Overview

Signal detection analyses of eyewitness identification usually reduce a lineup to a 2 x 2 table. Guilty-suspect identifications are hits, innocent-suspect identifications are false alarms, and filler identifications are pooled with rejections as “negative” responses. Lee and Penrod (2019) argue that this discards the information carried by filler choices. Their Multi-d’ model keeps the full 2 x 3 table (suspect ID, filler ID, rejection) for target-present (TP) and target-absent (TA) lineups, and computes several empirical d’ values from it.

library(r4lineups)

The indices

Write G, Fp and Rp for the guilty-suspect, filler and rejection rates in TP lineups, and I, Fa and Ra for the innocent-suspect, filler and rejection rates in TA lineups. Each index is a difference between two z-transformed rates.

The three filler indices combine to give d’(GI):

\[d'(GI) = d'(GFp) - d'(IFa) - d'(FaFp).\]

The filler terms cancel, so this holds for any data set. It is an accounting identity, not an empirical finding, and its value lies in showing where a given d’(GI) comes from. A lineup procedure can reach the same d’(GI) because witnesses separate the culprit well from the fillers, or because they rarely single out an innocent suspect.

A published example

Lee and Penrod’s Table 4 applies the model to the lineup data in Steblay et al.’s (2003) meta-analysis. The rates can be entered as counts per 100 lineups, and correction = "none" reproduces the published values.

md <- make_multid(counts = c(G = 45, Fp = 24, Rp = 31, I = 17, Fa = 26, Ra = 57),
                  correction = "none")
md
#> Multi-d' model (Lee & Penrod, 2019)
#>   TP lineups: 100   TA lineups: 100   Lineup size: 6
#>   Innocent suspect: designated   Filler rate: total   Correction: none
#> 
#> Response rates:
#>    G   Fp   Rp    I   Fa   Ra 
#> 0.45 0.24 0.31 0.17 0.26 0.57 
#> 
#>                                        description estimate
#>                 d'(GI): guilty vs innocent suspect     0.83
#>              d'(GFp): guilty suspect vs TP fillers     0.58
#>  d'(IFa): innocent suspect vs TA fillers (TA bias)    -0.31
#>                   d'(FaFp): TA vs TP filler appeal     0.06
#>              d'(RaRp): culprit presence vs absence     0.67
#>                 d'(GFp) - d'(RaRp): TP lineup bias    -0.09
#> 
#> Decomposition: d'(GI) = d'(GFp) - d'(IFa) - d'(FaFp): 0.83 = 0.58 - (-0.31) - (0.06)
#> Fair-lineup reference for d'(IFa): -1.10 (observed minus reference: 0.79)

The published values are d’(GI) = 0.83, d’(GFp) = 0.58, d’(IFa) = −0.31, d’(FaFp) = 0.06 and d’(RaRp) = 0.67. The decomposition plot shows how the components add up to d’(GI).

plot(md)

Interpreting d’(IFa) as lineup bias

Lee and Penrod use the total filler identification rate, so d’(IFa) compares the innocent suspect with all the fillers taken together. Consequently, a perfectly fair lineup does not give d’(IFa) = 0. If witnesses who choose spread their choices evenly over six members, the innocent suspect receives one sixth of the choices and the fillers five sixths, and d’(IFa) is z(1/6) − z(5/6) ≈ −1.93 when nobody rejects. The value also changes with the rejection rate. make_multid() therefore reports fair_reference, the d’(IFa) a perfectly fair lineup would give at the observed TA rejection rate, together with ifa_vs_fair, the observed d’(IFa) minus that reference. Positive values of ifa_vs_fair mean that the innocent suspect draws more than a fair share of choices. These two quantities are not part of Lee and Penrod’s model; we provide them as an aid to interpretation.

