ABSOLUTE THRESHOLD MEASUREMENTS WITH THE DIASTEREO TEST
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Tr shold Measurements with the
Absolute A
Diastereo Test
Dean H. W. Hofstetter, Ph.D., O.D., F.A.A.O., U.S.A.
Chairman: T. Standish Mason, F.B.O.A.
Dr. Hofstetter, Dean of the Division of Optometry
in Indiana University, a distinguished American
optometrist and educational administrator, is well known
in Europe for his numerous articles and works published
in optometric journals.
aluminium, they appeared black by reason of contrast
when the flashlight was turned on a
sheet To further diffuse the
awa lplacedr behind the flat
of thin white copy paper behind the
glass ]ens which, in turn, was directly
tranlucent plastic layer.
Recently Pardon' descrrbeo an uuu-- --- TRANSLUCENT
for screening out persons who do not demonstrate PLASTIC DISC
binocular stereopsis. He was able to demonstrate a
of separation of
bilit
li
y
a
virtually absolute validity and re
For the test PLANO GLASS LENS
subjects with and without stereopsis.
distance of 5 to 6 feet, corresponding to a stereopsis TRANSLUCENT
all subjects with
5l seconds
,
angle range of 36 to
WHITE PAPER
stereopsis made 100% correct responses whereas all
subjects without binocular vision failed to do so.
Because the criterion for passing was "
100% thresholds
sponses, the conventional or 11 50016 accuracy
were not determined. and Muellerand later
More recently Koetting an
Reismann,' essentially duplicated Pardon's results on
nse that
slightly rediabl versions of the
they separating those with binocular stereopsis
from m thole es those without. thout. This feature of the test prompted
its identification. as the "diastereo test "?
from thaab
iff
ers
The present study d
f test to explore i~
attempt is made to use the same type ersons who have
the absolute threshold values among p ) called
value e alsc
tbinocular stereopsis. The absolute is threshold,
the at which
he 501Y0 accuracy "
the correct and incorrect response probabilities are
equal. In the case of the diastereO test only one50
of three possible responses is correct, wee
accuracy " threshold corresponds to " 66; Y correct re
sponses ". This relationship can be represented by the
formula (1)
3y2x+ 100 . where y= % correct responses, and x=% accuracy.
Procedure pardon was further
The diastereo test described by tion
eatures
are in Fig. I f AnhordinnaryrRay O-O.-cell flash] ghtr was
Fig. forward from
used, one that had a shield protruding
the edge of the transilluminated face. This shield
served to prevent shadows from laterally ocated ambient
light sources and it also provided protection protruding discs mounted on the transilluminated face.
Two aluminium discs 0.5 mm. thick and 10 mm. in
diameter were cemented in direct contact with the trans-
lucent plastic disc serving as the transilluminated face.
A third aluminium disc of the same size was cemented
on one end of a transparent plastic rod 9 mm. long and
6 mm. in diameter, the other end of which was cemented
to the
face. T discs were plastic equidistant from theecentre of lthe
discs were in other then showconl our
1. Though the equidistant
L 59 --"I
Fig. i. Cutaway illustrating the diastereo test flashlight.
In the test procedure the examiner aimed the flash-
liht posed~the face of the flashlight efor aeperiod eofaoneeto
two seconds by temporarily removing a large card held
in front of the flashlight with the other hand. After
each exposure the subject was asked to report which of
rotated him . Prior to
examiner nearest
the three
each exp the flashlight
exposure or the spots,
e
randomly so that the proouding discright would be in 1 nt
of eight positions, up, left, up and f
Markers on tithe doutsiden of lthe shield o indicated these
positions exclusively to the examiner. Ordinary but
5-
Declass Review by NIMA/DOD
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consistent care in aiming the flashlight toward the sub-
ject's eyes during exposure seemed to be adequate to
prevent the subject's use of ally nonstereopsis clues. At
any time that the examiner suspected the influence of check
havin ubj ct coveroone ey e a techu quld ue whi h binvar ably re-
sulted sulted in complete loss of stereo judgment, and thus
assured the examiner that the correct binocular responses
were in fact attributable to binocular cities.
