Visual curve tracing properties

Subjects decided whether 2 dots were on the same curve or 2 different curves, and the curvature of the curves or the proximity of other (distractor) curves to the target curve was varied. Response time increased as the arc length of the curve connecting the 2 dots increased, suggesting that the curve was traced to perform the task. Tracing rate was faster for low- than high-curvature contours and was increasingly slower as distractor contours were increasingly proximal to the traced curve. Proximity results were predicted by a model in which response time depends on the ratio of the distance between the dots and the distance between adjacent lines. Curve tracing operations used to integrate information along contours are sensitive to several properties of the contours. The implications of the sensitivity of tracing operations to these curve properties are discussed.     

Size invariance in visual number discrimination

Summary This study deals with the observer's ability to discriminate the numerosity of two random dot-patterns irrespective of their relative size. One of these two patterns was a reference one that was always composed of 32 dots randomly distributed within aK ×K invisible square window (K = 1.92°). The second one was the test pattern with one of the five magnifications (K = 0.64°, 1.28°, 1.92°, 2.56°, 3.20°) and the relative number of dots varied on 11 levels (N = –15, –12, –9, –6, –3, 0, 3, 6, 9, 12, or 15 dots). The observer's task was to indicate which of the two patterns contained more dots. The results show that the stimulus size, as an irrelevant stimulus attribute, can be ignored in the judgements about relative numerosity. This means that the perceived numerosity is size invariant, at least for a 1.6-times magnification and a 3-times reduction of the test pattern. The size invariance observed constrains the range of potential models, since the perceived numerosity can be identified only by means of a feature of the stimulus that will remain invariant after any change in the absolute stimulus size.     

Mental curve tracing with elementary stimuli

It has been proposed that certain spatial relations are determined by an operation, or "visual routine," that can trace along a boundary (Ullman, 1984). This proposal was supported by Jolicoeur, Ullman, and Mackay's (1986) finding that the time required to determine if two Xs are on the same curve increased monotonically with the separation of the Xs along that curve. In the present study the generality of the curve tracing hypothesis was explored across four experiments by using elementary stimuli that eliminated interweaving curves, displaced the fixation point away from the curves and target Xs, and provided a simple alternative to curve tracing--namely, determining whether or not the Xs fell on the same side of the figure. Stimuli consisted of two curves (150 degrees arcs) and two Xs, and each stimulus was presented for 150 ms. In Experiments 1 and 2, subjects were instructed to decide as quickly as possible if the two Xs fell on the same curve or on different curves. Even for these elementary stimuli, mean reaction time (RT) for same trials increased monotonically with the distance separating the Xs. Mean RT for different trials, however, decreased with the distance separating the Xs. In Experiments 3 and 4 alternatives to curve tracing were tested. For same trials the evidence strongly favored curve tracing. However, different trials were apparently solved on the basis of judgmental processes presumably operating in parallel with curve tracing. Curve tracing rates fluctuated across experiments and seemed to be partially governed by the width of the "pathway" provided for the trace.     

Size and position invariance in the visual system

The encoding of simple features such as lines and angles leads naturally to size and position invariance when these features are analyzed symbolically in terms of structure and connectedness. No such analysis is possible for a spatial frequency encoding, however, as all a priori indications of pattern unity are lost. An additional transformation of spatial frequency information is therefore demonstrated that automatically derives a size invariant representation of form. The feasibility of the transform scheme in the visual system is outlined with a hypothetical processing sequence. It is found that the organization of cells in the striate cortex shows a marked similarity to an intermediate stage of the proposed sequence.     

A gradual spread of attention during mental curve tracing

The visual system has to segregate objects that are relevant to behavior from other objects and the background, if they are embedded in a visual scene. This segregation process can be time consuming, especially if the relevant object is spatially extended and overlaps with other image components, but the cause of the delays is presently not well understood. In the present study, we used a curve-tracing task to investigate processing delays during the grouping of contour segments into elongated curves. Our results indicate that contour segments that need to be grouped together are labeled with visual attention. Attention gradually spreads from contour segments that were labeled previously to other contours that are colinear and connected to them. The contour-grouping task is completed as soon as attention is directed to the entire curve. We conclude that processing delays during contour grouping are caused by a time-consuming spread of visual attention.     

