Share this article on:

Horizontal Fixation Disparity Measures With Nonius Lines


doi: 10.1097/01.opx.0000187843.46805.a4
Articles: Original Article

Purpose. It has been reported that changes in visual direction and retinal correspondence may limit the validity of nonius lines as a subjective measure of vergence, at least in particular viewing conditions, e.g., dynamic or forced vergence. Nonius lines may be valid at larger spatial separation between fusion contour and nonius lines. Therefore, we measured fixation disparity varying the amount of a vertical gap between nonius lines.

Methods. A static central fusion stimulus was given at a 100-cm viewing distance. The nonius lines were presented with various vertical gaps either continuously (and adjusted to subjective alignment) or flashed in a series of trials (in adaptive testing). Twelve subjects with normal binocular vision were tested repeatedly to identify individual vergence characteristics.

Results. When the vertical gap between nonius lines was increased up to 7.9 deg, the amount of subjective fixation disparity tended to increase in part of the subjects, predominantly in those with an exo fixation disparity. Most subjective fixation disparity measures were correlated with each other and with tonic vergence (i.e., dark vergence tested subjectively without fusion stimulus); however, flashed nonius lines at larger nonius gaps were an exception.

Conclusions. We found physiologically plausible measures of idiosyncratic fixation disparity with continuous nonius lines at any amount of nonius gap or with flashed nonius lines at small gaps. In these conditions, the intersubject variability of fixation disparity was much larger than effects of the spatial separation between fusion stimulus and nonius lines.

Institut für Arbeitsphysiologie, Dortmund, Germany

Received March 16, 2005; accepted May 25, 2005.

Fixation disparity is a condition of binocular vision in which a fused fixation point is not projected onto the center of the fovea in each eye. This means the principal visual directions (that are associated with the centers of the foveae) do not intersect at the fixation point but do intersect either in front (eso) or behind (exo). Such vergence errors may occur in subjects with normal binocular vision and typically amount to a few minutes of arc and thus are smaller than the Panum’s area and do not lead to double vision.1,2 Clinical testing of fixation disparity is an important diagnostic tool for impairments in binocular vision.3–8

Fixation disparity can be measured psychophysically with a pair of vertical nonius lines that are presented dichoptically, i.e., one to each eye; one determines the particular horizontal physical offset of the nonius lines at which they coincide with the principal visual directions and thus are perceived as aligned by the subject. Such “subjective tests” are technically simple and easy to administer. Ukwade9 reviewed test devices, including the Mallett unit and the Disparometer, which are used in research and clinic.10,11 In laboratory studies, purpose-made devices have been used with mechanical or electronic adjustment of nonius offset.1,12–15 These subjective tests differ in the arrangement and spatial dimensions of the fusion stimulus and nonius lines, which may be relevant for the test result.

Objective measurements of fixation disparity can be performed with binocular eye movement recordings. These include calibrations of each eye separately, i.e., monocular fixation targets are projected onto the center of each fovea relative to which the objective fixation disparity is calculated. An equivalent amount of fixation disparity can be detected by subjective tests only if the centers of the foveae agree with the principal visual directions. This normal correspondence is certainly not present in strabismus, where a deviated functional fovea has developed. However, research has demonstrated that changes in visual directions and retinal correspondence may also occur in subjects with normal binocular vision.16 The validity of nonius lines is reviewed by Howard.17 In this context, we discuss the following studies in which the spatial separation between fusion stimulus and nonius lines plays a role because this is one parameter of subjective fixation disparity tests.

Dynamic vergence was investigated by Erkelens and van Ee18,19 who presented a fusion target that moved sinusoidally in depth by 2.7 deg at a frequency of 0.75 Hz; the eyes performed vergence movements (with large errors), but a continuously presented monocular line (moving in amplitude and phase with the fusion stimulus in one eye) was perceived at a fixed position, provided the fusion contour was close to the monocular line. In this condition, the continuously presented monocular line did not indicate the vergence movement performed; rather, it received the visual direction of the close fusion stimulus. This effect of “capture of visual direction” gradually diminished with increasing separation between monocular line and fusion contour.

Shimono20 applied a random-dot stereogram with two depth planes, which each included a pair of dichoptic nonius lines. Although being embedded in different depth planes (corresponding to different vergence states), both pairs of nonius lines were perceived in alignment as long as the fusion contour was close to the nonius lines. Nonius lines reflected the vergence position better if the fusion contour was separated from the nonius lines by at least approximately 2 deg.

Fogt and Jones21 used a static fusion stimulus and increased the vergence stimulus up to 8 deg (in the convergent or divergent direction) relative to the baseline vergence position corresponding to a viewing distance of 71 cm. In these conditions of “forced vergence,” fixation disparity (measured subjectively with continuously presented nonius lines) was smaller than the objective fixation disparity; the difference increased with the extent of forced vergence in three of five subjects. Subjects with no difference between objective and subjective fixation disparity showed vergence adaptation, i.e., had flat forced vergence fixation disparity curves. Furthermore, Fogt and Jones22 measured the fixation disparity in one subject at a 9.1-deg forced convergence and varied the vertical gap between two nonius lines presented above and below a central fusion stimulus; the difference between subjective and objective fixation disparity became smaller as the nonius gap increased; at a separation between fusion stimulus and nonius lines of 2.3 deg, the nonius lines indicated a similar fixation disparity as the objective method.

