The double discrimination methods, Artykuły
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Food Quality and Preference 12 (2001) 507–513
www.elsevier.com/locate/foodqual
The double discrimination methods
Jian Bi
Sensometrics Research and Service, 9212 Groomfield Road, Richmond, VA 23236, USA
Received 24 October 2000; received in revised form 25 March 2001; accepted 10 April 2001
Abstract
Theoretical aspects of the double discrimination methods are investigated in this paper. The main aspects involve statistical
models, test powers, Thurstonian d
0
s and variance of d
0
s for the methods. The double discrimination methods can be treated
under the same framework as the conventional discrimination methods, i.e. under the same binomial model but with different
parameters. The double discrimination methods possess lower guessing probabilities and higher test powers than the conventional
discrimination methods. The relationship between the double methods and the conventional methods in Thurstonian d
0
s and
variances of d
0
s are derived in the paper. The tables of critical values for tests and the table of sample sizes required to reach 0.8 test
power for the double two-alternative forced choice (the double 2-AFC), the double three-alternative forced choice (the double 3-
AFC), the double triangular and the double duo-trio methods are given. # 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Discrimination methods; Double discrimination; Critical values; Sample sizes; Forced choice; Two-alternative forced choice; Three-
alternative forced choice
1. Introduction
unit, the double discrimination tests can be regarded as
replicated tests. Some models, e.g. beta-binomial model
(Ennis & Bi, 1998) and generalized linear model
(Brockhoff & Schlich, 1998) can be used for the repli-
cated testingdata.
The difference between the two approaches is not only
what is selected as the unit of analysis, but also the
assumption underlyingthe tests. In the first approach, it
is assumed that all panelists have the same probability of
correct responses and all the responses are independent
from each other. The binomial model is valid only under
this assumption. In the second approach, the parameter
in the binomial model is a variable under the assumption
that panelists have different discrimination abilities.
Both approaches are reasonable solutions under a
specified assumption. Detailed comparison of the two
approaches will involve discussion of the philosophy
behind the discrimination test, which is not the subject
of this paper. The present author will discuss the ratio-
nale of discrimination test in another paper. This paper
is focused mainly on the theoretical aspects for the first
approach under the same framework as the conven-
tional discrimination methods, i.e. under the same
binomial model but with different parameters. The
theoretical aspects of the methods discussed in the paper
involve statistical models for hypothesis tests, powers as
well as sample sizes for the tests, Thurstonian d
0
s and
variances of d
0
s for the methods.
The so-called double discrimination methods are the
variants of the conventional discrimination methods.
They are used in some companies. The often used double
discrimination methods are the double two-alternative
forced choice (the double 2-AFC), the double three-
alternative forced choice (the double 3-AFC), the double
triangular and the double duo-trio. The motivation of
usingthe double discrimination methods might be to
reduce the guessing probability and raise the test power.
However, rare references can be found from the literature
for theoretical discussion of the methods. The aim of this
paper is to discuss the theoretical aspects of the methods.
In the double discrimination methods, each panelist
executes two tests for the two sets of samples. A defined
response, not the direct observation for each sample set,
is used as an analysis unit. A response of a panelist is
counted as correct, if and only if the panelist gives cor-
rect answers for both of two sample sets. A response is
counted as incorrect, if one or both answers for the two
sample sets are incorrect. The binomial model with a
new parameter value is valid for the defined response.
There is another possible approach to deal with the
data from the double discrimination methods. If the
observation for each sample set is used as the analysis
E-mail address: bbdjcy@aol.com
0950-3293/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.
PII: S0950-3293(01)00045-3
508
J. Bi/Food Quality and Preference 12 (2001) 507–513
2. Critical values for statistical tests
The principles in Ennis (1993) and Bi and Ennis (1999)
are applicable to the double discrimination methods.
Based on the definition of correct response in the
double discrimination methods, accordingto the multi-
plicative law of probability, the probability of correct
responses is 1/4 in the double 2-AFC and the double
duo-trio methods because the probability is 1/2 in the
conventional 2-AFC and duo-trio methods. The prob-
ability of correct responses is 1/9 in the double 3-AFC
and double triangular methods, because it is 1/3 in the
conventional 3-AFC and triangular methods.
