Behavior Research Methods, Instruments, & Computers 2002, 34 (1), 37-45
Variance of d9 for the same–different method JIAN BI Sensometrics Research and Service, Richmond, Virginia Variance estimates of d9 are derived toward the same–different method based on Taylor-series expansion with one and two variables. The variance estimates can be used for statistical comparison of d9s obtained from various discrimination paradigms. Formulas and tables for estimating variance of d9 for the method are provided. One S-PLUS program, which can produce both d9 and variance of d9, is also provided.
The same–different method is one of the discrimination methods commonly used in sensory analysis and psychophysicalresearch. The regular two-intervalsame–different method involvesfour sample pairs, of which two are concordant (AA or BB) and two are discordant (AB or BA). The observer’s task is to report whether a sample pair that he or she receives on a trial is the same or different. The same– different method is different from the one-interval yes–no task (i.e., the A–not-A method), in which one of two samples A and B (not-A) is presented on each trial. The observer’s task is to report whether a sample is A or B (not-A). Macmillan, Kaplan, and Creelman (1977) developed a Thurstonian model for the two-interval same–different method. Kaplan, Macmillan, and Creelman (1978) provided tables of d9 for the method. In Thurstonian models or the theory of signal detection (TSD), d9 is a measure of sensory difference or discriminative sensitivity. It is theoreticallyunaffected by the criterion that subjects adopt or the method that is used to test them. Strictly speaking, d, which is often used in Thurstonian models, should denote a true value of the measure, and d9 is an estimate of d, just as m denotes a true mean and X denotes an estimate of m. As a statistic, d9 is also a variable and has its variance estimate. In order to compare statistically d9s from the same paradigm or different paradigms— for example,to compare the two d9s obtainedfrom the A–notA method and the same–different method, respectively— variances of the d9s are needed. Gourevitch and Galanter (1967) gave estimates of the variance of d9 for the yes–no task (i.e., the A–not-A method), and Miller (1996) obtained similar results using a different approach. Bi, Ennis, and O’Mahony (1997) provided estimates and tables for the variance of d9 from four forced-choice methods: two-alternative forced choice (2-AFC), 3-AFC, triangular, and duo-trio. The purpose of this paper is to provide variance estimates of d9 for the same–different method.
Estimating the Variance of d9 Macmillan et al. (1977) derived the Thurstonianmodel for the same–different method shown in Equations 1 and 2: æ ö Pss = 2 Fç k ÷ - 1, è 2ø
(1)
æ ö æ ö Psd = Fç k - d ¢ ÷ - F ç - k - d ¢ ÷ , è 2 ø è 2 ø
(2)
and
where Pss is the proportion of the “same” responses for the concordant sample pairs or ; Psd is the proportion of the “same” responses for the discordant sample pairs or ; k is a criterion; d9 is a measure of sensory difference or sensitivity; and F(.) is the standard normal distributionfunction. The estimate of d9 can be numerically obtained from Equations 1 and 2. The model is based on a monadic design, in that it assumes that the responses in an experiment are independent of each other. The estimate of variance of d9 can also be derived from Equations1 and 2. The main theoretical basis for obtaining the variance of d9 is Taylor-series expansion with one and two variables. According to the Taylor-series expansion for a univariable, Equation 1 can be expressed as follows: Pss p ss + Pss¢
k0
(k - k ) , 0
(3)
where pss is an observed value of Pss; P¢ss | k0 denotes the first derivative with respect to k evaluated at k0; æ p + 1ö k 0 = 2 F -1 ç ss ÷ è 2 ø from Equation 1; and æ p + 1ö F -1 ç ss ÷ è 2 ø denotes the p ss +1
The author thanks Jonathan Vaughan, Joan Gay Snodgrass, and two anonymous reviewers for their valuable comments on an earlier version of this paper. The comments led to improvement of the paper. Correspondence should be addressed to J. Bi, 9212 Groomfield Rd., Richmond, VA 23236 (e-mail:
[email protected]).
