Fig 1.
Definition of objective and subjective fixation disparity.
Eye position during monocular calibration of each single eye when the fellow eye is occluded and during binocular testing with stimulus shown in the inset, i. e., a central binocular target (XOX) and two vertical dichoptic nonius lines. The geometrically expected vergence angle V0 = 2 arc tan (p/2D) depends on the viewing distance disparity and the inter-pupillary distance p so that objective fixation disparity is the resulting vergence error oFD = V − V0. Subjective fixation disparity sFD = arc tan (d/D) is given by the angular amount of the nonius offset (Figure adopted from Schroth et al. [4]). In this case, the visual axes cross in front of the target which is referred to as eso fixation disparity with a positive sign. In other cases, the visual axes may intersect behind the target which means an exo fixation disparity with a negative sign. Note that the visual axes do not intersect with the nonius lines, since typically sFD is much smaller than oFD.
Fig 2.
Illustration of experimental setup and OLED display.
Upper photograph: Observer in the adjustable headrest with rests for the chin and the forehead; the cheekbones were fixed to prevent horizontal head movements; a flexible band around the head held the observer in the headrest. Shutter glasses provided dichoptic viewing of the nonius lines. The EyeLink II cameras had an unobstructed view of the eyes below the shutter glasses. The knob (at the left) allowed moving an occluder in front of each eye for the monocular calibration. The side-view camera (at the right) gave a video image that allowed placing the eye at the correct position with respect to viewing distance and height. Left lower photograph: visual stimulus with a central and peripheral binocular fusion target (XOX and outer frame) and dichoptic nonius lines¸ a single nonius line subtended 43 min arc. Right lower photograph: Two types of raw electronic devices with OLED displays before installation into the experimental setup.
Fig 3.
Mean subjective and objective fixation disparity as a function of viewing distance.
Viewing distance is given in the unit “1/meter”, which corresponds to the unit “meter angle”, i. e., a vergence angle between the visual axes, normalized to the vergence angle of 3.6 deg at the viewing distance of 1 meter. Positive and negative values mean an over-convergence (eso) or under-convergence (exo), respectively. Note that the scale is ten times larger for objective than for subjective fixation disparity and that the maximal values of the scales were chosen to agree with the larger ranges of the scatter of individual values in Figs 4–8. The black symbols at the viewing distances of 40, 30 and 24 cm refer to the data in the present study; for comparison, the grey symbols refer to results of previous studies (see Discussion).
Table 1.
Series of mixed-effects models (A, B, C, D) with increasing complexity for the dependent variables heterophoria, subjective and objective fixation disparity.
The goodness of fit of a model is given by the information criteria AIC and BIC; model improvements are tested by ANOVAs. Intercept (at the centered 30 cm viewing distance) and fixed effects are described by the mean ± standard error. Random effects are described by the Repeatability Index Ri = SDinter /SDintra, the ratio of inter- to intra-individual standard deviation. 120 observations were included (20 subjects, 2 sessions, 3 viewing distances); the 4 repetitions per condition were averaged in beforehand. The best model is highlighted: Model D for heterophoria, Model C for subjective and Model B for objective fixation disparity. If cells in the table are empty, these effects were not included in the corresponding model.
Fig 4.
Objective fixation disparity as a function of viewing distance for one session of an individual observer.
Viewing distance is plotted in the unit “1/meter” (see Fig 3). For each viewing distance, the four data points of the four runs per session were plotted and used to calculated a robust regression line that reduces the influence of outliers. The resulting y-intercept and slope are indicated.
Table 2.
Comparison of the two sessions with test-retest correlations and Bland-Altman analyses with respect to the mean value at 30 cm and the effect of viewing distance: For heterophoria (a), subjective (b) and objective fixation disparity (c) it is displayed (1) the group mean values ± SD of the two sessions, (2) the robust test-retest correlations (p ≤ 0.05 for r ≥ 0.38) and (3) mean differences ± SD between sessions.
Fig 5.
Robust regression between subjective and objective fixation disparity (intercept at 30 cm).
The results of Session 1 and Session 2 are shown separately by open and closed symbols and by broken and drawn lines, respectively. In the regression equations, asterisks indicate significant coefficients.
Fig 6.
Replotted data of subjective versus objective fixation disparity (see Fig 5).
Lines connect data points of Session 1 and Session 2 for each subject to illustrate the similarity of these two results; triangles refer to Session1 and circles to Session 2. Two subgroups were formed based on whether the mean objective fixation disparity was negative (exo, n = 12) or positive (eso, n = 8).
Fig 7.
Robust regressions showing the effect of heterophoria on subjective (A) and objective (B) fixation disparity (intercept at 30 cm). The results of Session 1 and Session 2 are shown separately by open and closed symbols and by broken and drawn lines, respectively. In the regression equations, asterisks indicate significant coefficients.
Fig 8.
Robust regression showing the effect of accommodation on subjective (A) and objective (B) fixation disparity (intercept at 30 cm). The accommodative performance is the product of the AC/A-ratio and the accommodative response at 30 cm. The results of Session 1 and Session 2 are shown separately by open and closed symbols and by broken and drawn lines, respectively. In the regression equations, asterisks indicate significant coefficients.