On Response Bias in the Face Congruency Effect for Internal and External Features

Some years ago Cheung et al. (2008) proposed the complete design (CD) for measuring the failure of selective attention in composite objects. Since the CD is a fully balanced design, analysis of response bias may reveal potential effects of the experimental manipulation, the stimulus material, and/or attributes of the observers. Here we used the CD to prove whether external features modulate perception of internal features with the context congruency paradigm (Nachson et al., 1995; Meinhardt-Injac et al., 2010) in a larger sample of N = 303 subjects. We found a large congruency effect (Cohen's d = 1.78), which was attenuated by face inversion (d = 1.32). The congruency relation also strongly modulated response bias. In incongruent trials the proportion of “different” responses was much larger than in congruent trials (d = 0.79), which was again attenuated by face inversion (d = 0.43). Because in incongruent trials the wholes formed by integrating external and internal features are always different, while in congruent trials same and different wholes occur with the same frequency, a congruency related bias effect is expected from holistic integration. Our results suggest two behavioral markers of holistic processing in the context congruency paradigm: a performance advantage in congruent compared to incongruent trials, and a tendency toward more “different” responses in incongruent, compared to congruent trials. Since the results for both markers differed only quantitatively in upright and inverted presentation, our findings indicate no change of the face processing mode by picture plane rotation. A potential transfer to the composite face paradigm is discussed.

Here, z D is the standard quantile of the criterion k relative to f (x|D), and z S is the standard quantile of k relative to f (x|S). Since z(p) = Φ −1 (p), with Φ −1 the inverse distribution function (quantile function) of the standard normal distribution, (1) is given by Using the symmetry property z(1 − p) = −z(p), (3) is often written as d = z(Hit) − z(F A).
Now, verify that standardization of x with respect to f (x|D) maps µ D → 0 and µ S → d , i.e.
The standardization z = (x − µ D )/σ may be shifted to a new origin, chosen as half the standardized distance of means, d : This scale is chosen to express the response criterion k on a transformed standard axis: Recall that z D = z(CR) = −z(F A). Then, by using (3), (4) reads (see MacMillan & Creelman, 2005, p. 29). On this scale, positive values of c mean that the response criterion is closer to µ S , and negative values mean that it is closer to µ D . The means transform z (µ D ) = −d /2, and z (µ S ) = d /2, respectively (see Fig. 1)  Figure 1. Likelihood functions f (x|D), f (x|S) as normal probability density functions with equal variance σ 2 , decision criterion k, and corresponding probabilities of "false alarm" (P ("same"|D)) and "miss" (P ("different"|S)) events resulting from the position of the decision criterion k on the latent sensory continuum x. The lower continuum represents a transformed standard axis with d /2 as the new origin. Expressed on this axis, positive values of the transformed decision criterion, c, correspond to more frequent "different" than "same" judgements, a bias towards the "different" response category (see arrow).

APPENDIX B: THE RESPONSE CRITERION C FOR MAXIMIZING PROPORTION CORRECT
Using a maximum a-posteriori probability decision rule for the states D and S, an observer responds "S" for a given value z of the standardized sensory continuum, if P (S|z) > P (D|z). Since the a-posteriori probabilities P (S|z) and P (D|z) are given as Bayes probabilities, this decision rule reads and "D" otherwise (see Green & Swets, 1966, chapter 1). After dividing both fractions, and "D" otherwise. Comparing (6) to (7) shows that the maximum Bayes probability decision rule is equivalent to a likelihood ratio decision rule, with LR SD (z) the likelihood ratio of the two likelihoods for z given S, and D, respectively. For convenience, one may set ϑ = P (D)/P (S) for the right hand side of (7). Assuming the likelihood functions to be Gaussians, the likelihood ratio becomes Considering the likelihood ratio at the response criterion z = c where LR SD (c) = ϑ, we take the logarithm in (8). This resolves to LR SD (c) = ln (ϑ) = cd Hence c = ln(ϑ) d .
Now, if we set P (S) = 0.25, P (D) = 0.75 and assume d = 2.5, inserting in (10) gives a value of c = 0.44. This means that the observer who uses a maximum a-posteriori probability decision rule would set the response criterion to a positive value on the standardized continuum of sensory states if D trials are more likely than S trials. This means that we should observe a "different" bias as a result of the attempt to compensate for a larger perceived likelihood of D trials compared to S trials.

APPENDIX C: SAMPLE SIZE CALCULATION FOR CE AND CB MEASURES
Because the CE and the CB are defined on individual differences obtained in congruent and incongruent conditions we performed a power calculation for a dependent sample (paired) ttest, using the power calculation module of STATISTICA 13.0. For calculating the critical sample size a power goal of 0.95 was chosen. The α level for testing the null hypothesis was set to α = 0.05. Figure 2 shows the critical sample size as a function of effect size d.  Table 1 of this appendix shows separate results for the four face identities. Because the average number of replications for each identity in "same" and "different" trials, n = 4, was too small to allow for a sound calculation of relative frequencies for each subject, the frequency data for Hit and CR were counted across subjects, and the rates were calculated afterwards. Signal detection measures were computed from these rates.