Let N be the number of panelists and X the number of
correct responses for the N panelists. Then X follows a
binomial distribution with parameter of p=1/4 (for the
double 2-AFC and double duo-trio methods) or p=1/9
(for the double 3-AFC and double triangular methods)
under the null hypothesis that the two compared pro-
ducts are identical. The critical value k for the two-sided
double 2-AFC method is the minimum whole number in
Eq. (1) and given in Table 1. The critical value k for the
one-sided double 2-AFC, double duo-trio methods is
the minimum whole number in Eq. (2) and given in
Table 2. The critical value k for the double 3-AFC and
double triangular is the minimum whole number in Eq.
(3) and given in Table 3.
3.1. Choice probability criterion
Accordingto choice probability criterion, on the basis
of the normal distribution as an approximate to the
binomial distribution, the power of the two-sided dou-
ble 2-AFC method can be determined by Eq. (4) for
specified significance level, a, a specified probability of
correct responses in an alternative hypothesis, p
1
and
sample size, n.
power ¼ 1
¼ 1
u
1
p
1
v
1
þ
1 u
1
p
1
v
1
ð4Þ
percentile of
a normal distribution with mean p
0
=1/4 and variance,
v
2
¼
16n
; v
1
¼
p
1
1pð
n
, which is the variance of p in an
alternative hypothesis.
The power of the one-sided double 2-AFC and the
double duo-trio methods can be determined by Eq. (5).
v
2
, which is the 1
2
X
1
4
x
nx
n
x
1
1
4
4
2
ð1Þ
power ¼ 1 ¼ 1 F
u
2
p
1
v
1
x¼k
ð5Þ
X
1
4
n
x
x
nx
1
1
4
where u
2
¼
4
þ F
1
1 ð Þv
2
.
The power of the double 3-AFC and the double tri-
angular methods can be determined by Eq. (6).
4
ð2Þ
x¼k
1
9
x
nx
X
n
x
1
1
9
power ¼ 1 ¼ 1 F
u
3
p
1
v
1
4
ð3Þ
ð6Þ
x¼k
here, n and x are realizations of N and X and a=0.05 is
the significance level. For example, 100 panelists parti-
cipated in a double duo-trio test. There are 35 correct
responses in the test because 35 panelists gave correct
answers for both of the two sample sets. From Table 2,
the critical value for sample size n=100 is 33. The con-
clusion is that the two products are significantly differ-
ent at a=0.05.
81n
.
For example, for n=100 and a=0.05, the power of
the double duo-trio method at p
1
=0.35 can be obtained
from Eq. (5). For v
2
¼
q
3
16100
q
0:25ð10:25Þ
100
¼ 0:0433, v
1
¼
¼ 0:0477, u
2
¼ 0:25 þ 1:6449 0:0433 ¼
0:3212, the power of the method is:
power ¼ 1 F
0:3212-0:35
0:0477
¼ 0:73
3. Test powers and sample sizes
3.2. Thurstonian criterion
Ennis (1993) discussed the power and sample size
determinations for the conventional discrimination
methods and gave general equations for power and
simple size. Bi and Ennis (1999) extended the power and
sample size determination to the replicated situation
and gave two criteria (the choice probability criterion
and the Thurstonian d criterion) for the determination.
Accordingto the Thurstonian d criterion, on the
assumption of the approximate normal distribution of
d
0
, the power can be determined based on d and the
variance of its estimate, d
0
. Usingthe new criterion, the
powers of the methods can be calculated from Eqs. (4
0
),
(5
0
) and (6
0
), which correspond to Eqs. (4), (5) and (6).
