Clinical experience suggests that patients with preoperative best-corrected visual acuities (BCVAs) that are better-than-average are more likely to lose BCVA after corneal refractive surgery than are patients with preoperative BCVAs that are worse-than-average. To obtain objective evidence in support of this view, researchers have looked for a significant correlation between preoperative and postoperative acuity. Erdem and Muftuoglu1 found that patients with poor preoperative BCVAs had large improvements in postoperative BCVA, whereas those with good preoperative BCVAs had less improvement in postoperative BCVA. Before accepting such evidence, however, it is important to exclude confounding factors unrelated to the surgery that might account for this clinical observation. For example, changes in retinal magnification caused by moving a myopic spectacle correction to the corneal plane can alter BCVA.2 Alternatively, making only single measurements of acuity at each visit while dividing the patient sample into groups based on preoperative BCVA may lead to mistaken conclusions because of a statistical artifact known as regression to the mean.3–6 This study investigates the above issues, taking potentially confounding factors into account, to determine if this “clinical wisdom” could be placed into the domain of evidenced-based clinical practice and risk management.
Retinal magnification effects are a well-known consequence of moving a spectacle correction from the spectacle plane to the corneal plane as happens with contact lens corrections. A change in retinal image size could be partly responsible for the clinical and experimental observations of changes in BCVA after refractive surgery, because moving a myopic correction from the spectacle plane to the corneal plane magnifies the retinal image. Consequently, measuring acuity of myopic eyes with spectacle corrections before refractive surgery and without spectacles after surgery is equivalent to moving the acuity chart closer to the patient after surgery for some subjects. The resulting improvement in acuity could be partly due to a change in retinal magnification rather than the surgery per se.
Regression to the mean is a statistical phenomenon that occurs when a population is sampled nonrandomly. Creating a subsample of patients with worse-than-average BCVA is an example of nonrandom sampling of the greater population of patients who underwent corneal refractive surgery. Even if the surgery had no effect on BCVA, the mean BCVA of this subsample would tend to increase after treatment because of random fluctuations in BCVA for any given individual. One way to avoid regression to the mean is to take multiple measurements of BCVA for each individual, but this is rarely done in clinical practice. Consequently, no estimate is typically available for the mean acuity of each individual in the sample at any given time point of measurement. If such a surgical population is subsampled based on whether their initial acuity is better or worse than the total sample mean, it is expected (on statistical grounds) that the subsample of better eyes will have a mean acuity on retest that is slightly worse and the subsample of worse eyes will have a mean acuity on retest that is slightly better. That outcome is called regression to the mean and is independent of any treatment effect that can occur for the entire test population as a whole. A particularly clear explanation of regression to the mean and how to correct for it can be found on the web at http://www.socialresearchmethods.net/kb/regrmean.php.
Understanding the mechanism(s) behind the clinically and experimentally observed changes in BCVA after corneal refractive surgery is important because the effects have broad implications. For example, clinical trials in the Food and Drug Administration approval process used to assess safety and efficacy could be biased by selecting patients based on their preoperative BCVA. If a clinical study selected patients with preoperative BCVAs that were worse-than-average and measured BCVA only once, the study could have an increased probability of indicating better safety and efficacy than if patients with a full range of preoperative BCVAs were included in the study. The primary goal of our study was to experimentally examine whether the common clinical observation described above can be verified (i.e., do eyes with better-than-average preoperative BCVA, when measured once, have a greater risk of losing BCVA after custom laser in situ keratomileusis (LASIK) than eyes with worse-than-average preoperative BCVA?). A secondary goal was to explore possible explanations for the effect.
One eye of 79 myopic subjects formed the sample for analysis. This sample was derived from two studies having identical experimental protocols except for the subject's level of preoperative myopia and the measurement of the vertex distance. Data for one eye of 64 subjects (N64 eye study group) were provided by the United States Navy (Navy Refractive Surgery Center, San Diego, CA) and had a mean age of 33.1 ± 6.9 years (range, 23–51 years). The mean preoperative spherical equivalent (SE) refractive error was −3.16 ± 1.30 D and ranged from −1.00 D to −5.38 D. In this group, the vertex distance for the preoperative phoropter correction at the time of measurement was not recorded, forcing an estimated vertex distance to be used in the calculation of magnification (see subsequent section on magnification effects below).
