Multiple-frame surveys for a multiple-data-source world
Section 4. Multiple-frame surveys and data integration

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Rao (2021) reviewed a number of data integration methods for combining information from a probability sample $S_1,$ assumed to come from a frame with complete coverage, with information from a nonprobability sample $S_2,$ often a census of part of the population as in Section 3.4. Rao considered two cases for making inferences about $y$: (1) $y$ is observed in both samples, and (2) auxiliary information $x$ is observed in both samples but $y$ is observed only in $S_2.$ In this section I examine various data integration methods from the perspective of the multiple-frame paradigm and the assumptions in Section 2.2.

4.1   Small area estimation

Small area estimation can be considered to be a special case of a dual-frame estimation problem in which Assumption (A6) is not met. Here, $S_1$ is a probability sample from Frame 1 and Frame 2 is often an administrative data source. Both frames are assumed to have complete coverage of the population, but the variable of interest $y$ is measured only in $S_1.$ Auxiliary information $x$ used to predict $y$ is measured in both samples. Beaumont and Rao (2021) discussed integrating probability and nonprobability samples through the use of the Fay-Herriot (1979) estimator with small area estimation techniques.

A composite small area estimator (Rao and Molina, 2015) of the population mean ${\eta }_a$ in area $a$ is of the form

$${\widehat{\eta }}_a\mathrm{<mo>=</mo>}{\theta }_a{\widehat{\eta }}_a^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}+\mathrm{<mo\; stretchy="false">(</mo>\; 1}-{\theta }_a\mathrm{<mo\; stretchy="false">)</mo>}{\widehat{\eta }}_a^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{,}$$

where ${\widehat{\eta }}_a^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}$ is the direct estimator for the sample mean in area $a$ from $S_1$ (which may have large variance or may not exist), ${\widehat{\eta }}_a^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{x}_a^T\widehat{\beta }$ is a predicted value from a regression model, and ${\theta }_a$ is a compositing factor. For the Fay-Herriot estimator, ${\theta }_a$ depends on the relative precision of the two estimators under an assumed regression model whose parameters are estimated from $S_1.$ For the estimator ${\widehat{\eta }}_a,$ the variable $y$ is measured differently in the two frames  $–$  predicted values are used for Frame 2  $–$  and different compositing factors are used in different areas.

4.2   Mass imputation and sample matching

Suppose that $S_1$ is a full-response probability sample from Frame 1, but the variable of interest $y$ is not measured in $S_1.$ However, $y$ is measured in $S_2$ from Frame 2, and auxiliary variables $x$ are measured in both samples. Let ${\tilde{y}}_i$ be the predicted value of $y_i$ from an imputation model, relating $y_i$ to $x_i,$ that is developed on $S_2$ and let ${\tilde{Y}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_1}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}{\tilde{y}}_i}$ and ${\tilde{Y}}_d^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_1}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}{\delta }_i\mathrm{<mo\; stretchy="false">(</mo>}d\mathrm{<mo\; stretchy="false">)</mo>}{\tilde{y}}_i}$ be the estimated population and domain- $d$ totals from $S_1$ using the imputed values.

Similarly to small area estimation, mass imputation fits into the dual-frame context by relaxing Assumption (A6) of no measurement error. Kim and Rao (2012) and Chipperfield, Chessman and Lim (2012) considered the situation where both frames are complete and $S_1$ and $S_2$ are both probability samples. The frames can differ  $–$  Frame 1, for example, might be an area frame and Frame 2 might be a population register  $–$  but both are assumed to have full coverage. Chipperfield et al. (2012) used a composite estimator

$${\widehat{Y}}_{\text{imp}}\mathrm{<mo>=</mo>}\theta {\tilde{Y}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}+\mathrm{<mo\; stretchy="false">(</mo>\; 1}-\theta \mathrm{<mo\; stretchy="false">)</mo>}{\widehat{Y}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{,}(4.1)$$

