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### DEPENDENT MEASURES

At the end of the experiment, the applet will measure your point of subjective equality under two conditions: a descending type of trial; and an ascending type of trial. The point of subjective equality is an estimate of how long the comparison line needs to be in order for you to perceive it as equal to the standard line. How this is done is best defined by examples. (You may want to print this page in order to refer to it after you are finished with the experiment).

### Ascending Trial

Presentation Comparison length Standard length Response
1 80 90 Shorter
2 85 90 Shorter
3 90 90 Equal Lower threshold
4 90 90 Equal
5 95 90 Equal
6 100 90 Equal
7 105 90 Longer Upper threshold

The table above is read from left to right: each row represents what happened during one presentation. For example, during presentation 1, the comparison line had a length of 80 pixels, the standard line had a length of 90 pixels, and the observer judged the comparison line to be shorter than the standard. After the "shorter" button was pressed, presentation 2 presented the observer with a new comparison line of length 85.

Using this type of data, let's compute an upper and lower threshold for an ascending trial. The lower threshold is the point where the observer changed from a "shorter" response to an "equal" response. The upper threshold is the point where the observer changed from an "equal" response to a "longer" response.

#### Lower Threshold

From the table, we see that the response changed from "shorter" to "equal" when the comparison line changed from 85 pixels (during presentation 2) to 90 pixels (during presentation 3). Think about what is happening here: when the comparison line was 85 pixels or less, it was perceived to be shorter than the standard. However, when it was changed to 90 pixels, its length was perceptually indistinguishable from the standard.

At what point did the comparison line length become indistinguishable from the standard line length? Well, we can't tell exactly because the applet changed the size of the comparison line by 5 pixels: the point where the comparison length became indistinguishable from the standard length could have been anywhere between 85 and 90 pixels; for example, the actual point could have been when the comparison line was 86 pixels, or 88 pixels--we don't know exactly.

It is here that we make an assumption: we assume that the comparison line became perceptually indistinguishable from the standard line half way in between 85 and 90 pixels, that is, the average of 85 and 90 pixels: (85 + 90)/2 = 87.5 pixels. So this length is our estimate of the lower threshold of uncertainty. (Note that as the step size between successive comparison line lengths becomes smaller, the more accurate this estimate becomes). Expressing this threshold as a percentage of the length of the standard (i.e., 87.5/90 x 100 = 97%), we say that the lower threshold of uncertainty is when the comparison line length is 97% of the standard line length: below this threshold we assume certainty that the comparison line is shorter than the standard line; above this threshold, we assume uncertainty--but: how far above this threshold does uncertainty exist? In other words, as the comparison line increases in length, how long would it need to become before the observer perceived it to be longer than the standard line? To estimate this, we need to compute an upper threshold:

#### Upper Threshold

From the table, we see that the response changed from "equal" to "longer" when the comparison line changed from 100 pixels (during presentation 6) to 105 pixels (during presentation 7). Using the same reasoning as above, we assume that this observer changed from an assessment of "equal" to an assessment of "longer" when the comparison line was half way between 100 and 105 pixels, that is: 102.5 pixels. Expressing this upper threshold length as a percentage of the standard line length, we get: 102.5/90 X 100 = 114%; we can now say that the upper threshold of uncertainty is when the comparison line length is 114% of the standard line length.

Thus the interval of uncertainty is <97%, 114%>: if the comparison line is within this interval (more than 97% but less than 114% of the length of the standard line), the two lines are indistinguishable. However, there is one final psychometric construct we can estimate, and it is based on the idea that there is one comparison line length that is more strongly perceived as being equal to the standard line length than any other comparison line length. We call the point of maximum uncertainty the point of subjective equality. The point of subjective equality is defined as the average of the upper and lower thresholds: (97% + 114%)/2 = 106%.

The point of subjective equality is the point where the comparison line length is such that it is maximally indistinguishable from the standard line length. By using this construct, we can observe how the Ponzo illusion affects your perception. First note, in this example, that the point of subjective equality is NOT 100%. The only way to get 100% is for the upper and lower thresholds to be symmetric about 100% (e.g., <95%, 105%>). This result would mean that the observer has perfect perception--that is, when the comparison line length is equal to the standard line length, the two lines are maximal indistinguishable to a perfect observer. This is what we would expect in a "normal" situation with depth and framing cues removed (i.e., the same task, but the 2 tilted lines are removed). While the applet does not have this control condition (i.e., the same task, but the 2 tilted lines are removed) to test this assumption, this is still a reasonable assumption. Given this, we can reasonably conclude that the illusion had the effect of shifting the point of subjective equality upward; this means that the standard line looked longer than it really was, which is the typical effect of this illusion.

### Descending Trial

For completeness sake, below is example data from a descending trial. Using this table, you can compute an upper and lower threshold: the upper threshold is the average point where the observer changed from a "longer" response to an "equal" response; the lower threshold is the average point where the observer changed from an "equal" response to a "shorter" response. Study this table and try to calculate from this data the interval of uncertainty (which should be: <95%, 108%>) and the point of subjective equality (which should be: 103%).

 Presentation Comparison length Standard length Response 1 110 95 Longer 2 105 95 Longer 3 100 95 Equal Upper threshold 4 100 95 Equal 5 95 95 Equal 6 90 95 Shorter Lower threshold

We now return to our original two questions: "can two dimensional depth cues affect your perception of the size of an object, and if so, how much of an affect can such cues exert?" Our example data answers "yes" to the first question: a point of subjective equality greater than 100% demonstrates that the standard line is perceived to be longer than it actually is. One theory, the depth cue theory, suggests that this is because the illusion contains depth cues (e.g., the tilted lines resembling a pair of railroad tracks receding into the distance); this would cause the standard line to appear "further away" because it is perceived as "further down the tracks." This means it might appear longer than an equal size comparison line because the comparison line is perceived as "closer." Such a theory can account for the above data. Further, by calculating a point of subjective equality, we can estimate how much of an affect these cues exert. Moreover, by manipulating the various independent variables, we can understand how various visual features of the illusion (e.g., angle of the tilted lines) might change the point of subjective equality. Using this applet, we can access the impact of type of trial.