**We studied a fundamental assumption in signal detection theory that is applied to motion discrimination. Using random-dot motion stimuli of 100% coherence, it is natural to represent the two directions as two normal distributions. Then a same–different task and a forced-choice task should give rise to consistent d′ estimates, because both tasks share the same underlying signal detection theory model. To verify this prediction, we used 4°, 8°, and 12° as the angular difference in motion discrimination. In a between-subjects design, we found the predicted result only with the 4° angular difference. With 8° and 12°, the estimated same–different d′ was 32% greater than the two-alternative forced-choice d′. In a subsequent within-subject design with counterbalancing, the first half of the data confirmed this finding. Interestingly, there was now within-subject consistency. Namely, the second task's d′ was comparable to the first task's, as if the first task's discrimination was carried over to the second task. This carryover effect diminished when the time gap between the two tasks was lengthened.**

*noise*and

*signal*. A participant's discriminating performance of these two categories is characterized by the distance between the two distributions divided by their shared standard deviation. This quantity is defined as the discrimination sensitivity

*d*′.

*d*′ is typically calculated without verifying the underlying assumptions. When the assumptions are verified, oftentimes only the equal-variance assumption is verified. In this study, we present a case in which the equal-variance assumption appears to hold but other aspects of the assumptions in the standard model appear to be violated.

*d*′ estimates from different experimental methods are self-consistent. This property is not restricted to motion discrimination, but applies generally to all psychophysical stimuli. According to Macmillan and Creelman, the standard model has provided natural paradigms to study the detection of weak signals. Whether or not it applies to stronger signals remains an empirical question. In the current study, we tested the standard model across a range of signal strengths.

*d*′. The two tasks we chose in this study were two-temporal-alternative forced-choice (2AFC) and temporal same–different. In the 2AFC task, two samples are randomly drawn

*without*replacement from the two directions. In the same–different task, two samples are randomly drawn

*with*replacement from the two directions. Figure 1 shows an example trial in a 2AFC task.

**Figure 1**

**Figure 1**

*d*′. Let

*S*

_{1}and

*S*

_{2}denote the two motion stimuli from the two directions in a 2AFC task. Then the stimulus sequence in a trial will be either <

*S*

_{1}

*S*

_{2}> or <

*S*

_{2}

*S*

_{1}>. According to the standard model,

*d*′ = [

*Z*(proportion correct responses|<

*S*

_{1}

*S*

_{2}>) +

*Z*(proportion correct responses|<

*S*

_{2}

*S*

_{1}>)]/ . However, this 2AFC task is not participant friendly, which can be illustrated in the following example. Let us assume that the two motion directions are either 43° and 47° or 133° and 137° (0° is upward). In the first case, the response “more clockwise” corresponds to a rightward change of motion direction. However, when the two directions are 133° and 137°, the response “more clockwise” corresponds to a leftward change of motion direction. As a result, the response “more clockwise” is not intuitive.

*d*′. Macmillan and Creelman (2005) worked out how to calculate this

*d*′ by assuming no bias in the same–different task. They offered two insights. The first is that the four possible stimulus combinations <

*S*

_{1}

*S*

_{1}>, <

*S*

_{2}

*S*

_{2}>, <

*S*

_{1}

*S*

_{2}>, and <

*S*

_{2}

*S*

_{1}> correspond to four distributions with centers at (0, 0), (

*d*′,

*d*′), (0,

*d*′), and (

*d*′, 0), respectively. These centers are located in the two-dimensional space spanned by two orthogonal axes that represent the two intervals (Figure 2). The optimal, unbiased decision criterion consists of two lines

*x*=

*d*′/2 and

*y*=

*d*′/2 that divide the two-dimensional space into four quadrants. Because of zero bias, the overall proportion correct

*p*(

*c*) is equal to the proportion correct with any of the four stimulus combinations. For example,

*p*(

*c*) is equal to the proportion correct in responding “different” when the stimuli are <

*S*

_{1}

*S*

_{2}>. This proportion correct is equal, when the stimuli are <

*S*

_{1}

*S*

_{2}>, to the probability volume in the top left quadrant plus the probability volume in the bottom right quadrant. The probability volume in the upper-left quadrant is where