Where, b is the probability of Type II error;
denotes the standard normal distribution function;
u
1
¼
4
þ F
1
1
2
n
n
n
where u
3
¼
g
þ
1
ð1 Þ
3
, here
3
¼
8
J. Bi/Food Quality and Preference 12 (2001) 507–513
509
power ¼ 1 ¼ 1
u
1
1
Þ
^ ðÞ
ð4
0
Þ
power ¼ 1 ¼ 1
uðÞ
1
^ ðÞ
ð6
0
Þ
power ¼ 1 ¼ 1
uðÞ
1
^ ðÞ
ð5
0
Þ
Where d(u
1
), d(u
2
) and d(u
3
) denote the d values cor-
respondingto u
1
, u
2
and u
3
, respectively. d
1
is a d value
in an alternative hypothesis for a given method and
^ ðÞdenotes the estimate of the standard variance of d
0
at d
1
. For the example in Section 3.1, usingthe methods
of estimates of d and variance of d
0
, that will be introduced
in Sections 4 and 5, we can get the d estimates corre-
spondingto u
2
and p
1
aswellasthevarianceestimateofd
0
at d
1
. d(u
2
)=d(0.3212)=0.89; d
1
(p
1
)=d
1
(0.35)=1.06 and
Table 1
The double two-sided two-alternative forced choice (2-AFC) and
preference method
n 0 1 2 3 4 5 6 7 8 9
p
¼ 0:2646. The power of the double duo-trio
method at p
1
=0.35 or d
1
=1.06 is then,
0:07
10 6 7 7 7 8 8 9 9 9 10
20 10 10 11 11 11 12 12 12 13 13
30 13 14 14 14 15 15 15 16 16 16
40 17 17 17 18 18 18 18 19 19 19
50 20 20 20 21 21 21 22 22 22 22
60 23 23 23 24 24 24 25 25 25 26
70 26 26 26 27 27 27 28 28 28 28
80 29 29 29 30 30 30 31 31 31 31
90 32 32 32 33 33 33 34 34 34 34
100 35 35 35 36 36 36 36 37 37 37
110 38 38 38 38 39 39 39 40 40 40
120 41 41 41 41 42 42 42 43 43 43
130 43 44 44 44 45 45 45 45 46 46
140 46 47 47 47 47 48 48 48 49 49
150 49 49 50 50 50 51 51 51 51 52
160 52 52 53 53 53 53 54 54 54 54
170 55 55 55 56 56 56 56 57 57 57
180 58 58 58 58 59 59 59 60 60 60
190 60 61 61 61 62 62 62 62 63 63
200 63 63 64 64 64 65 65 65 65 66
210 66 66 67 67 67 67 68 68 68 69
220 69 69 69 70 70 70 70 71 71 71
230 72 72 72 72 73 73 73 74 74 74
240 74 75 75 75 75 76 76 76 77 77
250 77 77 78 78 78 79 79 79 79 80
260 80 80 80 81 81 81 82 82 82 82
270 83 83 83 83 84 84 84 85 85 85
280 85 86 86 86 87 87 87 87 88 88
290 88 88 89 89 89 90 90 90 90 91
300 91 91 91 92 92 92 93 93 93 93
310 94 94 94 94 95 95 95 96 96 96
320 96 97 97 97 97 98 98 98 99 99
330 99 99 100 100 100 101 101 101 101 102
340 102 102 102 103 103 103 104 104 104 104
350 105 105 105 105 106 106 106 107 107 107
360 107 108 108 108 108 109 109 109 110 110
370 110 110 111 111 111 111 112 112 112 112
380 113 113 113 114 114 114 114 115 115 115
390 115 116 116 116 117 117 117 117 118 118
400 118 118 119 119 119 120 120 120 120 121
410 121 121 121 122 122 122 123 123 123 123
420 124 124 124 124 125 125 125 126 126 126
430 126 127 127 127 127 128 128 128 128 129
440 129 129 130 130 130 130 131 131 131 131
450 132 132 132 133 133 133 133 134 134 134
460 134 135 135 135 136 136 136 136 137 137
470 137 137 138 138 138 138 139 139 139 140
480 140 140 140 141 141 141 141 142 142 142
490 143 143 143 143 144 144 144 144 145 145
500 145 145 146 146 146 147 147 147 147 148
power ¼ 1
0:89 1:06
p
0:07
¼ 0:74
It agrees closely with 0.73 obtained by using the
choice probability criterion in Section 3.1.
Eqs. (4)–(6) and (4
0
)–(6
0
) can also be used for deter-
mination of a sample size for a specified method, power,
significance level a and P
1
or d
1
. Table 4 gives the sam-
ple sizes required for the four double discrimination
methods to reach 0.8 of power at a=0.05. Fig. 1 gives
comparisons of powers for the conventional and the
double discrimination methods. It confirms that the
powers of the double discrimination methods are larger
than the correspondingconventional discrimination
methods for the same number of panelists. It is not
surprisingbecause the double discrimination methods
utilize more information than the conventional dis-
crimination methods do.
