Table 1 Systems for full and partial matching, applicable to different identifier types. D, data; H, hypothesis that the proband and candidate under consideration are the same person. For each system, columns sum to 1, since the options for D are mutually exclusive. (A) Two-state system: a match occurs or does not occur. There is a probability pc (c, correct) that an identifier is correctly represented (is the same) when the proband and candidate are the same person, and a probability pe = pen (e, any error; en, error yielding no match) that an error or mismatch occurs. In the population, there is a probability pf (f, full match) that a randomly selected other person shares the proband’s identifier, and a probability pn (n, no match) that they do not. (B) Three-state (“fuzzy”) system. The nature of a partial match is specific to the identifier type. For example, for date of birth (DOB), the partial match is a DOB with 2/3 of year/month/day correct. Probabilities pc, pf, pen, and pn are as before, but now there is a probability pep (ep, error yielding partial match) that when the proband and candidate are the same person, the identifiers match only partially (so pe = pep + pen), and a probability ppnf that that a random other person will share the partial but not the full identifier. It may be easier to measure pp, the total probability of a partial or full match, than ppnf. (C) Four-state system. Partial matches now occur in two variants, hierarchically. (D) Adjustments for unordered pick-the-best comparisons between multiple identifiers of the same type (e.g. surnames, postcodes). “Positive” comparisons are those for which the log likelihood ratio is > 0. (E) Adjustments for ordered pick-the-best comparisons between multiple identifiers of the same type (e.g. forenames), using the probability po (o, ordered) that, given H, for ≥ 2 candidate identifiers, the candidate’s order strictly matches the proband’s, and its converse probability pu (u, unordered). Positive comparisons are strictly ordered when each proband identifier’s index matches the corresponding candidate’s identifier. † For P(D | ¬H), adjustments use the Bonferroni correction (see text)
Data, D | P(D | H, same person) | P(D | ¬H, different person) |
|---|---|---|
A. Two-state comparison | ||
Match | pc = 1 − pe = 1 − pen | pf |
No match | pe = pen | pn = 1 − pf |
B. Three-state comparison | ||
Full match | pc = 1 − pe = 1 − pep − pen | pf |
Partial (but not full) match | pep | ppnf = pp − pf |
No match | pen | pn = 1 − pp |
C. Four-state comparison | ||
Full match | pc = 1 − pe = 1 − pep1 − pep2np1 − pen | pf |
Partial match type 1 (but not full) | pep1 | pp1nf = pp1 − pf |
Partial match type 2 (but not full or partial type 1) | pep2np1 | pp2np1 |
No match | pen | pn = 1 − pp = 1 − pp2np1 − pp2nf |
D. Adjustments for unordered multi-identifier comparison | ||
For 1 ≤ c ≤ min(n, m) “positive” comparisons between proband identifiers 1…n and candidate identifiers 1…m † | × 1, no correction | \(\times \prod^{c-1}_{i=0} \left(m-i\right)\) |
E. Adjustments for ordered multi-identifier comparison | ||
For c ≥ 1 “positive” comparisons, m > 1, and strict order match | × po | × 1, no correction |
For c ≥ 1 “positive” comparisons, m > 1, and order mismatch † | × pu = 1 − po | \(\times \left(\left[\prod^{c-1}_{i=0} \left(m-i\right)\right]-1\right)\) |