Optimum allocation for a dual-frame telephone survey 1. IntroductionOptimum allocation for a dual-frame telephone survey 1. Introduction

• Previous
• Next

Modern random digit dial (RDD) telephone surveys in the U.S. use two samples: a sample of landlines and a sample of cell-phone lines. Wolter, Smith and Blumberg (2010) provide the statistical foundations for such dual-frame telephone surveys. The present article builds on that work and demonstrates the considerations and statistical methods for allocating the total survey resources to the two sampling frames.

Because it is less costly on a per-unit basis and has a longer history of use, the landline sample is often the larger sample and the survey interview is attempted for all respondents in this sample. The interviewing protocol for the smaller cell-phone sample is configured in one of two ways: (1) attempt to complete the survey interview for all responding persons, or (2) conduct a brief screening interview to ascertain the telephone status of the respondent, and then attempt to complete the survey interview only for respondents whose telephone status is classified as cell-phone-only (CPO) (i.e., respondents who report in the screening interview that they do not have a working landline in their household). (Within the screening approach there are variations, such as interviewing both CPO respondents and others who report that there is a landline in the household but they are not reachable through the landline.) As the size of the landline-only (LLO) population (i.e., persons who have a working landline telephone in the household but do not have access to a cell phone) declines over time (Blumberg and Luke 2010), survey statisticians may consider new designs in which the cell-phone sample is the larger sample and all respondents are interviewed, while the interviewing protocol for the smaller landline sample calls for screening or taking all respondents. Yet in this article, we focus on the prevailing circumstances in the last several years in which the cell-phone sample is typically the smaller sample and a take-all or screening protocol is used for respondents in this sample.

We shall develop the methods for optimum allocation under ideal assumptions that the sample sizes refer to completed cases (i.e., no nonresponse); that there is essentially a one-to-one relationship between the sampling units (telephone numbers) and the analytical units (e.g., households) in the landline population; that there is essentially a one-to-one relationship between the sampling units and the analytical units in the cell-phone population; and that all units in the target population are included in at least one of the two sampling frames. Given these assumptions, each and every specific analytic unit is linked to a landline, a cell-phone line, or both a landline and a cell-phone line, and is linked to at most one landline and at most one cell-phone line.

Most of the previous literature on dual-frame surveys studies estimation procedures rather than the question of allocation of the sample size to the various sampling frames, including Hartley (1962, 1974); Fuller and Burmeister (1972); Skinner and Rao (1996); and Lohr and Rao (2000, 2006). Biemer (1984) and Lepkowski and Groves (1986) looked at allocation when one frame is a subset of the other frame, as might be the case with an area sample supplemented by a special list.

To begin, we establish our notation and assumptions. Let $U^A$ be the landline population and $U^B$ the cell-phone population. The overall population of interest is $U=U^A\cup U^B.$ Some units have both a landline and a cell phone (the dual-user population), while others have only a landline (the LLO population) or only a cell phone (the CPO population), and thus the two populations overlap as follows: $U^{ab}=U^A\cap U^B,U^a=U^A-U^{ab},$ and $U^b=U^B-U^{ab}.$ $U^a$ is the LLO domain, $U^b$ is the CPO domain, and $U^{ab}$ is the dual-user domain. The population sizes are $N_A=\text{card}\left(U^A\right),$ $N_B=\text{card}\left(U^B\right),$ $N_{ab}=\text{card}\left(U^{ab}\right),N_a=\text{card}\left(U^a\right),$ and $N_b=\text{card}\left(U^b\right).$ We denote the proportions in the overlap (or dual-user) population by $\alpha =N_{ab}/N_A$ and $\beta =N_{ab}/N_B.$

Let $s_A$ be a simple random sample without replacement selected from $U^A,$ let $s_B$ be a simple random sample without replacement selected from $U^B,$ and let $n_A=\text{card}\left(s_A\right)$ and $n_B=\text{card}\left(s_B\right)$ be the sample sizes (i.e., completed interviews). We assume that domain membership $\left(a,ab,b\right)$ is not known at the time of sampling.

Let $Y_i$ be a variable of interest for the $i^{\text{th}}$ unit in the overall population. The population domain means and variance components are denoted by ${\overline{Y}}_A,{\overline{Y}}_B,{\overline{Y}}_{ab},{\overline{Y}}_a,{\overline{Y}}_b,S_A^2,S_B^2,S_{ab}^2,S_a^2,$ and $S_b^2.$ We take the goal of the survey to be the estimation of the overall population total $Y.$

In what follows, we derive the optimum allocation given the take-all protocol and the screening protocols in Section 2 and Section 3, respectively. Section 4 compares the two protocols in terms of efficiency and cost and attempts to provide guidance about the circumstances under which each protocol is better. The section also explores the optimum choice of a mixing parameter $p,$ which is used to combine the estimators from the two samples $\left(s_A\cap U^{ab}\text{ and }{s}_B\cap U^{ab}\right)$ that represent the dual-user population. Section 5 applies the methods to the National Immunization Survey, a large dual-frame telephone survey sponsored by the Centers for Disease Control and Prevention (CDC). The article closes with a brief summary in Section 6.

• Previous
• Next