density.lyx

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read_chunk("density_weave.R")
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\begin_layout Section
\begin_inset CommandInset label
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name "sec:results-number"

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Experiment 2: Changes in density versus number of elements
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Experiment 2
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%density-load
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@
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\begin_layout Standard
In 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

 I reported that judgements of motion direction were more sensitive to envelope
 motion when spacing is large.
 In that experiment, because spacing was determined by changing the number
 of elements, the same results could equally well be described as sensitivity
 being higher when there are fewer elements.
 Similarly, in the case of carrier motion, I modeled the sensitivity to
 carrier motion as being proportional to the number of identical elements,
 but it could also be described as inversely proportional to the spacing
 between elements.
 Mathematically this would be expressed by substituting 
\begin_inset Formula $N$
\end_inset

 for 
\begin_inset Formula $\frac{2\pi r}{S}$
\end_inset

 in the model formulation or vice versa.
 This leaves the question of which variable provides a better characterization
 of motion perception in this task.
 
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\begin_layout Standard
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

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 attempts to resolve the confounding of spacing and element number, to determine
 which provides a better characterization of motion perception in this task.
 The answer does not have to be the same for both types of motion; sensitivity
 to envelope motion may be determined by number while  sensitive to number
 whereas sensitivity to carrier motion is determined by spacing, or vice
 versa.
 Another way to frame the question is to ask whether the changes in sensitivity
 to envelope and carrier motion are due to influences from nearby flankers,
 or due to effects at a larger scale.
 For both carrier and envelope motion, a dependency on element spacing (as
 opposed to number) would imply a
\noun on
 local
\noun default
 process, sensitive to the immediate context of each element but insensitive
 to elements farther away.
 Conversely a dependence on element number would indicate a 
\noun on
global
\noun default
 process, which combines information across multiple elements even at large
 separations.
 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
We sketched an interpretation in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:model-visual-schematic"

\end_inset

 in which this term is described as a summation over a window surrounding
 one attentively tracked element.
 However as far as the data from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:results-basic"

\end_inset

 is concerned, this window of summation could be any size, from an interaction
 directly between an element and its neighbor to a summation over the entire
 visual field.
 We therefore wished to determine how global the summation of carrier motion
 was.
\end_layout

\end_inset

 To distinguish these alternatives, in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

 I used stimuli which decoupled the number of elements from the element
 spacing by creating a display where the elements cover only a portion of
 the scene.
 I did this by varying the 
\emph on
extent
\emph default
 of the stimulus, defined as the circumferential distance covered by the
 array of moving elements, as shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-stimulus"

\end_inset

.
 That is, the elements form a partial instead of a full circle.
 
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We also chose left versus right...
 It probably turns out we can't say anything useful on left vs.
 right, because we aren't measuring thresholds.
 Response rates aren't really accurate for quantifying whether there's more
 or less interference.
 Some subjects show differences, but we just explain any differences away
 as a bias.
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more discussion 
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\end_inset

It has been suggested that crowding results from 
\begin_inset Quotes eld
\end_inset

obligatory summation
\begin_inset Quotes erd
\end_inset

 of features within some distance; however another interpretation of crowding
 is that it results from a loss of information about spatial information,
 and the obligatory summation might be considered to be a separate phenomenon.
 MOVE TO DISCUSSION
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Our model also included terms controlling a repulsive effect.
 This effect did not appear to depend on the density 
\begin_inset Note Note
status open

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did I talk about this in the earlier section?
\end_layout

\end_inset

 it may either reflect a sort of normalized average of the motion energy
 across the visual field, or the influence of the carrier motion of an isolated
 element on itself.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
where am I going with this>:
\end_layout

\end_inset

 I would ignore this - getting lost in the trees!
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Another goal was to determine whether there were differences in performance
 dependent on the location of the elements in the visual field.
 This may give additional clues to the the nature of mechanisms underlying
 the perception of motion direction.
 Depending on which cortical areas are involved in generating a perception
 of motion, we may see more or less interference between moving objects
 For example, the numeric limit on object tracking appears to be independent
 between left and right hemifields 
\begin_inset CommandInset citation
LatexCommand citep
key "Alvarez:2005uq"

\end_inset

.
 In contrast, if the reversal shares mechanisms with crowding effects, then
 we might expect a lessened crowding effect for stimuli that span the horizontal
 meridian (, but see...)
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I don't find anything useful for crossing meridians.
 Move to Discussion if at all.
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This type of reasoning belongs in Discussion.
\end_layout

\end_inset

 There are a few overlapping reasons why we would wish to experimentally
 distinguish element spacing from element number.
 
