**Purpose.**:
To describe a novel methodology by which to measure the Hirschberg ratio (HR) in infants. The methodology does not require fixation on specific points, and measurements are made while infants look naturally at a display.

**Methods.**:
The HR is calculated automatically from measurements of the direction of the optical axis, the position of the pupil center, and corneal reflexes in video images from an advanced two-camera eye-tracking system. The performance of the novel fixation-free procedure (FFP) was evaluated in 43 adults by measuring the average difference and the 95% limits of agreement with the standard fixation-based procedure (FBP). Repeatability of the HR measurements was evaluated by assessing the 95% limits of agreement between two independent measurements. Performance of the FFP was also evaluated in five infants.

**Results.**:
In adults, the average HR was 12.89 ± 1.22°/mm for FFP and 12.81 ± 1.22°/mm for FBP. FFP and FBP measurements were highly correlated (*r* = 0.95; *P* < 0.001). The 95% limits of agreement between FFP and FBP were ±0.86°/mm. The 95% limits of agreement of repeated measurements were ±0.66°/mm for FFP and ±0.77°/mm for FBP. In infants, the 95% limits of agreement of repeated measurements by FFP were ±0.63°/mm.

**Conclusions.**:
In adults, the FFP provides accurate measurements of the HR that are in excellent agreement with measurements by FBP. In infants, measurements of HR by FFP show the same repeatability and consistency.

^{ 1–3 }By shining a penlight toward the patient's eyes, the displacement of the light reflex (first Purkinje image) from the center of the pupil can be observed, allowing an estimate of the amount of ocular misalignment. Originally, this displacement was described in terms of proximity of the corneal reflex to ocular landmarks (pupil, iris, limbus).

^{ 1,3 }More recently, the test has been interpreted in a more quantitative form, and ocular misalignment is expressed by the displacement of the light reflex multiplied by a simple proportionality constant. This proportionality constant, which expresses the ratio between ocular rotation and reflex displacement, is called the Hirschberg ratio (HR).

^{ 4,5 }It can be expressed in either degrees per millimeter or prism diopters (Δ) per millimeter

^{ 2,6 }[Δ = 100 × tan(°)].

^{ 2,7,8 }several recent investigations using photographic

^{ 6,9–11 }and videographic

^{ 12–14 }techniques have measured a mean value of approximately 12.5°/mm (22 Δ/mm) and an intersubject variability of more than ±20% of the mean value. To measure HR, subjects are required to accurately fixate multiple (at least two) targets with known spatial coordinates. Given that it is impossible to reliably complete such a procedure with infants or young children, ocular misalignment in these groups is calculated by multiplying the average HR (22 Δ/mm) by the measured/estimated displacement between the corneal reflex and the center of the pupil for each subject. The use of the average HR introduces inherent uncertainty of ±20% to the estimate of ocular misalignment of each infant/young child. As surgeons attempt to perform corrective surgery for infantile esotropia at younger and younger ages,

^{ 15 }it has become important to develop more exact measurements of ocular misalignment for infants and young children.

^{ 16 }for estimation of the position and orientation of the eye in space. Performance of the FFP for estimation of the HR is compared, in adults, with the performance of the standard fixation-based procedure (FBP). Performance of the FFP is also evaluated in a study with five infants.

^{ 16 }(Vision 2020-RB; El-Mar Inc., Toronto, ON, Canada) was used to determine the coordinates of the pupil center and corneal reflexes in images from the system's two video cameras (Fig. 1). With these coordinates, the direction of the optical axis of each eye was estimated without any user calibration procedure.