2 quantile of the standard normal distribution. On the basis of Equation 1, 37
Copyright 2002 Psychonomic Society, Inc.
38
BI
Pss¢
æ k ö = 2f ç 0 ÷ è 2ø º u,
k0
V ( d ¢) =
æ ¶P V ( Psd ) + ç sd è ¶k æ ¶Psd ç è ¶d ¢
where æk ö fç 0 ÷. è 2ø is the ordinate of the standard normal curve at
V (k )
V ( Psd ) =
pss (1 - pss )
(4) = , Pss¢ 2 N su 2 where N s is the sample size for the concordant sample pairs. According to Taylor-series expansion with two variables, Equation 2 can be expressed as follows: Psd p sd +
¶Psd
k 0 ,d 0¢
¶k
( k - k ) + ¶¶Pd ¢
sd
0
k 0 , d 0¢
(d ¢ - d ¢ ) , 0
(5A) or, ¶Psd ¶d ¢
k 0 ,d 0¢
(d ¢ - d ¢ ) ( P 0
sd
)
- p sd -
¶Psd ¶k
k 0 ,d 0¢
0
(k - k ) , 0
(5B) where psd is an observed value of Psd; ¶Psd ¶k
k 0 ,d 0¢
and ¶Psd k ,d ¢ ¶d ¢ 0 0 are partial derivatives with respect to k and d9 evaluated at k0 and d90. d90 is an estimate of d9. On the basis of Equation 2, ¶Psd ¶k
k 0 ,d 0¢
æ k - d 0¢ ö 1 æ - k - d 0¢ ö 1 = fç 0 +fç 0 ÷ ÷ è è 2 ø 2 2 ø 2 ºv
and æ k - d 0¢ ö 1 æ - k - d 0¢ ö 1 = -f ç 0 +fç 0 ÷ ÷ è è 2 ø 2 2 ø 2 º w. Under the assumption that Psd is independent of Pss, Psd is also independent of k. From Equation 5B, the variance of d9 can be obtained by the following: ¶Psd ¶d ¢
k 0 ,d 0¢
ö k 0 ,d 0¢ ÷ ø
2
.
(6)
Since
æ k0 ö ç ÷. è 2ø From Equation 3, the variance of k can be obtained by the following: V ( Pss )
2
ö k 0 , d 0¢ ÷ V ( k ) ø
(
p sd 1 - p sd Nd
),
where Nd is the sample size for the discordant sample pairs, the variance of d9 in Equation 6 can be estimated as follows: B B (7) V d¢ = d + s , Nd N s
( )
where Bd =
psd (1 - psd ) w2
and Bs =
v 2 pss (1 - pss )
. w 2 u2 Bs reflects the variation from the concordant sample pairs, whereas Bd reflects the variation from the discordant sample pairs. The d9 can be estimated from Equations 1 and 2 by using the two observed proportions pss and psd. The variances of d9 can be estimated from Equation 7 by using pss , psd , Ns , and Nd . It is assumed that Pss > Psd . If the observed proportion of Pss is smaller than the observed proportion of Psd, it is assumed that the true d9 = 0. One S-PLUS program is given in the Appendix for estimating both d9 and the variance of d9. There are four arguments for the program: xss , Ns , xsd and Nd, where xss is the number of “same” responses for the concordant sample pairs and xsd is the number of “same” responses for the discordant sample pairs. Here xss = pss Ns and x sd Nd
= psd .
Tables for the Variance of d9 Tables 1A–1B provide variance estimates of d9 for the same–different method. The tables can be used to get a variance estimate of d9 quickly, without one’s having to compute them. The tables give Bs and Bd values, the main components of variance of d9. The variance of d9 can be obtained easily by dividing the Bs and Bd values with Ns and Nd. Tables 1A–1B give the Bs and Bd values corresponding to different ranges of pss and psd.
VARIANCE OF d9
39
Figure 1. The weight of variation of d9 from discordant sample pairs. The weight is defined by w d = B d / (Bs + B d). B s and B d are components of variance of d9. B s reflects the variation from the concordant sample pairs, whereas Bd reflects the variation from the discordant sample pairs.