*N*() is the normal distribution. The probability volume of the bottom right quadrant is [Φ(−

*d*′/2)]

^{2}. The proportion correct is therefore

**Figure 2**

**Figure 2**

*H*=

*P*(“different”|<

*S*

_{1}

*S*

_{2}> or <

*S*

_{2}

*S*

_{1}>) and the false-alarm rate

*F*=

*P*(“different”|<

*S*

_{1}

*S*

_{1}> or <

*S*

_{2}

*S*

_{2}>). When there is no bias,

*H*=

*P*(“same”|<

*S*

_{1}

*S*

_{1}> or <

*S*

_{2}

*S*

_{2}>) = 1 −

*P*(“different”|<

*S*

_{1}

*S*

_{1}> or <

*S*

_{2}

*S*

_{2}>) = 1 −

*F*. Therefore,

*Z*(

*H*) =

*Z*(1 −

*F*) = −

*Z*(

*F*). The unbiased proportion correct

*p*(

*c*) is

*p*(

*c*) is a function of

*Z*(

*H*) −

*Z*(

*F*) in a same–different task for an unbiased observer, we can now introduce Macmillan and Creelman's second insight. They observed that the same–different receiver operating characteristic (ROC) is

*approximately*a straight line with a 45° slope in the Gaussian space,

*Z*(

*H*) =

*Z*(

*F*) + constant. Hence,

*Z*(

*H*) − Z(

*F*) remains approximately constant regardless of the decision criterion. Therefore, even if an observer is biased,

*Z*(

*H*) −

*Z*(

*F*) remains approximately the same as when there is no bias. The unbiased proportion correct

*p*(

*c*) can be calculated from Equation 4 by directly plugging in

*Z*(

*H*) and

*Z*(

*F*) from the data. Consequently,

*d*′ can then be recovered from Equation 3.

*S*

_{1}

*S*

_{2}> or <

*S*

_{2}

*S*

_{1}>. That is to say, the calculation assumes that

*P*(“different”|<

*S*

_{1}

*S*

_{2}>) =

*P*(“different”|<

*S*

_{2}

*S*

_{1}>). This is an assumption that can be verified empirically.

*p*(

*c*)

_{SD}and the bias-free yes–no proportion correct

*p*(

*c*)

_{yes-no}obey the equation

*d*′/2. Hence, the bias-free

*p*(

*c*)

_{yes‐no}=

*H*= Φ(

*d*′/2) (Equation 4); from Equation 2 we then obtain Equation 5. Starting from Equation 5, we have

*p*(

*c*)

_{SD}− 1/2 = 2(

*p*(

*c*)

_{yes‐no}− 1/2)

^{2}. Let us assume that

*p*(

*c*)

_{yes‐no}≥ 1/2, then 0 ≤ 2(

*p*(

*c*)

_{yes‐no}− 1/2) ≤ 1. Therefore,

*p*(

*c*)

_{SD}− 1/2 ≤

*p*(

*c*)

_{yes‐no}− 1/2. Hence,

*p*(

*c*)

_{SD}≤

*p*(

*c*)

_{yes‐no}. Given that

*p*(

*c*)

_{yes‐no}= Φ(

*d*′/2) (Equation 4) and

*p*(

*c*)

_{2AFC}= Φ(

*d*′/ ), we have

*p*(

*c*)

_{SD}≤

*p*(

*c*)

_{yes‐no}<

*p*(

*c*)

_{2AFC}. That is, the same–different task is always more difficult than the corresponding 2AFC task.

*d*′ can be recovered from the 2AFC and same–different tasks separately, we can verify the consistency between the two

*d*′ values thus computed. We used 2AFC and same–different tasks because these are common tasks in motion discrimination and motion perceptual learning in the literature. We compared these two tasks with both between-subjects and within-subject designs. We also used angular differences of 4°, 8°, and 12° in order to parametrically manipulate the signal strength.

*d*′, Macmillan and Creelman (2005) assumed the optimal decision rule. They also introduced a suboptimal rule, the differencing rule. In the case of motion discrimination, differencing takes the directional difference between the first and second stimulus. If the magnitude of this difference is below a certain threshold, a “same” response will be chosen; otherwise, “different” will be chosen. Geometrically, this policy is to carve out the space in Figure 2 by two lines:

*y*−

*x*= ±threshold. The region between these two lines corresponds to the stimuli with the “same” responses. The regions outside correspond to the “different” responses.

