Regression methods when the eye is the unit of analysis

Abstract

Purpose

The eye is often the unit of measurement for outcomes, and frequently also for covariates, in vision research, and measurements in the two eyes of the same person are often strongly but far from perfectly correlated. Advances have occurred in development and accessibility of analytic approaches to evaluate determinants of eye-specific outcomes including information from both eyes of some subjects.

Methods

We illustrate available regression approaches to analyze correlated outcomes from both eyes in data sets with both eye- and subject-specific exposures and potential confounding variables. We consider cross-sectional and longitudinal study designs, and discrete, continuous, and time-to-event outcomes.

Results

Across a range of study designs and measurement scales for the outcome variable, we show the under-estimation of P-values and widths of confidence intervals that occurs when the correlation between paired eyes in a person is ignored, and the reduced precision that occurs in separate analyses of right or left eyes, or in analyses of persons rather than eyes. By comparison, regression models with the eye as the unit of analysis and appropriate consideration of the correlation between paired outcomes generally offer maximal use of available data, enhanced interpretability of covariate-outcome associations, and efficient use of information from subjects who contribute only one eye to analyses.

Conclusions

For many studies in vision research, the now widely available regression models that appropriately treat the eye as the unit of analysis offer the best analytic approach.

Introduction

The unit of assessment and analysis in vision research is often the eye, and both covariates and outcomes in one eye will usually have a strong correlation with measures of these variables in the other eye of the same person. It has long been recognized that regression models that relate outcomes in an eye to covariates and include observations from both eyes of at least some study subjects must account for the correlation among these paired observations in a person.^1-5 The need to account for this correlation applies to models for continuous and discrete outcomes, to studies with cross-sectional and longitudinal designs, and, in particular, to analyses of time to event (survival) outcomes.

Recent advances in statistical methods and complementary software development have increased availability and accessibility of alternative regression approaches applicable for analysis of paired eye data. In this paper, we illustrate and contrast alternatives, and include an appendix with computer code to implement these approaches.

Data Sources and statistical models

We illustrate alternative analytic approaches with data from four studies, each based on human subjects research approved by institutional review boards at Harvard teaching hospitals.

(a) Percent of normal visual field (continuous outcome in a cross-sectional study)

One data set, used to compare alternative regression approaches with a continuous outcome, included measurements from 394 eyes in 197 older individuals (age ≥ 65 years) who were seen for treatment or were evaluated for possible diagnosis of glaucoma in the Glaucoma Consultation Service of the Massachusetts Eye and Ear Infirmary in 1987 and 1988. These patients participated in a study of falls associated with glaucoma⁶ and had bilateral perimetry via an Octopus perimeter. The percentage of normal visual field in an eye was the average threshold in the central 30° standardized by the normal value for a 65 year old person. Distance visual acuity was assessed with the use of spectacles, with and without pinhole, and the better of the two measures was taken as the Snellen acuity in an eye. Acuity was then transformed to a measure of percent impairment as described by Spaeth et al.⁷ Other measures assessed at glaucoma examination included phakic status in each eye and history of systemic hypertension.

To analyze these data, we fitted linear regression models to predict the percent of normal visual field based on characteristics of study subjects including their age (in years), sex, and whether they had a diagnosis of systemic hypertension; and characteristics of their eyes (possibly differing between right and left eyes), including whether an eye was phakic or not, and its percent of impairment. Thus, models took the form:

$$Y_{ij}=\alpha +{\beta }_{1}ag{e}_i+{\beta }_{2}se{x}_i+{\beta }_{3}hpt{n}_i+{\beta }_{4}ph{k}_{ij}+{\beta }_{5}pim{p}_{ij}+{∊}_{ij}$$

where Y_ij is the percent normal visual field in the jth eye of the ith subject, age_i is age in years, sex_i is gender (0=male, 1=female), hptn_i indicates whether systemic hypertension is present (1=yes, 0=no), phk_ij is the phakic status (0=phakic, 1=aphakic), pimp_ij is the percent impairment in visual acuity, and ε_ij is the error term.