Table 1A corresponds to psd from .01 to .11 with a step of .01 and from .11 to .19 with a step of .02, as well as corresponds to pss from .02 to .2 with a step of .02, from .25 to .8 with a step of .05, from .81 to .89 with a step of .02, and from .9 to .99 with a step of .01. Table 1B corresponds to psd from .2 to .80 with a step of .05 and from .8 to .98 with a step of .02, as well as corresponds to pss from .25 to .8 with a step of .05, from .81 to.89 with a step of .02, and from .9 to .99 with a step of .01. For values pss and psd that cannot be found in the tables, linear interpolation may be used to compute the B values. However, a good approximate can be obtained from the tables. From the tables we can find the trend that in most cases, especially when psd is smaller, Bd is larger than Bs. This relationship means that the variance of d9 in the same– different test is mainly determined by performance on the discordant sample pairs. Hence, in order to reduce variance of d9 in the test, sample size for the discordant sample pairs should generally be larger than that for the concordant sample pairs. However, when pss is close to 1, the situation is different. Here Bs is larger than Bd. In that situation, sample size for the concordant pairs should be more important than that of the discordant pairs in order to reduce variance of d9. Figure 1 shows the weight of vari-
ation from discordant sample pairs in the total variation. The weight, defined by Bd wd = , Bs + Bd measures the relative importance of Bd in the total B values. The weight (the wd axis in Figure 1) lies between zero and one, 0 < wd # 1. It is large when Psd (the sd axis in Figure 1) is close to zero, whereas it is small when Pss (the ss axis in Figure 1) is close to one. Examples In a same–different test, Ns = 100 concordant sample pairs (50 AA and 50 BB) and Nd = 200 discordant sample pairs (100 AB and 100 BA) are presented. xss = 17 and xsd = 18 are observed. Hence pss = .17 and psd = .09. From the tables in Kaplan et al. (1978) or from Equations 1 and 2, d9 = 1.61 can be obtained. From Equation 7, the exact estimate of variance of d9 at 1.61 is V(d9) = 0.162. The variance can also be obtained from Table 1A. We can find from Table 1A that for pss = .16 and psd = .09, the B values are Bs = 9.54 and Bd = 17.81. The variance of d9 is V ( d ¢ ) = 9.54 + 17.81 = 0.184. 100 200
.60
.55
.50
.45
.40
.35
.30
.25
.20
.18
.16
.14
.12
.10
.08
.06
.04
.01
70.75 142.86 17.41 71.47 8.88 55.36 5.71 47.77 4.12 43.22 3.20 40.14 2.62 37.90 2.22 36.18 1.94 34.83 1.74 33.74 1.43 31.75 1.28 30.45 1.22 29.57 1.22 28.96 1.26 28.55 1.33 28.28 1.43 28.11 1.56 28.00
pss
.02
