A method to correct for frame membership error in dual frame estimators
Section 1. Introduction

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In single frame surveys, all sample units are selected from the same frame. This is the preferred method when the frame covers the population of interest. But no single frame with adequate coverage exists for some applications, or it may be too expensive to sample from a complete frame that also includes units that are not part of the target population. In such cases, a multiple frame design can be an attractive alternative.

The dual frame design, in which two overlapping frames are used for sampling, was introduced by Hartley (1962). The frames, labeled $A$ and $B,$ collectively cover the population of interest $U,$ which is made up of three mutually exclusive subsets known as domains. Domain $a$ contains units in frame $A$ but not in $B;$ domain $b$ contains units in frame $B$ but not in $A;$ and domain $ab$ contains the units in the intersection of $A$ and $B.$ In dual frame surveys, samples are selected independently from frames $A$ and $B.$ The sampler typically does not know in advance which frame units are in which domains, but will ascertain this for the units that are sampled as part of the data collection process.

Several estimators are available for combining the information from the two samples to estimate parameters of $U.$ All require adjusting the weights of the sampled units based on domain membership. Thus, knowing the domain of sampled units is critical for estimation from dual frame sample designs. When domain membership can be determined accurately, unbiased estimators of means and totals can be constructed. However, when estimators are based on inaccurate domain information, they can be biased. Hartley (1962) developed a family of dual frame estimators. In this paper we propose a bias correction method for Hartley’s estimator that is available if either the misclassification probabilities are known, or more realistically, can be estimated from a random subsample of respondents. This estimator is compared to one proposed by Lohr (2011) for the same purpose, and the different assumptions under which each is justified are clarified.

The motivating application for this work was the Marine Recreational Information Program (MRIP), a data collection system carried out by the National Oceanic and Atmospheric Administration (NOAA), to provide bi-monthly estimates of fish catch by species in U.S. marine waters. One part of the program was the Coastal Household Telephone Survey (CHTS), which provided an estimate of the number of fishing trips made by recreational anglers in each coastal state in each time period. A National Research Council report (NRC, 2006) identified deficiencies of the original survey design, and pilot surveys tested improvements. Several of these pilots had a dual frame sample design, which supplemented a residential frame (either a telephone or address frame of all households in the state) with the state’s angler license registry. The telephone frame is inefficient; for example, only about 5% of CHTS sample contacts in urban areas reached an angler (NRC, 2006). The license frame is efficient but incomplete, due to omission of anglers for a variety of exemptions from licensing, as well as to illegal angling. Domain membership information for each angler sampled from the residential frame was requested from the survey respondent by asking if he or she was licensed. Andrews, Brick, Mathiowetz and Stokes (2010) demonstrated that the domain membership data obtained in this way is unreliable, as respondents both overreport and underreport license ownership. The goal of this research was to determine the effect of this error on estimation of number of trips and find a method to mitigate the damage.

The pilot study we used for this research was conducted by mail. The two frames were an address frame for all residents of the state, which was provided by US Postal Service, and the license frame listing all anglers licensed by the state, provided by the state natural resource agency. The address frame obviously did not contain information about whether the household members had fishing licenses, so that question was asked of all respondents sampled from the address frame in this study. The license frame did of course have the addresses so that it could be determined whether or not the anglers sampled from this frame resided in the state. Because we had address information, it was possible to link the households on the address frame with the addresses on the license frame fairly easily. However, determining whether the persons responding to the address frame survey were the licensed individuals in the household was laborious, since no names were requested. For this reason, it was difficult to determine whether a household was on both frames or not.

In Section 2, we review estimation for dual frame designs. In Section 3, we present examples to show how domain misclassication error affects estimation. In Section 4, we derive a bias correction method for Hartley’s dual frame estimator and compare it to an alternative (Lohr, 2011). In Section 5, we present results of a simulation designed to examine the accuracy of inference using the bias-corrected estimators. The method is illustrated in Section 6 by an application to data from one of NOAA’s pilot angler surveys. A discussion follows in Section 7.

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