The data for this report were collected by two high scho
Sen
iors
on a sen oors, and by a high chool fr(1 of other eshmanf on a high school
which included mostly children between ages 6 and (15
and several teachers.
In group I, 31 subjects were actually run through
the test, but for this report only the data of 19 were
used, those who showed acuity of 20/20, or better in
each eye and binocular stereopsis. Of these 19, 13 wore
glasses. Their ages ranged from 15 to 17, inclusive.
Only three of the 19 were females.
Each subject in group I was given six exposures or
trials at the test distance of 5 ft.,
six trials at 7 ft., and
six each at 9, 11, 13, 15, 17, 19 and 21 ft., respectively.
Then he was given six trials at 22 ft., six at 20 ft., and
six each at 18, 16, 14, 12, 10, 8, and 6 ft, respectively.
At each test distance the number of correct responses
out of six trials was recorded without informing the
subject as to the correctness of his answers.
The interpupillary distance of each subject was also
measured. This ranged from 57 mm. to 66 mm., with
a mean of 62 mm.
In group II, 45 subjects with binocular stereopsis were
tested, but the record sheets for 24 of the subjects were
inadvertently destroyed before all of the tallies and com-Completed
so th
a ofltthis gr up i s based on all 45at
sub ects andha partysis
only 21 subjects. Only six of the 45, and two of the 21,
wore glasses. The acuity was not measured, but the
relatively high socio-economic level of the population
for the school at which these tests were made and the
high attention given to proper vision care in the school
district strongly indicate that virtually all of the subjects
in group It had 20/20 vision. Approximatery half were
males and half females.
Each subject in group 11 was given five exposures or
trials successively at each of the test distances 6, 8, 10,
12, and 14 ft. The six adults in group [I were tested
also at 16 and 18 ft. At each test distance the number of
correct responses out of five trials was recorded without
informing the subject of the correctness of his answers.
Three subjects, aged 4, 8, and 9, who failed at six feet
also failed at four and three f
t
ee
and were not included
among the 45 in group II. The reasons for their failure
were not definitely ascertained, but there were indications
that the 8 and 9 year-olds were squinters and that the
4 Year-old did not understand the instructions.
For the purposes of this report all test distances were
computed in seconds of stereopsis angle according to the
following formula, in which the interpupillary distance
is assumed to be 64 mm.:
Stereopsis angle in seconds = 1280/(test distance in ft.)2
Results
The results for group I are shown in the 19 individual
graphs in Fig. 2. The ordinate values represent the
number of correct responses out of the total of twelve
trials at 5 and 6 feet averaged as 5.5 feet, at 7 and 8 feet
averaged as 7.5 feet, etc. The abscissa is the log value
of the seconds of stereopsis angle, whereby 5.5 feet
becomes 1.63 log seconds, 7.5 feet becomes 1.36 log
seconds, etc. The abscissa value in seconds is shown in
the scale at the top of the figure.
LOG SECONDS
Fig. 2. Stereopsis test response curves for 1S' subjects
in group 1, high school seniors. The horizontal
dashed line represents the 66 2/3% correct
accuracy hr shold which level. For corresponds
suto the 50%
bjc cts, Nos.
6, 11 and 12, the test was not carriec out far
enough to determine the 50% accuracy threshold
level. The test distances are represented on the
abscissa in log seconds of equivalent pa rrallactic
angle.
In the upper right curve, the per cen-: correct
restanponsesTh
ce areci
e for all subjects at each test
rcles
dis. represent serially approach-
ing (decreasing difficulty) tests; the dots represent
serially receding (increasing difficulty) tests.