*S*

_{1}

*S*

_{1}> and <

*S*

_{2}

*S*

_{2}> and between <

*S*

_{1}

*S*

_{2}> and <

*S*

_{2}

*S*

_{1}>.

*d*′ in the same–different task from the same behavioral data, the recovered

*d*′ from the optimal decision rule is always smaller than a

*d*′ recovered from a suboptimal model. This is because, in order for a suboptimal rule to produce the same level of behavioral performance as the optimal rule does, the corresponding

*d*′ necessarily has to be greater (i.e., the task has to be easier) in order to compensate for the inefficiencies of the suboptimal model (Petrov, 2009).

*d*′ estimates only for the weak signal case, at the 4° angular difference. When the angular difference increased to 8° and 12°, the same–different

*d*′ as recovered using the optimal model became greater than the 2AFC

*d*′.

*d*′ estimate in the second task was comparable with that in the first task, indicating that the participant adopted a consistent computation from the first to the second task. More specifically, when the angular differences were 8° and 12°, the participants whose first task was same–different gave rise to a higher average

*d*′ from both tasks than those participants whose first task was 2AFC. When the angular difference was 4°, the

*d*′ estimates remained consistent within and between subjects.

^{2}. A central red fixation disk had a diameter of 0.5° (16 pixels) and a luminance of 5.6 cd/m

^{2}. The background luminance was 22.0 cd/m

^{2}. In each stimulus, all dots moved along a single direction with a speed of 10°/s. The duration of each stimulus was 500 ms, and the interstimulus interval was 200 ms (Figure 1).

*d*′, it is nevertheless informative to visualize the proportion-correct data, whose calculation requires no assumptions. The 2AFC proportion correct was greater than the same–different one, consistent with Macmillan and Creelman's (2005) prediction, even though this proportion correct was the actual and not the unbiased one.

**Figure 3**

**Figure 3**

*d*′ for each of the two tasks (Figure 4), and using the optimal independent model and the suboptimal differencing model for the same–different task. If a participant's hit rate

*H*or false-alarm rate

*F*was 1 or 0, correction was made by subtracting 1/2

*n*from 1 or adding 1/2

*n*to 0, where

*n*is the number of trials used to calculate this particular rate. It turned out that 6% of the data needed correction, which will be addressed later. From

*Z*(

*H*) and

*Z*(

*F*), the same–different

*d*′ could be calculated using the independent model, which is the optimal model that assumes that the two stimuli in a trial are perceptually independent and separable (Macmillan & Creelman, 2005). For the 2AFC task, the

*d*′ calculation was straightforward:

*d*′ = (

*Z*(

*H*) −

*Z*(

*F*))/ .

**Figure 4**

**Figure 4**

*d*′ scores, ANOVA was performed with the following factors: 12 motion directions × 2 tasks × 3 angular differences. All effects except the three-way interaction—

*F*(22, 1122) = 1.06,

*p*= 0.39—were significant. Specifically, the main effect of angular difference was significant, as expected,

*F*(2, 102) = 121.19,

*p*= 1.51 × 10

^{−27}, with the mean

*d*′ for 4°, 8°, and 12° at

*d*′ = 0.73, 1.72, and 2.38, respectively. The main effect of motion direction was also significant,

*F*(11, 1122) = 112.28,

*p*= 3.54 × 10

^{−172}. This was also expected, since discrimination along cardinal directions (

*d*′ = 2.29) was better than along oblique directions (

*d*′ = 1.27). Importantly, the main effect of task was significant,

*F*(1, 102) = 18.41,

*p*= 0.00004. The same–different

*d*′ was greater than the 2AFC

*d*′ by 26% (1.80 vs. 1.42). The interaction between task and angular difference was also significant,

*F*(2, 102) = 3.68,

*p*= 0.03. This interaction revealed that, while the

*d*′ values of the two tasks at the 4° directional difference were comparable (

*d*′ = 0.75 vs. 0.71, 6% difference), their difference became greater at larger angular differences (8°:

*d*′ = 1.98 vs. 1.47, 35% difference; 12°:

*d*′ = 2.66 vs. 2.09, 27% difference; Figure 5).