The alternative, filler_rate = "per_filler", divides each filler rate by the number of fillers, so that the filler indices refer to an average single filler and a fair lineup gives d’(IFa) close to 0.

md$indices[md$indices$index %in% c("IFa", "fair_reference", "ifa_vs_fair"),
           c("index", "estimate")]
#>            index   estimate
#> 3            IFa -0.3108198
#> 8 fair_reference -1.1005728
#> 9    ifa_vs_fair  0.7897529
make_multid(counts = c(G = 45, Fp = 24, Rp = 31, I = 17, Fa = 26, Ra = 57),
            filler_rate = "per_filler", correction = "none")$indices[3, ]
#>   index                                       description  estimate
#> 3   IFa d'(IFa): innocent suspect vs TA fillers (TA bias) 0.6715981

When the TA lineups have no designated innocent suspect, the innocent-suspect rate is estimated as the TA filler rate divided by the nominal lineup size, as in make_fullroc(). Lee and Penrod call this the “average-filler” practice. It assumes the TA lineup to be fair, so in that case d’(IFa) carries no information about bias and moves only with the rejection rate.

Trial-level data, bootstrap intervals and confidence

With trial-level data, make_multid() takes the same target_present and identification columns as the other confidence-based functions in the package. Bootstrap percentile intervals are obtained by resampling within the TP and TA arms. With by_confidence = TRUE, the indices are also computed cumulatively: at each threshold, only responses made with at least that confidence are counted, over the full number of lineups in each arm, as in Lee and Penrod’s Table 9.

data(lineup_example)
md2 <- make_multid(lineup_example, by_confidence = TRUE,
                   n_bootstrap = 500, seed = 2026)
summary(md2)
#> Multi-d' model (Lee & Penrod, 2019)
#>   TP lineups: 700   TA lineups: 700   Lineup size: 6
#>   Innocent suspect: designated   Filler rate: total   Correction: loglinear
#> 
#> Response rates:
#>     G    Fp    Rp     I    Fa    Ra 
#> 0.593 0.312 0.096 0.126 0.587 0.287 
#> 
#>                                        description estimate    se ci_lower
#>                 d'(GI): guilty vs innocent suspect    1.379 0.079    1.226
#>              d'(GFp): guilty suspect vs TP fillers    0.726 0.096    0.526
#>  d'(IFa): innocent suspect vs TA fillers (TA bias)   -1.364 0.092   -1.556
#>                   d'(FaFp): TA vs TP filler appeal    0.711 0.072    0.580
#>              d'(RaRp): culprit presence vs absence    0.742 0.085    0.582
#>                 d'(GFp) - d'(RaRp): TP lineup bias   -0.017 0.128   -0.262
#>  ci_upper
#>     1.540
#>     0.920
#>    -1.206
#>     0.859
#>     0.906
#>     0.249
#> 
#> Decomposition: d'(GI) = d'(GFp) - d'(IFa) - d'(FaFp): 1.379 = 0.726 - (-1.364) - (0.711)
#> Fair-lineup reference for d'(IFa): -1.419 (observed minus reference: 0.054)
#> 
#> All indices:
#>           index estimate    se ci_lower ci_upper
#>              GI    1.379 0.079    1.226    1.540
#>             GFp    0.726 0.096    0.526    0.920
#>             IFa   -1.364 0.092   -1.556   -1.206
#>            FaFp    0.711 0.072    0.580    0.859
#>            RaRp    0.742 0.085    0.582    0.906
#>         tp_bias   -0.017 0.128   -0.262    0.249
#>       siphoning    2.090 0.136    1.827    2.382
#>  fair_reference   -1.419 0.022   -1.467   -1.379
#>     ifa_vs_fair    0.054 0.089   -0.129    0.214
#>               c    0.455 0.038    0.380    0.535
#>              dr    4.695 0.507    3.864    5.897
#> 
#> Cumulative indices by confidence (responses at or above threshold):
#>  threshold     G    Fp    Rp     I    Fa    Ra    GI   GFp    IFa  FaFp  RaRp
#>        100 0.098 0.011 0.001 0.001 0.029 0.001 1.895 1.006 -1.297 0.409 0.000
#>         90 0.143 0.019 0.001 0.006 0.044 0.001 1.423 1.004 -0.777 0.358 0.000
#>         80 0.189 0.039 0.001 0.015 0.065 0.001 1.289 0.878 -0.656 0.245 0.000
#>         70 0.260 0.076 0.001 0.024 0.128 0.001 1.343 0.788 -0.848 0.293 0.000
#>         60 0.352 0.109 0.001 0.031 0.180 0.001 1.490 0.850 -0.957 0.318 0.000
#>         50 0.436 0.163 0.001 0.055 0.270 0.001 1.437 0.819 -0.987 0.369 0.000
#>         40 0.507 0.218 0.001 0.078 0.369 0.001 1.438 0.798 -1.085 0.445 0.000
#>         30 0.556 0.265 0.001 0.103 0.479 0.001 1.402 0.769 -1.209 0.575 0.000
#>         20 0.593 0.312 0.096 0.126 0.587 0.287 1.379 0.726 -1.364 0.711 0.742
#>      c      dr
#>  2.242 137.000
#>  1.777  22.333
#>  1.526  12.619
#>  1.314  11.061
#>  1.126  11.465
#>  0.880   7.935
#>  0.701   6.523
#>  0.561   5.372
#>  0.455   4.695