We initiated a second study (the N15 eye study) to add eyes with higher preoperative refractive errors. The N15 eye study consisted of 15 eyes of 15 myopic subjects with a mean age of 29.9 ± 5.6 years (range, 20–40 years). The mean preoperative SE refractive error was −6.78 ± 1.45 D and ranged from −4.75 D to −10.38 D. These patients were recruited and underwent surgery at the same surgical center (Navy Refractive Surgery Center, San Diego, CA) as the N64 group. All patients signed an informed consent approved by one (N64 eye study) or both (N15 eye study) independent institutional review boards (University of Houston Committee for the Protection of Human Subjects and Naval Medical Center, San Diego Clinical Investigations Department), consistent with the Declaration of Helsinki.
For all eyes, ablation profiles were generated by importing the subject's wave aberration data (WaveScan aberrometer) into the VISX S4 excimer laser using CustomVue software (AMO, Abbot Park, IL). Before ablation, an IntraLase femtosecond laser (AMO) was used to cut a 9.5-mm LASIK flap. The minimum optical zone was set to 6.0 mm with a transition zone from 6.0 to 8.0 mm. The maximum optical zone was determined by the CustomVue software based on the relationship between sphere and cylinder and the total ablation zone (optical and transition zones). The goal of all treatments was emmetropia.
The following measurements and calculations were made (in order) both before and at 3 months after LASIK.
- Best correction: For all eyes, subjective refraction was measured once (using the technique of maximum plus to best visual acuity) to define the best correction for each subject. SE spectacle correction was defined as the sphere power plus half the cylinder power.
- Visual acuity: Acuity for each patient's study eye was measured once while viewing a back-illuminated (100 ± 8 cd/m2), translucent Early Treatment Diabetic Retinopathy Study (ETDRS)-type (Sloan letter) high contrast chart (Precision Vision, La Salle, IL) at 4 m through either a phoropter-based correction (N64 eye study) or trial frame correction (N15 eye study). For all eyes, credit was given for all letters read correctly until two letters were missed in any given line.
- Calculation of magnification effects on logarithm of the minimum angle of resolution (logMAR) acuity: For the N15 eye study, the vertex distance of the trial frame was recorded for each subject and used in conjunction with the SE spectacle correction to calculate the induced magnification. The change in magnification was calculated using the method described by Applegate and Howland2:
where ΔAcuity is the change in logMAR acuity due to moving the refractive correction from the spectacle plane to the corneal plane, V is the vertex distance (in meters), and SE is the preoperative SE refractive error (diopters).
To illustrate, consider a patient wearing a preoperative spectacle correction of −10 D at a vertex distance of 15 mm. When viewing a distant acuity chart during the preoperative acuity measurement, this eye will see a minified acuity chart (by 13%). If this eye views the same acuity chart after a successful emmetropic LASIK correction, it would experience an increase in acuity of −0.06 logMAR simply due to the magnification produced when shifting the spectacle correction to the corneal plane (13%). To fairly compare postoperative acuity to the preoperative acuity, we subtracted this magnification gain from the postoperative acuity. For example, if the measured postoperative acuity was −0.12 log MAR in our hypothetical eye with a preoperative best spectacle correction of −10 D (vertex = 15 mm), the magnification corrected postoperative acuity would be equal to −0.12 − (−0.06) = −0.06 logMAR.
As vertex distance was not recorded in the N64 eye study, the magnification correction was made assuming a 13-mm vertex distance. This vertex distance was selected after having three technicians at the Navy Refractive Surgery Center (San Diego) measure vertex distance on four patients using the same equipment and procedures as in the N64 eye study. The mean vertex distance of all measurements was 13 mm, with a range of 11 to 16 mm. Given that preoperative refractive errors were relatively low in this sample (average SE = −3.16 D), small variations in vertex distance (e.g., 11–16 mm) made only a small difference in final acuity (typically <1 letter).
- 4. Division into subsamples: The 79 study eyes were divided into two groups. One group comprised those eyes having better-than-average preoperative BCVA (<−0.11 logMAR, i.e., better than ∼20/16) and the other group comprised those eyes having average or worse-than-average preoperative BCVA (≥−0.11 logMAR, i.e., equal to or worse than 20/16).
- 5. Percentage regression to the mean: Given that the two subsamples from step 4 were not drawn from the total sample randomly, and acuity was only measured once preoperatively and once at the 3-month postoperative visit, the estimated percent regression to the mean (Prm) of the postoperative acuity for the two subsamples was calculated using the following formula:
where r is the correlation between preoperative and postoperative logMAR acuities of the total sample.