where the optimal value of the compositing factor $\theta$ minimizes the variance (considering both the sampling and imputation variability). Kim and Rao (2012) proposed adding a correction for bias with the estimator

$${\tilde{Y}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}+{\displaystyle \sum _{i\in S_2}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}{y}_i-{\tilde{y}}_i\mathrm{<mo\; stretchy="false">)</mo>\; <mo>;</mo>}}$$

this estimator is of the same form as (4.1) with $\theta \mathrm{<mo>=</mo>}1$ if the estimated parameters in the imputation model are required to satisfy ${\displaystyle {\sum }_{i\in S_2}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}{y}_i-{\tilde{y}}_i\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}0}.$

If the imputation model produces unbiased and accurate predictions for $y_i,$ combining the samples augments the effective sample size for calculating estimates. When both samples are probability samples with full coverage, it is possible to perform model diagnostics on $S_2.$ Chipperfield et al. (2012) suggested several diagnostics, including testing the imputation model on small areas, investigating whether it is possible to predict survey membership from the value of $y_i$ (for $S_2)$ or ${\tilde{y}}_i$ (for $S_1),$ and studying the sensitivity of the mean squared error to different levels of bias in ${\tilde{Y}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}.$ The sensitivity of the diagnostics, however, depends on the quality and size of $S_2.$ If $S_2$ is small relative to $S_1,$ $S_1$ may contain subpopulations that are not well represented in $S_2$ and are poorly fit by the imputation model.

The situation becomes more complicated when Frame 2 is incomplete or when $S_2$ has selection bias. When domain $\left\{1\right\}$ is nonempty as in Figure 2.2(a), then the composite estimator with imputed values becomes

$${\widehat{Y}}_{\text{imp}}\mathrm{<mo>=</mo>}{\tilde{Y}}_{\mathrm{<mo>\{</mo>\; 1\; <mo>\}</mo>}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}+\theta {\tilde{Y}}_{\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}+\mathrm{<mo\; stretchy="false">(</mo>\; 1}-\theta \mathrm{<mo\; stretchy="false">)</mo>}{\widehat{Y}}_{\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{.}(4.2)$$

The properties of the estimator in (4.2) depend on how well the imputation model predicts the values of $y_i$ in $S_1.$ Several imputation methods have been proposed. With sample matching (Rivers, 2007), ${\tilde{y}}_i$ for observation $i$ in $S_i$ is set equal to the value of $y_i$ of the observation’s nearest neighbor (with respect to the values of $x)$ in $S_2.$ Rivers (2007), considering the situation in which $S_2$ is a convenience sample, took $\theta \mathrm{<mo>=</mo>}1$ in (4.2) and used the information in $S_2$ for the sole purpose of finding the imputed values ${\tilde{y}}_i$ for $S_1.$ Yang, Kim and Hwang (2021) studied theoretical properties of mass-imputed estimators that employ nearest neighbor methods.

Chen, Li and Wu (2020), building on the work of Lee (2006), Lee and Valliant (2009), and Valliant and Dever (2011) on using propensity score weighting to estimate population characteristics from a nonprobability sample, proposed a “doubly-robust” estimator for the situation where $x_i$ is measured in both surveys but $y_i$ is measured only in nonprobability sample $S_2.$ Let $R_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}1$ if population unit $i$ is in $S_2$ and 0 otherwise. Under strong assumptions that (1) $R_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}$ and $y_i$ are independent given covariates $x_i,$ (2) ${\pi }_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}P\left(R_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}1\right)\mathrm{<mo>></mo>}0$ for all population units $i,$ and (3) $R_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}$ and $R_j^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}$ are conditionally independent given $x,$ they estimated ${\pi }_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}$ as a function of $x_i,$ using information in $S_1,$ and proposed the estimator

