An Adjusted Error Score Calculation for the Farnsworth-Munsell 100 Hue Test

Figures & data

Fig. 1. (Top) The distribution of caps of the FM-100 hue test in the a′b′ plane of CAM02-UCS (using the CIE 1964 10° standard observer) illuminated by CIE standard illuminant C. The caps have an average lightness correlate (J′) of 60.15 and an average colorfulness correlate (M′) of 19.73 (Luo et al. 2006). The dashed circle shows an equal colorfulness of 19.73 across all hues. Number labels indicate cap number. (Bottom) Top view of the four physical test trays of the FM-100 test. Tray A has 22 moveable caps between two fixed end caps; trays B, C, and D have 21.

Table 1. R_d for select light sources from the library of the IES TM-30-15 calculator. Multiple spectra of the same source type sometimes exhibited different R_d values. For example, there were multiple phosphor-based LED sources in the library that collectively exhibited all R_d values between 0 and 28.

Source type	Source name	Rd
Incandescent	75WA19 neodymium	12
Incandescent	Halogen/halogen MR16	4, 8
Incandescent	Filtered halogen	0
HID	HPS standard	40, 48
HID	HPS deluxe	36
HID	Super HPS	40
HID	Mercury	36, 44, 52
HID	CDM940, MHC100UMP4K–metal halide	0
HID	CDM830, MHC100/U/MP/3K	4, 8
LED	Mixed (experimental)	0–40
LED	Mixed (commercial)	36
LED	Hybrid (commercial)	0–16
LED	Phosphor	0–28
Fluorescent	Narrowband–F32T8/7XX	8, 12, 24
Fluorescent	Narrowband–F32T8/8XX	0–16, 24
Fluorescent	Narrowband–F32T8/9XX	0, 8
Fluorescent	Narrowband–F40T12/XXU	8, 12, 24
Fluorescent	Broadband	0, 4, 8, 20
Fluorescent	CIE C	0
Fluorescent	CIE F1–F12	0, 8, 12, 20
Model	CIE D-Series (5000–8000 K)	0
Model	Equal energy	0

Fig. 2. (Top left) A generalized comparison of standard error score and adjusted error score. Point 1 indicates that the standard error score equals the adjusted error score (that is, ES = ES_adj), which only occurs when a light source causes no cap transpositions (R_d = 0). Equation (1) assumes that the light source causes no cap transpositions and that all resulting participant error scores fall on the c = 1 line. For point 2, the adjusted error score is higher than a standard error score of zero (that is, ES_adj > ES = 0), which occurs when a light source causes many transpositions and the participant orders his or her caps in exact numerical order; in this scenario, many errors would be uncounted using the standard error score calculation. For point 3, the standard error score is higher than an adjusted error score of zero (that is, ES > ES_adj = 0), which occurs when the light source causes several transpositions and the participant exactly responds. In this scenario, errors are misapplied to the participant using the standard error score calculation because the correct order of caps is not the numerical order. (Top right/bottom left) Trays B and C error score comparison, respectively, for 480 responses (24 spectra × 20 participants) from Esposito and Houser (2017). (Bottom right) Total error score comparison for the same study. Seventeen of their 24 experimental spectra caused at least one hue transposition, which resulted in a large overall discrepancy between TES and TES_adj. It is clear that the assumption that all responses fall on the c = 1 line will result in significant error. Note that these panels are a new calculation from the data of Esposito and Houser (2017).

Table A1. Worked example where R_d = 0 and BES = 4.

j	Response order	CEj = |Cj − Cj − 1| + |Cj − Cj + 1|
1	21	= ABS(21–20) + ABS(21–22)	= 2
2	22	= ABS(22–21) + ABS(22–23)	= 2
3	23	= ABS(23–22) + ABS(23–24)	= 2
4	24	= ABS(24–23) + ABS(24–25)	= 2
5	25	= ABS(25–24) + ABS(25–26)	= 2
6	26	= ABS(26–25) + ABS(26–27)	= 2
7	27	= ABS(27–26) + ABS(27–28)	= 2
8	28	= ABS(28–27) + ABS(28–29)	= 2
9	29	= ABS(29–28) + ABS(29–31)	= 3
10	31	= ABS(31–29) + ABS(31–30)	= 3
11	30	= ABS(30–31) + ABS(30–32)	= 3
12	32	= ABS(32–30) + ABS(32–33)	= 3
13	33	= ABS(33–32) + ABS(33–34)	= 2
14	34	= ABS(34–33) + ABS(34–35)	= 2
15	35	= ABS(35–34) + ABS(35–36)	= 2
16	36	= ABS(36–35) + ABS(36–37)	= 2
17	37	= ABS(37–36) + ABS(37–38)	= 2
18	38	= ABS(38–37) + ABS(38–39)	= 2
19	39	= ABS(39–38) + ABS(39–40)	= 2
20	40	= ABS(40–39) + ABS(40–41)	= 2
21	41	= ABS(41–40) + ABS(41–42)	= 2
22	42	= ABS(42–21) + ABS(42–43)	= 2
23	43	= ABS(43–42) + ABS(43–44)	= 2

Table A2. Worked example where R_d,B=4 and BES_adj=4.

j	Source	ID	Response order	Pj	PEj = |Pj − Pj − 1| + |Pj − Pj + 1|
1	21			0	= ABS(0−(−1)) + ABS(0–1)	= 2
2	22	1	22	1	= ABS(1–0) + ABS(1–2)	= 2
3	23	2	23	2	= ABS(2–1) + ABS(2–3)	= 2
4	24	3	24	3	= ABS(3–2) + ABS(3–4)	= 2
5	25	4	25	4	= ABS(4–3) + ABS(4–5)	= 2
6	26	5	26	5	= ABS(5–4) + ABS(5–6)	= 2
7	27	6	27	6	= ABS(6–5) + ABS(6–7)	= 2
8	28	7	28	7	= ABS(7–6) + ABS(7–8)	= 2
9	29	8	29	8	= ABS(8–7) + ABS(8–10)	= 3
10	31	9	30	10	= ABS(10–8) + ABS(10–9)	= 3
11	30	10	31	9	= ABS(9–10) + ABS(9–11)	= 3
12	32	11	32	11	= ABS(11–9) + ABS(11–12)	= 3
13	33	12	33	12	= ABS(12–11) + ABS(12–13)	= 2
14	34	13	34	13	= ABS(13–12) + ABS(13–14)	= 2
15	35	14	35	14	= ABS(14–13) + ABS(14–15)	= 2
16	36	15	36	15	= ABS(15–14) + ABS(15–16)	= 2
17	37	16	37	16	= ABS(16–15) + ABS(16–17)	= 2
18	38	17	38	17	= ABS(17–16) + ABS(17–18)	= 2
19	39	18	39	18	= ABS(18–17) + ABS(18–19)	= 2
20	40	19	40	19	= ABS(19–18) + ABS(19–20)	= 2
21	41	20	41	20	= ABS(20–19) + ABS(20–21)	= 2
22	42	21	42	21	= ABS(21–20) + ABS(21–22)	= 2
23	43			22	= ABS(22–21) + ABS(22–23)	= 2

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