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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*).

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.

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*:

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.

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.