$${\widehat{Y}}_{\text{DR}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in S_1}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}{\tilde{y}}_i}+{\displaystyle \sum _{i\in S_2}\frac{1}{{\widehat{\pi }}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}}\mathrm{<mo\; stretchy="false">(</mo>}{y}_i-{\tilde{y}}_i\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}$$

where ${\tilde{y}}_i$ is an imputation prediction for the unknown values of $y$ in $S_1$ (developed using the information in $S_2).$ The estimator ${\widehat{Y}}_{\text{DR}}$ is approximately unbiased for $Y$ if either the imputation model or the model predicting ${\pi }_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}$ is correct. If the imputation model is correct, then the first term of ${\widehat{Y}}_{\text{DR}}$ is approximately unbiased for $Y$ and the second term has expected value 0. If the model predicting ${\pi }_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}$ is correct, then ${\displaystyle {\sum }_{i\in S_2}{y}_i}/{\widehat{\pi }}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}$ is approximately unbiased for $Y$ and $E\left[{\displaystyle {\sum }_{i\in S_1}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}{\tilde{y}}_i}-{\displaystyle {\sum }_{i\in S_2}{\tilde{y}}_i/{\widehat{\pi }}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}}\right]\approx 0.$ If neither model is correct, however, ${\widehat{Y}}_{\text{DR}}$ may have large bias.

Kim and Tam (2021) considered an extension of the situation in Section 3.4 in which $y_i$ is not measured in $S_1,$ or is measured differently than in $S_2,$ and proposed substituting an imputed value ${\tilde{y}}_i$ for $y_i$ in the estimators from $S_1$ in (3.6), obtaining the estimator in (4.2) with $\theta \mathrm{<mo>=</mo>}0;$ they calibrated this estimator to the known domain size $N_{\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}}.$

4.3   Imputation and the NSAF

The estimators in Section 4.2 impute a predicted value ${\tilde{y}}_i$ for the unknown value of $y_i$ in $S_1.$ All have the strong assumption that the imputation model developed on $S_2$ applies to the units in domain $\left\{1\right\}$. As Lu (2014b) noted when studying regression for dual-frame surveys, relationships between $x$ and $y$ may differ across domains. Thus, an imputation model developed on a sample from an incomplete frame, or on a sample with selection bias, may provide poor predictions for $y$ in other parts of the population. Moreover, without data on $y$ in the part of the population that is imputed, it may not be possible to assess the quality of the predictions.

A dual-frame survey was taken for the NSAF because of concern that characteristics of interest might differ for telephone and nontelephone households. Let $y_i=1$ if child $i$ is in a household that is below 200 percent of the poverty threshold, and 0 otherwise. Using the full sample from both frames (Urban Institute and Child Trends, 2007) an estimated 42.2 percent of children lived in households below 200 percent of the poverty threshold, with standard error 0.5 percent. The estimated percentage from the RDD sample was 38.6 percent and the estimated percentage from the area sample was 93.4 percent. Children in the nontelephone households, sampled from the area frame, were much more likely to be living in poverty.

Now suppose that the NSAF had not measured poverty and income variables in the area sample, and $y_i$ was imputed using regression relationships developed in the RDD sample. In many surveys, the only information available for developing an imputation model is demographic variables. Fitting a logistic regression model to the RDD sample that predicts $y$ from race (with categories white, black, and other), and assigning each child in the area sample to the category with highest predicted probability, results in an estimate of 30.5 percent of children in the area sample living in poverty  $–$  a lower value than in the RDD sample. Adding an indicator variable for living in a single-parent household to the model, the estimated percentage for the area sample goes up to 51.9 percent. Both of these estimates, and estimates calculated using cell-mean imputations, are far below the percentage of 93.4 percent from the real data.

The problem, of course, is that the auxiliary information is not rich enough to provide a good prediction of poverty in the area sample. The key feature of the data, and the reason that Waksberg and his colleagues used a dual-frame survey, is that being without a telephone is highly associated with poverty. That association cannot be estimated from the RDD sample where all households have telephones. It might be possible to develop an imputation model using information from other surveys such as the Current Population Survey, where both telephone and non-telephone households are sampled, but I could not find an imputation model predicting $y$ from non-income variables in the RDD sample that provided good predictions.

