Addendum to :

Hupé, J.M., & Rubin, N. (2003) The dynamics of bi-stable alternation in ambiguous motion displays: a fresh look at plaids. Vision Research 43: 531 - 548.

    RTtransp is a new and reliable measure of the probability of coherence in plaids. Since each measure is relatively fast to obtain (as long as the parameters are chosen so the probability of seeing the transparent percept is not too low), it is possible to test many parameters simultaneously within one experiment, using full-factorial designs. Such designs have also the methodological advantage of presenting to the observer many different stimulus configurations, so the observer cannot guess what the effects of the manipulated variables are while running the experiment (as long as the stimuli are presented in a random order, of course). Linear analyses are a powerful tool to study the effects of many variables. But their reliability depend on two major assumptions: that the noise in the data be normally distributed, and that the variances be homogenously distributed. RTtransp values have a log-normal distribution, so transforming them to their natural log allows us to use linear model analyses (see paper). The homogeneity of variances assumption can be checked in several ways. When doing a simple ANOVA, one can compare the variances of the different groups, and test if they are significantly different from each other. However, these tests do not indicate which group(s) is/are significantly different from the others, and we can not use them when several variables are tested, some of them being linear predictors (or covariates). A convenient way of visualizing the homogeneity of variances is to plot the residuals as a function of the values predicted by the linear model :
 

           Figure 1

    In the case presented in Figure 1, there is no obvious reason to suppose that the variances are different in the four different groups – the visual inspection can be confirmed here by doing a test of homogeneity of variances (p = 0.60, Hartley, Cochran, Bartlett). Residual plots allow us to check easily if there is any systematic relationship between the effects of parameters and variance. For instance, if we had not transformed the data to their natural log for the data presented in Figure 1, we would have observed that the variance systematicaly increases for larger predicted values of RTtransp :
 

        Figure 2

    Inspection of residuals plots allows us therefore to verify easily to what extent the conditions of validity of the linear model analysis were respected. Moreover, residuals plots can be generated whatever the complexity of the linear model – while in that case it does not allow to verify that the variances are homogenous between any pair of groups, it makes possible to detect any systematic bias in the data, as we can now observe for the analysis of the data of Experiment II presented in the paper.
 

        Figure 3

    The distribution of residuals after the ANCOVA of Exp. II is shown in Figure 3, and the inhomogeneity is clearly discernible: there is a shallow oblique cloud of higher density on the left side of the distribution. This cloud reflects the fact that as the predicted RTtransp value becomes smaller and smaller (moving left along the x-axis), the residuals tend to cluster at higher and higher values (up along the y-axis). In other words, as the predicted values become smaller, the errors, or discrepancies between the predicted values and the observed ones, become systematically positive. This is a clear indication of a floor effect, which is reasonable to expect given that response times cannot go below some lower limit due to processing and motor limitations (though what this lower limit is may vary between observers). Indeed, if we mark the points for which RTtransp had the lowest values (110-230 ms, bold dots in Figure 3), they lie below an oblique line which is roughly parallel to the dense cloud above it. (The exact orientation of the line drawn was chosen arbitrarily. We hypothesized that these very short RTtransp values were due to anticipatory responses or other response errors, and therefore excluded them from the analysis. This raised the total number of outliers in the analysis to 25, since points outside the two horizontal lines, i.e., for which the z-scores were below –3.5 or above 3.5, were also considered as outliers.) To validate this explanation of the floor effect further, we examined the distribution of residuals for observer 06, who had long response time (see Figure 8 of the paper), and further restricted the set to the four cardinal directions. The distribution of the residuals is shown in Figure 4: the variances are clearly homogeneously distributed.
 

   Figure 4

    The inhomogeneity of the distribution of the residuals in Figure 3 indicates that, even after removal of the outliers, the results of the ANCOVA are not exact. When coming to test small interaction effects, one therefore needs to be careful. A possible strategy is to restrict the analysis to subsets of the data for which there is no floor effect, as illustrated in Figure 4. This was done by selecting an observer with long response times. For this observer, the effect of manipulating alpha on Ln(RTtransp) was almost perfectly linear, in contrast to other observers. We argued in the paper that the linear effect of alpha was in fact the "true" effect of alpha, and that deviations from linearity were due to a floor effect. Control experiments in the "Experiment III’ part of the paper could confirm this hypothesis. Here we can show that within the set of data of Experiment II this linear relationship can be observed for each observer. We restricted the analysis to the 7 naive observers and to the 3 values of alpha which were used for all these observers. When we further restrict the set of data to parameters which lead to larger RTtransp values (component speed below 1.1 deg/sec, horizontal directions), the effect of alpha is almost linear for each observer (Figure 5), and the average curve is linear (solid line in Figure 6; the second order coefficient of the best fitting quadratic function is close to zero). On the other hand, when restricting the set of data to parameters which lead to smaller RTtransp values (component speed above 1.1 deg/sec, oblique directions), the second order coefficient of the best fitting quadratic function is not negligible (dashed line in Figure 6). The inspection of residuals for the high-RTtransp set of data confirms the relationship between floor effect, non-homogeneity of variances and observed non-linearity of alpha: the residuals presented in Figure 7 are homogenously distributed.
 

        Figure 5          Figure 6
 
 

        Figure 7