The combined per cent of correct responses for the
winthholee group at each test distance is sho vn in the curve
upper right corner of Fig. 2. Th. dots represent
the series at 5, 7? 8, . 21 feet in that (receding)
order of testing, while the circles represen: the subsequent
series at 22, 20, 18,
order of testin
g. he 6 feet in that (aPrpoaching)
Opposite to what aright have been xp ctcd easlalearning anj
effect.
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In all of the curves in Figs. 2 and 3 33; %, or 4 correct
responses out of 12, represent the frequency of correct
responses when the binocular clues are totally in-
adequate; 100% would represent the frequency when the
binocular clues are more than adequate; 662,%, or 8 out
of 12, would indicate the absolute threshold of stercopsis
at the 50% accuracy level, as computed from formula (1).
In Figs. 2 and 3 the absolute threshold response level,
66%0 (50% accuracy), is shown by a horizontal series
of dashes in each graph; the intersection of this with
the trend curve indicates the log second value of the
absolute threshold on the abscissa scale. It is readily
seen that the absolute threshold values for subjects
number 6, 11, and 12 were smaller than that included
within the maximum test distance of 22 feet. For
number 12 a few trials were made at 25 and 30 feet
without attaining the threshold.
The "log second " abscissa scale was adopted after
considerable experimental plotting to find a scale which
would give a normal increasing frequency of correct
answers as represented in the theoretical curve in Fig. 3.
Neither a "test distance" scale, a " stercopsis angle
scale, nor a "stereopsis angle reciprocal " scale gave
the symmetry of Fig. 3 as faithfully as did the " log
second " scale.
The distribution of threshold values for the 19
subjects in group I is shown in Fig. 4 on a rank scale.
The lowest curve in Fig. 4 represents the log second
equivalents of the greatest mean distance at which each
subject gave eight correct responses in 12 successive
trials. Since the tests were not carried out to this level
of performance for three subjects, the curve starts with
rank " 4 " and continues to rank " 19 ". This repre-
sentation of the stercopsis values on the ordinate in log
clo elys lreambles a curve which
oretical thcur e incomplete, coequal
cumulative area intervals of a normal curve as shown in
Fig. 5. From this it may be inferred that the designa-
tion of the stereopsis threshold in log seconds produces
a normal distribution. The plotting of these thresholds
on a " test distance ", " stereopsis angle," or " stereopsis
angle reciprocal " scale did not produce curves so
nearly like the corresponding theoretical curve in Fig. 5.
The middle and upper curves in Fig. 4 are derived in
the same way as the bottom curve except for the
adoption of a higher criterion of passing. This per-
mitted a ranking of
responses out aof 12, scrriterion, o and e all 0but the
two best performers for the 83, % (75% accuracy
threshold), 10 correct responses out of 12 criterion.
SCORE
Fig. 3. Curve showing per cent correct responses for
normally distributed hypothetical diastereo test
score values of decreasing difficulty.
These two additional curves for the same group indicate
that the lower ends of the curves have downward tails
like that in Fig. 5. The upper en.d of the absolute
threshold (66, % correct responses, or 50% accuracy)
curve has an upswing like that in Fig. 5, as does also
the 83,'-% correct responses, or 75 % accuracy, curve,
bu
This lack of t this feature is not apparent in the 100% curve.
tat stical artifact of the n 100% 1c criterion; t could be a
clustering of the several poorest performers at a single
level by reason of the large step to the next response
level; or it could represent the invasion of a secondary
clue at these poorer response levels. Whatever the
explanation or significance of this feature, it is not
eliminated by the choice of ordinate scale.
iv 1e 14 16 18 20
RANK
Fig. 4. Ranked stereopsis threshold values for 19 sub-
jects in group 1.