^{ 16 }Displacement of the central corneal reflex (CR) from the virtual image of the pupil center (P) in each eye was calculated by back-projecting the corresponding pupil center and corneal reflex in each image (Fig. 1, inset: pupil center is marked by a cross, corneal reflexes are enclosed by small boxes) to their three-dimensional positions inside the eye. As subjects look at video images on the computer monitor of the eye-tracking system, a graph showing the horizontal component of the displacement vector [CR(

*x*) − P(

*x*)] versus the horizontal component of the direction of the optical axis is created (Fig. 2). The absolute value of the slope of a line, which was fitted to the data points using a robust-fit algorithm

^{ 17 }(to remove outliers), is an estimate of the HR (in degrees per millimeter).

*x*) − P(

*x*)] was calculated in the same manner as that used for the FFP. The HR was determined by the slope (absolute value) of a line that was fitted to the data that described the changes in the horizontal angular coordinates of the points on the screen as a function of [CR(

*x*) − P(

*x*)]. Each measurement of the HR was repeated twice (to determine repeatability).

^{ 18 }The correlation coefficient and the 95% confidence interval for the difference between the measurements were calculated for the first measurement of the HR by each procedure. A paired

*t*-test (α = 0.05) was used to test for bias between the measurements of the two procedures. The repeatability of each procedure was tested by calculating the average difference between two independent measurements and the 95% confidence interval for the difference.

^{ 19 }shows that HRs measured by both procedures had normal distributions (α = 0.01). Mean ± SD of the HR was 12.81 ± 1.22°/mm (22.74 ± 2.13 Δ/mm) with FBP and 12.89 ± 1.22°/mm (22.88 ± 2.13 Δ/mm) with FFP. The range of HRs in each procedure was approximately 5.5°/mm or 10 Δ/mm.

*r*= 0.95,

*P*< 0.001; Fig. 4A). The difference versus mean analysis

^{ 18 }showed that the average difference between procedures was −0.08 ± 0.44°/mm and that the 95% limits of agreement were ±0.86°/mm (Fig. 4B). Paired

*t*-test (α = 0.05) showed no statistically significant bias between the two procedures.

*r*= 0.95,

*P*< 0.001; Fig. 6A). The average difference between the right and left eyes was 0.06 ± 0.42°/mm, and the 95% limits of agreement were ±0.82°/mm.

*r*= 0.83). The average difference between right and left eyes was −0.02 ± 0.36°/mm, and the 95% limit of agreement between the left and right eyes was ±0.70°/mm (Fig. 7B).

^{ 13 }12.93 ± 1.23°/mm,

^{ 14 }and 12.81 ± 1.22°/mm in this study).

^{ 15,20,21 }Given that the standard test for the measurement of ocular misalignment, the alternate prism and cover test, cannot be used reliably in infants and very young children, the angle of deviation is often determined using the Hirschberg test, which relies on the HR to accurately estimate the angle of misalignment.

^{ 9,12,13,22 }and in the present study), it can introduce significant errors in the measurement of eye misalignment and, consequently, significant errors in determining the surgical dose. For example, if one used the standard average HR for adults (22 Δ/mm

^{ 13,14 }) to calculate the eye misalignment of one of the infants in our study who had an HR of 17.5 Δ/mm, the error in the surgical dose for 40 Δ of eye misalignment would be 10.3 Δ. This error in surgical dose would compromise the ability to achieve a postoperative alignment within 8 Δ of orthotropia, which is considered to be a favorable outcome for infantile esotropia.

^{ 23–25 }

^{ 13 }described an automated procedure that was successfully used to estimate angle kappa in infants and young children. In this procedure, infants and young children looked at a bright light in their central visual field, and the displacement of the corneal reflex of the light from the pupil center was measured in both eyes (the displacement in the nondeviating eye divided by the HR is angle kappa). By assuming mirror symmetry of angle kappa in the two eyes, the displacement of the corneal reflex from the pupil center in the deviating eye could be adjusted to account for angle kappa. This adjusted value multiplied by the personal HR of each infant or young child can improve significantly the accuracy of the Hirschberg test. A more accurate Hirschberg test should help in the planning of strabismus surgery or other interventions to correct eye misalignment in infants and young children.