OVERALL RESULTS BY GLOBAL STATISTICS: NO OBSERVED INFLUENCE OF VISUAL AWARENESS IN REGRESSION EFFECTS

Figure 15b shows the distribution of responses by orientation and condition: in condition 1, the mean orientation of unmasked and masked Gabors is -24o _{and 24}o_,

respectively; in condition 2 it is the opposite. As suggested by visual inspection of the plots, we found no evidence for an effect of the global statistics of visible Gabors on responses. To formally ascertain this, we selected the orientations common to both distributions, D1 and D2 (see ‘Methods: Experimental blocks’ section), i.e., -12o_{, 0}o _and

12o_{, and ran a Bayesian ANCOVA with response as dependent variable, orientation as}

covariate, condition as fixed factor and participant’s ID as random factor. There was strong evidence against any effect on ‘condition’ on responses by orientation, in all trials pooled as well as when analysing responses to unmasked trials separately (BF10<0.100

in both cases); when selecting masked trials only, evidence was neither for nor against an effect of ‘condition’ (BF10=0.946).

In summary, according to our analyses, influence of global statistics on responses (regression effect) was not mediated by Gabor visibility. This may suggest that high level processing is not necessary to generate a global prior. Nevertheless, it is also possible that the absence of observable effect was due to our experimental design. It is very likely

that the length of the blocks (84 Gabors, 42 visible) was insufficient to ‘learn’ a prior for the global statistics (at least until very late in the trial sequence) -in fact, it may even be the case that during the first part of the block participants exhibited regression effects toward the condition learnt in the previous block. Another possible issue is that the linear response scale, centred on the global mean (0o_{), might have produced a strong}

regression effect toward 0o _{that prevailed over other global effects.}

2.4. SERIAL DEPENDENCE IN ORIENTATION: REPORTS ARE ATTRACTED TO N-1 GABOR ORIENTATION

For ascertaining serial dependence, we selected current trials with unmasked (visible) Gabors and analysed whether the reported orientation of the stimulus was influenced by previous history. Previous studies on serial dependence in orientation have described an attractive bias exerted by previous stimuli, whose effect size depends on the similarity between previous and current stimulus magnitudes (Cicchini et al., 2017; Fischer & Whitney, 2014; Fritsche et al., 2017). This relationship is similar, but of opposite sign, to a well-known property of negative adaptation (for example, the tilt after-effect, TAE (Gibson & Radner, 1937; Magnussen & Kurtenbach, 1980)). In both cases, the size of the effect is maximal when the difference between previous and current stimulus is relatively small, and declines or disappears for larger differences (or even reverts for very large differences (Fritsche et al., 2017; Gibson & Radner, 1937; Wenderoth & Johnstone, 1988)). In our experiments on variance, we also found such relationship with stimulus similarity, although results were compromised due to floor and ceiling effects given the particularities of the experimental design (see Chapter 1, section 2.2.2).

Considering all this, we explored serial dependence in our data by analysing response errors in function of stimulus dissimilarity. Response errors were calculated as REn=Rn-

ORn, where Rn is the current response and ORn is the orientation of the Gabor in the

current trial. In our experiment, convention has it that clockwise (CW) orientations have negative sign and counter-clockwise (XCW) orientations have a positive sign. Therefore, a positive REn indicates that the reported orientation was more XCW than veridical, and

a negative REn indicates the that the report was more CW.

Figure 15c presents the average REn (and between-participants standard error) plotted

by orientation difference between previous and current Gabor: DORn-1,n=ORn-1-ORn. The

current Gabor is always unmasked, but the previous trial could be unmasked or masked -all cases are pooled together in Figure 15c. A positive DORn-1,n indicates that the

previous Gabor orientation was more XCW compared to the current one, while a negative DORn-1,n indicates that the previous one was tilted more CW. Therefore, an

attractive bias (a pull exerted on the current response by the previous trial Gabor) would be represented by REn reports, on average, of the same sign as DORn-1,n: a more XCW

Gabor in recent history would determine a XCW bias on orientation estimation, and vice versa. A repulsive bias (such as negative adaptation effects) would translate into REn of

opposite sign as DORn-1,n.