Predicting the shape of distance functions in curve tracing: Evidence for a zoom lens operator

Jolicoeur, Ullman, and MacKay (1986) showed that the time to confirm that two dots are on the same curve increases monotonically, but nonlinearly, as the curve distance between the two dots increases. These displays contained two curves and two dots. On same trials, the two dots were on the same curve (target curve), while the other curve served as a foil (distractor curve). The monotonically increasing effects of curve distance on response times for same trials suggested that the intervening curve segment was traced. In the present investigation of the source of the nonlinearity in these distance functions, it was hypothesized that differences in the distractor curves may have allowed a curve tracing operator with zoom lens properties to widen its receptive field while tracing parts of certain target curves. The wider receptive field may have allowed faster tracing over certain segments, owing to a reduced number of shifts required by the operator to scan the curve. The consequence of training certain segments of the curve more quickly than other segments of the curve would be a nonlinear effect of distance. A new set of stimuli was created for testing this hypothesis directly. Fairly linear distance effects were found for stimuli that contained a distractor curve that constrained the breadth of the postulated curve tracing operator, whereas stimuli that contained a distractor curve that could allow for a larger receptive field yielded nonlinear distance functions. The results are compared with the predictions of three quantitative models: pixel-by-pixel tracing; Jolicoeur, Ullman, and MacKay's (1991) bipartite operator; and a new zoom lens model, analogous to the zoom lens model of visual attention. The results were fit best by the latter model, in which tracing is accomplished by tracking the curve with a variably sized local operator.     

Size-distance invariance: Kinetic invariance is different from static invariance

The static form of the size-distance invariance hypothesis asserts that a given proximal stimulus size (visual angle) determines a unique and constant ratio of perceived-object size to perceived object distance. A proposed kinetic invariance hypothesis asserts that a changing proximal stimulus size (an expanding or contracting solid visual angle) produces a constant perceived size and a changing perceived distance such that the instantaneous ratio of perceived size to perceived distance is determined by the instantaneous value of visual angle. The kinetic invariance hypothesis requires a new concept, an operating constraint, to mediate between the proximal expansion or contraction pattern and the perception of rigid object motion in depth. As a consequence of the operating constraint, expansion and contraction patterns are automatically represented in consciousness as rigid objects. In certain static situations, the operation of this constraint produces the anomalous perceived-size-perceived-distance relations called the size-distance paradox.     

Size-distance invariance: kinetic invariance is different from static invariance.

The static form of the size-distance invariance hypothesis asserts that a given proximal stimulus size (visual angle) determines a unique and constant ratio of perceived object size to perceived object distance. A proposed kinetic invariance hypothesis asserts that a changing proximal stimulus size (an expanding or contracting solid visual angle) produces a constant perceived size and a changing perceived distance such that the instantaneous ratio of perceived size to perceived distance is determined by the instantaneous value of visual angle. The kinetic invariance hypothesis requires a new concept, an operating constraint, to mediate between the proximal expansion or contraction pattern and the perception of rigid object motion in depth. As a consequence of the operating constraint, expansion and contraction patterns are automatically represented in consciousness as rigid objects. In certain static situations, the operation of this constraint produces the anomalous perceived-size-perceived-distance relations called the size-distance paradox.     

Measurement invariance,factor analysis and factorial invariance

Several concepts are introduced and defined: measurement invariance, structural bias, weak measurement invariance, strong factorial invariance, and strict factorial invariance. It is shown that factorial invariance has implications for (weak) measurement invariance. Definitions of fairness in employment/admissions testing and salary equity are provided and it is argued that strict factorial invariance is required for fairness/equity to exist. Implications for item and test bias are developed and it is argued that item or test bias probably depends on the existence of latent variables that are irrelevant to the primary goal of test constructers.Presidential address delivered at the Annual Meeting of the Psychometric Society in Berkeley, California, June 18–20, 1993.     

Factorial invariance and significance

It is shown that invariance requirements remove the indeterminacy in factor determination and lead to an integration of factorial studies with promise of considerable reduction in computational labor. The selection of significant primary factors is discussed, with special reference to Thurstone's simple structure criterion.     