The observations of Erkelens and van Ee,18,19 Shimono,20 and Fogt and Jones21,22 refer to different viewing conditions but they come to a similar conclusion; monocular nonius lines do not indicate the objective vergence position as long as the fusion contour is close to the nonius lines. The authors suggest that a valid fixation disparity is only indicated if the nonius lines are presented at a separation from the fusion stimulus of approximately 2.5 deg or more.

However, the viewing conditions used by these authors (dynamic vergence, stereoscopic depth planes, or forced vergence) do not apply to a static fusion target in a single depth plane without forced vergence, i.e., with equivalent stimuli for vergence and accommodation. Thus, in this clinically relevant test condition of fixation disparity, the reported objections against the subjective nonius method may not necessarily apply, e.g., Ukwade9 measured subjective fixation disparity for a static fusion target and did not find an effect when the separation between fusion stimulus and nonius line was increased up to 0.6 deg.

The nonius method has been shown to be clinically successful; groups of subjects with a larger exo fixation disparity in near vision (tested subjectively) have stronger asthenopic symptoms.4,14,23–26 In view of the small visual angle of fixation disparity (a few minutes of arc), objective recordings require sophisticated instrumentation and test procedures that are mostly confined to research laboratories. Thus, subjective testing of fixation disparity with nonius lines seems to be (and to remain) the only method applicable for clinical purposes. For these reasons, it is important to know the most appropriate test conditions in subjective testing. Because of possible artifacts resulting from changes in visual directions, the spatial separation between fusion stimulus and nonius lines appears to be an important parameter. Therefore, it was the aim of the present study to measure subjective fixation disparity as a function of the vertical gap between nonius lines (up to 7.9 deg). We used a central fusion stimulus because the resulting fixation disparity is more stable27 and seems to be ecologically more valid, because most visual tasks (reading printed text or visual tasks at computer screens) involve central fusion stimuli. Furthermore, many studies reporting on the relation between fixation disparity and asthenopic complaints used a central fusion stimulus.14,23–26

An additional experimental factor in the present study was the temporal mode of presenting the nonius lines. In all clinically used devices for measuring fixation disparity, both the fusion stimulus and the nonius lines are presented continuously, thus they are permanently visible for the subject. However, in many research studies, adaptive psychophysical test procedures are used. These computer-controlled methods have the advantage that individuals can be tested repeatedly without effects that might be introduced by the expectation of the experimenter or subject. Adaptive psychophysical procedures use a series of repeated short presentations (trials) of test stimuli. In the case of fixation disparity, the nonius lines are typically presented, e.g., 50 times for a short period of time, e.g., 100 ms with different amounts of the horizontal nonius offset; the subject indicates the perceived position of one line relative to the other and the nonius offset of subjective alignment is calculated from these responses. It is an open question whether such adaptive test procedures with flashed nonius lines give different results compared with the adjustment of the perceived offset with continuously presented nonius lines.

Our study included measurements of the nonius bias (also referred to as binocular vernier constant error). This is the physical nonius offset required to perceive two coinciding nonius lines without dichoptic separation, i.e., when both nonius lines are visible for both eyes.28 The individual nonius bias can amount to a few minutes of arc (if nonius lines have a vertical gap) and may be a more appropriate reference of fixation disparity than the physical coincidence of nonius lines.29 Thus, our analyses also include the corrected fixation disparity, i.e., the fixation disparity minus nonius bias.

We also measured the resting position of vergence (without any effective stimulus for vergence or accommodation) that is known to determine—at least at longer viewing distances—the direction of fixation disparity.29–34 We measured dark vergence without a fusion stimulus in a dark visual field with flashed dichoptic nonius targets.35

Back to Top | Article Outline


Apparatus and Stimuli

For measuring fixation disparity, we used a special setup with the central fusion stimulus on a thin-film transistor (TFT) screen and the nonius lines on a cathode ray tube (CRT) screen; these two images were superimposed by a half-silvered mirror (30 × 40 cm) and appeared as shown in Figure 1. This arrangement is not relevant for the present study (it was used for purposes of a different experiment made in parallel). The relative position of the two monitors was precisely adjusted and mechanically fixed so that the images were exactly superimposed with identical viewing distances adjusted by motion parallax. The TFT screen was operated with bright central fusion stimulus on black background, so that the frame of the TFT screen could be made invisible (relative to the bright background of the CRT screen) with a black cardboard. The frame of the CRT screen remained visible (which cannot fully be avoided with a bright background); however, because a central fusion stimulus was applied, the effect of the additional fusion contour at the 10-deg eccentricity will have negligible effect, if any.