34.73 70.75 14.41 44.69 8.48 35.46 5.81 30.61 4.35 27.55 3.45 25.44 2.86 23.88 2.44 22.68 2.14 21.73 1.69 20.04 1.45 18.94 1.34 18.21 1.31 17.70 1.32 17.34 1.37 17.11 1.46 16.95 1.58 16.85
.02
83.58 112.49 22.76 46.74 11.92 33.08 7.71 27.00 5.56 23.50 4.29 21.21 3.48 19.58 2.92 18.35 2.52 17.40 1.92 15.75 1.61 14.71 1.45 14.01 1.38 13.53 1.37 13.19 1.41 12.96 1.48 12.81 1.60 12.70
.03
38.83 59.30 16.80 34.74 10.07 26.33 6.95 22.01 5.22 19.36 4.14 17.55 3.41 16.23 2.90 15.23 2.14 13.53 1.76 12.48 1.55 11.79 1.45 11.31 1.42 10.98 1.44 10.75 1.51 10.59 1.61 10.48
.04
86.21 104.41 24.71 40.56 13.24 27.56 8.67 21.87 6.29 18.64 4.87 16.56 3.95 15.09 3.31 14.00 2.38 12.20 1.91 11.12 1.65 10.41 1.52 9.92 1.47 9.59 1.48 9.35 1.53 9.19 1.63 9.08
.05
40.27 54.64 17.90 30.83 10.89 22.77 7.59 18.68 5.72 16.19 4.55 14.51 3.76 13.29 2.62 11.35 2.06 10.21 1.76 9.47 1.60 8.98 1.52 8.63 1.52 8.39 1.56 8.22 1.64 8.11
.06
86.57 99.83 25.56 37.44 13.94 24.84 9.22 19.37 6.73 16.29 5.24 14.31 4.26 12.93 2.88 10.79 2.22 9.57 1.86 8.81 1.67 8.29 1.57 7.93 1.55 7.68 1.58 7.51 1.66 7.39
.07
40.74 51.81 18.45 28.58 11.35 20.76 7.97 16.82 6.04 14.43 4.82 12.82 3.17 10.43 2.39 9.13 1.97 8.31 1.74 7.78 1.63 7.40 1.59 7.15 1.61 6.97 1.68 6.84
.08
psd
86.06 96.47 25.91 35.41 14.31 23.12 9.54 17.81 7.02 14.84 5.48 12.93 3.48 10.23 2.57 8.81 2.08 7.95 1.82 7.38 1.68 6.99 1.63 6.72 1.63 6.54 1.70 6.41
.09
40.76 49.74 18.71 27.03 11.62 19.41 8.21 15.58 6.26 13.26 3.83 10.16 2.76 8.60 2.21 7.67 1.90 7.07 1.74 6.67 1.67 6.38 1.66 6.19 1.72 6.05
.10
Table 1A B Values (B s and Bd) for Variance of d9 From the Same–Different Method
85.15 93.70 26.00 33.90 14.50 21.89 9.74 16.71 7.20 13.82 4.22 10.19 2.97 8.46 2.33 7.47 1.98 6.83 1.79 6.40 1.71 6.11 1.69 5.90 1.74 5.76
.11
84.02 91.25 25.92 32.67 14.57 20.92 9.85 15.87 5.19 10.58 3.46 8.40 2.62 7.22 2.17 6.49 1.92 6.02 1.79 5.69 1.75 5.46 1.78 5.31
.13
82.75 89.01 25.75 31.62 14.56 20.12 6.52 11.47 4.06 8.58 2.96 7.15 2.38 6.31 2.06 5.76 1.89 5.40 1.82 5.15 1.82 4.98
.15
81.43 86.92 25.50 30.69 8.49 13.09 4.84 9.03 3.36 7.23 2.62 6.23 2.21 5.61 1.99 5.20 1.89 4.92 1.87 4.73
.17
80.03 84.91 11.75 16.07 5.89 9.80 3.86 7.46 2.90 6.26 2.39 5.54 2.11 5.07 1.97 4.76 1.93 4.54
.19
40 BI
1.73 27.93 1.94 27.90 2.22 27.88 2.60 27.88 2.69 27.88 2.91 27.88 3.18 27.87 3.52 27.87 3.96 27.88 4.23 27.88 4.56 27.87 4.96 27.87 5.45 27.87 6.09 27.87 6.95 27.88 8.19 27.88 10.15 27.87 13.80 27.87 23.67 27.87
.65
1.74 16.79 1.94 16.75 2.22 16.73 2.60 16.73 2.70 16.72 2.91 16.72 3.18 16.72 3.52 16.72 3.96 16.72 4.23 16.72 4.56 16.72 4.96 16.72 5.45 16.72 6.09 16.72 6.95 16.72 8.19 16.72 10.15 16.72 13.80 16.72 23.67 16.72
.02