The fact that the use of a log second scale results not
only in a normal distribution of the responses for in-
dividual subjects as shown i,n Fig. 2, but also in a
normal distribution of the threshold values for the
group, as shown in Fig. 4, permits an evaluation of
test reliability by conventional statistical methods. In
order to incorporate the test results of all 19 subjects
at the 66; % response level (50% accuracy level) in the
computation of a reliability coefficient two such
thresholds were derived for each subject, one from the
series of receding test trials (5, 7, 9. . 21 feet), and
one from the series of approaching test trials (22, 20, 18,
subject6 ft). T
the in dieach st nce series which112
correct responses were obtained in 18 trials. Thus, a
subject who gave six correct .responses out of six at
both 20 and 22 feet could be considered as having given
at least 12 correct responses out of 18 even if he gave
all. wrong responses in six trials at 24 feet (at which he
was not tested), hence his threshold would be identified
as 22 feet or 0.41 log seconds. Notwithstanding the
imposition of such limitations for deriving threshold
values, the product moment coefficient of correlation
for reliability was 0.49?0.17 s.d. The scatterplot of
these values is shown in Fig. 6.
A scatter plot of the threshold values for the subjects
in group I against the interpupillary distances showed
no apparent relationship, but the limited number of
subjects does not exclude the possibility of such a
correlation in a larger sample.
The distribution of threshold values for 21 subjects
in group lI is shown in Fig. 7 on a rank scale. The
lowest curve (70% correct responses, or 55% accuracy
threshold) represents the log second equivalents of the
greatest mean distance at which each subject gave seven
correct responses in 10 successive trials. Since the tests
were not carried out to this level of performance for
10 of the 21 subjects, the curve starts with rank " 11 "
and continues to rank " 21 ". The representation of the
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w
0
U
In
Fig. 5. Theoretical curve of normally distributed scores
of a random sample of subjects plotted accord-
ing to rank. The five dots on the curve represent
the mean and the first and second standard
deviations on either side of the mean.
662/3 ?l? Correct Responses
I.._I I I I I ?J_ J
RANK
SECONDS
2 a .-Br--- SO
(12 Correct responses out of 18)
GROUP I t
w l 0.50 12
X
00
W 04
1.0
~~ ? ? 8~
'no ? O
l-w ? aw
(D U' 0.6 % ? ? /)
1
0.2 O4 06 0.8 1.0 1.2
APPROACHING TEST SERIES
LOG SECONDS
subjects accuracy threshold stereopsis in group 1.
Fig. 6. Seautesrpl t 19 of
stereopsis values on the ordinate in log seconds pro-
duced a curve which, though only half complete, closely
resembles the theoretical curve of equal cumulative area
intervals of a normal curve, as shown in Fig. 5th at.
this it may again be inferred, as for Fig.
designation of the stereopsis threshold in log seconds
produces a normal distribution. Similarly, the plotting
of these thresholds on a " test distance ", " stereopsis
angle ", or "reciprocal of stereopsis angle" scale did
not produce curves so nearly like the theoretical curve
in Fig. 5.
The middle (80% correct answers, or 70% accuracy,
threshold) and upper (100% correct answers, or 1005
accuracy, threshold) curves in Fig. 7 are derived in the
same way as the bottom curve excepttfor enthe ds ao ophem
of higher criteria of passing. lower
two curves clearly resemble the lower end of the
ecurve in nd of the Fig. 100 The lack of an
theoretical
curve corresponds tog the
the upper
same characteristic in Fig. 4
The combined per cent of correct each pons that each
test distance is shown in Fig. 8 for subgroups of group II. The average of the six adults
shows a 50% accuracy threshold of less than four
seconds; the same threshold for the 20 Le
enagerst is
eight seconds, and the corresponding threshold
6 to 10 year olds is 11 seconds. It is noteworthy that
the older teenagers in group I gave a corresponding
z
01.2
U
1.0
20Uj
0
100
6U
w
5(11
4
0.4 GROUP II
4
11 1 1 1 r 1 1 1 1 a_L_ I r I(
02 0 2 4 6 8 R10 12 ANK 14 16 18 20 22
/ AGE 1315
N2O--
Fig. 7. Ranked stereopsis threshold values for 21
subjects in group II.
