A Model-Based Statistical Classification Analysis for Karamattepe Arrowheads

Tutku Tuncali Yaman

doi:10.5334/jcaa.20

Figures & Tables

Table 1

Distribution of arrowheads according to the typology developed by Baykan (2012).

Typology	Frequency	%
Type 1	289	59.8
Type 2	129	26.7
Type 3	28	5.8
Type 4	13	2.7
Type 5	8	1.7
Type 6	5	1
Type 7	2	0.4
N/A	9	1.9
Total	483	100

Table 2

Descriptive statistics of the arrowheads.

		Total		Type
		Total		Type 1		Type 2		Type 3		Type 4		Type 5		Type 6		Type 7		N/A
		Mean	n	Mean	n	Mean	n	Mean	n	Mean	n	Mean	n	Mean	n	Mean	n	Mean	n
Weight		8.92	456	8.05	274	11.87	123	5.08	28	8.70	12	4.38	6	4.43	2	10.68	2	11.12	9
Length		5.67	471	5.79	286	5.59	123	5.74	28	5.62	13	3.56	8	3.75	4	4.40	2	5.99	7
Width		1.05	324	1.02	268	1.15	24	1.37	3	0.96	12	1.24	7	0.97	3	1.50	1	1.57	6
Thickness		0.80	220	0.80	128	0.94	57	0.45	18	0.77	3	0.66	6	0.40	1	0.85	2	0.66	5
Case width		1.13	110		0	1.16	102		0		0	0.70	4	.070	3		0	1.2	1
Helve thickness		0.55	295	0.55	248	1.34	5	0.40	24	0.50	12	0.80	1		0	0.30	1	0.75	4
Stem width		1.43	28	0.90	1		0	1.44	25		0	1.30	1		0	1.70	1		0
Driller length		3.65	425	4.12	261	2.59	116	4.52	26	2.57	12	2.50	4	2.35	2	4.40	1	2.73	3
Elevation		437.2	419	436.9	257	437.5	110	439.4	23	435.6	8	436.9	8	434.9	3	437	2	438.1	8
Abs	Yes		103		46		38		3		7		5			0			2
No		338		225		77		20		4		3			2			6
Material	Iron		468		289		129		28		13		0			0			8
Bronze		15		0		0		0		0		8			2			1

Table 3

Descriptive statistics before and after data completion.

	Original Data			Missing Data		Completed Data (N:483)
	N	Mean	Standard Deviation	Frequency	%	Mean	Standard Deviation
Weight	456	8.91	3.82	27	6	8.92	3.86
Length	471	5.67	1.30	12	2	5.67	1.30
Width	324	1.04	1.19	159	33	1.16	1.20
Thickness	220	0.80	0.24	263	54	0.78	0.33
Case Width	110	1.12	0.25	373	77	1.00	0.31
Helve Thickness	295	0.55	0.75	188	39	0.60	0.72
Stem Width	28	1.42	0.18	455	94	1.51	0.29
Driller Length	425	3.64	1.03	58	12	3.62	1.05
Elevation	419	437.20	3.32	64	13	437.22	3.38

Table 4

Goodness-of-fit criteria.

Criteria	Value
AIC (ideal)	882.815
AIC (restricted)	453.427
Cox and Snell’s Pseudo R²	0.647
Nagelkerke R²	0.750
McFadden	0.524
Pearson Chi – Squared	11443.137
Relative Chi – Squared	417.427

Table 5

Measures of good compliance with the multi-territorial logistic regression model.

	GoF (Restricted Model)			Likelihood Ratio Test
	AIC	BIC	–2logλ	χ²	df	p
Intercept	453.427	527.030	417.427^a	0.000	0
WEIGHT * LENGTH	581.545	642.881	551.545^b	134.118	3	0.000
MATERIAL	518.147	579.482	488.147	70.719	3	0.000
DRILLER LENGTH	744.169	805.504	714.169^b	296.742	3	0.000
ABS	456.826	518.162	426.826^b	9.399	3	0.024
ELEVATION	461.745	523.081	431.745^b	14.318	3	0.003

[i] ^a This reduced model is equivalent to the final model because omitting the effect does not increase the degrees of freedom.

^b Unexpected singularities in the Hessian matrix are encountered. This indicates that either some predictor variables should be excluded or some categories should be merged.

Table 6

Parameter estimates of the multinomial logistic model.

Type^a	Variables	B	St. Error	Wald	df	p.	Exp(β)	%95 CI for Exp(B)
Type^a	Variables	B	St. Error	Wald	df	p.	Exp(β)	Lower Bound	Upper Bound
2	Intercept	–91.247	35.72	6.522	1	0.01
	WEIGHT * LENGTH	0.057	0.008	52.05	1	0.00	1.058	1.042	1.075
	[MATERIAL=IRON]	–1.278	0.000		1		0.279	0.279	0.279
	[MATERIAL=BRONZE]	0^b			0
DRILLER LENGTH	–2.745	0.275	99.64	1	0.00	0.064	0.037	0.110
[ABS=YES]	0.961	0.436	4.843	1	0.02	2.613	1.111	6.147
[ABS=NO]	0^b			0
ELEVATION	0.223	0.082	7.393	1	0.00	1.250	1.064	1.467
3	Intercept	–106.93	38.35	7.774	1	0.00
	WEIGHT * LENGTH	–0.125	0.024	26.65	1	0.00	0.882	0.842	0.925
	[MATERIAL=IRON]	–0.112	0.000		1		0.894	0.894	0.894
	[MATERIAL=BRONZE]	0^b			0
DRILLER LENGTH	2.281	0.434	27.56	1	0.00	9.787	4.177	22.934
[ABS=YES]	0.826	0.756	1.193	1	0.27	2.284	0.519	10.054
[ABS=NO]	0^b			0
ELEVATION	0.228	0.087	6.803	1	0.00	1.256	1.058	1.490
4	Intercept	–43.918	1228	0.001	1	0.97
	WEIGHT * LENGTH	0.045	0.010	20.21	1	0.00	1.047	1.026	1.068
	[MATERIAL=IRON]	–20.958	1227	0.000	1	0.98	7.907E-10	0.000	.^c
	[MATERIAL=BRONZE]	0^b			0
DRILLER LENGTH	–2.280	0.374	37.24	1	0.00	0.102	0.049	0.213
[ABS=YES]	1.637	0.595	7.567	1	0.00	5.138	1.601	16.491
[ABS=NO]	0^b			0
ELEVATION	0.153	0.127	1.461	1	0.22	1.166	0.909	1.495

[i] ^a The reference category is: Type 1.

^b This parameter is set to zero because it is redundant.

^c Floating point overflow occurred while computing this statistic. Its value is therefore set to system missing.

Table 7

Classification results.

	Type	Predicted
	Type	1	2	3	4	Percent Correct (%)
Observed	1	255	14	2	0	94.1%
	2	20	95	0	0	82.6%
	3	16	0	7	0	30.4%
4	5	12	1	14	43.8%
Overall Percentage %	67.1%	27.4%	2.3%	3.2%	84.1%