The double discrimination methods.pdf

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-

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

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

1 1

ð1Þ

power ¼ 1 ¼ 1 F u 2 p 1

v 1

x¼k

ð5Þ

1 1

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).

ð2Þ

x¼k

1 1

power ¼ 1 ¼ 1 F u 3 p 1

v 1

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

16100

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

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

^ 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

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

Þ 2 =2 ðÞ m1 dz for the m-

^ ð =

510

J. Bi/Food Quality and Preference 12 (2001) 507–513

d ðÞ

for the triangular method (David & Trivedi, 1962; Ura,

2=3

þ z 3

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

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

100

101

102

350

102

103

104

360

105

106

107

370

107

108

109

110

380

110

111

112

390

113

114

115

400

115

116

117

118

410

118

119

120

420

12].

121

122

123

430

123

124

125

126

440

126

127

128

450

129

130

131

460

131

132

133

134

470

134

135

136

480

137

138

139

490

139

140

141

142

500

142

143

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

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

410

420

430

440

450

460

470

480

490

500

Minimum number of correct responses for signiﬁcant at a=0.05

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