4. Psychometric functions
Psychometric function for a specified discrimination
method gives the relationship between P
c
, the prob-
ability of correct responses, and d (or its estimator d
0
),
the measure of sensory difference. Ennis (1993) gave an
explanation and tables for the psychometric functions
for the m-alternative forced choice (m-AFC), the trian-
gular and the duo-trio methods.
Because the probability of correct responses for the
double discrimination is the product of two prob-
abilities of correct responses in conventional dis-
crimination methods, the psychometric functions for the
double discrimination should be (7).
P
c
¼ gdðÞ¼fdð
2
ð7Þ
where g(d
0
) denotes a psychometric function for a dou-
ble discrimination method, f(d
0
) is a psychometric
function for the conventional discrimination method.
fðd
0
Þ¼
1
2
p
Ð
1
Minimum number of correct responses for significant at a=0.05
1
exp½ z d
0
ð
Þ
2
=2 ðÞ
m1
dz for the m-
ð
^ ð =
510
J. Bi/Food Quality and Preference 12 (2001) 507–513
d ðÞ
for the triangular method (David & Trivedi, 1962; Ura,
p
p
2=3
þ z
3
p
p
2=3
transferringthe proportion in the double discrimination
methods to that in the conventional discrimination
methods. For example, for the data in Section 1, the
proportion of correct responses in the double duo-trio
method is 0.35. The transferred proportion of correct
responses correspondingto the conventional duo-trio
method is fdðÞ¼
þ d
0
d
0
1960) and fðd
0
Þ¼1 ðd
0
=
p
Þðd
0
=
p
Þþ
p
Þ for the duo-trio method (David &
Trivedi, 1962; Ura, 1960). Based on Eq. (7), the d
0
, can
be obtained from the tables correspondingto the con-
ventional discrimination methods in Ennis (1993) by
p
Þðd
0
=
gdð
p
¼ 0:35 ¼ 0:5915. From Table 4
Table 2
The double two-alternative forced choice (2-AFC) and duo-trio methods
n
0
1
2
3
4
5
6
7
8
9
10
6
6
7
7
7
8
8
8
9
9
20
9
10
10
10
11
11
11
12
12
12
30
13
13
13
13
14
14
14
15
15
15
40
16
16
16
17
17
17
17
18
18
18
50
19
19
19
20
20
20
20
21
21
21
60
22
22
22
23
23
23
23
24
24
24
70
25
25
25
25
26
26
26
27
27
27
80
27
28
28
28
29
29
29
30
30
30
90
30
31
31
31
32
32
32
32
33
33
100
33
34
34
34
34
35
35
35
36
36
110
36
36
37
37
37
38
38
38
38
39
120
39
39
39
40
40
40
41
41
41
41
130
42
42
42
43
43
43
43
44
44
44
140
45
45
45
45
46
46
46
47
47
47
150
47
48
48
48
48
49
49
49
50
50
160
50
50
51
51
51
52
52
52
52
53
170
53
53
53
54
54
54
55
55
55
55
180
56
56
56
57
57
57
57
58
58
58
190
58
59
59
59
60
60
60
60
61
61
200
61
61
62
62
62
63
63
63
63
64
210
64
64
65
65
65
65
66
66
66
66
220
67
67
67
68
68
68
68
69
69
69
230
69
70
70
70
71
71
71
71
72
72
240
72
72
73
73
73
74
74
74
74
75
250
75
75
75
76
76
76
77
77
77
77
260
78
78
78
78
79
79
79
80
80
80
270
80
81
81
81
81
82
82
82
83
83
280
83
83
84
84
84
84
85
85
85
85
290
86
86
86
87
87
87
87
88
88
88
300
88
89
89
89
90
90
90
90
91
91
310
91
91
92
92
92
93
93
93
93
94
320
94
94
94
95
95
95
95
96
96
96
330
97
97
97
97
98
98
98
98
99
99
340
99
100
100
100
100
101
101
101
101
102
350
102
102
102
103
103
103
104
104
104
104
360
105
105
105
105
106
106
106
107
107
107
370
107
108
108
108
108
109
109
109
109
110
380
110
110
111
111
111
111
112
112
112
112
390
113
113
113
114
114
114
114
115
115
115
400
115
116
116
116
116
117
117
117
118
118
410
118
118
119
119
119
119
120
120
120
120
420
12].
