Optimal Trait Scoring for Age Estimation

Lyle W. Konigsberg and Susan R. Frankenberg

Options for ordinal traitsLogit, probit or exponential transitions

on log or straight scaleCumulative (common standard deviation)

Unrestricted cumulative (separate standard deviations)

Continuation ratios (forward or backward)

Stopping rules (forward or backward)Kernel densitiesSugeno fuzzy integral

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.

0 20 40 60 80 100

0.0

000

.005

0.0

100

.015

0.0

20

Age

Den

sity

Materials

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

A little historyI II III IV V VI VII VII

IIX X

2 4.5 6 7.5 9 10

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

I II III IV V VI VII VIII

IX X

1 2 3 4 5 6

P-values from goodness-of-fit tests

Todd 10 phases

Suchey-Brooks 6 phases

Males 0.0027 0.0052

Females <0.0001 <0.0001

Combined

<0.0001 <0.0001

Collapsing three ordered states

“open” “closing” “closed”

“open” “closing” “closed”

“open” “closing” “closed”

Collapsing four ordered states

1 2 3 4

1 2 3 3

1 2 2 3

1 1 2 3

1 2 2 2

1 1 1 2

1 1 2 2

Collapsing five ordered states1 2 3 4 51 2 3 4 4

1 2 3 3 4

1 2 2 3 4

1 1 2 3 4

1 2 3 3 3

1 2 2 2 3

1 1 1 2 3

1 2 2 3 3

1 1 2 3 3

1 1 2 2 3

1 2 2 2 2

1 1 1 1 2

1 1 2 2 2

1 1 1 2 2

1+1+1+2 = 5

1+1+3 = 5

1+2+2 = 5

1+4 = 5

2+3 = 5

Forming all compositions of an integerForm 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,…, = 10Form all unique permutations for each

partition (Knuth’s “algorithm L”)

111115, 111151, 111511, 115111, 15111, 511111

> 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 510Down to how many stages? 1:

The “R” script “smoosh”

> smoosh(5)[1] # of # of Total[1] phases ways # ways[1] 4 4 4[1] 3 6 10[1] 2 4 14Down 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

1 2 3 4 51 2 3 4 4

1 2 3 3 4

1 2 2 3 4

1 1 2 3 4

1 2 3 3 3

1 2 2 2 3

1 1 1 2 3

1 2 2 3 3

1 1 2 3 3

1 1 2 2 3

1 2 2 2 2

1 1 1 1 2

1 1 2 2 2

1 1 1 2 2

Females

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Probability (Straight Scale)

Pro

ba

bili

ty (

Lo

g S

cale

)

Males

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Probability (Straight Scale)

Pro

ba

bili

ty (

Lo

g S

cale

)

Males

Number of Phases

Go

od

ne

ss-o

f-fit

p-v

alu

e

0.0

0.2

0.4

0.6

0.8

1.0

2 3 4 5 6 7 8 9 10

I, II, III, IV, V, VI, VII, VIII-X

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

Number of Phases

Go

od

ne

ss-o

f-fit

p-v

alu

e

0.0

0.2

0.4

0.6

0.8

1.0

2 3 4 5 6 7 8 9 10

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

Number of Phases

Go

od

ne

ss-o

f-fit

p-v

alu

e

0.0

0.2

0.4

0.6

0.8

1.0

2 3 4 5 6 7 8 9 10

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage I

Age

Pro

babi

lity

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage II

Age

Pro

babi

lity

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage III

Age

Pro

babi

lity

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage IV

Age

Pro

babi

lity

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage V

Age

Pro

babi

lity

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage VI

Age

Pro

babi

lity

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage VII

Age

Pro

babi

lity

20 60 100

0.0

0.2

0.4

0.6

0.8

1.0

Stage VIII-X

Age

Pro

babi

lity

Females

Age

De

nsi

ty

20 40 60 80 100

Males

Age

De

nsi

ty

20 40 60 80 100

Males & Females

Age

De

nsi

ty

20 40 60 80 100

> 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

Moorrees, Fanning and Hunt (1963)?

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.

AcknowledgmentData collection supported by NSF SBR-9727386 to LWK