**Figure 5**

**Figure 5**

*d*′ of the same–different task was that

*P*(“different”|<

*S*

_{1}

*S*

_{2}>) =

*P*(“different”|<

*S*

_{2}

*S*

_{1}>). In order to check whether this assumption held, we compared these two measures from all 54 participants. With all three angular differences collapsed,

*t*(53) = 0.78,

*p*= 0.44. Given that the 4° data already showed consistency, we also looked at the 8° and 12° data separately. There,

*t*(35) = 0.59,

*p*= 0.56, meaning that there was little asymmetry between the two “different” responses (but see later the more detailed analysis using the methods from Petrov, 2009).

*F*(2, 102) = 1.62,

*p*= 0.20—were statistically significant, with the largest

*p*= 0.02,

*F*(2, 102) = 4.17, for the two-way Task × Angular Difference interaction. This means that the task difference increased with the angular difference for cardinal and oblique directions alike.

*d*′. The differencing model is suboptimal with two fixed stimuli but optimal when the average of the two stimuli changes from trial to trial in a roving experiment (Dai, Versfeld, & Green, 1996). We performed an ANOVA as we did for the independent-model analysis, and found that the only difference from the previous ANOVA was the three-way interaction. Using the differencing model, this effect was significant:

*F*(22, 1122) = 1.63,

*p*= 0.033, as compared to

*F*(22, 1122) = 1.06,

*p*= 0.39, in the independent model. The main message that the task difference became substantial with larger angular differences, however, remained robust. We should also mention that, using the differencing model, the mean

*d*′ was 2.21, which was significantly greater than the 1.80 in the independent model. This confirms that the optimal decision rule gives rise to a smaller estimate of

*d*′ than a suboptimal rule.

*Z*scores. Since the 2AFC accuracy was greater than the same–different accuracy, this correction was expected to be more frequent for the 2AFC task. In order to avoid the corrections that were arbitrary and asymmetric between the two tasks, rather than calculating for each of the 12 directions we recalculated the hit and false-alarm rates for two directions only: oblique and cardinal. This new calculation turned out to be sufficient to avoid any corrections. We then repeated the ANOVA with motion direction, task, and angular difference as the main factors. The results were qualitatively very similar as before. Namely, all effects except one were significant, with the largest significance being

*p*= 0.001. The exception was the Angular Difference × Task interaction,

*F*(2, 102) = 2.13,

*p*= 0.12. At 4°, the

*d*′ estimates were 2AFC

*d*′ = 0.63, same–different

*d*′ = 0.97,

*t*(34) = 2.61,

*p*= 0.013. At 8° and 12°, the

*d*′ estimates were more different: 2AFC

*d*′ = 1.63, same–different

*d*′ = 2.22,

*t*(70) = 4.80,

*p*= 0.000009. Therefore, although the interaction effect was only marginal, we maintain that the analysis was reasonably consistent with the rest of the analyses.

*Z*(

*H*) and

*Z*(

*F*) were calculated because the bias-free proportion correct was needed in order to calculate

*d*′. Here, as an approximation, we used the actual proportion correct to calculate

*d*′ in order to avoid the corrections when

*H*= 1 or

*F*= 0. An ANOVA was performed similarly as when we used the independent model with 12 directions. Completely consistent results were obtained. Namely, all effects except the three-way interaction were significant. In particular, the Task × Angular Difference interaction was significant,

*F*(2, 102) = 3.10,

*p*< 0.05.

*P*(“different”|<

*S*

_{1}

*S*

_{1}>) =

*P*(“different”|<

*S*

_{2}

*S*

_{2}>) and whether or not

*P*(“different”|<

*S*

_{1}

*S*

_{2}>) =

*P*(“different”|<

*S*

_{2}

*S*

_{1}>). Whenever possible, we also calculated the corresponding

*d*′ values.