The CRT 17-inch color monitor (Belinea 107035 with 120-Hz refresh rate, resolution 800 × 600 pixel, 0.40-mm pixel size) produced a bright background of 1.8 cd/m2 as viewed by the subject through the activated LCD shutter glasses (Elsa). The latter were used for dichoptic separation; the right and left eye viewed the upper and lower nonius line, respectively. Nonius lines were not black but had a luminance 0.9 cd/m2 to eliminate visible crosstask between the eyes in dichoptic separation with the shutter glasses. The nonius lines (10 minarc wide and 57 minarc high) had different vertical gaps in different experimental runs: 1.0, 3.3, 5.6, and 7.9 deg. The step size of the horizontal nonius offset (during adapting testing, see subsequently) was chosen to increase with nonius gap size (i.e., 1.0, 2.6, 4.2, and 5.8 minarc, respectively), because at larger gaps, it was more difficult to detect a difference in horizontal nonius position. We used an antialiasing method for the precise placement of the nonius lines; the right and left border pixel of a nonius line was programmed to assume one of 13 luminance levels to shift the center of gravity of the nonius line to 13 positions within a pixel width. For this purpose, the luminance of the monitor was linearly calibrated with a “gamma correction.”

A continuous central fusion stimulus (a row of the characters OXOXOXOXO, 17 minarc high and 160 minarc wide) was generated on a TFT screen (ADI A715) and appeared centered between the two nonius lines on the CRT screen (Fig. 1). The fusion characters had a luminance of 11 cd/m2 measured through the activated shutter glasses.

For measuring dark vergence (Jaschinski 2001), two points of light (one red, one white) appeared dichoptically (using polarizing filters) on a CRT screen at 100-cm viewing distance in an otherwise completely dark suround. In a series of 100 presentations of 100-ms duration, the two points of light were presented with different horizontal offset to find the point of subjective coincidence with the adaptive procedure Best PEST.36 These conditions reduce possible influences of the subject’s knowledge of the test distance and residual stimulus effects during dark vergence testing.37 For all tests, a chin and forehead rest was used to minimize head movements.

Back to Top | Article Outline

Experimental Design

To describe the test–retest reliability, two sessions were made on separate days that each comprised two test series. In each series, four experimental runs were made with different amounts of nonius gap (in random order). The experimental factors were 1. The effect of the vertical gap between nonius lines (1.0, 3.3, 5.6, 7.9 deg) with a fixated fusion stimulus in the center of the gap (see Fig. 1); and 2. The temporal mode of presenting the nonius lines, i.e., continuously or flashed using appropriate psychophysical procedures.

Back to Top | Article Outline

Psychophysical Procedures for Measuring Fixation Disparity and Nonius Bias

Each experimental run comprised three phases with continuous or flashed nonius lines using the following psychophysical procedures.

Back to Top | Article Outline

Phase 1a (nonius bias with continuous nonius lines, first test)

Without dichoptic separation, the nonius lines appeared continuously and the horizontal positions of the two nonius lines were controlled by the subject using the buttons of the computer mouse. At the start, the nonius lines were physically aligned. Subjects were instructed to adjust the nonius lines to subjective alignment and to readjust whenever subjective deviations may arise again as a result of fluctuations of fixation disparity. This was done for a 20-second period. Nonius offset data of the first 10 seconds were discarded (because this period included the rough approach to alignment) and data of the following 10 seconds were averaged to represent the result.

Back to Top | Article Outline

Phase 1b (fixation disparity with continuous nonius lines, first test)

The procedure of phase 1a was immediately started again; however, dichoptic separation was applied for measuring fixation disparity.

Back to Top | Article Outline

Phase 2 (nonius bias and fixation disparity with flashed nonius lines)

The nonius lines were flashed for 100-ms duration in a series of presentations with an interval of 2.5 seconds, which is a convenient quick test procedure without time pressure. The nonius presentation time of 100 ms appeared to be long enough for most adult subjects to precisely judge the nonius offset and shorter than the latency of vergence of approximately 150 ms (in case that even dichoptic nonius lines should have residual effects on vergence). The test was controlled by two separate adaptive psychometric procedures Best PEST36; the nonius lines were flashed 50 times (trials) with dichoptic separation (for fixation disparity) and 50 times without dichoptic separation (for nonius bias). These two modes were randomly interleaved from trial to trial; subjects were not able to perceive the difference. Subjects were instructed to respond after each nonius presentation whether the upper line was perceived to the right or to the left of the lower line. According to Best PEST, each amount of presented nonius offset is an estimation of the actual result based on the previous responses; the mean of the last 45 trials was taken as the average of a run. The first five trials were discarded because in the beginning, the adaptive procedure uses large nonius offsets to approach the point of subjective coincidence.

Back to Top | Article Outline

Phase 3a, b (fixation disparity and nonius bias with continuous nonius lines, second test)

To test possible time effects during a run (which have been reported at short viewing distances of 40 to 30 cm38), the procedure of phase 1 was repeated, but fixation disparity was tested first (as in phase 1b) followed by a nonius bias test (like in phase 1a).

Back to Top | Article Outline

Statistical Analyses

We used analyses of variance with repeated measures on the group level and on the individual level because subjects appeared to have reliably different effects of vertical nonius gap (partly in opposite directions). Most statistical tests require independent data, i.e., data of different subjects in a group analysis. On the individual level, repeated measurements are independent if one observation does not depend on previous ones. It has been shown that at a 100-cm viewing distance systematic changes in fixation disparity do not occur during experimental sessions.38,39 Furthermore, any possible confounding effects of serial tests were eliminated because the different amounts of vertical nonius gap size were administered randomly during each sessions (on separate days).