As mentioned before (Chapter 1, section 2.2.2), the non-linear relationship between stimulus (dis)similarity and effect size of the induced bias has been characterized by a derivative of Gaussian function (DoG), both for negative adaptation (Heron et al., 2012) and (positive) serial dependence (Fischer & Whitney, 2014; Fritsche et al., 2017). We therefore followed previous studies and fitted our data to a DoG function of the shape:

(1)

𝑦 = 𝑎𝑥 √2𝜋𝜎-𝑒

/01

231

where x is the difference between previous and current stimulus (DORn-1,n), y is the

representing the amplitude and the width (standard deviation) of the function, respectively. Fitting was achieved by non-linear least-squares method.

The best-fitting function is shown in Figure 15c along with the experimental data and the 95% prediction intervals for the value of the function computed non-simultaneously at each DORn-1,n. The coefficient estimates and 95% confidence intervals for the function

are: a=6477 (1586 – 11370), s=29.48o _{(18.28 – 40.67). In other words, the amplitude of}

the serial dependence effect exerted by trial n-1 was maximal when the difference between the orientation of the previous and current trial was 29.48o_{. For that value of}

DORn-1,n, the bias on the current response was 1.80o toward the previous trial

orientation -an attractive bias, as indicated by the positive sign of the amplitude a coefficient. The fact that the 95% confidence intervals for a did not contain zero indicates that the effect was statistically significant. This is also illustrated on Figure 15c by the 95% prediction intervals for the value of the function, which show a clear attractive effect even at the prediction boundary that is closest to the horizontal axis. The width of the function (s) is of similar magnitude as that reported by Fischer and Whitney (27.78o_{) (Fischer & Whitney, 2014); however the amplitude of the effect is}

much lower in our case (1.80o _{versus 8.19}o_{). This could be related to our employment of}

monocular stimuli, half of them masked by CFS; nevertheless, effect sizes reported by Fritsche and colleagues are similar to our own -in fact slightly lower (1.15o_{), but more}

narrowly tuned (s=17o_{) (Fritsche et al., 2017).}

In order to remove the influence of systematic individual biases, we repeated the analysis with response errors normalized within each participant and current orientation (zREn). A positive zREn indicates that the report was more XCW,

comparatively, than other reports given by the same participant for the same (unmasked) Gabor orientation; a negative zREn indicates a comparatively more CW

estimate, compared with others given by the same participant for the same stimulus. Thus, normalization explores response biases by attending only to trial-by-trial variability within participant and stimulus. As with absolute response errors, an

attractive bias by previous orientation would be reflected by DORn-1,n and zREn of the

same sign: a more CW Gabor in trial n-1 would produce a more CW report for trial n Gabor, compared with other reports provided by the same individual and for the same veridical orientation, and vice versa.

Figure 15d shows the distribution of average zREn (and between-participant standard

errors) as a function of DORn-1,n. The best-fitting DoG function and 95% prediction

intervals are also depicted. Its coefficient estimates and 95% confidence intervals are: a=218 (75.15 – 360.8), s=19.78o _{(13.29 – 26.27), representing a peak attractive effect of}

0.1348 z-scores toward the n-1 Gabor orientation, when the difference between this and the current Gabor orientation was 19.78o_{. Again, the positive sign of a coefficient}

and its confidence intervals demonstrate a significant attractive bias. The width of the function is narrower than for unnormalized reports.

For trial n-2 position, there was no longer evidence for an attractive bias: the amplitude of the effect of the best-fitting derivative of Gaussian function (fitted on all current unmasked trials, with both unmasked and masked trials at n-2 position, and by using normalized response errors, zREn as dependent variable) was a =5668 (-7.069*105 –

7.182*105_{), with a width of s = 196.7}o _{(0 – 8741}o_{). The confidence intervals for a}

indicated that even at its peak, the effect was non-significantly different than zero.

2.5. SERIAL DEPENDENCE BY PREVIOUS GABOR VISIBILITY: ONLY UNMASKED (VISIBLE)