- 6. Corrected mean BCVA after surgery due to regression to the mean for the two subsamples: The predicted mean BCVAs for both subsamples after correction for regression to the mean (BE and WE) were calculated as follows:
where CF is the correction factor in logMAR acuity due to regression to the mean; Prm is the estimated percentage regression to the mean; Δ is the difference between the mean value of the measured preoperative logMAR acuity for the subsample of interest and the value of the mean measured preoperative acuity for the total sample; BE is the predicted mean postoperative logMAR acuity corrected for regression to the mean for the subsample with better-than-average preoperative acuity; WE is the predicted mean postoperative logMAR acuity corrected for regression to the mean for the subsample with average or worse-than-average preoperative acuity; BENC is the mean postoperative logMAR acuity not corrected for regression to the mean for the subsample with better-than-average preoperative acuity; and WENC is the mean logMAR acuity not corrected for regression to the mean for the subsample with average or worse-than-average preoperative acuity.
- 7. Test for a significant treatment and/or learning effect: A two-tailed paired t-test (p ≤ 0.05) was performed by comparing the mean magnification corrected acuity before and after surgery for the total sample.
- 8. Test for a significant difference between mean acuities measured before and after refractive surgery: We corrected the preoperative mean acuity of each subgroup for regression to the mean (i.e., we estimated the mean acuity of each subsample assuming that acuity had been measured a second time before refractive surgery). We then tested whether the mean preoperative acuity (corrected for regression to the mean) was significantly different than the actual postoperative acuity (corrected for only magnification effects) by using the standard error of the mean and assuming the variances remained the same pre- to postoperatively.
Whole Sample Results
Basic statistics for preoperative logMAR BCVA are provided in Table 1. The average preoperative BCVA for the N15 (−0.093 logMAR) and N64 groups (−0.118 logMAR) is consistent with other larger population studies of normal subjects.7 An unpaired t-test revealed no significant difference in average preoperative BCVA (p = 0.2692) between the N15 and N64 studies. We therefore combined the two samples to form the N79 sample for subsequent analyses.
Table 2 displays basic statistics for the study sample for preoperative acuity, postoperative acuity not corrected for magnification effects, and postoperative acuity corrected for magnification effects. The apparent improvement in visual acuity is reduced but not eliminated once the magnification correction is applied. Before correction for magnification, there was a small average gain in acuity of −0.05 logMAR (2.5 letters on an ETDRS-type chart) after surgery for the sample as a whole. A paired Student's t-test revealed this change in acuity to be significant (p < 0.0001). After compensating for the magnification effects, the mean gain was smaller (as expected). Nonetheless, there remained a small acuity benefit of −0.03 logMAR (∼1.5 letters on an ETDRS-type chart; p = 0.0009).
To evaluate the clinical observation that motivated our study, the N79 sample was subdivided into two groups based on preoperative BCVA. The first subsample consisted of 29 eyes with a preoperative visual acuity better than the mean acuity of the whole sample (<−0.11 logMAR). The 50 remaining eyes formed the second subsample consisting of eyes with a preoperative visual acuity equal to or worse than the mean acuity of the whole sample (≥−0.11 logMAR).
Table 3 specifies each subsample's statistics before correcting for regression to the mean. The difference in mean preoperative SE for eyes with better vs. worse-than-average preoperative acuity was not statistically significant (p = 0.576, unpaired t-test). We found a slightly larger than one line difference (0.114) in average preoperative BCVA between subsamples with better- (Table 3, A) and worse- (Table 3, B) than-average preoperative BCVA. As expected, this difference was statistically significant (p < 0.0001, unpaired t-test). Magnification effects accounted for a one letter decrease in the magnification corrected postoperative mean acuity (0.022 and 0.020 logMAR acuity for better- and worse-than-average eyes, respectively). Without magnification correction, eyes with better-than-average preoperative acuity experienced a mean change in acuity of −0.003 logMAR (<1 letter on an ETDRS-type chart), whereas eyes with worse-than-average preoperative acuity experienced an average gain of −0.077 logMAR (∼3 letters on an ETDRS-type chart). A similar observation was noted when incorporating magnification correction, as eyes with better-than-average preoperative acuity experienced a mean decrease in acuity of 0.018 logMAR (∼1 letter on an ETDRS-type chart), whereas eyes with worse-than-average preoperative acuity still experienced an average gain in acuity of −0.057 logMAR (∼3 letters on an ETDRS-type chart).