The nontelephone households were a small part of the population for the NSAF, but the differences between the multivariate relationships in the telephone and nontelephone households were so great that the imputation only slightly reduced bias. If poverty had not been measured for the nontelephone sample, however, and the published statistics had relied only on the imputations, there would have been no way to detect the bias.

4.4   Domain misclassification

One major challenge for combining data using a multiple-frame approach is identifying the domain membership (or multiplicity) of units in the data sources. This is challenging even for surveys that are designed to make use of multiple frames.

The NSAF was designed as a screening survey where telephone households were excluded from the area sample. All households sampled from Frame 2, the RDD frame, were correctly classified since they were contacted by telephone. The more difficult part was obtaining the correct domain classification for households in the area-frame sample. Initial prescreening questions asked whether the household had any working telephones; those that answered no were transferred to the telephone interviewer who conducted the detailed interview. The telephone interviewer administered another brief screening interview and asked again about telephone service. An additional 7 percent of households were excluded after answering the more detailed questions about telephone ownership. Some had told the in-person interviewer that they did not have a telephone because they thought the interviewer wanted to borrow it. Others had misunderstood the question about telephone ownership  $–$  one respondent, answering the prescreening questions in the living room, thought they applied only to telephones in the living room and did not mention the telephone in the bedroom (Cunningham, Shapiro and Brick, 1999). Although the second screening interview may have corrected for misclassification from respondents who mistakenly said they did not have a telephone during the prescreening, there was no remedy for potential misclassification from respondents who responded in prescreening that they had a telephone when in fact they did not. Misclassification in this direction may have been part of the reason the investigators had a smaller sample size of nontelephone households than they had anticipated.

In dual-frame telephone surveys, the domain for Figure 2.2(b) (cell only, landline only, or both) is usually determined by asking the respondent about other available telephones and, sometimes, the relative amount each type of telephone is used. Brick, Flores-Cervantes, Lee and Norman (2011) found that their landline samples and cell samples both had smaller estimated proportions of dual users than expected from statistics collected on telephone ownership in the National Health Interview Survey. They conjectured that this was because of persons who had access to both types of telephones but rarely used one of them.

Domain membership may be unknown or difficult to estimate when combining existing data sources. In some cases, as when administrative lists are combined, it may be possible to link records, or the data files may contain information that indicates whether the unit is in other frames. In others, there may be little or no information available on domain membership. How can one know whether a participant in an opt-in panel survey is also in a frame of Medicare recipients if no questions about Medicare are asked in the survey?

Lohr (2011) found that even a small amount of domain misclassification could create large biases in dual-frame estimators; moreover, calibration to domain counts that were based on misclassifications could worsen the bias. She proposed a method for adjusting for bias due to domain misclassification, assuming that misclassification probabilities $P$ (observation classified in domain $d^{\prime }\mathrm{<mo>|</mo>}$ observation actually in domain $d)$ are known or can be accurately estimated for different population subgroups. Lin, Liu and Stokes (2019) studied a similar method using misclassification probabilities $P$ (observation actually in domain $d\mathrm{<mo>|</mo>}$ observation classified in domain $d^{\prime }).$

It may be possible to use multiple-frame methods when domain membership is unknown if the probability that unit $i$ is in domain $d$ can be estimated from auxiliary information $x_i$ known for all sampled units. Kim and Tam (2021) proposed substituting an estimator for the unknown domain membership for the situation in Section 3.4 where $S_2$ is a census of a subset of the population. They set ${\tilde{\delta }}_i\left(\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}\right)\mathrm{<mo>=</mo>}1$ if the predicted probability that unit $i\in S_1$ was in domain $\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>},$ $\widehat{P}\left[{\delta }_i\left(\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}\right)\mathrm{<mo>=</mo>}1\mathrm{<mo>|</mo>}{x}_i\right],$ exceeded 1/2, and estimated the population total for domain $\left\{1\right\}$ as ${\displaystyle {\sum }_{i\in S_1}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}\left[1-{\tilde{\delta }}_i\left(\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}\right)\right]y_i}.$