Regression coefficients are interpretable as the difference in percent normal field per unit difference in a covariate, adjusted for other variables in the model. We compared five alternative regression models: ordinary least squares regression analysis including all eyes and ignoring the correlation between right and left eyes, separate analysis of right eyes, separate analysis of left eyes, a mixed effects model with a random subject effect added to the above model to account for the correlation between eyes, and a linear regression model including all available eyes fitted by a generalized estimating equation approach that assumed an exchangeable correlation between eyes. For this dataset, all models were fitted in SAS, using the REG, MIXED, and GENMOD procedures, and sample code is included in the Appendix.⁸

(b) Impaired visual acuity (dichotomous outcome in a cross-sectional study)

To illustrate methods for analysis of paired dichotomous outcomes, such as arise in cross-sectional or cumulative incidence studies with the eye as the unit of analysis, we used measurements from 888 eyes in 444 individuals with retinitis pigmentosa who were seen at the Berman-Gund Laboratory of the Massachusetts Eye and Ear Infirmary, Boston, MA, from 1970 until 1979. These 444 individuals were all unrelated. An outcome of interest in this study⁹ was whether best-corrected visual acuity in an eye was 20/50 or worse. The main exposure of interest was genetic type of retinitis pigmentosa, which was determined on the basis of a detailed family history as autosomal dominant, autosomal recessive, sex-linked, or isolate retinitis pigmentosa. Potential confounding variables were age, sex, and diagnosis of posterior subcapsular cataract.

We fitted logistic and related polytomous logistic regression models to predict whether or not an eye had a visual acuity deficit. Person-specific covariates included in the models were age in years, sex, and genetic type of retinitis pigmentosa (autosomal dominant, autosomal recessive, sex-linked or isolate), whereas presence of posterior subcapsular cataract was an eye-specific covariate. Thus, the logistic regression model took the form:

$$\log [p_{ij}∕(1-p_{ij})]={\beta }_0+{\beta }_{1}DO{M}_i+{\beta }_{2}A{R}_i+{\beta }_{3}S{L}_i+{\beta }_{4}ps{c}_{ij}+{\beta }_{5}ag{e}_i+{\beta }_{6}se{x}_i$$

Coefficients are interpretable as the log odds of a visual acuity deficit in an eye per unit increase in the covariate, adjusted for other variables in the model. The five models we compared, each including the same covariates, were: multiple logistic regression with all study eyes and without consideration of the correlation between outcomes in right and left eyes; separate multiple logistic regression models in all right eyes and in all left eyes; a logistic regression model including all available eyes fitted by a generalized estimating equation approach that assumed an exchangeable correlation between outcomes in each eye; and a polytomous logistic regression model to predict pairs of outcomes and with constraints so that it is equivalent to the beta-binomial model described by Rosner, and also to a conditional logistic regression model that estimates the effects of covariates conditional on the outcome status of the other eye.^4,10 We used Proc LOGISTIC in SAS for the first three approaches,⁸ Proc Genmod in SAS for the fourth,⁸ and the MLOGIT command in STATA for the final model.¹¹ The Appendix includes sample code.

(c) Progression of diabetic retinopathy (time-to-event analysis)

To illustrate methods for analysis of paired survival outcomes, based on independent evaluations of time until disease occurrence in each eye, we used data on the time until progression of diabetic retinopathy in each eye of 478 patients with insulin-dependent diabetes mellitus who participated in the Sorbinil Retinopathy trial.¹² These subjects were aged 18-56 years at study entry, had begun insulin treatment prior to age 40 years, and had received 1-15 years of insulin treatment. Eligible patients were required at screening to have a total glycosylated hemoglobin level of more than 9.25 percent (normal range for the study control laboratory was 6.0-8.8 percent) and absent or very mild retinopathy (five or fewer microaneurysms and no other abnormalities in each eye). Examinations of patients included fundus photographs of each eye at a screening visit, at randomization, and then 12 months after randomization, and subsequently at 9-month intervals. During the last 4 months of the trial, all participants were scheduled for a final visit including bilateral fundus photographs. The Fundus Photograph Reading Center graded the photographs of each eye independently based on the Early Treatment Diabetic Retinopathy Study (ETDRS) adaptation of the modified Airlie House classification.¹³ The primary endpoint of the Sorbinil Retinopathy trial was a two or more step progression of retinopathy during follow-up, based on a person-specific composite measure of retinopathy formed by combining the levels in the two eyes. In addition to the time until such progressions, we also consider here a two-step progression in each eye separately, as this perspective is required to evaluate the strength of association between rates of progression in each eye.