Correlations between repeated measurements and between different vergence measures were tested with one-tailed levels of significance, because directional hypotheses apply in all cases. We used the computer program nlme-3.1 of the software package R.40

Back to Top | Article Outline


We recruited a sample of 12 subjects (aged 20–31 years, mean 22 years) meeting the criteria of a high visual acuity without glasses. The monocular visual acuity was 0.8 or better (median 1.25) and binocular visual acuity was 1.0 or better (median 1.6) in decimal units (tested with Landolt rings). Stereoacuity (TNO test) was at least 30 secarc and 60 secarc for crossed and uncrossed disparity (median 15 and 30 secarc), respectively. Forced vergence fixation disparity curves were measured with the Disparometer at 40 cm viewing distance in the range of ±6 prism diopters. These optometric variables were measured in a preliminary session, which also included two measurements of dark vergence.

Back to Top | Article Outline


Stability of Measurements

In each experimental run, subjects adjusted the continuous nonius lines twice (phases 1 and 3). Possible differences between these two phases we tested with a statistical mixed-effects model that included the gap size as fixed effect, a subject factor as random effect, and the phase1/phase3 factor nested in subjects; the variance induced by the two phases was <1% of the variance resulting from individual differences among subjects. Apparently, no relevant changes in these measures occurred during an experimental run, confirming the assumption that repeated tests within individuals are independent.

To describe the stability of the measures, test–retest correlations between the repeated tests were calculated. Each of the two experimental sessions (on separate days) included two tests with flashed nonius lines (thus, four data were available in all) and four tests with continuous nonius lines (eight data in all). We used the following bootstrap procedure for a complete description of test–retest correlations; independently for each of the 12 subjects, random combinations of a result of one experimental day with a result of the other experimental day were drawn from the sample. In this way, an arbitrary large number (i.e., 1024) of correlation coefficients was randomly drawn separately for each experimental condition. The resulting distribution is described by median, first and third quartile in Table 1. For the nonius bias, the median test–retest correlations reached an amount of up to approximately 0.75. At the largest nonius gap (7.9 deg), correlations were smaller (median values of 0.45 and 0.30), probably because the judgment of nonius offset was more difficult with a large nonius gap. The low test–retest correlation of only 0.29 with continuous nonius at the small gap of only 1 deg (when judgment is easy) is probably a result of the small interindividual variation in this particular condition (see Fig. 2). Irrespective of the amount of vertical nonius gap or the temporal mode of nonius presentation (i.e., over all conditions), we found a 90% confidence interval from 0.24 to 0.81 (median = 0.50) for the test–retest correlation of the nonius bias. For fixation disparity, the test–retest correlation had a 90% confidence interval ranging from 0.27 to 0.90 (median = 0.71) over all conditions. Thus, we can assume reliable measures in most of the conditions applied. The amount of these test–retest correlations can be used as a reference to evaluate the correlations between different conditions reported subsequently. Further analyses refer to the mean of the repeated measurements for each subject, separately for continuous and flashed nonius lines. In these individual mean values, the random variability between single experimental runs (also resulting from uncertainty in judging peripheral nonius lines) is largely eliminated, but systematic effects of the vertical nonius gap remain.





Back to Top | Article Outline

Continuous vs. Flashed Nonius Lines at Different Vertical Gap of Nonius Line

Figure 2 shows the dependent variables separately for each individual as the vertical nonius gap was increased. In some subjects, the nonius bias and the fixation disparity were unaffected by the nonius gap. However, other subjects showed a change to a larger amount (in the positive or negative direction) as the nonius gap was increased. Therefore, we computed the t-values of the slope for each individual. When the absolute value of t tended to be significant (p < 0.10, including a Bonferroni correction for the 12 individuals tested), bold lines are marked in Figure 2. The nonius bias showed four individual trends with continuous nonius lines and three with flashed nonius lines. For fixation disparity, flashed nonius lines revealed more individual effects than continuous nonius lines, five versus two cases. For corrected fixation disparity (i.e., the fixation disparity minus nonius bias), no individual trends occurred with continuous nonius lines but five with flashed nonius lines.

The changes in dependent variables as a function of nonius gap can also be described by linear slopes for each individual. These slope values were significantly correlated between nonius bias and fixation disparity for both continuous (r = 0.82, p < 0.001) and flashed (r = 0.66, p < 0.025) nonius lines, i.e., subjects with a positive (or negative) slope in nonius bias tended to have a positive (or negative) slope in fixation disparity. For continuous nonius lines, the rather high correlation of 0.82 resulted in a corrected fixation disparity that did not depend on vertical nonius gap in any subject. The pattern was different for flashed nonius lines; five cases with individually significant effects of nonius gap remained in the corrected fixation disparity. The different nature of the nonius gap effect for the two temporal modes of nonius presentation is also reflected by the fact that relatively small correlations appeared between the slopes of continuous versus flashed nonius lines for the nonius bias (r = 0.49), the fixation disparity (r = 0.38), and the corrected fixation disparity (r = 0.50; p < 0.05 if r > 0.50, one-tailed).