1.75 12.64 1.95 12.60 2.22 12.58 2.60 12.58 2.70 12.57 2.92 12.57 3.18 12.57 3.52 12.57 3.96 12.57 4.23 12.57 4.56 12.57 4.96 12.57 5.45 12.57 6.09 12.57 6.95 12.57 8.19 12.57 10.15 12.57 13.80 12.57 23.67 12.57
.03 1.76 10.42 1.96 10.38 2.22 10.36 2.60 10.35 2.70 10.35 2.92 10.34 3.18 10.34 3.52 10.34 3.96 10.34 4.23 10.34 4.56 10.34 4.96 10.34 5.45 10.34 6.09 10.34 6.95 10.34 8.19 10.34 10.15 10.34 13.80 10.34 23.67 10.34
.04 1.77 9.01 1.96 8.97 2.23 8.95 2.60 8.94 2.70 8.94 2.92 8.93 3.18 8.93 3.52 8.93 3.96 8.93 4.23 8.93 4.56 8.93 4.96 8.93 5.45 8.93 6.09 8.93 6.95 8.93 8.19 8.93 10.15 8.93 13.80 8.93 23.67 8.93
.05 1.78 8.04 1.97 7.99 2.23 7.97 2.60 7.96 2.70 7.95 2.92 7.95 3.18 7.95 3.52 7.95 3.96 7.95 4.23 7.95 4.56 7.95 4.96 7.95 5.45 7.95 6.09 7.95 6.95 7.95 8.19 7.95 10.15 7.95 13.80 7.95 23.67 7.95
.06 1.79 7.32 1.97 7.27 2.23 7.24 2.61 7.23 2.70 7.23 2.92 7.23 3.18 7.22 3.52 7.22 3.96 7.22 4.23 7.22 4.56 7.22 4.96 7.22 5.45 7.22 6.09 7.22 6.95 7.22 8.19 7.22 10.15 7.22 13.80 7.22 23.67 7.22
.07 1.80 6.76 1.98 6.71 2.24 6.68 2.61 6.67 2.70 6.67 2.92 6.66 3.18 6.66 3.52 6.66 3.96 6.66 4.23 6.66 4.56 6.66 4.96 6.66 5.45 6.66 6.09 6.66 6.95 6.66 8.19 6.66 10.15 6.66 13.80 6.66 23.67 6.66
psd .08 .09 1.81 6.32 1.99 6.27 2.24 6.24 2.61 6.22 2.70 6.22 2.92 6.22 3.19 6.21 3.52 6.21 3.96 6.21 4.23 6.21 4.56 6.21 4.96 6.21 5.45 6.21 6.09 6.21 6.95 6.21 8.19 6.21 10.15 6.21 13.80 6.21 23.67 6.21
.10 1.83 5.96 2.00 5.91 2.25 5.87 2.61 5.86 2.71 5.85 2.92 5.85 3.19 5.85 3.52 5.85 3.96 5.85 4.23 5.84 4.56 5.84 4.96 5.84 5.45 5.84 6.09 5.84 6.95 5.84 8.19 5.84 10.15 5.84 13.80 5.84 23.67 5.84
.11 1.84 5.67 2.00 5.61 2.25 5.57 2.61 5.55 2.71 5.55 2.92 5.54 3.19 5.54 3.52 5.54 3.96 5.54 4.23 5.54 4.56 5.54 4.96 5.54 5.45 5.54 6.09 5.54 6.95 5.54 8.19 5.54 10.15 5.54 13.80 5.54 23.67 5.54
.13 1.87 5.20 2.02 5.13 2.26 5.09 2.62 5.07 2.71 5.07 2.93 5.06 3.19 5.06 3.52 5.06 3.96 5.06 4.23 5.06 4.56 5.06 4.96 5.05 5.45 5.05 6.09 5.05 6.95 5.05 8.19 5.05 10.15 5.05 13.80 5.05 23.67 5.05
.15 1.90 4.86 2.04 4.78 2.27 4.73 2.62 4.71 2.72 4.71 2.93 4.70 3.19 4.70 3.52 4.69 3.96 4.69 4.23 4.69 4.56 4.69 4.96 4.69 5.45 4.69 6.09 4.69 6.95 4.69 8.19 4.69 10.15 4.69 13.80 4.69 23.67 4.69
.17 1.93 4.60 2.06 4.51 2.28 4.46 2.63 4.43 2.72 4.42 2.93 4.42 3.19 4.41 3.53 4.41 3.96 4.41 4.23 4.41 4.56 4.41 4.96 4.41 5.45 4.41 6.09 4.41 6.95 4.41 8.19 4.41 10.15 4.41 13.80 4.41 23.67 4.41
.19 1.97 4.40 2.09 4.30 2.30 4.24 2.64 4.21 2.73 4.20 2.94 4.19 3.20 4.19 3.53 4.18 3.96 4.18 4.23 4.18 4.56 4.18 4.96 4.18 5.45 4.18 6.09 4.18 6.95 4.18 8.19 4.18 10.15 4.18 13.80 4.18 23.67 4.18
Bs B + d , N s Nd
where Ns is the number of concordant sample pairs and N d is the number of discordant sample pairs.
