Previous research has shown that not all segments of a square frame are necessary to produce the illusory tilt of an enclosed vertical line. Indeed, the presence of a single tilted line is often sufficient to induce the illusory tilt of a nearby vertical line (Carpenter & Blakemore, 1973). How do the four segments of a quadrilateral frame contribute to the illusory tilt if any one of them is sufficient to induce the illusion? Response classification (RC) was used in two experiments to examine the independent contributions of the four segments of a quadrilateral frame to judgments of the direction of tilt of an enclosed vertical line. Orientation perturbations were added independently to the four frame segments. The orientation of the top segment contributed most systematically to these judgments, whereas the orientation of the bottom segment contributed very little. Individual differences were observed with two of the four observers showing the largest apparent tilt of the test line for shear configurations of the quadrilateral in which the top and bottom segments were rotated in a direction opposite to the right and left segments. Logistic regression was used with a double-pass technique to estimate the relative importance of the four segments. Interactions between the segments were not systematically related to the observers' judgments. The results are discussed in terms of the utility of RC and logistic regression for studying perceptual phenomena whose mechanisms are thought to lie at levels such as orientation that are different from those typically examined with RC and pixel noise.

^{2}, segment luminance: 183.6 cd/m

^{2}). Two of the frame sides, the top and bottom, had canonical orientations of horizontal at 0 deg, and the right and left sides had canonical orientations of vertical at 90 deg. The rod and frame segments were 0.08 deg wide × 1.1 deg long. Unlike in standard RFI studies, the frame segments did not intersect. The frame segments were separated from the central rod by a center-to-center distance of 0.75 × segment length or 0.825 deg. The central rod was always presented vertically. For each trial, the frame segments were allowed to deviate in 5-deg increments from 0 deg (canonical orientation) to 25 deg deviation in either direction. The classification image methodology requires Gaussian noise distributions. Because of discreteness issues regarding variation in the orientations of small line segments, we used noise values that were chosen instead of a discrete, uniform distribution that ranged from −25 to +25 deg in 5-deg steps. Although this violates one of the assumptions of the classification image methodology, we would not expect this difference to have a critical impact on the results reported below. Additionally, we primarily relied on binary logistic regression to estimate decision weights rather than using the standard method of differencing response-sorted noise vectors. CCW rotations were assigned negative values, and CW rotations were assigned positive values. The deviation for each frame segment was independent of the deviation for the other segments across trials. Frame segments were rotated about their center points that remained fixed in position.

*r*= .402; A.A. 59.9%,

*r*= .198; A.R. 63.7%,

*r*= .268; K.G. 75.8%,

*r*= .537. Responses on the two passes were the same at least 60% of the time or more for these four observers.

*r*

^{2}from a standard linear regression.

*p*) = ln[

*p*/(1 −

*p*)]. For example, with the other segments set to produce a net proportion of CCW responses of 50%, rotating the top segment 10 deg CW by itself would increase the proportion of CCW responses to approximately 62% with a coefficient of −0.05 for the top segment.

Observer | Segment | r | Mean difference (deg)^{a} |
---|---|---|---|

M.L. | Top | −.566* | 17.20 |

Right | −.022 | 0.75 | |

Bottom | −.141* | 4.42 | |

Left | −.081* | 2.64 | |

A.A. | Top | −.240* | 7.41 |

Right | −.165* | 5.14 | |

Bottom | −.177* | 5.57 | |

Left | −.207* | 6.54 | |

A.R. | Top | −.351* | 10.87 |

Right | .291* | −9.09 | |

Bottom | −.082* | 2.58 | |

Left | .220* | −6.98 | |

K.G. | Top | −.358* | 11.05 |

Right | .348* | −10.85 | |

Bottom | −.027 | 0.85 | |

Left | .414* | −13.13 |

*p*< .05) interactions out of the set of 44. If we correct for multiple tests (0.05 / 11 = 0.004), only one of these interactions will remain significant. The observed number of significant interactions does not exceed what would be expected by chance. We conclude that to a first approximation, the segments independently behaved in influencing the observer's tilt judgments.

*r*= .495; A.A. 60.2%,

*r*= .201; A.R. 74.5%,

*r*= .483; K.G. 72.2%,

*r*= .448. These reliability estimates are similar to, and slightly higher than, those in Experiment 1. Once again, they show that despite the constant vertical orientation of the central test line, identical perturbations of the orientations of the four line segments produced reasonably similar responses for the perceived tilt of the test line.

*X*-axis in Figure 6) are both near 50%. Other configurations produce proportions of CCW responses that substantially deviate from 50%; hence, this is not simply insensitivity to the added noise deviations. It is interesting to note that the two mean stimulus configurations used in this experiment had neighboring segments orthogonally oriented to each other, as shown in Figure 4. It is as if these two observers are immune to the illusion when the segments are configured to resemble a square frame regardless of its orientation.

*p*< .05) interactions out of the set of 44 possible interactions across the four observers. After correcting for multiple tests, only two of these remained significant. Once again, the observed number of significant interactions did not exceed what would be expected by chance. As in Experiment 1, we conclude that to a first approximation, changes in the orientations of the frame segments for the most part additively contribute to judgments of the perceived tilt of the test line.

Observer | Segment | r | Mean difference (deg)^{a} |
---|---|---|---|

M.L. | Top | −.5221* | 16.35 |

Right | −.0180 | −0.61 | |

Bottom | −.0478* | 0.59 | |

Left | −.0862* | 2.72 | |

A.A. | Top | .3426* | −10.74 |

Right | −.1307* | −0.79 | |

Bottom | −.0231 | 0.74 | |

Left | −.1958* | 3.74 | |

A.R. | Top | −.5464* | 17.23 |

Right | .2324* | −7.14 | |

Bottom | −.0151 | 0.49 | |

Left | .2301* | −7.3 | |

K.G. | Top | −.3274* | 10.26 |

Right | .3083* | −9.65 | |

Bottom | −.1071* | 3.42 | |

Left | .3320* | −10.42 |