Previous research on effects of fusion contour on the visual direction of nearby nonius lines suggests that (1) the full amount of objective fixation disparity can be estimated with nonius lines having a large vertical gap and that (2) the subjective fixation disparity (in either the eso or exo direction) decreases in amount as the gap between nonius lines becomes smaller. To test these hypotheses, we consider the result at the 7.9-deg nonius gap as the best subjective estimation of true fixation disparity and use this measure to divide our sample in two groups with eso or exo fixation disparity; this was made separately for continuous and flashed nonius lines and implies that a few subjects had different directions (eso vs. exo) of fixation disparity in these two temporal modes of nonius presentation. Figure 3 shows the mean of these two groups. The eso groups showed a trend of increasing fixation disparity with increasing nonius gap (t = 1.84, p = 0.0671 for continuous nonius lines; t = 1.92, p = 0.0589 for flashed nonius lines), whereas the exo groups showed clearly significant effects (t = −4.76, p < 0.001 and t = −6.92, p < 0.001, respectively). However, this analysis inevitably overestimates the level of significance; because the groups have been formed on the basis of the 7.9-deg nonius gap, it is likely to find a smaller group difference at smaller nonius gaps simply by chance (regression to the mean). Therefore, the effect of the nonius gap is convincing only in the exo groups, particularly for flashed nonius lines in which four of the seven subjects had an individually significant trend; thus, the effect in the exo groups is very unlikely a result of regression to the mean.



Following Fogt and Jones,21 vergence adaptation is an individual parameter that might be related to the change in subjective fixation disparity as a function of vertical nonius gap. However, the steepness of the forced vergence fixation disparity curves was not significantly correlated with the effect of vertical nonius gap on fixation disparity (r < 0.30).

In the following analyses, we use correlations between fixation disparity in different experimental conditions to see which test conditions can be used to detect interindividual differences in fixation disparity (in view of effects of nonius gap that may occur in some subjects). This is important for the clinical application of subjective fixation disparity tests.

The correlations between continuous and flashed presentation of nonius lines are described in Table 2 based on the individual data averaged across the repeated measurements. The nonius bias showed significant correlations of 0.58 and 0.63 at the two small amounts of nonius gap, but not at larger gaps. The two fixation disparity measures had generally highly significant correlations in the range 0.69 to 0.88; only at the largest nonius gap, the correlation was insignificant. Thus, continuous and flashed nonius lines gave well-correlated results at smaller vertical nonius gaps, but not at large ones.



Figure 4 shows the correlations between small (1-deg) and large (7.9-deg) nonius gaps. For continuous nonius lines, high correlations of 0.92, 0.82, and 0.76 were found for the nonius bias, the fixation disparity, and the corrected fixation disparity, respectively. The corrected fixation disparity data cluster closely around the identity line; the regression line has slope that does not differ significantly from 1.0 (95% confidence interval [CI], 0.32 to 1.31) and the intercept does not differ significantly from 0.0 (95% CI, −2.7 to 1.0); this basically holds also for the fixation disparity; however, the slope tends to deviate from 1.0. Thus, the individual amount of these measures can be determined with small or large nonius gaps when nonius lines are presented continuously.



However, with flashed nonius lines, these correlations between small and large nonius gaps were insignificant for fixation disparity and corrected fixation disparity (r = 0.47 and r = 0.11, respectively), reflecting the individual shifts with increasing nonius gap in some subjects (Fig. 2). As already shown in Figure 2, the range of individual nonius bias values increased with the nonius gap; the range was much larger at 7.9 deg than at 1 deg.

For testing the physiological plausibility of fixation disparity measures, we calculated the correlations with dark vergence (using the mean of the two dark vergence tests). We expected that no fixation disparity occurs if dark vergence agrees with the actual viewing distance of 100 cm and that an eso or exo fixation disparity is found if the viewing distance corresponding to dark vergence is closer or more distant relative to 100 cm, respectively. As shown in Figure 5 for the corrected fixation disparity, such a correlation was found to be significant with small and large nonius gaps when continuous nonius lines were used (r = 0.50 at 1 deg and r = 0.54 at 7.9 deg, p < 0.05). However, with flashed nonius lines, a significant correlation with dark vergence was found for a 1-deg nonius gap (r = 0.66, p < 0.025), but not with a 7.9-deg nonius line gap (r = −0.19). For each of the three significant correlations in Figure 5, we found that the 95% confidence interval of the regression line at a dark vergence of 1.0 meter angle (corresponding to the 100-cm viewing distance) includes the corrected fixation disparity of zero. Table 3 shows the complete set of correlation coefficients for the two measures of fixation disparity and all amounts of the vertical nonius offset; these were generally significant, except for those with flashed nonius lines at large nonius separation.





To summarize, previous results (reviewed at the beginning of this article) suggest that continuous nonius lines at large gaps are likely to agree with objective fixation disparity, i.e., to be free of possible effects as a result of a nearby fusion stimulus. However, in the present conditions of a static fusion stimulus, continuous nonius lines at small and at large nonius gaps were well correlated, i.e., both measures reflect the individual amount of fixation disparity. Flashed nonius lines tended to produce deviating results at large nonius gaps, but the result with flashed nonius lines at the small 1-deg nonius gap was highly correlated with the result of continuous nonius lines at the large 7.9-deg nonius gap. This was found for the fixation disparity (r = 0.79, p < 0.001) and for the corrected fixation disparity (r = 0.78, p < 0.001).

