Optimal Trait Scoring for Age Estimation

1. Optimal Trait Scoring for Age Estimation Lyle W. Konigsberg and Susan R. Frankenberg

2. Options for ordinal traits • Logit, probit or exponential transitions on log or straight scale • Cumulative (common standard deviation) • Unrestricted cumulative (separate standard deviations) • Continuation ratios (forward or backward) • Stopping rules (forward or backward) • Kernel densities • Sugeno fuzzy integral

3. Testing the normality assumption • Johnson PA. 1996. A test of the normality assumption in the ordered probit model. Metron 54:213-221. • Glewwe P. 1997. A test of the normality assumption in the ordered probit model. Econometric Reviews 16:1-19. • Weiss AA. 1997. Specification tests in ordered logit and probit models. Econometric Reviews 16:361-391.

4. Materials Todd scores from: 422 males (Terry Collection) 332 females (Terry Collection) 163 females (Gilbert and McKern)

5. A little history Katz and Suchey (1986) collapsed the Todd (1920) ten phase system into a “T2” system of six stages.

6. P-values from goodness-of-fit tests

7. Collapsing three ordered states

8. Collapsing four ordered states

9. Collapsing five ordered states 1+1+1+2 = 5 1+1+3 = 5 1+2+2 = 5 1+4 = 5 2+3 = 5

10. Forming all compositions of an integer • Form all partitions of the integer (Hindenburg’s algorithm) 2+8, 3+3+4, 2+2+3+3, 2+2+2+2+2, 1+1+1+1+1+5,…, = 10 • Form all unique permutations for each partition (Knuth’s “algorithm L”) 111115, 111151, 111511, 115111, 15111, 511111

11. The “R” script “smoosh” > smoosh(10) # of # of Total phases ways # ways 9 9 9 8 36 45 7 84 129 6 126 255 5 126 381 4 84 465 3 36 501 2 9 510 Down to how many stages? 1:

12. > smoosh(5) [1] # of # of Total [1] phases ways # ways [1] 4 4 4 [1] 3 6 10 [1] 2 4 14 Down to how many stages? 1: 2 [,1] [,2] [,3] [,4] [,5] [1,] 1 2 3 4 4 [2,] 1 2 3 3 4 [3,] 1 2 2 3 4 [4,] 1 1 2 3 4 [5,] 1 2 3 3 3 [6,] 1 2 2 2 3 [7,] 1 1 1 2 3 [8,] 1 2 2 3 3 [9,] 1 1 2 3 3 [10,] 1 1 2 2 3 [11,] 1 2 2 2 2 [12,] 1 1 1 1 2 [13,] 1 1 2 2 2 [14,] 1 1 1 2 2

13. Females

14. Males

15. Males I, II, III, IV, V, VI, VII, VIII-X

16. Females I, II, III, IV, V, VI, VII, VIII-X

17. Males & Females I, II, III, IV, V, VI, VII, VIII-X

19. Females

20. Males

21. Males & Females

22. Moorrees, Fanning and Hunt (1963)? > smoosh(14) [1] # of # of Total [1] phases ways # ways [1] 13 13 13 [1] 12 78 91 [1] 11 286 377 [1] 10 715 1092 [1] 9 1287 2379 [1] 8 1716 4095 [1] 7 1716 5811 [1] 6 1287 7098 [1] 5 715 7813 [1] 4 286 8099 [1] 3 78 8177 [1] 2 13 8190

23. Some comments about “smooshing” • Not possible to “un-smoosh” data that is already “smooshed” (e.g., from Suchey-Brooks to Todd or Demirjian et al. to Moorrees, Fanning and Hunt). • The specification test provides goodness-of-fit to normal or log normal transitions. • If the fit is poor, stages can be “smooshed” until the fit is adequate. • For the Todd phases, “smooshing” showed that phases I, II, III, IV, V, VI, VII, and VIII-X fit to log normal transitions with a common log variance.

24. Acknowledgment Data collection supported by NSF SBR-9727386 to LWK