Regression estimation in multiple frames when there are measurement and response errors

Introduction

Multiple frames surveys

Estimation in elusive population in which a direct sampling frame may not exist can be done by use of multiple frames (MF) surveys. Multiple frames are useful for reducing costs for a given precision constraint and improving coverage i.e. under-or over-coverage. To overcome the problem of under coverage the original frame is supplemented with additional frames so that the union of all frames covers the target population. In this case some units may be present in more than one frame, leading to over-coverage. Duplication in the same frame may also occur leading to over-coverage. Thus imperfect frames can be dealt with by use of MF surveys where two or more listings of units is simultaneously used for sample selection. For instance, we can have a three-frame sampling design by carrying out a MF survey of homeless population where eligibility is from homeless shelters, soup kitchens and street locations. Thus MF set up corresponds to sampling individually at locations and has been applied in European migration studies. MF surveys allows use of more flexible strategy than conventional single frame survey as well as use of different modes of data collection. For example, in one frame, one can use face-to-face interviews and in another frame, we use email-questionnaires with an aim of controlling coverage survey costs, response rates and estimation accuracy. MF surveys have significant challenges in efficient estimation of population parameters of interest due to possible overlap among the frames at the selection stage leading to multiple opportunities of selecting the same unit in the final sample.

Therefore, the multiplicity of selected units and their possible duplication at the estimation stage should be dealt with. There is limited use of MF designs though very appealing due to difficulty in a simple interpretation and implementation of MF estimates including variance estimation brought about by the natural complexity of the multiple frame estimation framework. The estimation is difficult due to lack of a standard software. Singh and Mecatti (2009, 2011) [15,14] provided a simplified and unified review of available MF estimation methods by using the multiplicity approach to MF estimation when there no measurement and response errors.

Estimation in dual frames and multiple frames

The concept of multiple frames arises in surveys when no complete sampling frame comprising all the population units exists. It is assumed that by considering various aspects of the population it is possible to have at least two or more incomplete lists of the population units whose union is the whole population. As an illustration consider a survey where the sampling frame is those with mail boxes. Not all families may have mail boxes but there are those with mobile phones and no mail boxes and also in the groups there are those with mail boxes and mobile phones. The list of mail box owners is one frame and the list of mobile telephone owners is another frame. Considering each frame, it is possible to partition the list of units into disjoint sets determined by the membership of the units in other frames. These disjoint sets are called domains of the frame. For instance, if there are two frames say A and B the disjoint sets or the domains of A are two, those in A and those in both A and B i.e. a(A) and a(AB). Similarly, B has two domains b(B) and b(AB). Of course a(AB) and b(AB) are same set. If there are three frames say A, B, C then A will have four domains as a(A), a(AB), a(AC) and a(ABC). The same with B as b(B), b(AB), b(BC) and b(ABC) and the same with C as c(C), c(AC), c(BC) and c(ABC). In general, if there are say L frames, then frame A will have

domains. This is the same for all other frames. The estimation in Multiple Frames (MF) is done by partitioning of all frames into disjoint subsets (Singh and Mecatti, 2014) [6] . For example, if we have three frames A, B and C, where A represents the list of people with telephones, B represents the list of people with electricity and C represents the list of people with piped water, the three frames can be partitioned as follows,

The number of domains in a frame is equal to

Regression estimation under multiple frames

In regression estimation we assume that different domains have different characteristics and hence the regression estimation will be based on the domains of a frame.

Let the auxiliary variables be p and the observed vector be

  

The estimator can be written as,

  

Taking expectation with respect to the measurement error process we get,  

The variance is, ˆˆ( ) ( ) ( )

. Under Taylor linearization approximation, the approximate variance is,

The response variance component is,

The estimate of the variance is,

This is approximated under Taylor linearization as,

Taking expectation under sampling gives,

The bias introduced by measurement errors is, 11 11 ()

The variance is, ˆˆ( ) ( ) ( )

The response variance component is,

The estimate of the variance is,

Two-phase regression estimation under multiple frames with errors Let

  

This estimator can be written as,

The expectation of the estimator is now based on three levels, namely, first phase, second phase and the measurement error model phase. The expectation of the estimator is therefore, 12 ˆ( ) ( )

Under the measurement error model,

Under Taylor linearization approximation, ()

  

Taking expectation under the 2 nd phase now gives,

Again to get the approximated variance,

To get the component 12 ()

Under Taylor linearization approximation,

Special case: Two-phase regression estimation under multiple frames with errors

The regression estimator of the frame f total under simple random sampling without replacement can be written as,

  

This estimator can be written as,

The expectation of the estimator is therefore,

Under the measurement error model, Et  can be written as,

  

Taking expectation under the 2 nd phase now gives,

To get the approximated variance,

To get the component 12 ()

Under Taylor linearization approximation,

Results and Discussion

Regression estimation under multiple frames

The table below represents the distribution of units of the generated population 300 with the specified regression parameters and population size for each frame. X represents the auxiliary variable. Y is the variable of interest, modeled as a linear function of X with some random error. In frame 1 the variable X has an average of 60.19399779 while Y has an average of 185.5474753. Frame 1 has a total of 126 units.

In frame 2 the variable X has an average of 59.75562628 while Y has an average of 184.3415139. Frame2 has a total of 100 units. In frame 3 the variable X has an average of 59.58270185 while Y has an average of 183.9812518. Frame 3 has a total of 114 units. ˆtdq hovers consistently around 18,430 and shows remarkable stability. Tiny variations (<0.1) across conditions. 0 b increases with k, e.g., 14 at k=0.01 to 20.9 at k=0.2 but is the least biased of all frames. Intercept estimates are closest to the true value, though still slightly overestimated at higher k.

Single-Phase Regression Estimation under Multiple Frames

1 b nearly stable, with only minor decreases at higher k, e.g., 2.85 at k=0.01 to 2.74 at k=0.2.

Slope estimates are the most accurate across frames, with minimal bias.

ˆtdq highly stable, unaffected by k or p. The mean estimate is consistently reliable, similar to other frames. 0 b shows moderate increase with k but best performance overall. p=0.05 produces lower intercepts than p=0.01 1 b most stable of all frames (2.74-2.88). Minimal decline with increasing k. p=0.05 shows slightly better performance ˆtdq insensitive to parameter changes. Variation <0.05 across all conditions.

Two-Phase Regression Estimation under Multiple Frames when there are response and measurement errors A subsample of 100 units is taken from the 300 units of the first phase units. The

Conclusion

Regression estimation in multiple-frame surveys provides a flexible and efficient method for enhancing survey estimates by integrating auxiliary information across overlapping frames and complex sampling designs. When combined with multiplicity adjustment and two-phase sampling strategies, regression estimators can achieve substantial gains in precision and coherence across frames. However, the presence of measurement and response errors poses serious challenges to the validity of these estimators, as such errors introduce bias and inflate variability, thereby weakening the efficiency gains typically associated with regression-based methods. These effects are often magnified in multiple-frame contexts due to frame overlap, weighting adjustments, and the need to reconcile information from different sources. Evidence from studies on complex survey designs and regression estimation shows that failure to account for these errors can lead to misleading inferences, even when advanced estimation techniques are employed. Consequently, it is essential that measurement and response errors be explicitly incorporated into both the design and analysis stages of multiple-frame surveys, through improved data collection procedures, calibration, model-assisted regression techniques, and adjustments for two-phase sampling. Addressing these error sources enhances the robustness, accuracy, and policy relevance of regression estimates, particularly in official statistics and applied research settings where multiple-frame surveys are increasingly used to achieve comprehensive population coverage.

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