Back to Top | Article Outline


It was the purpose of the present study to investigate the validity of nonius lines for measuring fixation disparity in test conditions that are typical for some clinical tests of fixation disparity, i.e., a static central fusion stimulus with corresponding stimuli for both vergence and accommodation (without forced vergence). In these conditions, fixation disparity amounts to only several minutes of arc and, therefore, is technically very difficult to measure objectively. Therefore, our interpretation relies on subjective measures of fixation disparity under conditions that are critical for the validity of nonius lines as suggested by the following previous studies (including objective measurement).

Erkelens and van Ee18,19 and Shimono20 had shown that nonius lines receive the visual direction of closely presented fusion stimuli (capture of visual direction) and, thus, are not valid indicators of vergence eye position (in dynamic vergence or different depth planes, respectively). Fogt and Jones21,22 had reported changes in retinal correspondence using a static fusion stimulus when forced vergence was applied. From these studies, it was suggested that a fusion stimulus close to the nonius lines may give invalid results of nonius tests; only if a sufficient spatial separation from the fusion stimulus is maintained, nonius lines may allow for a valid estimation of the objective vergence position (or movement). This conclusion is the basis of the following interpretation of our subjective measurements of fixation disparity as a function of the vertical gap between nonius lines and the temporal mode of nonius presentation, i.e., continuous versus flashed.

At larger vertical nonius gaps, the data include a larger measurement error because it is more difficult to accurately judge the nonius offset; this leads to some nonsignificant test–retest correlations between single measurements at large gaps (Table 1). This random measurement error is expected to be averaged to zero in the series of repeated measurements, whereas systematic effects will remain in the individual mean values. This interpretation is supported by the findings that in most cases, smooth curves as a function of vertical nonius gap were found and that the individual measures changed in a particular direction (in most subjects exo) as the vertical nonius gap increased. Furthermore, the slopes as a function of vertical nonius gap were correlated between the nonius bias and fixation disparity. These observations suggest that the described changes in subjective fixation disparity with vertical nonius gap represent a physiological effect, the nature of which is discussed subsequently.

When continuous nonius lines were used, the amount of the nonius gap effect was small relative to the range of individual differences so that, nevertheless, the resulting subjective fixation disparity measures were well correlated between different amounts of the nonius gap. Thus, a similar individual amount of subjective fixation disparity could be found with continuous nonius lines irrespective of the actual gap. However, a small vertical nonius gap has the advantage that the horizontal nonius offset can be judged easier.

With flashed nonius lines, a different pattern of result was observed. Fixation disparity was affected by the nonius gap in five of the 12 subjects; in most cases, the shift was in the exo direction. As a result of these shifts, the correlation between small and large nonius gaps was very low for flashed nonius lines. The changes in fixation disparity occurred only at amounts of the nonius gap exceeding 3.3 deg and not at smaller gaps (see Fig. 2). The constant amount of fixation disparity in the present study for small amounts of nonius gap agrees with the finding of Ukwade,9 who did not find any variation in fixation disparity for gaps between nonius lines up to 1.3 deg (using flashed nonius lines).

The origin of the individually different effects of the vertical nonius gap could not fully be identified. We only observed that subjects with an exo fixation disparity at a large nonius gap tended to have larger effects of the vertical nonius gap in the exo direction. Fogt and Jones21 had suggested that vergence adaptation is a relevant individual parameter for differences between subjective and objective fixation disparity. However, the slope of prism fixation disparity curve did not explain the individual effects in the present study.

As a control variable, we used a measure of vergence that does not include any fusion stimulus; dark vergence was tested with dichoptic points of light flashed in an otherwise dark visual field; thus, capture of visual direction cannot occur. We assume that fixation disparity measures are physiologically plausible if they are correlated with dark vergence (at least at a viewing distance of 100 cm, which is close to the population mean of dark vergence). Like in a previous study,29 we found that subjects tended to have an eso or exo fixation disparity if the viewing distance corresponding to dark vergence was closer or more distant than the test distance of fixation disparity (100 cm). This correlation with dark vergence was observed at all gaps with continuous nonius lines and at 1.0- and 3.3-deg gaps with flashed nonius lines. These were the conditions in which the fixation disparity measures had high correlations among each other, including the following three conditions that are important for the interpretation and practical application: 1. Continuous nonius lines at the large 7.9-deg nonius gap are assumed to give the best subjective estimation of objective fixation disparity. 2. Continuous nonius lines with a small nonius gap are used in clinical instruments as the Mallett unit.10 3. Flashed nonius lines with a small nonius gap are used in computer-controlled adaptive test procedures.41

The practically applied methods (2) and (3) have the advantage that the nonius offset can be judged easily with the small nonius gap and that compensation for nonius bias (which is unlikely to occur in the clinical situation) is much less important when the separation between the nonius lines is small. Computer-controlled adaptive testing avoids effects of expectation by the subject or experimenter and, furthermore, flashed presentation prevents the suppression of one of the two nonius lines that sometimes occurs with continuous nonius lines.

