Restricted Independent-Observations/MAX Model for Lineup Counts

Fits the equal-variance, independent-signal, single-criterion MAX model to aggregate simultaneous-lineup counts using minimum-Pearson chi-squared estimation. Estimates d' (sensitivity) and lambda (criterion) simultaneously, supporting one or two conditions and optional cross-condition constraints.

Usage

fit_max_sdt(
  n_hit,
  n_tp_choose,
  n_fa,
  N_tp,
  N_ta,
  n_hit_2 = NULL,
  n_tp_choose_2 = NULL,
  n_fa_2 = NULL,
  N_tp_2 = NULL,
  N_ta_2 = NULL,
  n = 6,
  constrain_d = FALSE,
  constrain_c = FALSE,
  start = NULL,
  nboot = 0,
  ci_level = 0.95,
  seed = NULL
)

Arguments

n_hit: Number of correct suspect identifications in target-present (TP) lineups (condition 1).
n_tp_choose: Total number of participants who chose any lineup member in TP lineups (condition 1). Must be >= n_hit.
n_fa: Number of participants who chose any lineup member in target-absent (TA) lineups (condition 1).
N_tp: Total number of participants in TP lineups (condition 1).
N_ta: Total number of participants in TA lineups (condition 1).
n_hit_2, n_tp_choose_2, n_fa_2, N_tp_2, N_ta_2: Corresponding counts for an optional second condition. All five must be supplied together or not at all.
n: Lineup size (number of members). Default 6.
constrain_d: Logical. If TRUE and two conditions are supplied, constrains d' to be equal across conditions. Default FALSE.
constrain_c: Logical. If TRUE and two conditions are supplied, constrains lambda to be equal across conditions. Default FALSE.
start: Named numeric vector of starting values for the optimizer. If NULL (default), reasonable starting values are chosen automatically. Names should be "d1", "lambda1" (and "d2", "lambda2" for two conditions).
nboot: Number of parametric bootstrap replicates for confidence intervals. Default 0 (no bootstrap). Set to e.g. 1000 to obtain bootstrap CIs.
ci_level: Confidence level for bootstrap CIs. Default 0.95.
seed: Optional integer seed for reproducible bootstrap.

Value

An object of class "max_sdt_fit" containing:

dprime_1, lambda_1: Estimated d' and criterion for condition 1.
dprime_2, lambda_2: Estimated d' and criterion for condition 2 (if supplied).
chisq: Chi-squared goodness-of-fit statistic.
df: Degrees of freedom for the GoF test.
p_value: P-value for the GoF test.
observed: Named vector of observed counts.
predicted: Named vector of predicted counts.
convergence: Convergence code from optim() (0 = success).
constrain_d, constrain_c, n_conditions, n: Input metadata.
boot_ci: Bootstrap confidence intervals (if nboot > 0).
boot_params: Full bootstrap parameter matrix (if nboot > 0). Rows for replicates that failed to converge are NA, so summarise with na.rm = TRUE.
boot_n_used, boot_n_failed: Number of bootstrap replicates that converged and that did not (if nboot > 0).

Details

This is a restricted Independent Observations/MAX model for simultaneous, fair lineups. Fillers are IID $N(0, 1)$, the culprit signal is $N(d', 1)$, memory signals are independent, and a single criterion is applied to the largest signal. Target-absent n_fa therefore means a choice of any lineup member, not only a designated innocent suspect.

For a lineup of $n$ members with culprit signal strength $d'$ and response criterion $\lambda$: $$P(\text{correct ID}) = \int_{\lambda}^{\infty} \phi(x - d') \Phi(x)^{n-1}\,dx$$ $$P(\text{choose anyone} \mid \text{TP}) = 1 - \Phi(\lambda - d')\Phi(\lambda)^{n-1}$$ $$P(\text{choose anyone} \mid \text{TA}) = 1 - \Phi(\lambda)^n$$

Parameters are estimated by minimizing the Pearson chi-squared discrepancy between the mutually exclusive response cells: target-present correct ID, filler ID and rejection; and target-absent choice and rejection. This is not maximum-likelihood estimation and is not the multi-criterion likelihood model of Wixted et al. (2018). Nested comparisons (equal d' or equal lambda across conditions) are supported via compare_max_sdt.