121
121
122
122
122
122
123
123
123
430
123
124
124
124
124
125
125
125
126
126
440
126
126
127
127
127
127
128
128
128
128
450
129
129
129
130
130
130
130
131
131
131
460
131
132
132
132
132
133
133
133
134
134
470
134
134
135
135
135
135
136
136
136
136
480
137
137
137
138
138
138
138
139
139
139
490
139
140
140
140
140
141
141
141
142
142
500
142
142
143
143
143
143
144
144
144
144
Minimum number of correct responses for significant at a=0.05
2
Ð
0
½ z
3
AFC methods (Hacker & Ratcliff 1979); fdðÞ¼
2ðd
0
=
J. Bi/Food Quality and Preference 12 (2001) 507–513
511
in Ennis (1993), the corresponding d
0
or d value is about
1.06.
of d and the methods used. An estimate of the variance
of d
0
is essential for statistical inference for d
0
s obtained
from the same or different discrimination paradigms.
Gourevitch and Galanter (1967) gave an estimator of
the variance of d
0
for the A-not-A method. Bi, Ennis
and O’Mahony (1997) provided estimates and tables for
the variances of d
0
s for four commonly used discrimina-
tion methods. Bi and Ennis (1998, 2001) extended the
5. Variance of d
0
s
Because d
0
is an estimator of d, it is a variable and
involves variation related with sample size, the magnitude
Table 3
The double three-alternative forced choice (3-AFC) and triangular methods
n
0
1
2
3
4
5
6
7
8
9
10
4
4
4
4
5
5
5
5
5
6
20
6
6
6
6
6
7
7
7
7
7
30
7
8
8
8
8
8
8
8
9
9
40
9
9
9
9
10
10
10
10
10
10
50
10
11
11
11
11
11
11
11
12
12
60
12
12
12
12
12
13
13
13
13
13
70
13
13
14
14
14
14
14
14
14
15
80
15
15
15
15
15
15
16
16
16
16
90
16
16
16
17
17
17
17
17
17
17
100
17
18
18
18
18
18
18
18
19
19
110
19
19
19
19
19
20
20
20
20
20
120
20
20
20
21
21
21
21
21
21
21
130
22
22
22
22
22
22
22
22
23
23
140
23
23
23
23
23
24
24
24
24
24
150
24
24
24
25
25
25
25
25
25
25
160
26
26
26
26
26
26
26
26
27
27
170
27
27
27
27
27
27
28
28
28
28
180
28
28
28
29
29
29
29
29
29
29
190
29
30
30
30
30
30
30
30
30
31
200
31
31
31
31
31
31
32
32
32
32
210
32
32
32
32
33
33
33
33
33
33
220
33
33
34
34
34
34
34
34
34
34
230
35
35
35
35
35
35
35
35
36
36
240
36
36
36
36
36
37
37
37
37
37
250
37
37
37
38
38
38
38
38
38
38
260
38
39
39
39
39
39
39
39
39
40
270
40
40
40
40
40
40
40
41
41
41
280
41
41
41
41
41
42
42
42
42
42
290
42
42
42
43
43
43
43
43
43
43
300
43
44
44
44
44
44
44
44
45
45
310
45
45
45
45
45
45
46
46
46
46
320
46
46
46
46
47
47
47
47
47
47
330
47
47
48
48
48
48
48
48
48
48
340
49
49
49
49
49
49
49
49
50
50
350
50
50
50
50
50
50
51
51
51
51
360
51
51
51
51
52
52
52
52
52
52
370
52
52
53
53
53
53
53
53
53
53
380
54
54
54
54
54
54
54
54
55
55
390
55
55
55
55
55
55
55
56
56
56
400
56
56
56
56
56
57
57
57
57
57
410
57
57
57
58
58
58
58
58
58
58
420
58
59
59
59
59
59
59
59
59
60
430
60
60
60
60
60
60
60
61
61
61
440
61
61
61
61
61
62
62
62
62
62
450
62
62
62
63
63
63
63
63
63
63
460
63
64
64
64
64
64
64
64
64
65
470
65
65
65
65
65
65
65
65
66
66
480
66
66
66
66
66
66
67
67
67
67
490
67
67
67
67
68
68
68
68
68
68
500
68
68
69
69
69
69
69
69
69
69
Minimum number of correct responses for significant at a=0.05
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