V (d ¢) =
Note—pss = The proportion of the “same” responses for the concordant sample pairs (AA or BB). psd = The proportion of the “same” responses for the discordant sample pairs (AB or BA). There are two B values, Bs and Bd , corresponding to a same pair of pss and psd . The variance is then obtained by dividing the B values with corresponding sample sizes,
.99
.98
.97
.96
.95
.94
.93
.92
.91
.90
.89
.87
.85
.83
.81
.80
.75
.70
.01
pss
Table 1A (Continued)
VARIANCE OF d9 41
14.35 18.54 6.57 10.36 4.16 7.65 3.06 6.31 2.49 5.52 2.17 5.03 2.01 4.69 1.95 4.47 1.99 4.31 2.10 4.21 2.31 4.15 2.64 4.11 2.73 4.11 2.94 4.10 3.20 4.09 3.53 4.09
.25
.87
.85
.83
.81
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.20
pss
13.96 17.29 6.51 9.54 4.18 6.98 3.12 5.73 2.57 5.00 2.27 4.54 2.13 4.23 2.10 4.03 2.17 3.89 2.35 3.80 2.67 3.75 2.75 3.75 2.96 3.74 3.21 3.73 3.54 3.72
.25
13.50 16.21 6.40 8.88 4.17 6.47 3.16 5.29 2.63 4.61 2.36 4.18 2.25 3.90 2.27 3.72 2.41 3.60 2.70 3.53 2.78 3.52 2.98 3.51 3.23 3.50 3.55 3.49
.30
12.99 15.24 6.25 8.32 4.13 6.04 3.17 4.93 2.69 4.30 2.46 3.91 2.40 3.66 2.49 3.50 2.75 3.40 2.82 3.39 3.01 3.36 3.25 3.35 3.56 3.34
.35
12.46 14.35 6.08 7.81 4.08 5.67 3.19 4.64 2.76 4.05 2.59 3.69 2.61 3.47 2.81 3.34 2.88 3.32 3.05 3.29 3.28 3.26 3.59 3.24
.40
11.91 13.51 5.90 7.36 4.03 5.35 3.22 4.39 2.86 3.85 2.76 3.53 2.90 3.34 2.96 3.31 3.11 3.26 3.33 3.23 3.62 3.20
.45
11.36 12.71 5.72 6.93 3.99 5.06 3.27 4.17 3.00 3.68 3.03 3.40 3.08 3.36 3.20 3.30 3.39 3.25 3.67 3.21
.50
10.80 11.93 5.55 6.54 3.97 4.80 3.36 3.99 3.23 3.55 3.25 3.49 3.33 3.39 3.49 3.32 3.74 3.26
.55
.65
10.25 11.18 5.39 9.71 6.17 10.44 3.99 5.27 4.57 5.82 3.54 4.09 3.84 4.37 3.52 3.99 3.74 4.20 3.54 3.88 3.59 3.93 3.64 3.90 3.47 3.73 3.85 4.03 3.38 3.59
.60
9.20 9.72 5.22 5.51 4.92 5.14 4.53 4.59 4.35 4.22 4.35 3.95
.70
8.72 9.02 7.49 7.73 6.05 6.15 5.32 5.23 4.98 4.66
psd .75
36.38 36.65 12.86 13.00 8.36 8.34 6.63 6.40
.80
34.85 35.02 12.49 12.51 8.27 8.08
.82
33.31 33.36 12.15 12.01
.84
Table 1B B Values (Bs and B d) for Variance of d9 From the Same–Different Method
31.74 31.65
.86
.88
.90
.92
.94
.96
.98
42 BI
3.96 4.09 4.24 4.08 4.56 4.08 4.96 4.08 5.45 4.08 6.09 4.08 6.95 4.08 8.19 4.08 10.15 4.08 13.80 4.08 23.67 4.08
.89