\end_layout

\begin_layout Enumerate
Studies of the 
\begin_inset Quotes eld
\end_inset

multiple object tracking
\begin_inset Quotes erd
\end_inset

 task 
\begin_inset CommandInset citation
LatexCommand citep
key "Pylyshyn:1988fk"

\end_inset

 have shown that when observers attentively track a subset of objects moving
 among distractors, there appears to be a limit to the number of objects
 whose changes in position can be tracked for later identification.
 Adding more elements to our display might then interfere with the ability
 to individuate and track each element, leading to the inability to correctly
 discern their change in position over time.
 Although the number of elements whose global motion can be correctly discerned
 in our display is substantially larger than the limits usually found in
 MOT studies, the elements also move coherently, consistent with a structural
 rotation of the stimulus, which may increase the capacity for tracking
 
\begin_inset CommandInset citation
LatexCommand citep
key "Yantis:1992fk"

\end_inset

.
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\begin_layout Enumerate
For similar reasons, element spacing in section 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:results-basic"

\end_inset

 is also confounded with the aggregate amount of motion energy; reducing
 spacing requires adding more elements, which contributes more motion energy.
 Some motion mechanisms may sum up short-range motion information over large
 regions of the visual field; for example, global optic flow patterns such
 as rotations produce aftereffects in quadrants of the visual field that
 did not contain the adapting stimulus
\begin_inset CommandInset citation
LatexCommand citep
key "Snowden:1997vn"

\end_inset

, suggesting a mechanism tuned to optic flow that integrates large areas
 of the visual field.
 Since the patterns of carrier and envelope motion in our stimulus are (consider
ed separately) consistent with a global rotation, channels sensitive to
 global rotations are likely to contribute to the appearance of our stimuli.
 If these detectors are more sensitive to first-order than than higher order
 motion, 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
ugh, here's one place where the short-range / long-range distinction is
 confusing, because I'm suggesting that the mechanism with huge receptive
 fields is all about 
\begin_inset Quotes eld
\end_inset

short-range
\begin_inset Quotes erd
\end_inset

 motion (has anyone looked at this?).
 It does make sense that the optic flow computed in MST is based on a large-scal
e integration of first-order motion, though.
 Note, that there are disputes of Bertne and Faubert, finding integration
 of second-order optic flow; but I don't think the 
\begin_inset Quotes eld
\end_inset

long-range
\begin_inset Quotes erd
\end_inset

 mechanism is 
\begin_inset Quotes eld
\end_inset

second order
\begin_inset Quotes erd
\end_inset

 anyway.
\end_layout

\end_inset

 motion
\begin_inset CommandInset citation
LatexCommand citep
key "Bertone:2003zr"

\end_inset

, merely adding more elements my tip the balance toward the short-range
 component of the stimulus.
\end_layout

\begin_layout Enumerate
A third reason concerns the induced motion effect we observed at larger
 element spacings.
 Do these effects depend on element number???
\end_layout

\end_inset


\end_layout

\begin_layout Standard
If the array of elements covers only a portion of the scene, spacing and
 number can be varied independently.
 This involves varying the 
\emph on
extent
\emph default
 of the stimulus, defined as the circumferential distance covered by the
 array of moving elements.
 Any two of these variables (spacing, number, and extent) determine the
 third, so I will somewhat arbitrarily focus on spacing and element number
 as the relevant variables.
\end_layout

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status collapsed

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We also chose left versus right...
 It probably turns out we can't say anything useful on left vs.
 right, because we aren't measuring thresholds.
 Response rates aren't really accurate for quantifying whether there's more
 or less interference.
 Some subjects show differences, but we just explain any differences away
 as a bias.
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Methods
\begin_inset CommandInset label
LatexCommand label
name "sub:density-methods"

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\backslash
includegraphics[width=4in]{numdensity-stimulus.pdf}
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\end_inset


\begin_inset Caption

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\begin_inset CommandInset label
LatexCommand label
name "fig:density-stimulus"

