General Discussion of Experiments 5 and 6

Experiments 5 and 6 clearly show that the distortional fields of single land-marks do not combine linearly. To show this, two linear models were tested, one su-perposition model stating that distortional fields of two landmarks simply add, and a more general two-parameter linear model that allows for weighting of single-landmark fields with some constants. The major weaknesses of the superposition model are that it overestimates the amount of distortion in most areas of the display, that it pre-dicts distortion where none occurs, and that it prepre-dicts a bias towards the center po-sition between the landmarks rather than away from it. Weighting the individual fields before summing cannot fix any of these problems because the distortions near a landmark are the same irrespective of whether or not a second landmark is present, while distortions farther away from single landmarks follow different patterns. There-fore, any single-landmark field needs a weight close to 1 in the half containing the landmark but a weight close to 0 in the opposite half. This makes plausible why fitting a linear model with both weights as free parameters leads to a very unsatisfactory averaging model which largely underestimates the distortional effects everywhere in the display. Therefore, it can not only be concluded that distortional fields fail to com-bine as a superposition system - they do not form any other kind of linear system ei-ther.

In marked contrast to the linear models, a simple partition model without any free parameters fits the data surprisingly well. An additional advantage of the partition

73 model over the linear models is that variance predictions follow very naturally from it while the linear models yield variance predictions only under a set of strong assump-tions. One of these assumption is that landmarks are the only sources of variance in the distortional field, which is proved wrong by Experiments 1 and 2 of this thesis:

landmarks serve to diminish positional uncertainty, not to generate it.

However, the partition model does not consistently predict distortion away from the midpoint; in addition, it overestimates variable error at the partition bound-ary. Indeed, reproductions of targets located directly on the midpoint show very little scatter, seemingly forming a region of very low positional uncertainty. This strongly suggests that the distortion near the midpoint might arise from strategies of encoding the mid-position as an additional spatial reference or anchor point. The midpoint would then function like a virtual landmark, creating its own pattern of distortion. This concept of virtual landmarks is in line with a psychophysical model by Hollands and Dyre (2000) showing that when observers strategically code stimuli relative to (vir-tual) anchoring points on a psychophysical continuum, this creates a cyclical pattern of biases similar to the one observed here. Moreover, Bryant and Subbiah (1993) have shown that strategic differences have large effects on spatial memory distor-tions.

It is therefore important that adding a landmark to an existing configuration gives rise to geometrical properties that were not present before. When a second landmark is provided, not only the landmark itself can be used as a spatial reference, but also the virtual midpoint between the landmarks, the virtual line connecting them, and perhaps additional geometrical cues constructed in short-term memory. The po-tential relevance of such geometrical auxiliaries is nicely illustrated by Psotka (1978).

Participants had the simple task of arbitrarily placing a single dot anywhere in an out-line figure presented on a sheet of paper. Surprisingly, with large groups of subjects, very systematic patterns emerged depending on the geometry of the outline figure:

people tended to place their dots on imaginary lines, e.g., connecting corners, lines connecting midpoints, and the perpendiculars to these lines. These findings are in general agreement with our data showing minimal departures from the virtual line connecting two landmarks and from the perpendicular one halving this line. It is also in agreement with Experiments 7 to 9 of this thesis using configurations of three landmarks.

The results reported here are consistent with attentional models like Suzuki and Cavanagh's (1997) which assume that distortional effects are strictly local (see General Discussion for a more detailed treatment). They are not predicted by either the Nelson and Chaiklin (1980) model nor the Huttenlocher et al. (1991) model. Al-though Nelson and Chaiklin's model correctly predicts that distortional fields combine after being weighted with a nonlinear function that declines with increasing distance from the landmark, it does not recognize that this weighting function must be a sym-metrical step function, leading to a partitioning of visual space. Worse, the model in-correctly assumes that spatial memory is biased towards a landmark, not away from it, and that the distortional effect becomes stronger with increasing distance from the landmark. With the weighting function approximating a simple step function, the de-crease in the weighting function would not be able to counteract this effect near the landmarks, so that the predictions of the model are clearly falsified by the data.

Although the Huttenlocher et al. (1991) model does not make any straightfor-ward predictions about how two landmarks combine, its prediction of spatial biases jointly depends on the geometry of the stimulus and the coordinate system used by the participants, which usually cannot be determined beforehand. Although partition-ing of visual space is not predicted by this model, it makes plausible why additional distortional effects are present near the midpoint between landmarks: adding a sec-ond landmark leads to new category boundary along the vertical midline and along the line connecting the landmarks, so that the model could easily account for distor-tion away from the midpoint. However, some of the main assumpdistor-tions of the model are inconsistent with the data. The model would explain repulsion from the midpoint by assuming a) that category membership of this point is unclear and b) that the lo-cation of the point itself is uncertain. Because a target might randomly be assigned to one or the other spatial category, memory would be unbiased in the long run but should be associated with high spatial uncertainty. The same reasoning holds for regions near the landmarks. In contrast, we find that the midpoint is a region of very low spatial uncertainty, which is just the opposite of what Huttenlocher et al.'s (1991) model predicts.

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