3.97 3.72 4.24 3.72 4.56 3.72 4.96 3.71 5.45 3.71 6.09 3.71 6.95 3.71 8.19 3.71 10.15 3.71 13.80 3.71 23.67 3.71
.25
3.97 3.48 4.24 3.48 4.57 3.48 4.96 3.48 5.46 3.48 6.09 3.47 6.95 3.47 8.19 3.47 10.15 3.47 13.80 3.47 23.67 3.47
.30
3.98 3.33 4.25 3.32 4.57 3.32 4.97 3.32 5.46 3.32 6.10 3.32 6.96 3.32 8.19 3.32 10.15 3.32 13.80 3.32 23.67 3.32
.35 4.00 3.23 4.26 3.23 4.58 3.22 4.97 3.22 5.46 3.22 6.10 3.22 6.96 3.22 8.19 3.22 10.15 3.22 13.80 3.22 23.67 3.22
.40 4.02 3.18 4.28 3.18 4.60 3.17 4.98 3.17 5.47 3.17 6.10 3.16 6.96 3.16 8.19 3.16 10.15 3.16 13.80 3.16 23.67 3.16
.45 4.05 3.18 4.31 3.17 4.62 3.16 5.00 3.16 5.48 3.15 6.11 3.15 6.97 3.14 8.20 3.14 10.15 3.14 13.80 3.14 23.67 3.14
.50 4.10 3.22 4.35 3.20 4.65 3.19 5.02 3.18 5.50 3.17 6.12 3.17 6.97 3.16 8.20 3.16 10.15 3.16 13.80 3.16 23.67 3.16
.55 4.18 3.31 4.41 3.29 4.70 3.27 5.06 3.25 5.53 3.24 6.14 3.23 6.99 3.22 8.21 3.22 10.16 3.22 13.80 3.22 23.68 3.22
.60
.75 4.94 4.28 5.03 4.14 5.19 4.03 5.45 3.94 5.82 3.87 6.36 3.81 7.14 3.77 8.30 3.74 10.21 3.73 13.82 3.72 23.68 3.71
4.31 4.52 3.48 3.76 4.51 4.69 3.44 3.69 4.78 4.92 3.41 3.64 5.13 5.24 3.38 3.59 5.58 5.66 3.36 3.55 6.18 6.24 3.34 3.52 7.01 7.05 3.33 3.50 8.22 8.25 3.32 3.49 10.16 10.18 3.32 3.48 13.80 13.81 3.32 3.48 23.68 23.68 3.32 3.47
psd .70
.65
.80 5.92 5.38 5.78 5.04 5.78 4.78 5.90 4.58 6.17 4.42 6.61 4.30 7.31 4.21 8.42 4.15 10.27 4.11 13.85 4.09 23.69 4.08
.82 6.74 6.26 6.39 5.71 6.23 5.31 6.24 5.01 6.42 4.78 6.80 4.60 7.44 4.48 8.50 4.39 10.32 4.33 13.87 4.30 23.69 4.29
.84 8.25 7.84 7.43 6.83 6.97 6.14 6.77 5.64 6.80 5.28 7.07 5.02 7.63 4.83 8.63 4.70 10.40 4.61 13.91 4.56 23.70 4.54
.86 11.86 11.52 9.57 9.06 8.35 7.62 7.70 6.69 7.45 6.06 7.52 5.63 7.94 5.32 8.83 5.11 10.51 4.98 13.97 4.90 23.72 4.87
.88 30.15 29.89 16.13 15.71 11.66 11.05 9.64 8.76 8.67 7.44 8.32 6.61 8.46 6.06 9.16 5.70 10.71 5.47 14.06 5.34 23.74 5.29 28.55 28.06 15.63 14.90 11.65 10.60 10.01 8.52 9.49 7.35 9.78 6.64 11.07 6.21 14.23 5.97 23.80 5.86
.90
27.00 26.17 15.35 14.12 12.06 10.23 11.16 8.41 11.81 7.44 14.59 6.93 23.90 6.70
.92
25.64 24.20 15.70 13.44 13.82 10.08 15.48 8.65 24.18 8.07
.94
.98
25.11 22.26 18.76 13.19 25.09 32.12 10.80 21.47
.96
Bs B + d , N s Nd
where Ns is the number of concordant sample pairs and Nd is the number of discordant sample pairs.