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Example frame from a partial-circle stimulus.
 A number of moving elements are present on one side of the display.
 Two additional flanking elements are added, which always have carrier strength
 of 0 (i.e.
 they flicker in counterphase.) The flanking elements were positioned so
 as to be 
\begin_inset ERT
status open

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$
\backslash
Sexpr{paste(numbers$min.distance, collapse=" to ")}
\backslash
degree$
\end_layout

\end_inset

 away from the nearest moving elements at their closest approach.
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset Note Note
status collapsed

\begin_layout Itemize
Show stimulus design; covarying element number and density
\end_layout

\begin_layout Itemize
Mention flankers, which are not counted as moving elements
\end_layout

\begin_layout Itemize
Show demo movie, numbers linked to stimulus space
\end_layout

\begin_layout Itemize
Mention how displacement and direction content is chosen, based on data
 from Experiment 1, in an ad-hoc manner.
 Illustrate with comparisons of Exp 1 and Exp 2 data from illustrative subjects
 (NJ, PBM)
\end_layout

\begin_layout Itemize
We also test in alternate hemifields blockwise (but this is less important.)
\end_layout

\begin_layout Itemize
SUBTLE BUT IMPORTANT POINT: It's the slope of the lines that matters most,
 not their height or the change in height with element number (which is
 explained away as full-field sum of motion-energy).
 Whether the lines collapse together plotted against spacing, as PBM, or
 collapse plotted versus number, as NJ, doesn't actually matter.
 What matters is that In particular, it's that the slope does not change
 with element number, that distinguishes this model from e.g.
 one in which element number causes collapse due to failure of attentive
 tracking or whatever.
 Work through why the model predicts this.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The task was identical to that in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

.
 The motion elements were fixed at 
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status collapsed

\begin_layout Plain Layout

$
\backslash
Sexpr{sprintf("%.2f",numbers$segment.ecc)}^
\backslash
circ$
\end_layout

\end_inset

 eccentricity.
 The stimulus consisted of evenly spaced moving elements, identical to those
 in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

, but instead of being distributed around the entire circle, the elements
 were distributed along a partial-circle on either the left or right wide
 of the display.
 Stimuli were positioned either left or right of the fixation point.
 In pilot experiments observers found it difficult to judge the direction
 of motion if the stimulus appeared in a random location on each trial,
 so stimuli alternated between hemifields in blocks.
 Each session used an equal number of stimuli positioned in the left and
 right hemifields, with the position alternating in blocks of 100 to 200
 trials.
\end_layout

\begin_layout Standard
I varied both the spacing and the number of elements used; accordingly,
 the total spatial extent of the stimulus varied from trial to trial.
 The stimuli used are plotted in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-conditions"

\end_inset

.
 This stimulus set was constructed so that each value of spacing was tested
 using more than one value for the element number, and vice versa.
 I chose a set of stimuli that was symmetrically arranged in element number
 and spacing, spanning a similar range for both variables (covering a factor
 of 
\begin_inset ERT
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\backslash
Sexpr{chain(numbers$tested$trial_extra_nVisibleTargets, max(.)/min(.),latex.format(
digits=3))}
\end_layout

\end_inset

 in element number and a factor of 
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\backslash
Sexpr{chain(numbers$tested$spacing, max(.)/min(.),latex.format(digits=3))}
\end_layout

\end_inset

 in element spacing.)
\end_layout

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\backslash
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\begin_inset Caption

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\begin_inset CommandInset label
LatexCommand label
name "movie:density-stimuli"

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(
\family typewriter
demo_segment.mov
\family default
) Example stimuli for 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
 In this movie, the elements are shown in the left hemifield; in the experiment,
 stimuli alternated across hemifields, blockwise.
 Two stationary flankers with no carrier motion (showing counterphase flicker)
 are present at either end of the array of elements.
 Stimulus pairs 1-2 and 3-4 show stimuli with identical spacing, with different
 numbers of targets; pair 2-3 holds target number constant while reducing
 spacing.
 The numbers index the stimulus set given in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-conditions"

\end_inset

.
\end_layout

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\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="2">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
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%density-conditions
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<<density-conditions,fig.width=4,fig.height=4,out.width="3.5in",out.height="3.5in">>=
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%density-specifications
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<<density-specifications,fig.width=6,fig.height=4,out.width="5.25in", out.height="3.5i
n", echo=FALSE, message=FALSE>>=
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\begin_inset CommandInset label
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name "fig:density-conditions"

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The combinations of element spacing and element number used in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
 
\series bold
A.
 