Fading-figure tracing in Williams syndrome

Williams syndrome (WS) is a neurodevelopmental disorder characterized by severe impairment of visuospatial abilities. Figure-drawing abilities, which are thought to reflect visuospatial abilities, have yet to be fully investigated in WS. The purpose of the present study was to clarify whether drawing abilities differ between WS individuals and typically developing children (TD). We compared the performance of two groups of subjects (WS, mean age 16 years; TD, 5-6 years of age) using a fading-figure tracing task that requires subjects to trace a target figure that is gradually disappearing from a PC screen. Although the TD group exhibited clearly improved performance with long fading time, the WS group did not. Moreover, the TD group exhibited poor performance for figures with more than six angles, regardless of the figure type (e.g. closed or open), whereas the WS group exhibited generally poor performance for figures with more than five angles but relatively preserved performance for open figures. These findings indicate that a combination of decreased visuospatial span associated with incomplete development of visual scanning and disproportionate development of global processing may cause drawing disabilities in WS.     

Notes on factorial invariance

Lawley's selection theorem is applied to subpopulations derived from a parent in which the classical factor model holds for a specified set of variables. The results show that there exists an invariant factor pattern matrix that describes the regression of observed on factor variables in every subpopulation derivable by selection from the parent, given that selection does not occur directly on the observable variables and does not reduce the rank of the system. However, such a factor pattern matrix is not unique, which in turn implies that if a simple structure factor pattern matrix can be satisfactorily determined in one such subpopulation the same simple structure can be found in any subpopulation derivable by selection. The implications of these results for parallel proportional profiles and factor matching techniques are discussed.     

Developmental invariance in distinctiveness effects in memory

Improvements in 5- and 7-year-olds' acquisition and retention of related concept pairings were examined when additional similarities and differences between pair members were provided. Using a standard paired-associate learning paradigm, children learned 18 related picture pairs; some of the children either were given or produced additional similarities or differences between pair members at the time of learning. Three weeks after learning was complete, children attempted to recall the pairs. Using a model to determine the storage and retrieval loci of these effects, the results showed that (a) all children benefited from self-generated elaborations, regardless of whether these were similarities or differences, and these benefits were storage related, and (b) difference elaborations improved children's retention regardless of whether they were self- or experimenter-generated, and these effects were primarily retrieval based. These results are consistent with theories that (a) view retrieval as the locus of distinctiveness effects and (b) view storage as the locus of self-generated memory improvements.     

Factor structure, measurement invariance and structural invariance of the MSCEIT V2.0

This study explored factor structure and measurement and structural invariance of the MSCEIT V2.0 across two age groups: 258 young (18-31 years) and 262 older adults (32-79 years). Results supported a three-factor solution reflecting the Experiential Emotional Intelligence area, and Understanding Emotions and Managing Emotions branches. There was evidence of measurement invariance of factor structure and factor loadings, and partial support for invariance of the intercepts. Comparisons of latent factor means suggested that older adults have significantly higher mean scores on two of the three factors: Understanding and Managing Emotions. Implications of the invariance tests and latent means analyses are discussed.     

Timescale invariance and Weber's law in choice

Pigeons were exposed to concurrent schedules for which reinforcement was alternately available at different times for each of two choices. In Experiment 1 (in which reinforcement times progressed arithmetically), overall, but not relative, response rate was timescale invariant. In Experiment 2 (in which reinforcement times progressed geometrically and were more spaced out), there was temporal control at all reinforcement times, but the amplitude of left-right response alternation decreased as time in the trial increased. These results indicate that the temporal regulation of both overall and relative response rates conforms to Weber's law although relative rate is heavily influenced by processes other than timing. It also adds support to the idea that overall and relative response rate reflects the operation of two independent processes.     

Factorial invariance in multilevel confirmatory factor analysis

This paper presents a procedure to test factorial invariance in multilevel confirmatory factor analysis. When the group membership is at level 2, multilevel factorial invariance can be tested by a simple extension of the standard procedure. However level‐1 group membership raises problems which cannot be appropriately handled by the standard procedure, because the dependency between members of different level‐1 groups is not appropriately taken into account. The procedure presented in this article provides a solution to this problem. This paper also shows Muthén's maximum likelihood (MUML) estimation for testing multilevel factorial invariance across level‐1 groups as a viable alternative to maximum likelihood estimation. Testing multilevel factorial invariance across level‐2 groups and testing multilevel factorial invariance across level‐1 groups are illustrated using empirical examples. SAS macro and Mplus syntax are provided.