When domain membership is imputed, the mean squared error depends on the accuracy of the domain imputations as well as design features and nonresponse bias in $S_1.$ More research is needed to establish statistical properties of estimators when domain membership is estimated. It may also be desired to study alternative estimators that use the predicted probabilities directly to estimate the total in domain $\left\{1\right\}$ as ${\displaystyle {\sum }_{i\in S_1}{w}_i^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}\widehat{P}\left[{\delta }_i\left(\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}\right)\mathrm{<mo>=</mo>}0\mathrm{<mo>|</mo>}{x}_i\right]y_i}.$

Dever (2018) used sample matching to evaluate the frame overlap for a probability sample $S_1,$ taken from an address-based sampling frame, and a nonprobability sample $S_2$ recruited from social media sites. She investigated the percentage of respondents in $S_1$ who had no close match in $S_2.$ Although this procedure does not provide an unbiased estimate of the size of domain $\left\{1\right\}$ a large percentage of unmatched cases for large samples can indicate that $S_2$ represents a different population than $S_1.$

4.5   Indirect sampling and capture-recapture estimation

Sections 4.2 to 4.4 looked at extensions of multiple-frame estimators that relaxed Assumptions (A2), (A4), and (A6). All of these, though, assumed that at least one of the frames, or their union, had full coverage. Let’s now look at an example where Assumption (A1) of full coverage is relaxed, and the multiple frames are used to estimate the population size.

In indirect sampling, the target population consists of units that are linked to units in the sampling frame but are not necessarily in the frame (Lavallée, 2007) $–$  units in the target population are sampled indirectly through the links to the sampling units in the frame. Lavallée and Rivest (2012) extended the idea to multiple-frame sampling. As an example, suppose the target population consists of home care workers, who provide paid care for elderly, ill, or disabled persons in their homes. Frame 1 might be a list of persons receiving Medicare benefits, and Frame 2 might be a list of home health care aides from employment or licensing agencies. Persons in the Frame-1 sample are asked to identify workers who provide them with home care, who are then interviewed. A sample of workers from Frame 2 is also interviewed. The home care workers identified from the Frame-1 sample may have links to multiple persons in Frame 1 and may also be in Frame 2. Similarly, persons in the Frame-2 sample may also have links to units in Frame 1. An example of linkage structure is shown in Figure 4.1.

With indirect sampling, the $Q$ frames can contain different types of units (the situation with different types of units was also considered by Hartley, 1974). We are not interested in overlap of the sampling frames (shown as nonoverlapping in Figure 4.1 because they contain different types of units) but in the overlap for the units in the target population. Sampled units in the target population have multiple chances of selection if they are linked to multiple units in one or both sampling frames.

Let $l_{j\mathrm{,}k}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}1$ if unit $j$ from Frame $q$ is linked to unit $k$ in the target population, and let $L_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}$ be the total number of links between unit $k$ in the target population and Frame $q$ (assumed to be knowable from asking unit $k).$ Then an estimator ${\widehat{Y}}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}$ can be found for each frame using the links as

$${\widehat{Y}}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\in S^{(q)}}{w}_j^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}{\displaystyle \sum _k}\frac{l_{j\mathrm{,}k}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}}{L_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}}{y}_k}\mathrm{<mo>=</mo>}{\displaystyle \sum _k{u}_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}{y}_k}\mathrm{,}$$

where

$$u_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\in S^{(q)}}{w}_j^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\frac{l_{j\mathrm{,}k}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}}{L_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}}}\mathrm{.}$$

