665	14665
$\mathop{i.}\nolimits$	40.60	43.25	33.12
$\mathop{iz}\nolimits$	339.74	292.80	499.72
	Repetitions of units in groups/clusters
Fj	1	2	3
Enrollees	54437	15993	798
	Sampling strategies (sampled 1% of eligibility groups)
	SRS	PPS	PPSW
$n_{i}$ (1%)	435	307	147
Variance/MSE	11.732	6.417	6.415
	Comparison of strategies
	PPS w.r.t. SRS	PPSW w.r.t. SRS	PPSW w.r.t. PPS
Relative efficiency (%)	182.835	182.891	100.031
	Sampling strategies (sampled 0.5% of eligibility groups)
	SRS	PPS	PPSW
$n_{i}$ (0.5%)	217	153	73
Variance/MSE	12.597	7.059	7.057
	Comparison of strategies
	PPS w.r.t. SRS	PPSW w.r.t. SRS	PPSW w.r.t. PPS
Relative efficiency (%)	178.453	178.502	100.028

Figure 2.

Overlap of three major enrollment groups in the WTC Health Registry (its overlap is described in text and Table 1).

8. Practical illustration of sampling strategies for comparison

Using data from the World Trade Center Health Registry as described above, we dropped the student and staff group and categorized 71,228 enrollees (including students and staff who qualified) into three main categories, as follows:

(1) (1)
Rescue and recovery workers and volunteers,
(2)
Building occupants, passersby, and people in transit south of Chambers Street, and
(3)
Residents south of Canal Street.

The above three large groups were considered in this study as an enrollee may be associated with more than one eligibility groups.

Twenty-six percent of enrollees met more than one eligibility criterion. The greatest overlap was among building occupants, passersby, and people in transit who were also either rescue/recovery workers ( $n=$ 7,690) or residents ( $n=$ 7,867) or both ( $n=$ 798) as shown in Table 1. Apart from above overlap, there were 436 workers who were also residents(Fig. 2).

The size of the eligibility groups were 14,665 for residents, 30,655 for rescue and recovery workers and volunteers, and 43,487 for building occupants, passersby, and people in transit. For the present illustration, only these three large eligibility groups were used to compare three sampling strategies.

The description of overlapping in eligibility groups for this population has been shown in Fig. 2. The percentage of overlapping (OL) was calculated as $100(M-N)/N$ and was 24.68% for this population. At the first stage, a sample of two eligibility groups was taken from the population of three eligibility groups. One percent of enrollees from each selected eligibility group were taken at the second stage of sampling. The final sample was comprised of 147 residents, 307 rescue and recovery workers and volunteers, and 435 for building occupants, passersby, and people in transit (Table 1).

To illustrate we estimated ‘age on 9/11’ (rounded to the nearest integer) comparing three sampling strategies in overlapping eligibility groups. Average age was imputed separately for missing values for 88 residents, 82 rescue and recovery workers and volunteers, and 257 building occupants, passersby, and people in transit. Table 1 compares the two sampling strategies SRS and PPS of Singh (1988) with the one PPSW of Osahan (1997).

Some other important parameters like WTCHR eligibility group size, sample sizes taken, and means and variances of Zi’s are also shown in the table to shed light on their computations. A second set of comparisons formed by selecting samples of 0.5% from the population of the same three main eligibility groups is also shown.

The percent relative efficiency (RE) of the compared strategies varied from 100.03 to 182.89. PPSW was nearly 83% more efficient at generating a stable sample for estimation of age on 9/11 in the three overlapping eligibility groups as compared to SRS. The WTCHR population of the three main overlapping eligibility groups is exceptionally large ( $N=$ 71,228), however the size of population containing repeated units is the sum of the eligibility groups combined ( $M=$ 88,817). This large population contributed to only a small improvement in PPSW sampling strategy as compared to PPS strategy (0.031%) as $M^{2}$ appears in the denominator of difference in two variance expressions Eqs (2) and (3). For smaller populations this increase in efficiency will be substantial.
9. Conclusion

The present empirical investigation is unique in its nature, as it is the only one that considers the real overlap in eligibility groups of the WTCHR population. In earlier studies, very small populations were constructed to compare variances of the sampling strategies. For example, Tracy and Osahan (1994) created two hypothetical populations of $N=$ 9 to explain the results. Similarly Osahan (1997) manipulated overlapping clusters of contacts of tuberculosis cases by combining four clusters in each of the two populations, respectively, of sizes $N=$ 27 and $N=$ 247. In contrast, this present empirical investigation considers overlapping in eligibility groups in a large, real world population ( $N=$ 71,228). Efficiency comparison results lead to the conclusion that in any study, whenever we need to select a random sample in overlapping eligibility groups of WTCHR population, the PPSW sampling strategy will be the best choice followed by PPS and SRS strategies. PPSW provides more stable samples that yield more precise results for estimation of characteristics like age.

There is a realistic limitation of unknown population size in such studies dealing with overlapping clusters, though this is not the case in the present study where population size is known in advance.

Footnotes

Acknowledgments

I would like to thank World Trade Center Health Registry, NYC Department of Health and Mental Hygiene, for allowing me to use WTCHR Wave 1 dataset to prepare this manuscript and reprint Fig. (Farfel et al., 2008). I also wish to thank Cheryl Stein, Robert Brackbill, Mark Farfel, Sharon Perlman and Charon Gwynn for their critical review. The comments of the referees on an earlier version were most helpful in improving this manuscript.

Conflict of interest

The author has no conflicts of interest to disclose.

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