*d*′ estimated from a suboptimal strategy is necessarily greater than that from the optimal model, when the same behavioral data are used. This is because a task has to be easier (with a greater

*d*′) in order to compensate for the inefficient suboptimal strategy. Given that our main result so far was that for larger angular differences, the same–different

*d*′ estimated from the optimal model was greater already than the 2AFC

*d*′, then the even greater same–different

*d*′ estimated from a suboptimal strategy would not change the result. In the interest of space, we skip the details but note that, for the 4° angular difference, the same–different

*d*′ values re-estimated from the suboptimal strategies were still statistically comparable to the 2AFC

*d*′.

*Z*(

*H*) and

*Z*(

*F*) such that arbitrary correction was needed.

*d*′ was calculated where the optimal independent model was used for the same–different task. We first verified that there was no significant difference between the proportion of

*P*(“different”|<

*S*

_{1}

*S*

_{2}>) and

*P*(“different”|<

*S*

_{2}

*S*

_{1}>) responses for each of the three angular differences. The smallest

*p*value was

*p*= 0.19,

*t*(19) = 1.37, for the 8° angular difference. This confirmed that the independent model was applicable.

*F*(2, 54) = 55.40,

*p*= 8.27 × 10

^{−14}, as expected (4°:

*d*′ = 0.46; 8°:

*d*′ = 1.16; 12°:

*d*′ = 2.19). The main effect of task was also significant,

*F*(1, 54) = 5.29,

*p*= 0.025. In fact, the

*d*′ values (same–different:

*d*′ = 1.42; 2AFC:

*d*′ = 1.11) were comparable to those in Experiment 1 averaged along the eight oblique directions (

*d*′ = 1.69 and 1.15). The interaction effect was marginally significant,

*F*(2, 54) = 3.12,

*p*= 0.052. That is to say, when the angular difference was 4°, the

*d*′ values of the two tasks were comparable (same–different:

*d*′ = 0.43; 2AFC:

*d*′ = 0.49). When the angular differences were greater, the

*d*′ differences between the two tasks became larger (8°: 1.28 and 1.04; 12°: 2.56 and 1.81). These numbers were again consistent with the

*d*′ values along oblique directions in Experiment 1 (Figure 6).

**Figure 6**

**Figure 6**

*F*(2, 54) = 66.79,

*p*= 2.50 × 10

^{−15}. The main effect of task was marginally significant,

*F*(1, 54) = 3.78,

*p*= 0.057. Interestingly, however, the sign of the task difference was reversed from the first half. That is, while in the first half the same–different

*d*′ (1.42) was greater than the 2AFC

*d*′ (1.11), in the second half the two

*d*′ values became 1.16 (same–different) and 1.41 (2AFC). Another way to describe the results is that the participants who ran the same–different task first and 2AFC second had

*d*′ values of 1.42 and 1.41, whereas those participants in the opposite task order had

*d*′ values of 1.11 (2AFC) and 1.16 (same–different). This indicates that, when running the second task, the participants may have carried over their strategy from the first task, such that the recovered

*d*′ values of the two tasks were comparable within subjects. It is also interesting to note that this carryover effect from one task to the next was also between directions that were 90° away. Due to this carryover effect, the first participant group (task order: same–different, 2AFC) had a greater overall

*d*′ than the second group when the two tasks were combined,

*t*(38) = 3.49,

*p*= 0.001 (Figure 7).

**Figure 7**

**Figure 7**

*F*(2, 108) = 121.20,

*p*= 2.50 ×10

^{−28}, as expected. The Task × Task Order interaction was significant,

*F*(1, 108) = 9.03,

*p*= 0.003. The three-way interaction was also significant,

*F*(2, 108) = 3.83,

*p*= 0.025. All the remaining effects were nonsignificant, with

*F*values smaller than 1.

*d*′ was greater than when it was the first task run by the other half of participants:

*d*′ = 2.44 versus 1.81,

*t*(18) = 2.49,

*p*< 0.025. This enhanced performance did not seem simply due to a sequential learning effect, because a similar comparison for the same–different task gave rise to the opposite effect. That is, the same–different task as second gave rise to a

*d*′ smaller than as first:

*d*′ = 1.94 versus 2.56,

*t*(18) = 2.36,

*p*< 0.03. In order to take into consideration that the same–different

*d*′ depends on the underlying decision rules (Petrov, 2009), we looked at the accuracies of the same–different task that are unchanged for all the assumed decision rules. We found that the same–different task as first was 0.81 in proportion correct, as compared to 0.72 as the second task,

*t*(18) = 2.23,

*p*< 0.05. This confirms that the effects were not simply due to the sequential learning effect. A similar comparison for the 2AFC task yielded 0.88 versus 0.94,

*t*(18) = 1.89,

*p*= 0.07.

