Function for generating Tredoux's transformed index-of-diversity effective size.
Details
Tredoux (1998) proposed using Agresti and Agresti's (1978) index of diversity for lineup choices, $$I = 1 - \frac{\sum_i o_i^2}{N^2},$$ where \(o_i\) is the observed count for lineup member \(i\) and \(N\) is the total number of choices. The index is then transformed back to an effective-size scale: $$E' = \frac{1}{1-I} = \frac{N^2}{\sum_i o_i^2} = \frac{1}{\sum_i p_i^2}.$$ This formula is algebraically equivalent to a reciprocal concentration form.
Small-sample bias
The expression above is a plug-in estimator: sample proportions are substituted for the unknown \(p_i\). The plug-in concentration \(\sum_i \hat p_i^2\) is biased upward. Its reciprocal \(E'\) is biased downward for fair and near-fair lineups, but the direction is not guaranteed: when one member draws most choices the bias can be upward (two members, \(p = (.9, .1)\), \(N = 10\): mean 1.230 against a true 1.220, by exact enumeration). For \(N\) choices spread evenly over \(k\) members the bias is approximately $$E[\hat{E}'] \approx \frac{Nk}{N + k - 1}, \qquad \mathrm{bias} = -\frac{k(k-1)}{N + k - 1}.$$ A perfectly fair six-person lineup assessed with 50 choices therefore returns about 5.47 rather than 6, and with 20 choices about 4.87. The bias is depends on both nominal size and the choice distribution; no single sample size makes it negligible for every lineup.
With unbiased = TRUE the function substitutes the unbiased estimator
of the concentration, \(\sum_i o_i(o_i - 1) / (N(N-1))\), and returns its
reciprocal,
$$\hat{E}'_{\mathrm{corr}} = \frac{N(N-1)}{\sum_i o_i(o_i - 1)}.$$
The reciprocal of an unbiased estimate is not itself unbiased. For
near-fair lineups this can reduce bias, with increased variability; in concentrated
distributions it can overshoot (mean 1.269 in the two-member example).
When fewer than two choices are available the correction is undefined and
returns NA with a warning. When every choice is on a different member,
the corrected reciprocal is Inf, retained with a warning. If
\(N \le k\), that event can have positive probability, so the corrected
estimator can have an infinite sampling mean. It is not uniformly superior.
Which to use depends on what is being reported. For a description of the
choice distribution actually observed, the plug-in value is the quantity of
interest. For inference about the lineup that produced those choices, the
framing of Tredoux (1998), the corrected form usually has less bias when
choices are spread across members. Neither is uniformly better.
Note that the other functions in this family (esize_T_boot,
esize_T_ci_n, i_esize_T) use the plug-in form.
References
Malpass, R. S. (1981). Effective size and defendant bias in eyewitness identification lineups. Law and Human Behavior, 5(4), 299-309.
Agresti, A., & Agresti, B. F. (1978). Statistical analysis of qualitative variation. Sociological Methodology, 9, 204-237.
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.