SECONCS
a 5 8 _10-
V)
.0100
ADULT
00.. 90 N=6\ /x
N
LLI 80
H 70
U
ce 60
rr
0 50
0 40
AGE 6-10
N=191
-____
1 ' I '-Tf-I-r-rTr1
__l1
06 0.7 08 0.9 1.0 1.1 1.2 1.3
LOG SECONDS
I 1_-1
1.4 1.5 1,6
Fig. 8. Per cent correct responses at each test level for
the group II subjects n each of three age sub-
groups. The horizontUd clashed line represents
the 50% accuracy threshold.
O.
-(19)
Q(20)
e o
z
6.300
W
5 (n
0 0.5.--- - r5 20 25 30
MEAN AGE OF GROUPS
Fig. 9. Average 50% accuracy thresholds for each of
four groups of subjects plotted against their
average ages. The numbers in parentheses
represent the number of subjects in each group.
e
mean threshold value of 5.5 seconds, as shown in the
upper right curve of Fig. 2.
interpolated quite accurately from the trends with age
in group H. These average values for the four age
groups are plotted in Fig. 9
Discussion
To provide a basis for the comparison of these results
with those of other investigtors, it is possible to derive
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a broad statement of the absolute threshold values for
the whole group of subjects in this study by inspection
of the bottom curves in Figs. 4 and 7. In Fig. 4 the
bottom or 509,6 accuracy curve centers at about 0.6
log seconds and shows about two-thirds of the sample
between 0.4 and 0.9 log seconds. In Fig. 7, the bottom
curve, which can be presumed to be just slightly higher
than a 50% accuracy curve, centers at about 0.8 log
seconds and shows about two-thirds of the sample
between about 0.4 and 1.1 log seconds. A combination
of these two observations suggests a mean absolute
threshold value of about 0.7 log seconds with a standard
deviation of about 0.3 log seconds. This range, 0.7?0.3
log seconds, would represent a mean of 5.0 seconds
and a range from 2.5 to 10 seconds. The inclusion of
two standard deviations from the mean would give a
range in seconds from 1.3 to 20.
This range of results compares very favourably with
the 2 to 4 seconds thresholds obtained by Berry' on
three subjects. Howard" obtained a range of values
between 1.8 and 7.3 seconds for 85 of his 106 subjects,
while the other 21 showed a range from 10.6 to 136.2
seconds. Howard believed the latter poor scores to be
attributable to physical factors interfering with the
subject's vision, presumably inadequate visual acuity or
absence of binocular vision.
The presently reported results also compare favourably
with those of Bourdon' (5"), Crawley' (2.3" and up),
Anderson and Weymouth' (1.64" and up), Frubose and
Jaensch' (3.2" to 6.6"), Langlands" (1.8" to 7.3"), and
Munster" (5"), all of whom carried out their testing
in well-controlled laboratory settings.
The results obtained by the more typical screening
techniques are not so impressive, however. Probably
the most inclusive collection of such data are those of
Sloan and Altman." On both the standard and
a modified Stereopter they obtained a continuum of
scores on 68 subjects ranging from 10 seconds to 132
seconds, with modes at about 25". These were based on
a 7 out of 8 correct responses or 81 % accuracy instead
of 50% accuracy. On the Armed Forces Vision Tester
they obtained a mode value of 16 seconds for 42
subjects with 40% of the subjects failing the easiest
test plate, which represented a parallactic angle value
of 39-41 seconds. Weymouth and Hirsch'' obtained
similarly high thresholds for a large share of the 65
subjects on a telebinocular stereopsis test. Even the
"100% 11
performance level on the scales devised by
Shepard and Fry" for use with stereoscope test slides
represents 16 seconds of parallactic angle.