*d*′ using the optimal model or in proportion correct, was worse as the second task than as the first task, we wondered whether there was any difference in the decision rules used by the participants who ran the same–different task as the second task compared to the other half, who ran it as the first task. We again used the methods of Petrov (2009) to categorize the decision rules the participants used, and found that the majority of the participants were in two categories of suboptimal decision rules, for both task orders (28/30 and 27/30). However, in terms of the number of participants in each of these two categories, there was hardly any difference between task order—Category 1: 15 (same–different as Task 1) and 12 (same–different as Task 2); Category 2: 13 and 15. Unfortunately, therefore, using the methods of Petrov (2009) to narrow down the possible same–different strategies used by the participants did not help answer the question of what strategies participants might have used in the same–different task after performing the 2AFC task compared to before the 2AFC task. It is also an open question exactly how the first task influenced the second task such that the second task's

*d*′ became comparable to the first's.

*d*′ values for both tasks would depend less on the task order. Experiment 3 tested this prediction. Given that the

*d*′ difference was biggest between the two task orders for the 12° angular difference, we focused on this angular difference in this experiment. This experiment was otherwise identical to Experiment 2, with 24 fresh participants similarly recruited.

*d*′ values, which were calculated identically as in Experiment 2, using the optimal model for the same–different task. A two-way ANOVA with task and task order as the main factors showed that there was a significant main effect of task: 2AFC

*d*′ = 1.91, same–different

*d*′ = 2.28,

*F*(1, 22) = 7.75,

*p*= 0.011. The main effect of task order was not significant,

*F*(1, 22) = 1.06,

*p*= 0.32. The interaction was not significant either,

*F*(1, 22) = 1.44,

*p*= 0.24.

**Figure 8**

**Figure 8**

*d*′ was numerically greater than when it was first:

*d*′ = 2.13 vs. 1.70,

*t*(22) = 1.72,

*p*= 0.05 (one-tailed). A similar comparison for the same–different task did not reach significance. However, since this difference was only marginally significant using a one-tailed

*t*test, and since the interaction was nonsignificant, the implication of the asymmetric transfer requires further study.

*d*′ values that were computed from same–different and 2AFC or yes–no tasks. In most of these studies, the same–different

*d*′ differed from the yes–no or 2AFC

*d*′ by only −14% to 3%, suggesting reasonable applicability of the standard model of SDT (Chen & Macmillan, 1990; Creelman & Macmillan, 1979; Hautus & Irwin, 1995; Macmillan, Goldberg, & Braida, 1988). However, in an auditory study, Creelman and Macmillan (1979) found that their same–different

*d*′ was 50% higher, where the underlying model failed (see also Taylor, Forbes, & Creelman, 1983). In visual motion perception, the applicability of the standard model of SDT has been little studied, as far as we know.

*d*′ might have been calculated incorrectly in some studies of visual motion perception. For example, Ball and Sekuler (1987) calculated their same–different

*d*′ in their pioneering study of motion-discrimination learning as follows: “Hit rates and false alarm rates for identifying ‘different' trials in each block were converted by standard methods into

*d*′, to provide a measure of discrimination performance (Green and Swets, 1966)” (p. 954). To our knowledge, nevertheless, Green and Swets (1966) did not discuss the same–different design or how to calculate

*d*′ in that design. According to R. Sekuler (personal communication, February 2016), the

*d*′ values for Ball and Sekuler (1987) may have been calculated by defining

*d*′ =

*Z*(hit) −

*Z*(false-alarm), where hit =

*P*(“different”|<

*S*

_{1}

*S*

_{2}> or <

*S*

_{2}

*S*

_{1}>) and false-alarm =

*P*(“different”|<

*S*

_{1}

*S*

_{1}> or <

*S*

_{2}

*S*

_{2}>). To be fair, in some of the studies by Liu (Liu, 1999; Liu & Weinshall, 2000),

*d*′ was incorrectly calculated as

*d*′ =

*Z*(hit) −

*Z*(false-alarm).