V (d ¢) =
Note—pss = The proportion of the “same” responses for the concordant sample pairs (AA or BB). psd = The proportion of the “same” responses for the discordant sample pairs (AB or BA). There are two B values, Bs and Bd , corresponding to a same pair of pss and psd . The variance is then obtained by dividing the B values with corresponding sample sizes,
.99
.98
.97
.96
.95
.94
.93
.92
.91
.90
.20
pss
Table 1B (Continued)
VARIANCE OF d9 43
44
BI
For pss = .18 and psd = .09, the B values are Bs = 7.02 and Bd = 14.84. The variance of d9 is
mate of d9 at 2.96 is 0.022. It is very close to the exact estimate value (0.023).
V (d ¢ ) = 7.02 + 14.84 = 0.144. 100 200 Hence a good approximate of the variance of d9 at 1.61 is a value between 0.184 and 0.144. It is about 0.164, which is close to the exact estimate value 0.162. In another same–different test, Ns = 200 and Nd = 400, xss = 134, and xsd = 52 are observed. We get pss = .67 and psd = .13. From the tables in Kaplan et al. (1978) or from Equations 1 and 2, d9 = 2.96. The exact variance estimate of d9 is 0.023 from Equation 7. In Table 1A, the closest proportions are pss = .65 and psd = .13, the corresponding B values are Bs = 1.87 and Bd = 5.2. The approximate esti-
REFERENCES Bi, J., Ennis, D. M., & O’Mahony, M. (1997). How to estimate and use the variance of d9 from difference tests. Journal of Sensory Studies, 12, 87-104. Gourevitch, V., & Galanter, E. (1967). A significance test for one parameter isosensitivity functions. Psychometrika, 32, 25-33. Kaplan, H. L., Macmillan, N. A., & Creelman, C. D. (1978). Tables of d9 for variable-standard discrimination paradigms. Behavior Research Methods & Instrumentation, 10, 796-813. Macmillan, N. A., Kaplan, H. L., & Creelman, C. D. (1977). The psychophysics of categorical perception. Psychological Review, 84, 452-471. Miller, J. (1996). The sampling distribution of d9. Perception & Psychophysics, 58, 65-72.
APPENDIX S-PLUS Code for Estimating d9 and Variance of d9 for the Same–Different Method Function Name sddvn Description Returns estimates of d9 and variance of d9 for the same–different method. Usage sddvn (sn, n1, dn, n2) Required Arguments sn: The number of “same” responses for concordant sample pairs n: The number of concordant sample pairs dn: The number of “same” responses for discordant sample pairs n2: The number of discordant sample pairs Output Values of d9 and variance of d9 Examples > sddvn(17,100,18,200) [1] 1.607 0.162 > sddvn(134,200,52,400) [1] 2.963 0.023 S-PLUS Code sddvn function(sn, n1, dn, n2) { # d9 and variance of d9 for same±different method. s