\series default
Tick marks on symbols mimic variations in spacing and number of the motion
 elements (which are also shown on the two axes.) Numbered markers denote
 stimuli demonstrated in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "movie:density-stimuli"

\end_inset

.
 
\series bold
B.
 
\series default
The set of spacings and element counts was fixed for all experiments, but
 the carrier strength and displacement were selected by the experimenter
 for each observer.
 These parameters were chosen so that responses might span a useful range
 of response rates.
 Here example settings chosen for observer 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{toupper(specification.subject)}
\end_layout

\end_inset

 are shown.
 Background color indicates model fit using data from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

, while symbols illustrate values of spacing that would be used in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In addition to the moving elements, the display included two stationary
 flanking elements, indicated in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-stimulus"

\end_inset

.
 These served to hide the displacement of outermost moving elements, which
 would otherwise be easy to discern, as elements on the edge of an array
 suffer less crowding than elements within the array 
\begin_inset CommandInset citation
LatexCommand citep
key "Felisbert:2005ve"

\end_inset

.
 The flankers had no envelope movement and zero carrier strength (i.e.,
\begin_inset space \space{}
\end_inset

 they flickered in counterphase.) The space between the moving elements and
 the flankers changes over the course of the trial as the elements move;
 the flankers were positioned so that they were between 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

$
\backslash
Sexpr{min(numbers$min.distance)}
\backslash
degree$
\end_layout

\end_inset

 and 
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status collapsed

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$
\backslash
Sexpr{max(numbers$min.distance)}
\backslash
degree$
\end_layout

\end_inset

 away from the adjacent moving element at its closest approach, which prevented
 conspicuous gaps opening or closing between the flankers and the moving
 elements during the brief motion stimulus.
 The flankers spanned a partial circle that was at least 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

$
\backslash
Sexpr{floor(numbers$min.flanker.angle)}
\backslash
degree$
\end_layout

\end_inset

 and no more than 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

$
\backslash
Sexpr{ceiling(numbers$max.flanker.angle)}
\backslash
degree$
\end_layout

\end_inset

 measured around the fixation point.
 The movie shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "movie:density-stimuli"

\end_inset

 demonstrates these stimuli for the four configurations indicated with numbers
 in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-conditions"

\end_inset

.
\end_layout

\begin_layout Standard
While element spacing and number varied across trials, carrier and envelope
 motion were held constant during each session.
 I selected values for direction content and envelope motion for each subject
 based on preliminary findings of 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

; with the intent to find values such that that changes in spacing or number
 would produce a distinct change in response rate.
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-predictions"

\end_inset

 illustrates the choice of stimulus parameters for one observer, with reference
 to the model fit from Experiment 1 shown as a heatmap as in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:results-contours"

\end_inset

.
 Some observers were tested using more than one value of envelope displacement
 or carrier strength, but results were qualitatively similar in these cases,
 so they are aggregated in the graphs shown in this thesis; the original
 values were preserved in modeling.
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:stimuli"

\end_inset

 shows the selections of carrier strength and envelope displacement used
 for each observer.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Discussion: similarities and differences between our stimulus and 
\begin_inset Quotes eld
\end_inset

crowding
\begin_inset Quotes erd
\end_inset

 stimuli: in both cases flanker distance determines the percept, but in
 crowding stimuli the flankers are irrelevant, however in ours the flankers
 are necessarily relevant.
 They are 
\begin_inset Quotes eld
\end_inset

necessarily
\begin_inset Quotes erd
\end_inset

 relevant because position itself is a relevant feature that subjects are
 trying to discriminate.
 Our stimuli change position over time; so that both target and flanker.
 Accordingly, there is no distinct 
\begin_inset Quotes eld
\end_inset

target
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

flanker
\begin_inset Quotes erd
\end_inset

 in this stimulus; all elements serve both roles.
 This requires the stimulus to be distributed all the way around the visual
 field; or else hacks like our non-moving flanker.
 