Estimates are conditional on the model assumptions. Correlated signals, unequal variances, confidence criteria, sequential presentation, unfair lineups, and the Ensemble and Integration decision variables are outside the scope of this function.

Parametric bootstrap CIs are obtained by simulating data from the fitted model and re-fitting. Replicates that fail to converge are dropped and counted in boot_n_failed.

Provenance

The implementation follows R scripts written by Andrew M. Smith (Iowa State University) for the analyses in Reisberg and Pezdek (2024), which fit what he describes as “the MAX SDT model (sometimes called the compound-decision model)”. The default example counts are the Profile-lineup condition from that work. The constrained fits (constrain_d, constrain_c) and the nested chi-square comparison in compare_max_sdt follow the procedure described there.

Relation to Smith's scripts

Estimates from this function will not match those scripts exactly, and the reason is the set of cells entering the discrepancy function.

Smith's scripts fit culprit identifications, total target-present choices and total target-absent choices, together with the rejection counts. Culprit identifications therefore enter twice — once on their own and again inside total target-present choices — so the cells do not partition the sample, and the filler count is never fitted directly.

This function uses five mutually exclusive cells per condition: correct identification, filler identification, target-present rejection, target-absent choice, target-absent rejection. A Pearson chi-square is defined for a partition, so this form keeps the statistic and its degrees of freedom interpretable, and it fits the observed counts better on that criterion.

On the Profile-lineup counts used in the examples below, the two give d' = 1.94 (this function) against d' = 1.78 (Smith's script) for the first condition. Neither is a coding error; they minimise different discrepancies.

References

Duncan, M. (2006). A signal detection model of compound decision tasks. DRDC Toronto TR 2006-256.

Reisberg, D., & Pezdek, K. (2024). Data analysed with the MAX SDT model in scripts supplied by A. M. Smith (personal communication, 4 April 2024).

Kaesler, M., Dunn, J. C., Ransom, K., & Semmler, C. (2020). Do sequential lineups impair underlying discriminability? Cognitive Research: Principles and Implications, 5, 35. doi:10.1186/s41235-020-00234-5

Wixted, J. T., Vul, E., Mickes, L., & Wilson, B. M. (2018). Models of lineup memory. Cognitive Psychology, 105, 81–114. doi:10.1016/j.cogpsych.2018.06.001

Examples

# Single condition
fit1 <- fit_max_sdt(
  n_hit = 69, n_tp_choose = 82, n_fa = 64,
  N_tp = 96, N_ta = 106, n = 6
)
print(fit1)
#> Restricted Independent-Observations/MAX SDT Model
#>   Lineup size: 6 | Conditions: 1 | Free parameters: 2
#>   Convergence: OK
#> 
#> Parameter estimates:
#>   d' = 1.944 | lambda = 1.146
#> 
#> Goodness of fit: chi-sq(1) = 4.408, p = 0.0358
#> 
#> Observed vs predicted:
#>           n_hit n_filler_tp n_reject_tp n_fa n_reject_ta
#> Observed   69.0        13.0        14.0 64.0        42.0
#> Predicted  66.4        19.2        10.4 58.7        47.3

# \donttest{
# Two conditions, free parameters (each two-condition fit takes a few seconds)
fit_free <- fit_max_sdt(
  n_hit = 69, n_tp_choose = 82, n_fa = 64,
  N_tp = 96, N_ta = 106,
  n_hit_2 = 68, n_tp_choose_2 = 78, n_fa_2 = 42,
  N_tp_2 = 96, N_ta_2 = 96, n = 6
)

# Two conditions, equal d' (constrained)
fit_eqd <- fit_max_sdt(
  n_hit = 69, n_tp_choose = 82, n_fa = 64,
  N_tp = 96, N_ta = 106,
  n_hit_2 = 68, n_tp_choose_2 = 78, n_fa_2 = 42,
  N_tp_2 = 96, N_ta_2 = 96, n = 6,
  constrain_d = TRUE
)

# Compare models
cmp <- compare_max_sdt(fit_free, fit_eqd)
print(cmp)
#> MAX SDT Model Comparison (chi-squared difference test)
#>   Free model:       chi-sq = 6.619 (4 free params)
#>   Constrained model: chi-sq = 7.039 (3 free params)
#>   Constraint: d' equal
#>   Delta chi-sq(1) = 0.420, p = 0.5170
#>   Interpretation: constraint does not significantly worsen fit.
# }