*d*′ was the same as the 2AFC

*d*′ when the underlying stimuli were the same. We also tested our hypothesis with weak signals (angular difference = 4°), when the standard model was expected to work (Macmillan & Creelman, 2005), and with stronger signals (8° and 12°), when it was unknown whether the standard model would work or not.

*d*′ estimates. However, when the angular differences were 8° and 12°, the recovered

*d*′ values from the same–different task were greater than those from the 2AFC task. Here the same–different

*d*′ values were recovered using the optimal decision rule. If suboptimal decision rules were used instead, the recovered same–different

*d*′ values were numerically even greater. Hence, the discrepancy between the same–different

*d*′ and the 2AFC

*d*′ was even greater for 8° and 12° (for 4°, the consistency between the two tasks remained). That is to say, for stronger signals, the standard model of SDT could not apply no matter which decision rule, optimal or otherwise, was used.

*d*′ values with stronger signals is that the same–different ROC in the Gaussian space could no longer be approximated by a 45° straight line. This possibility can be verified experimentally, using a rating experiment in a same–different task. We will investigate this possibility in the future.

*d*′ can no longer be estimated as shown earlier in this article, the participants still likely performed the two tasks with different

*d*′ values. This is because, as shown in Experiment 2, the same

*d*′ recovery method (the optimal model) for the same–different task gave rise to very different results for 8° and 12° depending on the order of the two tasks. This means that the underlying computations used in these two tasks were different when they were not run one immediately after another. When the two tasks were run one immediately after another, carryover occurred such that the second task's

*d*′ was comparable to the first task's. This result is intriguing, because it suggests that these comparable within-subject

*d*′ values may not be coincidental. It further indicates that the same–different

*d*′ as recovered by the optimal model per Macmillan and Creelman (2005) may be reasonably accurate, and that the same–different ROC might not be severely nonlinear after all. This is another motivation for us to carry out an empirical study on the linearity of the same–different ROC in the future.

*d*′ than the sequence <2AFC, same–different>. Recall that in the 2AFC task, the two motion directions were symmetric about either 45° or 135°. This means that the two probability distributions corresponding to the two directions in the standard SDT model were likely to be identical in shape and differed only in their respective means. If one makes an additional and reasonable assumption that both distributions are Gaussians, then

*d*′ is definable. This means that the 2AFC task may have a definable

*d*′ (and not just ) that is also straightforward to calculate. Therefore, the 2AFC

*d*′ values calculated in the current study are likely to be reasonably accurate.

*d*′ values were indeed accurate. Then—as shown in Experiment 2 for the 8° and 12° angular differences—when the 2AFC task was run first, the average

*d*′ was 1.42. However, when the 2AFC task was run after the same–different task,

*d*′ = 1.91, probably because the preceding same–different task gave rise to a

*d*′ = 1.92. This was a remarkable result in that the 2AFC

*d*′ increased by 35% when it immediately followed the same–different task, as compared to following no task at all. This suggests that in the original 2AFC task, there was internal noise that could be substantially reduced after only 360 trials of the same–different task. This type of learning appears different from the motion perceptual learning that exhibits gradual improvement through days of training (Epstein, 1967; Fahle & Poggio, 2002; Gibson, 1969; Sagi, 2011; Sasaki, Nanez, & Watanabe, 2009). This difference from traditional perceptual learning is further illustrated when we look at the opposite task order. When the same–different task followed the 2AFC task, the same–different

*d*′ = 1.57, which was close to the preceding 2AFC task's

*d*′ = 1.42. As a second task, this same–different

*d*′ = 1.57 was lower than when it was the first task, with

*d*′ = 1.92.

*d*′ values were comparable.

*d*′ was substantially greater as the second task than the first task.

*d*′ under a number of suboptimal decision rules. We also thank Dr. Bob Sekuler for his kind correspondence regarding the same–different

*d*′ calculations used by Ball and Sekuler (1987). This research was supported in part by the Natural NSF of China to ZL (NSFC 31228009) and to YZ (NSFC 31230032). This research was also supported by a fellowship to ZL at the Hanse Institute for Advanced Study, Delmenhorst, Germany.

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