\end_layout

\end_inset


\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Move this to later? Since the strength of crowding is typically not uniform
 across the visual field and varies between hemifields 
\begin_inset CommandInset citation
LatexCommand citep
key "Feng:2007lr"

\end_inset

, 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset ERT
status open

\begin_layout Plain Layout

%density-table
\end_layout

\begin_layout Plain Layout

<<density-table,results="asis">>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:stimuli"

\end_inset

Stimulus configurations used in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
 The envelope displacement and carrier strengths were chosen by the experimenter
; spacing and element number were prescribed by the stimulus set in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-conditions"

\end_inset

A.
 For some observers, multiple values of stimulus parameters were tried,
 noted as ranges here.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Results 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Things look different to start out with, but recasting in terms of the model,
 we will find that things are consistent between subjects.
 NJ and PBM have the most leverage in this experiment, in different directions,
 but consistent with the underlying core, so we will focus on them for examples.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

%density-measurements
\end_layout

\begin_layout Plain Layout

<<density-measurements, fig.width=5, fig.height=2.5>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:density-measurements"

\end_inset

Example data from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
 Here data for three observers is shown.
 The vertical axis shows proportion of responses clockwise, while the horizontal
 axis measures the spacing between elements.
 Lines connect stimuli with the same number of elements, indexed by color
 and symbol.
 Lines show how the proportion of responses clockwise varies as a function
 of element spacing, with other parameters (envelope displacement, carrier
 strength and element number) held constant.
 The error bar at right spans twice the maximum standard error over all
 data points.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Of the observers who had previously participated in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{length(unique(segment$subject))}
\end_layout

\end_inset

 participated in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
 Example data for three observers is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-measurements"

\end_inset

.
 Each plot shows the rate at which subjects responded 
\begin_inset Quotes eld
\end_inset

clockwise
\begin_inset Quotes erd
\end_inset

 to a stimulus at a particular value of spacing and of element number, with
 the spacing values indicated on the horizontal axis and element numbers
 indicated by color and symbol.
 The response patterns of these two observers appear different.
 NJ's direction judgements change little with changes in element spacing,
 while being greatly affected by changes in element number.
 In contrast, PBM's responses are sensitive to changes in spacing, while
 being little affected by changes in element number.
 NS shows a slight increase in proportion of responses clockwise with increasing
 spacing, while PBM shows a steep decrease.
 However, despite these differences, there are commonalities in these data
 sets.
 For none of the observers does there appear to be much interaction between
 spacing and element number; the lines traced out by changing spacing at
 a smaller number of elements are roughly parallel to the lines traced out
 by changing spacing at a larger number of elements.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%density-extent-interaction
\end_layout

\begin_layout Plain Layout

<<density-extent-interaction>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
No interaction between element number and spacing
\end_layout

\begin_layout Standard
I checked whether there was an interaction between spacing and element number
 using a logistic regression.
 Considering only the data from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

, the data may be empirically fit by a logistic regression; for fixed values
 of carrier and envelope motion, a simple regression model is 
\begin_inset Formula 
\[
\Pr\left(\mathrm{clockwise}\right)=\mathrm{logit}^{-1}\left(\beta_{N}CN+\beta_{S}CS+\beta_{E}CNS+\beta_{0}C+I\right),
\]

\end_inset

where, 
\begin_inset Formula $\beta_{N}$
\end_inset

 controls the change in motion sensitivity as a function of element number,
 
\begin_inset Formula $\beta_{S}$
\end_inset

 the sensitivity as a function of spacing, 
\begin_inset Formula $\beta_{E}$
\end_inset

 the interaction term due to the stimulus extent.
 an interaction term.
 
\begin_inset Formula $\beta_{0}$
\end_inset

 and 
\begin_inset Formula $I$
\end_inset

 are intercept and bias terms.
 Note that each term includes a factor of 
\begin_inset Formula $C$
\end_inset

, whose sign randomly alternates from trial to trial; the regression is
 modeling the response to carrier motion strength in terms of the number
 of elements and the spacing.
 (The spatial displacement is disregarded for this regression, because it
 always has the same sign relation to carrier motion in this experiment.)
 