plot(md2, type = "indices")

Lee and Penrod report that d’(GI) is fairly stable across confidence thresholds while its components change. High-confidence witnesses earn their d’(GI) mainly by distinguishing the culprit from the fillers; low-confidence witnesses earn much of theirs because their choices spread over the fillers and so seldom land on the innocent suspect.

plot(md2, type = "confidence")

In lineup_example, rejections were recorded only at the lowest confidence level, so d’(RaRp) is uninformative above that threshold.

Comparing two conditions

compare_multid() gives the difference between two conditions on each index, with bootstrap standard errors, intervals and p-values when both analyses were bootstrapped. Lee and Penrod’s Table 3 reports simultaneous and sequential lineups from Steblay, Dysart and Wells (2011); here the rates are entered as counts per 100 lineups.

sim <- make_multid(counts = c(G = 52, Fp = 25, Rp = 24, I = 28, Fa = 26, Ra = 46),
                   n_bootstrap = 500, seed = 1)
sqn <- make_multid(counts = c(G = 44, Fp = 19, Rp = 39, I = 15, Fa = 17, Ra = 68),
                   n_bootstrap = 500, seed = 2)
compare_multid(sim, sqn, labels = c("simultaneous", "sequential"))
#> Multi-d' comparison: simultaneous minus sequential
#> 
#>        index simultaneous sequential difference    se ci_lower ci_upper p_value
#>           GI        0.613      0.850     -0.237 0.278   -0.789    0.298   0.393
#>          GFp        0.711      0.709      0.002 0.319   -0.697    0.583   0.995
#>          IFa        0.060     -0.080      0.140 0.337   -0.494    0.804   0.678
#>         FaFp        0.038     -0.061      0.099 0.282   -0.499    0.633   0.724
#>         RaRp        0.606      0.759     -0.153 0.258   -0.652    0.369   0.553
#>      tp_bias        0.105     -0.050      0.155 0.392   -0.654    0.855   0.693
#>    siphoning        0.652      0.790     -0.138 0.466   -1.082    0.797   0.768
#>  ifa_vs_fair        1.274      0.912      0.362 0.347   -0.296    1.048   0.296

Because these are rates per 100, the intervals reflect samples of 100 lineups per arm, not the much larger samples of the meta-analysis. With real data, the counts should be the observed frequencies.

Cautions

The Multi-d’ indices are resultant measures: they require the responses of real witnesses to TP and TA lineups. They can describe or compare lineups after an experiment, but, unlike mock-witness measures such as esize_T() or func_size(), they cannot be used to vet a lineup before it is shown to a witness. Lee, Mansour and Penrod (2024) found that only resultant measures, including the Multi-d’ bias indices, reliably predicted eyewitness performance across 40 lineup sets, while mock-witness measures, perceptual ratings and face-recognition algorithms did not.

Each index is an equal-variance d’ computed from a single pair of rates. Lee and Penrod present the model as a descriptive framework, a “series of thought experiments”, and do not claim that the indices estimate underlying memory parameters. Models fitted to the full response distribution, such as fit_max_sdt() or fit_winter_2ht(), serve that purpose. Finally, the indices are sensitive to how extreme rates are handled. The default log-linear correction keeps them finite when a cell is empty, but it moves them slightly away from values computed from raw proportions.

References

Further Reading

• ?make_multid, ?compare_multid, ?plot.multid
• vignette("sdt_summary_comparisons") for d’ and criterion from 2 x 2 counts
• vignette("msdt_model") and vignette("lineup_memory_models") for process models of the full response breakdown