In the present study, the EyeLink II video eyetracker was used which - as similar video systems - has several advantages: (1) it is not invasive compared to, e. g. the search coil system, (2) it is mechanically easy to install, and (3) binocular recordings are directly possible without the need to double the system. Due to these features, video systems may not only be used in elaborated research laboratories, but potentially also in applied optometry, which is one aim of the present research. However, video eyetrackers might have limitations when investigating fixation disparity which typically is smaller than 1 degree of visual angle. Further, it is a general concern that without a bite bar no stable high resolution eye movement measures can be obtained. However, we wished to have a convenient test procedure where many subjects are willing to participate, since we aim to test large samples of subjects in order to investigate individual differences. Therefore, we did not use a bite bar in our experiments.

For these reasons, we made the following control experiments in order describe resolution, validity, and stability of data obtained with the EyeLink II system and our test procedures. Additionally, we will discuss the limitation that video eyetrackers are not able to differentiate between rotations and displacements of the eyes.

The theoretical noise limited resolution of the EyeLink II equipment is specified to amount to 0.6 min arc for the dark pupil system that we applied. But, the practical question refers to the smallest change in eye position that realistically can be measured in the actual experimental conditions. Therefore, we made a control experiment (Appendix 1 in Jainta /et al/., 2009), where subjects performed saccades between targets separated by 3.0 deg plus an additional offset separation of 0.0, 3.2, 6.5, 9.7, or 13.0 min arc. It was shown that changes in saccade amplitude of 4 - 6 min arc could be recorded as the average across 29 – 41 single saccades. This finding suggests that - in the binocular case - vergence changes in the order of 8 – 12 min arc could be resolved.

The present study refers to fixation disparity, which is defined as the difference in vergence between the binocular observation of a target and a monocular reference observation (during calibration in the present study); i. e. fixation disparity is a relative measurement. Thus, to specify the validity with respect to fixation disparity, it is required to test the system in response to small vergence changes.

Validity is conventionally tested by using a gold-standard recording system for comparison. Since such an objective eyetracker is not available in our laboratory, we tested whether the EyeLink II system is able to provide measures of vergence changes that are geometrically expected if we change the stimulus for vergence by small amounts. For this purpose, the haploscope was modified to have half-silvered mirrors so that the subject was able to view straight ahead onto a flat screen where the fusion target was presented. The viewing distance was changed by defined small amounts, so that the stimulus for vergence and accommodation was varied. The following series of test conditions was applied. As a starting point, we measured the fixation disparity at an initial baseline viewing distance of 600 mm which corresponds to a baseline vergence angle of 6.0 deg. i.e . 2 arc tan (PD/2D), with an average interpupillar distance PD=63 mm and the viewing distance D from the target to the plane of the centre of rotation of the eyes. For subsequent tests, we shifted the target by -28, -14, 13.5, 26 mm relative to the initial viewing distance of 600 mm to vary the vergence stimulus by small amounts of ± 8 min arc and ± 16 min arc; minus (plus) signs refer to a shift to more distant (closer) positions. To allow for such small range adjustments, the flat monitor was mounted on a mechanical stand that could be shifted back and forth in a purpose-made slide. The following 6 experimental conditions were presented in two different orders to reduce possible effects of sequential testing: (1) 0, -16, +16, -8, +8, 0 min arc and (2) 0, +16, -16, +8, -8, 0 min arc. We made six of these runs (three of each orders) in a single session and four sessions within each subject. The fixation stimulus was a central cross of 18 min size, surrounded by 12 crosses of 30 min arc size in an circular area of 8 deg diameter to assist fusion. The following time scheme was applied. The fusion stimulus was presented for 3 s. After this period, the stimulus was removed from the screen for 1 s, during which the experimenter shifted the screen to the subsequent position. The subjects were instructed to blink during this period, in order stimulate a fusional response to the new target position. Before and after the presentation of these 6 target positions, a monocular calibration was performed. The monocular calibration targets were presented in a haploscope which allowed to view the calibration targets monocularly on two monitors left and right of the mirrors; the fusion target at eye level was observed straight ahead through the mirrors. Although a bite bar was not used, the subject’s head was stabilized in a chin and forehead rest; additionally, translation of the head was minimized by applying firmly adjusted pads for the cheeks and by fixing the head with a flexible band around the back of the head.