\end_layout

\begin_layout Standard
The quality of fit for 
\begin_inset Formula $\beta_{N}$
\end_inset

 and 
\begin_inset Formula $\beta_{S}$
\end_inset

 are good, achieving 
\begin_inset Formula $R_{L}^{2}$
\end_inset

 scores of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{latex.format(r2.number, digits=2)}
\end_layout

\end_inset

 and 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{latex.format(r2.spacing, digits=2)}
\end_layout

\end_inset

, respectively, aggregated over all observers (as described in 
\begin_inset CommandInset label
LatexCommand label
name "sub:results-goodness"

\end_inset

, these measures compare the given model against a null model with constant
 effect and a full model saturated over each value of the given variable.)
 In the absence of an interaction term, 
\begin_inset Formula $\beta_{S}$
\end_inset

 was found significantly different from zero at the 
\begin_inset Formula $p<0.05$
\end_inset

 level for 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{sum(null.pvals$`content:target_number_shown` < 0.05)}
\end_layout

\end_inset

/
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{nrow(null.pvals)}
\end_layout

\end_inset

 observers, and 
\begin_inset Formula $\beta_{N}$
\end_inset

 significantly differed from zero for 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{sum(null.pvals$`content:spacing` < 0.05)}
\end_layout

\end_inset

/
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{nrow(null.pvals)}
\end_layout

\end_inset

 observers.
 Thus the response rates depend on both spacing and number.
 The support for an interaction term 
\begin_inset Formula $\beta_{E}$
\end_inset

 is weaker.
 When models included the interaction term, the resulting coefficients varied
 in sign, and aggregated over all observers only obtained a 
\begin_inset Formula $R_{L}^{2}$
\end_inset

 score of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{latex.format(r2.interaction, digits=2)}
\end_layout

\end_inset

.
 Thus it appears to be a satisfactory qualitative description of the data
 to say that motion direction judgements covary with spacing and element
 number, without much interaction.
 In the next section I will evaluate whether a single model fit may account
 for data from both this experiment and 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

.
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
At first glance 
\begin_inset Note Note
status open

\begin_layout Plain Layout
this is so discussion
\end_layout

\end_inset

 it would appear that these two observers are using different rules to perform
 the task.
 However, interpreting these patterns of behavior in detail requires correlating
 them with the measurements from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:results-basic"

\end_inset

; since the earlier experiment suggested more than one mechanism that could
 contribute to an observer's perceived direction of motion, what appear
 to be different patterns of response may be a result of differing strengths
 of contribution of each respective mechanism.
\end_layout

\end_inset


\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
\begin_inset Note Note
status open

\begin_layout Plain Layout
There's actually not a consistent interaction (which is good.)
\end_layout

\end_inset

In addition to the clear dependence on the spacing and the number of elements,
 observers' responses appear to show an interaction with the overall size
 of the stimulus.
 This is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-extent"

\end_inset

.
 This figure plots the same response rates as 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-measurements"

\end_inset

 but the horizontal axis now measures the circumferential spatial extent
 of the stimulus (defined for convenience as the number of elements times
 the inter-element spacing.) With this plotting it appears that there is
 an interaction between the spatial extent and the number of elements in
 the; lines connecting identical numbers of elements are clumped together
 at the left side of the graph and diverge at the right side of the graph.
 In other words, adding an element to a large stimulus appears to cause
 a greater effect than adding an element to a smaller stimulus.
\end_layout

\begin_layout Plain Layout
With reference to this regression I find that interaction term is significant
 for and positive for 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
Sexpr{"all"}
\end_layout

\end_inset

 of 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
Sexpr{"howmany"}
\end_layout

\end_inset

 observers (largest p-value 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
Sexpr{"all"}
\end_layout

\end_inset

 for observer 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
Sexpr{"someone"}
\end_layout

\end_inset

.) While this provides a descriptive account of responses in 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

, showing that there is an interaction between number and spacing, the next
 section will evaluate how the data from this experiment compares against
 the model presented in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:modeling"

\end_inset

.
\end_layout

\begin_layout Plain Layout
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

%density-extent
\end_layout

\begin_layout Plain Layout

<<density-extent, fig.width=5, fig.height=2.5>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:density-extent"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The lack of interaction can be predicted from the model sketched in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:model"

\end_inset

.
 This model predicts that there will be a progressive change in sensitivity
 to envelope motion as a function of spacing (or number) and there will
 be a change in sensitivity to carrier motion based on changes in the number
 or elements (or their spacing).
 Thus, as either spacing or number changes, the proportion of responses
 clockwise will progressively change, with the amount of change dependent
 on the observer's particular sensitivity to envelope and carrier motion
 and the particular stimulus parameters.
 Thus, the model of 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:model"