It is apparent that the diastereo test, even when used
as a rapid screening instrument, measures stereopsis at
a much more critical threshold level.
The matter of scaling stereopsis scores does not seem
to have been given very analytical treatment except that
skewness of typical data has been pointed out by
Weymouth and Ilirsch" who represented their data in
relation to separation and/or parallactic angle thresholds.
Similar skewing can be observed in virtually all pub-
lished data, whether they are the frequency of correct
response data on a single subject, as in Figs. 2 and 8,
or the rank distribution of threshold values in a group
of subjects as in Figs. 4 and 7. The transformation of
such data to log second scales show substantial if not
virtually complete elimination of skewness in the data
of Howard", Crawley', Anderson and Weymouth',
Langlands", Sloan and Altman", and Hirsch and
Weymouth'".
Such skewness appears to have prevented meaningful
statistical correlation computation, although Weymouth
and Hirsch'' did attempt to derive correlation co-
efficients for some of their samples by omitting extreme
scores. By this technique they derived reliability co-
efficients from which they concluded that, " . the
less-time-consuming rod-test (Howard-Dolman) and the
telebinocular test are unreliable and invalid, respectively
. In the same vein Sloan and Altman " reported
for the Howard-Dolman and the Stereopter test that.
" The data suggest, however, that within the group
showing good depth perception there is no close agree-
ment in relative r a n k i n g on the two tests ".
Unfortunately, the data from both reports are not pre-
sented in raw form and so do not lend themselves to
re-evaluation on a transformed log scale as was done in
the present study (Fig. 6) showing a test-retest reliability
coefficient of 0.5 for a group of 19 subjects all of whom
showed good scores.
The indication of improved stereopsis with age
appears to be practically uninvestigated. Tiffen1e 1'
showed an increasing percentage of passing of a
stereopsis test among adults up to about the age of 40.
Twenty subjects in Crawley's' report, ranging in age
from 4 to 70, showed an average of about 10 seconds
around age 8 and a decrease to about 4 seconds at age
35. It is quite possible that the apparent agreement of
these two reports with the present data is purely
fortuitous, but it certainly justifies further investigation.
weenulilnterpup llary h disance and statistical
be
stereopsis relationship
have
been found. The theory that larger interpupillary
distances should give better stereopsis scores is not con-
firmed in the presently reported data. Neither is the
large apparent increase of stereopsis with age
quantitatively attributable to an increase in inter-
pupillary distance with age. Rather, these results suggest
that a continuous stereopsis learning process may be
involved, right up to full adulthood.
Summary
Disastereo test thresholds were determined on two
groups of subjects, one a group of 31 high school
students and the other a group of 45 subjects of a sub-
stantial range of ages, mostly children, all of whom had
binocular vision. The two groups gave mean threshold
values of 0.7 log seconds (5 seconds) and a standard
deviation of ?0.3 log seconds, representing a standard
deviation range from 2.5 to 10 seconds of parallactic
angle. The test-retest coefficient of reliability for one
group was 0.5. The stereopsis scores showed no
apparent trend with the interpupillary distances, but they
showed a marked improvement with the increase of
age into adulthood. The sample was not large enough
to establish the statistical significance of the latter
relationship. .
Analysis of the data in terms of the relative fre-
quency of correct responses about the absolute threshold
and in terms of the distribution of individual subjects
threshold values clearly indicate the justification of a
log second scale to represent stereopsis data. In other
words, the log second scale produces the distribution
characteristics of normal data and so permits the
application of conventional statistical correlation
formula. A review of previously reported stereopsis
data supports the log second technique.
The diasterco test, though simple and quick in applica-
tion, gives results fully comparable with the best
stereopsis data previously reported for rigorous and
time consuming laboratory techniques. The diastereo
test results reported here appear substantially more valid
and more reliable than those reported for other popular
stereopsis screening instruments.
I
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BIBLIOGRAPHY
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