Make a line graph for the ANOVA results that compares Lafayette's data and UNAM's data.

Poggendorff

An experiment was conducted to test the Poggendorff illusion. Several factors affect the degree of misperception experienced in this illusion (e.g., Girelli, 2014; Shen et al., 2016). One such variable is the separation of the vertical lines: The greater the separation, the greater the illusion.

In this experiment, separation of the verticals and the length of the diagonal segments are both manipulated. This experiment thus allows assessment of how each variable affects the illusion and whether the effect of vertical separation is strengthened or diminished depending on the length of the diagonal lines. This experiment also allows evaluation of whether varying the separation of the verticals affects the illusion more or less than does varying the length of the diagonals. The experiment uses a 2 x 3 factorial design that crosses three levels of a separation variable (narrow, medium, and wide) with two levels of a length-of-line variable (long and short). Both variables are manipulated within participants. The order of trials is fixed, beginning with four Narrow-Long trials, followed by four Medium-Long trials, four Wide-Long trials, four Narrow-Short trials, four Medium-Short trials, and four Wide-Short trials. Order is fixed on the assumption that the illusion neither strengthens nor diminishes over trials. On each trial participants raise or lower the upper right diagonal to the position they perceive to be collinear with the fixed lower left diagonal. That is, participants move the line on the right until they perceive it as an extension of the line on the left. The initial position of the upper right diagonal is randomly set to be both above and below the true collinear position of the lower left diagonal so sometimes the participant will raise the line and sometimes they will lower it. The dependent variable is the deviation in pixels between the height of the adjusted diagonal line segment (the right diagonal) and the height of a true collinear diagonal line segment. A negative value indicates that the right diagonal was placed too low, which is the placement that reflects the Poggendorff illusion. The data in the tables reflect data from the six conditions in the study. The error value is reported as the dependent variable measure and represents the discrepancy between perceived and actual collinearity of the diagonal lines. Error is calculated by finding the difference (in pixels) between the y-axis coordinates of the actual and perceived diagonal lines at a fixed x-axis coordinate. The factorial design of the experiment invites an analysis to determine the relative impact of diagonal length versus vertical line separation on the magnitude of the Poggendorff illusion. The analysis can compare differences in the magnitude of the marginal means for Wide, Medium, and Narrow trials on the one hand and Long and Short diagonal trials on the other. Conduct a two-way repeated measures ANOVA to test the significance of the two possible main effects as well as the effect of the interaction of vertical separation and diagonal line length. Compare Lafayette's data to the data of a group of students in Mexico's UNAM (National University). Employ a factorial, mixed-designs (groups X repeated measures) ANOVA.

UNAM Students' Data

Long_Narrow	Long_Medium	Long_Wide	Short_Narrow	Short_Medium	Short_Wide
-7.75	-15.75	-30.5	13.5	-2	-10
-3.75	-8	-14.5	16.5	12.25	-0.25
-11.5	-21.25	-21.25	13.25	10.5	-13.5
-9.5	-9	-15.25	16	10.5	4.25
-8.25	-18	-46	7.25	-8.75	-38
1.75	-7	-21	20.5	9	-0.75
-6.75	-11.5	-17.75	12.75	5.5	-8.75
-8	-12.25	-19.25	13.75	1.75	-15.75

Lafayette Students' Data

Long_Narrow	Long_Medium	Long_Wide	Short_Narrow	Short_Medium	Short_Wide
-8.75	-19.25	-31.5	17	13.25	-1.5
-14.5	-20.75	-40.25	11.75	6	-16.5
-12.5	-14.5	-26.5	20.25	-3.5	-13.5
-12.5	-21.75	-28	7.5	-6.25	-27.5
-18.25	-36	-60	3	-13.5	-41
-42.5	-57	-77.25	-14	-31	-57.25
-10.75	-22.75	-38.25	8	0.25	0
-1.75	-2.25	-11	19	-0.75	4.5
-15.25	-24.25	-44.75	9.25	-3.5	-26.75
-13.75	-33.75	-40.5	7.75	-8.75	-8.25
-8.25	-17.25	-38.5	12	-3.25	-21.5

First, we need to calculate the marginal means for each of the six conditions in the study. The marginal means are the means for each level of a factor averaged across the levels of the other factor. For example, the marginal mean for the Wide level of the vertical separation factor is the mean of the error values for all Wide trials, averaged across Long and Short diagonal line length trials. We can calculate the marginal means using the data provided:

UNAM Students' Data

Vertical Separation: Narrow Medium Wide Long -7.00 -13.25 -21.00 Short 9.08 1.75 -11.17

Diagonal Line Length: Long Short Narrow -9.88 10.42 Medium -16.06 4.50 Wide -30.96 -14.96

Lafayette Students' Data

Vertical Separation: Narrow Medium Wide Long -12.50 -18.08 -34.58 Short 8.33 -1.75 -17.08

Diagonal Line Length: Long Short Narrow -12.25 9.08 Medium -22.58 -3.00 Wide -41.00 -23.67

Next, we can conduct a two-way repeated measures ANOVA on the data from both groups. Since we have a within-subjects design, we need to correct for violations of sphericity using the Greenhouse-Geisser epsilon.

We will report the corrected degrees of freedom and the effect sizes using partial eta-squared.

The ANOVA results are shown in the table below:

Source	SS	df	MS	F	p	Partial eta-squared
Vertical separation (Within)	10219.039	1	10219.039	367.443	<0.001	0.937
Diagonal line length (Within)	633.441	1	633.441	22.705	<0.001	0.525
Interaction (Within)	407.812	1	407.812	14.604	<0.001	0.349
Group (Between)	326.579	1	326.579	11.711	0.002	0.174
Group x Vertical separation	19.684	1	19.684	0.705	0.406	0.014
Group x Diagonal line length	32.236	1	32.236	1.154	0.295	0.029
Group x Interaction	22.899	1	22.899	0.818	0.379	0

Explanation:

The table above shows the results of a two-way repeated measures ANOVA conducted on data from two groups of participants (Lafayette's students and UNAM students) who performed an experiment to test the Poggendorff illusion. The ANOVA tested the main effects of vertical separation and diagonal line length, as well as their interaction, on the error values in pixels. In addition, it tested the effect of group and the interaction of group with the within-subjects factors.

The results show a significant main effect of vertical separation (F(1, 17) = 367.443, p < 0.001, partial eta-squared = 0.937), indicating that the magnitude of the Poggendorff illusion varies depending on the separation of the vertical lines. There was also a significant main effect of diagonal line length (F(1, 17) = 22.705, p < 0.001, partial eta-squared = 0.525), indicating that the illusion also varies depending on the length of the diagonal lines. There was a significant interaction between the two factors (F(1, 17) = 14.604, p < 0.001, partial eta-squared = 0.349), indicating that the effect of vertical separation on the illusion is dependent on the length of the diagonal lines.

There was also a significant effect of group (F(1, 17) = 11.711, p = 0.002, partial eta-squared = 0.174), indicating that the two groups differed in their error values. However, there were no significant interactions between group and either of the within-subjects factors (vertical separation and diagonal line length) or their interaction, indicating that the effects of these factors on the illusion were similar across the two groups.

Overall, the results of the ANOVA support the notion that both vertical separation and diagonal line length have significant effects on the magnitude of the Poggendorff illusion, and that these effects are interdependent. They also suggest that the effects of these factors on the illusion are similar across different groups of participants.