\end_inset

 would not predict much interaction between spacing and element number.
 Note that a lack of interaction also implies that there is not a limited
 spatial range of summation coming into play; that is, if element number
 determines carrier sensitivity, an element far away from the locus of attention
 adds as much as an element nearby.
 If carrier motion were integrated over a spatial scale within the range
 tested by this experiment, we would see summation trail off for stimuli
 with larger spatial extents, which would have produced an interaction in
 this data.
\end_layout

\begin_layout Subsubsection
Extending model predictions
\begin_inset CommandInset label
LatexCommand label
name "sub:density-modeling"

\end_inset

 from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

%density-extent-interaction
\end_layout

\begin_layout Plain Layout

<<density-extent-interaction>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 I extended the model from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:model"

\end_inset

 to the data from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

 and fit it to data from both experiments.
 There are four general scenarios under which the model may be extended
 to these stimuli, based on whether the sensitivity to envelope motion was
 determined by spacing (envelope local) or element number (envelope 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:density-modeling"

\end_inset

global,) and by whether the sensitivity to carrier motion was determined
 by spacing (carrier local) or by element number (carrier global.) These
 four models make identical predictions for the data from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

, but differ in their predictions for 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
\end_layout

\begin_layout Standard
The 
\begin_inset Quotes eld
\end_inset

global
\begin_inset Quotes erd
\end_inset

 models calculated the sensitivities to motion as a function of the number
 of elements falling within a hemifield.
 That is, the trials from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-basic"

\end_inset

 are considered to have an 
\begin_inset Formula $N$
\end_inset

 equal to half the number of elements that were actually displayed, while
 trials from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

 use the full count.
 This appeared to fit better than using the complete element count for both
 trial types, and is consistent with the observation that the effect of
 crowding does not cross the vertical meridian 
\begin_inset CommandInset citation
LatexCommand citep
key "Liu:2009ly"

\end_inset

.
\end_layout

\begin_layout Standard
For some observers, the responses appeared to differ between left and right
 hemifields; this is consistent with the observation that the critical spacing
 of crowding is not uniform across the visual field 
\begin_inset CommandInset citation
LatexCommand citep
key "Petrov:2011ve"

\end_inset

.
 Another confound may be the influence of the stationary flankers located
 on either end of the array of objects.
 The intention behind including these flankers was to avoid giving away
 the envelope motion by the motion of the outermost elements, since effective
 crowding requires flankers on either side of a target 
\begin_inset CommandInset citation
LatexCommand citep
key "Toet:1992db,Pelli:2004km"

\end_inset

, and is weaker for targets closer to the ends of an array 
\begin_inset CommandInset citation
LatexCommand citep
key "Felisbert:2005ve"

\end_inset

.
 While the flankers prevented the envelope motion from becoming uncrowded,
 they may have induced other side effects, since being positioned at the
 ends of the array makes them more salient than the moving elements and
 their lack of carrier or envelope motion is likely to have weakened both
 types of motion signal.
\end_layout

\begin_layout Standard
I added two parameters to account for possible effect of endpoints and for
 differences in carrier motion sensitivity between hemifields, making the
 substitution
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\beta_{S}\rightarrow\left(1+\delta_{e}e+\delta_{h}h\right)\beta_{S}\textrm{,}
\]

\end_inset

where 
\begin_inset Formula $e$
\end_inset

 indicates whether endpoint flankers are present, and 
\begin_inset Formula $h$
\end_inset

 takes on values of 
\begin_inset Formula $-1$
\end_inset

, 0, or 1 based on which hemifield the stimulus is shown in.
 Note that this adjustment was the same for all four models considered here.
 The ranking of local and global scenarios was similar with or without this
 adjustment.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Being a little cheeky here and hammering on what are the the actually correct
 usages of 
\begin_inset Quotes eld
\end_inset

local
\begin_inset Quotes erd
\end_inset

 and 
\begin_inset Quotes eld
\end_inset

global.
\begin_inset Quotes erd
\end_inset

 Local is a process that operates over a smaller region of space, and global
 describes a process that operates over a larger region of space.
 Here's the reason I refuse to use 'local' and 'global' anywhere else; many
 people (well, Mike and Tony) start out presuming that the envelope motion
 is global and the carrier is local.
 