The results in Figure A1 show on the x-axis the shift in vergence stimulus and on the y-axis the vergence response, both relative to the geometrically expected baseline vergence angle of 6.0 deg corresponding to the baseline viewing distance of 600 cm. Thus, a line with a slope of one with zero y-intercept (heavy line) indicates the expected result if a subject had a zero fixation disparity and a change in vergence response as expected from geometry. This is approximately the case in subject WP. The y-intercept of the regression lines (dotted line) represents the fixation disparity, which was negative indicating an exo (uncrossed) fixation disparity in three of these four subjects, up to an amount of 40 min arc in subject TB. A statistical mixed-effects model estimated a standard deviation of the fixation disparity (y-intercept) of 6.6 minutes of arc, as average across these four observers. Most important, the slope is close to one in all four subjects, which indicates that the present instrumentation and test procedure are able to detect small vergence changes in the range of ± 16 min arc with the appropriate quantitative amount. It should be noted that the reasoning of this control experiment is based on the assumption that the subjects fully performed the stimulated vergence responses (similar as any calibration procedure assumes that the subject actually fixates the target presented). It seems extremely unlikely that inappropriate responses might have been occurred and that these would just have been compensated by some unknown error in the recording system or signal analysis, so that an artificial slope close to 1.0 would have appeared in the four subjects tested. We conclude that our recording system operates with the appropriate gain, i. e. changes in recorded responses quantitatively correspond to the change in stimulus. The gain of the recording system is important for a correct quantitative description of fixation disparity. We see no reason to suspect that the large amounts of objective fixation disparity could be a result of an overestimation of vergence changes.

The validity of the zero value of objective fixation disparity is a further question which is more difficult to test than the validity of gain. In our procedure, a zero fixation disparity means that the left and right visual axes have the same orientation in the binocular (test) and in the monocular (calibration) condition. The level of zero fixation disparity determines the distribution of subjects with an eso or exo fixation disparity. In the present study, we can consider subjective fixation disparity: although it was not well correlated with objective fixation disparity, both measures had the same direction (eso or exo) in the majority of subjects (/Figure 8/) when for the subjective measure the individual nonius bias was used as a reference. The same direction of objective and subjective measures is physiologically plausible based on the reasoning in /Figure 1/. This arguments suggests that the actual zero level of objective fixation disparity is appropriate.

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In the present experiment, the subjective measures of fixation disparity require a considerable period of time until sufficient subjective responses to the nonius lines are collected. Particularly, the objective recordings of the 10 short fixation periods of each of the three targets took a period of 45 s (before and after which a calibration was performed). Thus, the question arises whether the head position remains sufficiently fixed to allow for stable eye movement recordings over 45 s. To test whether the resulting data might have been contaminated by drifts due to residual head movements, we calculated a comparative data set based on only 10 s directly after calibration. As shown in Figure A2, we found a high correlation (/r/ = 0.93, /p/ < 0.0001, /n/ = 17) between objective fixation disparity from the reduced 10 s period and the full 45 s period (both calculated with the trimmed mean procedure). This suggests that residual head movements did not have a detrimental effect on the dataset as a whole, at least if we regard correlations between different optometric measures which is the main aim of the present study. Detrimental effects of movements of the head are reduced in our procedure by analysing the difference between the signals of the left and right eye; the resulting full vergence angle is unaffected by small translational head movements (head position is much more critical if monocular components of fixation disparity are to be measured).

