Estimating the false negatives due to blocking in record linkage
Section 3. Neighbours and errors

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When examining the potential errors, it helps to look at how many register records form an accepted (by the blocking criteria) pair with a given file record. In what follows, these records are called neighbours of the file record, and their number is denoted by $n_i$ for record $i$ on the file. The empirical $n_i$ distribution provides much information about the errors because in the current setting, each file record would have exactly one neighbour if the blocking strategy were error-free, i.e. no false negatives or false positives. Note that the satisfaction of this condition does not imply the absence of errors. As an example, consider the situation shown in Figure 3.1, where the two sources are registers of a population with $N=6$ individuals, such that individual $i$ is associated with record $i$ in each register, i.e. the record pair $(i,i)$ is matched for $i=1,\dots ,6.$ Before looking at the $n_i’\text{s,}$ it is known that the number of false negatives is either 0 or 1, while the number of false positives is between 0 and 5, for each record in the first register. However the $n_i’\text{s}$ provide additional error information. Indeed, when $n_i=0$ (e.g. with record 2), it is known with certainty that there is a false negative but no false positives. When $n_i=1,$ one of two cases may occur, including a first case (as with record 5) where the neighbour is the matched record such that there are no errors, and a second case (as with record 3) where the neighbour is an unmatched record such that there are two errors including a false negative and a false positive. In summary, when $n_i=1,$ the number of false negatives is 0 or 1, while the number of false positives is also 0 or 1. Thus there is no additional information about the false negatives since it was known to be 0 or 1 prior to looking at $n_i.$ However, much information is gained about the false positives, since it was known to be in a wider interval (0 to 5) before looking at $n_i.$ This observation confirms the relative ease with which the false positives may be estimated.

Table 3.1 summarizes the general connection between the number of neighbours and the linkage errors at a given record, in the current setting where each file record is matched with exactly one register record.

The above table clearly demonstrates that the number of neighbours provides much error information, including in the case $n_i=0$ and the more unlikely case $n_i=N,$ where this information is complete. When the blocking decision about two records depends on no other record, with a high probability when $m$ and $N$ are large, some file records are bound to have no neighbour. In theory one could design the blocking criteria to ensure a positive $n_i$ for each file record, but this would violate the assumption made in Section 2 that the blocking decision about two records depends on no other record. Under this assumption, the number of neighbours does provide valuable error information, but some uncertainty remains in the case $1\le n_i\le N-1,$ where a statistical model is needed to predict the errors based on $n_i.$

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