\emph on
This turns out to be completely backwards.
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%density-pred-modelfiles
\end_layout

\begin_layout Plain Layout

<<density-pred-modelfiles, include=FALSE>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%density-pred-load
\end_layout

\begin_layout Plain Layout

<<density-pred-load,include=FALSE,cache.extra=file.info(quad.modelfiles$modelfile)$
mtime>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%density-predict
\end_layout

\begin_layout Plain Layout

<<density-predict,include=FALSE>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%density-predict-likelihoods
\end_layout

\begin_layout Plain Layout

<<density-predict-likelihoods,include=FALSE>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset ERT
status open

\begin_layout Plain Layout

%density-predict-plot
\end_layout

\begin_layout Plain Layout

<<density-predict-plot, fig.width=6, fig.height=8, out.width="5.4in", out.height="7.2i
n">>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:density-predictions"

\end_inset

Alternate predictions for Experiment 2.
 Data for three observers are shown, one in each column.
 The top row shows observed data.
 Each of the subsequent rows shows the model prediction under a particular
 interpretation of envelope and carrier motion processing: either local
 or global, so four combinations in all.
 Width of shaded region in predictions shows 
\begin_inset Formula $\pm$
\end_inset

 standard error of fit; error bar in top row shows 
\begin_inset Formula $\pm$
\end_inset

 standard error of observed response rates.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Example results are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-predictions"

\end_inset

.
 These show the model predictions corresponding to four scenarios (labeled
 on the right side.) For comparison the observed data is shown in the top
 row.
 Visually, these predictions confirm that both the entirely-local (row C)
 and entirely-global scenarios (row D) cannot capture the behavior of every
 observer.
 When both carrier and envelope processing depends on global element number,
 there is no change in response rate with spacing (row D), and when both
 carrier and envelope motion sensitivity depend on local spacing, the change
 in response rate with element number is not captured (row C).
\end_layout

\begin_layout Standard
The two remaining alternatives are carrier global, envelope local (row B)
 and carrier global, envelope local (row E).
 Between these alternatives, carrier global, envelope local appears to fit
 the data better.
 Aggregated across observers, it fits with the highest likelihood out of
 the four models, with a log-likelihood ratio of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{floor(sum(quad.sum.lls$lp[1:2]*c(1, -1)))}
\end_layout

\end_inset

 over the next best fit.
 However, the comparison in likelihoods does not hold for every observer.
 Given local envelope motion, global carrier motion gave a better fit, in
 terms of likelihood, for 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{sum(!carrier.better$carrier.local)}
\end_layout

\end_inset

 out of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Sexpr{nrow(carrier.better)}
\end_layout

\end_inset

 observers.
 Results for envelope motion were more equivocal; given global carrier motion,
 local envelope motion fit better for 4 out of 8 observers.
 However as mentioned earlier, unknown effects of the flankers may be present
 which would tend to make strict likelihood comparisons unreliable.
 Visually comparing the complete set of model predictions (
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:everything-density-predictions"

\end_inset

) I find that carrier global, envelope local (column E) does the best job
 at capturing the directions of effect; models with global envelope motion
 often have the wrong direction of effect for changes in element number
 (see observers JE, MC, ML.) It is possible that a mixture of local and global
 effects determines envelope motion visibility.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%density-combined-model-plot
\end_layout

\begin_layout Plain Layout

<<density-combined-model-plot, fig.width=5, fig.height=5>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
Model fits using the carrier-global, envelope-local model, plotted under
 data from 
\begin_inset CommandInset ref
LatexCommand nameref
reference "sec:results-number"

\end_inset

.
 Connected lines with symbols show observed data, while shaded lines are
 model predictions.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The model fits to the data from this experiment for all the observers tested
 are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:density-conditions"

\end_inset

.
 In summary, as discussed in more detail below, results from Experiment
 2 suggest that the summation of carrier motion is determined over a large
 spatial scale, large enough to encompass several targets, while the decline
 in sensitivity to envelope motion might be determined within a smaller
 spatial scale (though this point needs additional confirmation.)
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Discussion -- It is also possible that envelope motion sensitivity is determined
 by a mixture of local and longer-range spatial interactions.
 There are mixed reports of the how the strength of crowding changes with
 the number of flankers in an array; in some stimuli adding more flankers
 causes more crowding, while in other stimuli there is a recovery, and in
 some experiments the effects are nonmonotonic.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

%density-save
\end_layout

\begin_layout Plain Layout

<<density-save>>=
\end_layout

\begin_layout Plain Layout

@
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "bibliography"
options "apalike"

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\end_body
\end_document