Effective Size with Confidence Intervals from Normal Theory (Tredoux, 1998)

Computes effective size E' (Tredoux, 1998) together with a confidence interval derived from the normal-theory variance formula in Tredoux (1998).

Usage

esize_T_ci_n(lineup_tab, alpha)

Arguments

lineup_tab: A table of lineup choices.
alpha: Confidence level as a decimal (e.g., 0.95 for a 95-percent CI).

Details

The variance of the diversity index I is estimated via Tredoux's (1998) closed-form expression: $$\mathrm{Var}(I) \approx \frac{4}{N}\left(\sum_i p_i^3 - \left(\sum_i p_i^2\right)^2\right).$$ Normal quantiles are applied to I and the resulting bounds are back-transformed to the E' scale via $E' = 1/(1-I)$. For small samples or skewed distributions, all interval methods require qualification. In particular, plug-in bootstrap intervals also fail at exact uniform fairness; see esize_boot_dist. Bayesian credible intervals describe a stated posterior rather than guaranteeing fixed-parameter frequentist coverage.

Note that this concerns the interval. The point estimate itself is also biased downward near uniform choice distributions at small samples, although the direction is not universal, since $E'$ is a plug-in estimator; see the “Small-sample bias” section of esize_T. This function uses the plug-in form. This first-order interval is unreliable in small samples and at uniform or concentrated boundary distributions. A zero estimated variance yields a degenerate interval, not evidence of zero population uncertainty. An ordinary bootstrap is also not a general solution to these boundary problems.

References

Malpass, R. S. (1981). Effective size and defendant bias in eyewitness identification lineups. Law and Human Behavior, 5(4), 299-309.

Malpass, R. S., Tredoux, C., & McQuiston-Surrett, D. (2007). Lineup construction and lineup fairness. In R. Lindsay, D. F. Ross, J. D. Read, & M. P. Toglia (Eds.), Handbook of Eyewitness Psychology, Vol. 2: Memory for people (pp. 155-178). Mahwah, NJ: Lawrence Erlbaum Associates.

Tredoux, C. G. (1998). Statistical inference on measures of lineup fairness. Law and Human Behavior, 22(2), 217-237.

Tredoux, C. (1999). Statistical considerations when determining measures of lineup size and lineup bias. Applied Cognitive Psychology, 13, S9-S26.

Wells, G. L.,Leippe, M. R., & Ostrom, T. M. (1979). Guidelines for empirically assessing the fairness of a lineup. Law and Human Behavior, 3(4), 285-293.

Examples

#Data:
lineup_vec <- round(runif(100, 1, 6))
lineup_tab <- table(lineup_vec)

#Call:
e_ci <- esize_T_ci_n(lineup_tab, .95)