Fig 1.
Conditions of measuring fixation disparity.
Without prisms, the objective fixation disparity (oFD) is the difference between the observed vergence angle V during binocular recording and the stimulus vergence angle V0, which is geometrically given by the viewing distance D and the interpupillary distance p, i.e. V0 = 2 arctan (p/2D). In this example of an over-convergent (eso) oFD, V is larger than V0. The monocular components of V0 are measured during the eye tracker calibration that is made separately for the left and right eye: the eye position during monocular fixation represents the zero position for the subsequent binocular recording period. The covered eye assumes the heterophoria resting state. To correct an eso fixation disparity (as in this example), base-out prisms are applied. These prisms turn the visual axes optically outward (drawn lines), which requires the eye muscles to converge more (broken lines) to maintain fusion (see Fig 2). When prisms are applied, V0P = Prismpower + V0 is the stimulus vergence angle and Vp is the vergence angle. The fixation disparity with prisms is measured relative to the reference condition of monocular fixation when the prisms are worn. The subjective fixation disparity (sFD = arctan (dNon/D)) is illustrated by the amount of nonius offset dNon, which is typically smaller than the objective fixation disparity. Note that both types of fixation disparity are smaller with prisms than without prisms, as suggested by the study results. The graphs show the case of visual axes that intersect in front of the fixation point; this over-convergent state is referred to as eso fixation disparity with a positive sign. In the opposite under-convergent state, the visual axes intersect behind the fixation point (exo fixation disparity with a negative sign); in the latter case, base-in prisms are applied.
Fig 2.
Forced-vergence fixation disparity curves, replotted from Fogt and Jones [30].
Subjective and objective fixation disparity (sFD and oFD) are measured as a function of the forced-vergence angle for two individuals (B and E2). The zero position on the x-axis represents the baseline vergence angle, corresponding to the viewing distance of 73 cm in [30]. Relative to this baseline, the vergence stimulus is varied in the convergent (positive) and divergent (negative) direction, corresponding to base-out and base-in prisms, respectively. For fixation disparity, positive and negative values mean an over-convergence (eso) or under-convergence (exo), respectively.
Table 1.
Refractions and prisms for the 24 participants.
BO and BI refers to prisms base-out and base-in, BU and BD refers to prisms base-up and base-down.
Fig 3.
The MCH-method for prescribing prisms.
Six different test plates are applied at a 6 m viewing distance. Black parts are binocularly visible fusion targets, grey parts are presented dichoptically (i.e., separately to the two eyes). For scale: each of the four lines in the Cross test is 1.2 deg long.
Fig 4.
Testing of the dependent variables.
The three tests (Cross test, Mallett tests and Nonius bias tests) were used for measuring subjective and objective fixation disparity, without and with prisms, respectively. For scale: each of the four lines in the Cross test is 1.2 deg long.
Fig 5.
Example of a 1 minute recording period of fixation disparity.
The 22 green markers indicate when the observer had finished shifting the nonius targets to perceived alignment which represents a measure of subjective fixation disparity. At the same moments in time, the objective fixation disparity was sampled from the continuous stream of data (blue trace). This procedure provides simultaneous recordings of both types of fixation disparity. Regressions across the sampled data allow analysing the time trend within the 60 second recording period.
Table 2.
Results of the statistical linear mixed effects model, separately for (1) the objective and subjective measures and (2) the Cross test, Mallett tests, and Nonius bias tests.
Fig 6.
Subjective and objective measures for the three types of tests measured with and without prisms.
Box-plots are presented separately for base-out and base-in prisms. Positive and negative values mean an over-convergence (eso) or under-convergence (exo), respectively. In each box, the line indicates the median and the dotted line the mean. Significant mean prism effects are indicated by an asterisk and a line that connects the two mean values. Each box represents 12 participants.
Fig 7.
Relation between subjective and objective fixation disparity.
Robust regression analyses between these two measures in the two conditions with and without prisms, separately for the Cross test and the Mallett tests and participants who received base-in or base-out prisms (n = 12, respectively). Positive and negative values mean an over-convergence (eso) or under-convergence (exo), respectively. These robust correlations were significant as indicated by asterisks with one-tailed p < 0.005 if r > 0.70.
Fig 8.
Relation between individual prism effects sizes of subjective versus objective fixation disparity, i.e di(sFD) versus di(oFD).
Data are presented separately for base-in and base-out prisms (n = 12 participants, respectively) and for the Cross test and the Mallett tests. The grey quadrants indicate where individual data points are expected based on the hypotheses, that shifts should be in the positive (eso) direction with base-in prisms and in the negative (exo) direction with base-out prisms. The inner circle has a radius of 1.0 and indicates that individual data points near or outside the circle are statistically large. Regression lines, correlation coefficients and p-values (p < 0.05, two- tailed, indicated by asterisks) are based on robust regression analyses.
Fig 9.
Effect of the fusional vergence reserve on the individual prisms effect sizes.
Regression analyses are presented for subjective [di(sFD)] and objective [di(oFD)] prism effect size, separately for base-in and base-out prisms. The data of the Cross test and the Mallett tests were combined. The regression equations, correlation coefficients and two-tailed p-values are based on robust statistical procedures (n = 12).
Fig 10.
Interpretation of the relation between individual prism effects sizes of subjective versus objective fixation disparity, i.e. di(sFD) versus di(oFD).
The combined data of the Cross test and the Mallett tests are presented separately for base-in and base-out prisms. The grey quadrants indicate where individual data points are expected based on the hypotheses, that shifts should be in the positive (eso) direction with base-in prisms and in the negative (exo) direction with base-out prisms. The inner circle has a radius of 1.0 and indicates that individual data points near or outside the circle are statistically large. The regression equations, correlation coefficients and two-tailed p-values are based on robust statistical procedures (n = 12). The blue insets describe the working hypothesis of the underlying physiological mechanisms (see Discussion).