In the context of our example, person $j$ in $S_1$ would say they receive paid home care from provider $k,$ resulting in $l_{j\mathrm{,}k}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}1.$ Then the linked home care provider would be asked about how many other persons they work for who receive Medicare (assume they would know this or it could be determined from other sources), giving the value $L_k^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}.$ The quantity $u_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}$ sums the weights of the units in $S_q$ with links to unit $k,$ adjusting for the multiplicity of the links to that frame. If $w_j^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}1/{\pi }_j^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}},$ then

$$E\mathrm{<mo\; stretchy="false">[</mo>}{u}_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">]</mo>}\mathrm{<mo>=</mo>}E\left[{\displaystyle \sum _{j\in S^{(q)}}{w}_j^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\frac{l_{j\mathrm{,}k}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}}{L_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}}}\right]\mathrm{<mo>=</mo>}{a}_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}$$

where $a_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}1$ if target population member $k$ is linked to at least one unit in Frame $q$ and 0 otherwise.

Multiple-frame methods may then be used to estimate characteristics of the population of home care providers, assuming that unit $k$ linked from $S_q$ provides accurate information on (1) the number of links to members of Frame $q\mathrm{<mo\; stretchy="false">(</mo>}{L}_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>},$ needed for multiplicity adjustments with Frame $q,$ and (2) whether they are also linked to the other frame(s) $(a_k^{\mathrm{<mo\; stretchy="false">(</mo>}f\mathrm{<mo\; stretchy="false">)</mo>}}$ for $f\ne q),$ needed to adjust for the multiplicity of linkage from different frames.

Lavallée and Rivest (2012) noted that if the union of the two frames has incomplete coverage  $–$  Assumption (A1) is violated  $–$  the samples from the two frames can be used to estimate the size of the target population. Let ${\widehat{T}}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_k{u}_k^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}}$ for $k\mathrm{<mo>=</mo>}\mathrm{1,}2.$ Then $E\left[{\widehat{T}}^{\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}}\right]$ is the number of target population members that can be linked from Frame $q$ . Each sample also provides an estimate of the number of target population units that can be linked from both frames: ${\widehat{T}}_{\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_k{u}_k^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}{a}_k^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}}$ and ${\widehat{T}}_{\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_k{u}_k^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}{a}_k^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}}.$ These can be composited to obtain an estimator ${\widehat{T}}_{\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}}$ of the number of persons in the target population who can be captured from both frames.

The Lincoln-Petersen capture-recapture estimator of population size can then be used. Under the strong assumption that being captured by Frame 1 is independent of being captured by Frame 2, the total number of home care providers can be estimated by ${\widehat{T}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 1\; <mo\; stretchy="false">)</mo>}}{\widehat{T}}^{\mathrm{<mo\; stretchy="false">(</mo>\; 2\; <mo\; stretchy="false">)</mo>}}/{\widehat{T}}_{\mathrm{<mo>\{</mo>\; 1,}\mathrm{2\; <mo>\}</mo>}}.$ In some cases where the independence assumption is not met for the entire population, it may be approximately met on subpopulations whose estimated numbers can be summed. If there are more than two frames, loglinear models may be used to explore associations among the frames (Lohr, 2022, Chapter 14); Zhang (2019) presented a model for the situation in which frames may contain misclassified units.

Alleva, Arbia, Falorsi, Nardelli and Zuliani (2020) proposed using indirect multiple-frame sampling to estimate the number of people infected by SARS-CoV-2 during the early stages of the COVID-19 pandemic in 2020  $–$  information needed for estimating transmissibility and infection parameters in epidemiologic models. In this application, Frame 1 consists of persons with verified infections (perhaps obtained from hospitals, quarantine centers, or clinics), and Frame 2 consists of other persons; the persons in $S_2$ are administered a test for SARS-CoV-2. The linked sample consists of persons who had contact within the past 14 days with anyone in $S_1$ or with a member of $S_2$ who tested positive.

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