Correction for Regression to the Mean
To correct for regression to the mean, it is necessary to calculate the correlation coefficient (r) between preoperative acuity and magnification corrected postoperative acuity for the sample as a whole. This correlation was 0.427 for all eyes in the combined N79 sample. Thus, the percentage regression to the mean (Prm) for both the better-than-average and worse-than-average subsamples was calculated to be 57.24% using Eq. 2. Given the postoperative acuity measurement was a second measure of acuity; we can assume it includes the regression to the mean. Consequently, to properly compare preoperative acuity to postoperative acuity, the preoperative data need to be corrected for regression to the mean (i.e., we wish to compare preoperative eyes that have not experienced regression to the mean to postoperative eyes that have experienced regression to the mean). Given a percentage regression to the mean of 57.24%, the mean preoperative acuity of the better-than-average BCVA subsample regressed to a worse acuity by 0.042 logMAR and the mean preoperative acuity of the worse-than-average BCVA subsample regressed to a better acuity by −0.024 logMAR. These results are summarized in Table 4, which compares the preoperative mean acuities corrected for regression to the mean and the postoperative mean acuities corrected for magnification. The average gains in acuity for both subsamples were small (−0.023 logMAR and −0.033 logMAR for the better- and worse-than-average subsamples, respectively) but significant (p = 0.040 and p = 0.0018, respectively). These gains are most likely attributable to surgical and/or learning effects experienced by the test sample as a whole and are not unique characteristics of the subsamples.
Although the “clinical wisdom” was confirmed in our study, our analysis demonstrates that the majority of the observed effect is accounted for by the statistical phenomenon called regression to the mean. Nonetheless, correcting for regression to the mean did not eliminate the significant change in acuity for the sample as a whole. We calculated a statistically significant surgical and/or learning effect of −0.03 logMAR (∼1.5 letters on an ETDRS-type chart) in the total sample where eyes, on average, saw slightly better postoperatively than preoperatively. This gain cannot be accounted for by eyes having better- or worse-than-average preoperative acuity once the data are corrected for regression to the mean. This conclusion is supported by the observation that adding the learning and/or surgical effect to mean preoperative logMAR acuities (corrected for regression to the mean for both subsamples) yields predicted postoperative mean acuities that are not significantly different from the magnification corrected postoperative acuities. Future studies should include a control group to test for a possible learning effect so that determination of a treatment effect vs. a learning effect could be made. Although it is exciting to ponder a true treatment effect in BCVA after custom LASIK, it is debatable whether this statistically significant gain in acuity has a significant amount of clinical relevance.
It is worth emphasizing that the issues associated with regression to the mean can be minimized by measuring acuity pre- and postsurgery several times to establish a good estimate of each patient's average acuity before and after surgery. If multiple measures were routinely gathered, the change in each patient's mean acuity in the sample would be more predictive without the need to correct for regression to the mean. It is also important to note that retinal image quality at any given acuity can vary widely. As a consequence of this variance in quality of vision, a given patient may not lose or gain acuity after treatment and yet may experience an increase or decrease in retinal image quality that makes the patient pleased or dissatisfied with their surgical outcome.8,9
In conclusion, the clinical observation that eyes with better-than-average preoperative acuity are at risk of losing acuity and eyes with worse-than-average preoperative acuity stand to gain acuity is accounted for by two factors: retinal magnification due to moving the refractive correction from the spectacle plane to the cornea and regression to the mean. After appropriate corrections for these two factors, the mean acuity for the study sample improved a small but significant amount (∼1 letter on an ETDRS-type chart) after custom LASIK.
We thank the members of the Navy and Marine Corps and their families who volunteered to participate in our research. We also thank Dr. Harold Bedell for statistical advice and reading/commenting on the manuscript, and Dr. Ying Sheng Hu for confirming the regression to the mean calculations. This work was supported by the United States Air Force, Air Force Institute of Technology Scholarship (to MTA). This work was also supported by grants from the National Eye Institute, National Institutes of Health, Department of Health and Human Services, Bethesda, MD (R01 EY08520 and R01 EY019105; to RAA), University of Houston, College of Optometry Core grant P30 EY07551, and the Borish Endowment funding for the Chair of Optometry (to RAA).
The views expressed in this article are those of the authors and do not reflect the official policy or position of the Department of the Air Force, Department of the Navy, Department of Defense, or the United States Government. None of the authors has a conflict of interest with the contents of this manuscript.
Michelle Thomas Aaron
USAF School of Aerospace Medicine
2507 Kennedy Circle
Brooks City Base, TX 78235-5116