From the data reported in the result section, we can deduce some statistical properties of the objective fixation disparity measures. The mixed-effects model in /Table 1/ estimated an average intra-individual standard deviation /SD_intra / due to the /n/ repeated measurements (each of these represent the average resulting from the trimmed-mean procedure). These were /SD_intra /= 14.4 min arc , /n = /8 for short fixation periods and /SD_intra /= 10.6 min arc , /n /= 4 for long fixation periods. From these standard deviations, we calculate a confidence interval CI = /z/(0.05) */SD_intra / / ?/n, /where /z/(0.05) = 1.96 is the critical value corresponding to the area of 0.95 below the normal probability function for a two-tailed significance level of 0.05. For short and long fixation periods, we find figures of CI = 9.9 min arc and CI= 10.4 min arc, respectively. Thus, generally for our data set, we can conclude that an objective fixation disparity with an absolute amount of about 10 min arc (or larger) is significantly different from 0.0. If two subjects are compared, their average objective fixation disparity must differ by at least 10 * ? 2 = 14 min arc to be significantly different. Such interindividual differences occurred in a number of cases of the present sample which resulted in the signicant correlations reported above.

Video eye trackers (or other techniques based on the image of the eye or the cornea reflex) do not differentiate whether the eye has rotated or a displacement has occurred. In particular viewing conditions, the eyes may translate in the nasal/temporal direction (lateral) and/or may shift forward or backward in the orbit. E. g., temporal shifts of the eyes of 0 – 200 µm were precisely measured by Enright (1980) in a task were subjects changed monocular fixation between viewing distances of 4 m an 15 cm; although vision was monocular, a 20 deg change in accommodative vergence was expected. Effects of such ocular displacements on vergence measures have been investigated by Tani /et al/. (1956) using photographic recordings of the cornea reflex: the authors measured vergence, while the stimulus was first increased until the break point of fusion was reached and then reduced towards the recovery point and to fusion again. In these conditions of large amounts of forced vergence, Tani /et al/. (1956) interpreted own results and earlier findings in that the eyes did not rotate round a fixed point in the orbit, rather displacements of the eyes can occur: thus, the center of rotation may follow a curve when the eyes turn by large angles. However, a quantitative description of these patterns of shifts in eye position did not appear from these studies; contradictory observations from different studies were also noted; e. g., Alpern (1957) found forced vergence fixation disparity curves measured with an electro-oculogramm to be similar as those based on the cornea reflex method and concluded that eye displacements are extremely unlikely during vergence eye movements. Regarding that Tani /et al/. (1956) noted that convergence of the eyes within 10 degrees usually does not induce ocular displacements, we see no evidence that any relevant displacements of the eyes had occurred in the present experiment where we used a constant comfortable vergence state of 6 deg, i. e. without high vergence load or particular near vision

The control experiments and additional analyses made in this Appendix provide evidence that our objective fixation disparity measures are appropriate for the aim of the present study. Certainly, the instrumentation and procedures used in the present study have some inherent limitations (e. g., using no bite bar and only horizontal calibration) that will have introduced some measurement error. However, these sources of error are random, independent and unbiased and therefore are reduced by averaging the recorded data over time during each recording period and across a number of repeated measurements in two experimental sessions. Apparently, the measurement noise was small enough so that statistical properties of the data set suggest that – on the average in the sample - objective measures of fixation disparity of about 10 min arc or larger are significantly different from zero fixation disparity (which represents the monocular reference condition). Further, differences in objective fixation disparity between individuals of at least 14 min arc are significantly, as average in our data set. Due to these statistical properties, the resulting individual differences in objective fixation disparity are reflected in significant correlations between the two sessions (about /r/ = 0.84) and between heterophoria and objective fixation disparity (about /r/ = 0.82). Further, the main results of the study were very similarly found in the two conditions of fixation (short and long fixation periods with intercorrelations of about /r/ = 0.85); this replication shows the internal consistency of our findings. Therefore, the Eye Link II instrumentation, the experimental procedures and data analyses appear to be sufficient for the purposes of the present study.