We conclude that these three conditions (1–3) gave physiologically plausible results shown by high intercorrelations and the correlation with dark vergence. It is true that this argument is based on subjective measurements alone, but it is supported by the objective measurements of Kertesz and Lee,42 and their subjective tests with central continuous nonius lines for a central fusion stimulus and a 4-deg peripheral surround at a 51-cm viewing distance. The authors emphasized that objective and subjective measures differed significantly in seven of 12 cases (four subjects, each tested at three levels of vergence demand), but they also mentioned that the two measures had the same direction (eso or exo) in 11 of these 12 cases. Thus, the subjective and objective measures were correlated. From the data published by Kertesz and Lee,42 we calculated a correlation of 0.60 for the condition without forced vergence and of 0.88 and 0.94 for a 4-deg forced vergence in the divergent and convergent direction, respectively. With only four subjects, only the highest correlation of 0.94 is significant, but finding three correlations of 0.60 and higher is unlikely the result of chance. A correlation between subjective and objective measures is sufficient for the clinical purpose to detect subjects with a large exo fixation disparity, because these subjects tend to complain of asthenopic symptoms.

Compared with the fixation disparity measures (1–3), flashed nonius lines at vertical gaps larger than 3.3 deg resulted in deviating results that were physiologically implausible since not correlated with dark vergence. This different pattern of result is not the result of a larger random measurement error at larger vertical nonius gaps, as argued previously. Rather, the following observations suggest that visual directions might be coded in a different way for nonius lines flashed in the periphery. The present nonius gap effect was stronger with flashed nonius lines than with continuous nonius lines. The relation between nonius bias and fixation disparity was different for the two modes of temporal nonius presentation; with continuous nonius lines, the corrected fixation disparity (i.e., fixation disparity minus nonius bias) was unaffected by the nonius gap in all subjects, whereas with flashed nonius lines, five of the 12 subjects showed a change of corrected fixation disparity in the exo direction with increasing nonius offset. The specific physiological origin of the present effects cannot be specified from the present data. However, the nature of the effect appears to differ from “capture of visual direction” as observed in dynamic vergence18,19 or in different depth planes20; the latter capture effect was present in all subjects and declined already at small amounts of eccentricity of nonius lines, whereas the present effect with flashed nonius lines occurred only in part of the subjects and was observed only at larger amounts of nonius eccentricity.

The present study investigated two extreme conditions of temporal nonius presentation: continuous nonius lines and those flashed shortly for only 100 ms. After differences between these conditions have been shown in the present study, it is interesting to explore intermediate conditions, e.g., nonius lines flashed for durations longer than 100 ms or nonius lines with a smooth on- and offset compared with the abrupt temporal profile used in the present study. This would allow for a systematic investigation of the temporal properties involved and to describe which temporal conditions are essential for the deviating results at large vertical gaps found with flashed nonius lines. It is sometimes discussed that nonius lines are flashed to avoid attempts on the part of the subject to fuse the nonius lines. This would require vertical fusion that is very unlikely if a strong central fusion stimulus is present like in the present study. Attempts of vertical fusion are impossible at large vertical gaps and therefore cannot account for the deviating results in this condition.

It should be noted that our conclusions are based on the present sample of subjects with good binocularity, that is, good stereoacuity and good ocular motility. It cannot be excluded that heterophoric subjects with sensory adaptation located only at the fovea might show more pronounced effects of the vertical nonius gap.43,44

Back to Top | Article Outline


This study was supported by the German Research Council (DFG Ja/747/4-1). The authors thank Ewald Alshuth and Matthias Bonacker for the software and technique.

Wolfgang Jaschinski

Ardeystr. 67, D-44139 Dortmund, Germany


Back to Top | Article Outline


1. Ogle KN, Martens TG, Dyer JA. Oculomotor Imbalance in Binocular Vision and Fixation Disparity. Philadelphia: Lea and Febiger; 1967.
2. Schor CM. Fixation disparity and vergence adaptation. In: Schor CM, Ciuffreda KJ, eds. Vergence Eye Movements: Basic and Clinical Aspects. Boston: Butterworth; 1983:465–516.
3. Carter DB. Parameters of fixation disparity. Am J Optom Physiol Opt 1980;57:610–7.
4. Sheedy JE, Saladin JJ. Validity of diagnostic criteria and case analysis in binocular vision disorders. In: Schor, CM, Ciuffreda KJ, eds. Vergence Eye Movements: Basic and Clinical Aspects. Boston: Butterworth; 1983:517–40.
5. Birnbaum MH. Optometric Management of Nearpoint Vision Disorders. Boston: Butterworth-Heinemann; 1993.
6. Scheiman M, Wick B. Clinical Management of Binocular Vision: Heterophoric, Accommodative, and Eye Movement Disorders. Philadelphia: Lippincott; 1994.
7. Evans BJW. Pickwell’s Binocular Vision Anomalies: Investigation and Treatment. Boston: Butterworth-Heinemann; 2002.
8. Griffin JR, Grisham JD. Binocular Anomalies: Diagnosis and Vision Therapy, 4th ed. Boston: Butterworth-Heinemann; 2002.
9. Ukwade MT. Effects of nonius line and fusion lock parameters on fixation disparity. Optom Vis Sci 2000;77:309–20.
10. Mallett RJ. The investigation of heterophoria at near and a new fixation disparity technique. Optician 1964;148:547–51.
11. Sheedy JE. Actual measurement of fixation disparity and its use in diagnosis and treatment. J Am Optom Assoc 1980;51:1079–84.
12. Carter DB. Studies of fixation disparity. II. Apparatus, procedure and the problem of constant error. Am J Optom Arch Am Acad Optom 1958;35:590–8.
13. Bonacker M, Schubert-Alshuth E, Jaschinski W. Precise placement of nonius lines on a personal computer screen for measuring fixation disparity. Ophthal Physiol Opt 1994;14:317–9.
14. Yekta AA, Pickwell LD, Jenkins TC. Binocular vision, age and symptoms. Ophthal Physiol Opt 1989;9:115–20.
15. Jaschinski W. Relation between static and dynamic aspects of vergence, estimated with a subjective test using flashed dichoptic nonius lines. Ophthal Physiol Opt 2004;24:399–410.
16. Wick B. Stability of retinal correspondence in normal binocular vision. Optom Vis Sci 1991;68:146–58.
17. Howard IP. Seeing in Depth, vol 1. Basic Mechanisms. Thornhill, Ontario, Canada: I. Porteous; 2002.
18. Erkelens CJ, van Ee R. Capture of visual direction: an unexpected phenomenon in binocular vision. Vision Res 1997;37:1193–6.
19. Erkelens CJ, Van ER. Capture of the visual direction of monocular objects by adjacent binocular objects. Vision Res 1997;37:1735–45.
20. Shimono K, Ono H, Saida S, Mapp AP. Methodological caveats for monitoring binocular eye position with nonius stimuli. Vision Res 1998;38:591–600.
21. Fogt N, Jones R. Comparison of fixation disparities obtained by objective and subjective methods. Vision Res 1998;38:411–21.
22. Fogt N, Jones R. The effect of forced vergence on retinal correspondence. Vision Res 1998;38:2711–9.
23. Jenkins TC, Pickwell LD, Yekta AA. Criteria for decompensation in binocular vision. Ophthal Physiol Opt 1989;9:121–5.
24. Pickwell LD, Kaye NA, Jenkins TC. Distance and near readings of associated heterophoria taken on 500 patients. Ophthal Physiol Opt 1991;11:291–6.
25. Jaschinski W. Fixation disparity at different viewing distances and the preferred viewing distance in a laboratory near-vision task. Ophthal Physiol Opt 1998;18:30–9.
26. Jaschinski W. The proximity–fixation–disparity curve and the preferred viewing distance at a visual display as an indicator of near vision fatigue. Optom Vis Sci 2002;79:158–69.
27. Wildsoet CF, Cameron KD. The effect of illumination and foveal fusion lock on clinical fixation disparity measurements with the Sheedy Disparometer. Ophthal Physiol Opt 1985;5:171–8.
28. Jaschinski W, Brode P, Griefahn B. Fixation disparity and nonius bias. Vision Res 1999;39:669–77.
29. Jaschinski W. Fixation disparity and accommodation for stimuli closer and more distant than oculomotor tonic positions. Vision Res 2001;41:923–33.
30. Owens DA. The resting state of the eyes. Am Scientist 1984;72:378–87.
31. Owens DA, Leibowitz HW. Perceptual and motor consequences of tonic vergence. In: Schor CM, Ciuffreda KJ, eds. Vergence Eye Movements: Basic and Clinical Aspects. Boston: Butterworth; 1983:25–74.
32. Jaschinski-Kruza W. Eyestrain in visual display unit users: viewing distance and the resting position of ocular muscles. Hum Factors 1991;33:69–83.
33. Jaschinski-Kruza W. Dark vergence in relation to fixation disparity at different luminance and blur levels. Vision Res 1994;34:1197–204.
34. Rosenfield M. Tonic vergence and vergence adaptation. Optom Vis Sci 1997;74:303–28.
35. Jaschinski W. Fixation disparity and accommodation as a function of viewing distance and prism load. Ophthal Physiol Opt 1997;17:324–39.
36. Lieberman HR, Pentland AP. Microcomputer-based estimation of psychophysical thresholds: the Best-PEST. Behav Res Meth Instr 1982;14:21–5.
37. Jaschinski-Kruza W. Effects of stimulus distance on measurements of dark convergence. Ophthal Physiol Opt 1990;10:243–51.
38. Jaschinski W. Methods for measuring the proximity fixation–disparity curve. Ophthal Physiol Opt 2001;21:368–75.
39. Jaschinski-Kruza W. Fixation disparity at different viewing distances of a visual display unit. Ophthal Physiol Opt 1993;13:27–34.
40. Ihaka R, Gentleman R. A language for data analysis and graphics. J Comput Graph Stat 1996;5:299–314.
41. Jaschinski W. The eye-test PC. Available at: Accessed July 21, 2005.
42. Kertesz AE, Lee HJ. Comparison of simultaneously obtained objective and subjective measurements of fixation disparity. Am J Optom Physiol Opt 1987;64:734–8.
43. Lie I, Opheim A. Long-term acceptance of prisms by heterophorics. J Am Optom Assoc 1985;56:272–8.
44. Lie I, Opheim A. Long-term stability of prism correction of heterophorics and heterotropics; a 5 year follow-up. Part I: heterophorics. J Am Optom Assoc 1990;61:491–8.

fixation disparity; nonius lines; visual directions; retinal correspondence

© 2005 American Academy of Optometry