We fitted six different proportional hazards models, each including five person-specific covariates assessed at the beginning of follow-up: glycosylated hemoglobin (percent), duration of diabetes in years, diastolic blood pressure in mmHg, serum total cholesterol in mg/dL, and randomized Sorbinil assignment (1=yes, 0=no). One model additionally included the time-varying progression status of the contralateral eye.¹⁴ Thus, survival models took the form

$$\log \left[h\left(t_{ij}\right)\right]=\log \left[h_0\left(t_{ij}\right)\right]+{\beta }_{1}tg{h}_i+{\beta }_{2}dur\_d{m}_i+{\beta }_{3}db{p}_i+{\beta }_{4}cho{l}_i+{\beta }_{5}sor{b}_i,$$

where, h(t_ij) is the hazard of progression at time t in the jth eye of the ith person, h₀ is an unspecified baseline hazard, tgh_i is glycosylated hemoglobin, dur_dm_i is duration of diabetes, dbp_i is diastolic blood pressure, chol_i is total cholesterol, sorb_i is an indicator of treatment assignment, and an additional model added a term β₆ times a time-varying, eye-specific indicator of whether the contra-lateral eye had progressed at that time. Model parameters are interpretable as the log-relative hazard of progression associated with a 1 unit higher level of a covariate, controlling for the other variables in the model. The six models were: (1) a proportional hazards model including all available eyes without consideration of the association between progression in right and left eyes; (2,3) separate proportional hazards models in all right eyes and in all left eyes; (4) a proportional hazards model with the subject as the unit of analysis and progression defined on the basis of a two-step progression on a person-specific composite measure of retinopathy in both eyes (as indicated in the trial protocol); (5) a proportional hazards model that includes all available eyes but uses a robust estimate of the variance of estimated parameters to account for correlation between right and left eyes; (6) and a similar model with robust estimates of variance that adds the time-varying status of the contralateral eye as a covariate. We used Proc PHREG in SAS⁸ to fit all survival models, and obtained robust variance estimates according to the approach of Lee et al,¹⁵ as illustrated in the example SAS code in the Appendix.

(d) Change in Humphrey field in a clinical trial setting (continuous outcome in a longitudinal study)

In a recent clinical trial, 240 retinitis pigmentosa (RP) patients were randomized to either lutein(12 mg/day) and vitamin A (15,000 IU per day) or placebo and vitamin A (15,000 IU per day) and were followed annually for 4 years.¹⁶ The inclusion criteria were: (a) age 18-60 (b) non-smoking at baseline (c) Humphrey 30-2 total point score of >= 250 dB, and (d) visual acuity of 20/100 or better. The latter two criteria were applied separately for each eye. Thus, some patients had 1 or 2 eligible eyes at baseline. Patients were seen annually for 4 years and the primary outcome variable was the rate of change in Humphrey 30-2 central visual field (VF) over 4 years. The secondary outcome was the rate of change in Humphrey 60-4 mid-peripheral VF over 4 years. For simplicity, we focus here on analysis of change in Humphrey 60-4 mid-peripheral VF.

One possible approach to analyzing these data is to use the person as the unit of analysis and make the primary outcome variable the average VF over all available eyes at a given visit (i.e., VF_OU). However, there are sometimes substantial differences between both the level and rate of change of VF_OD and VF_OS. Also, one may not have the same set of eyes available at each follow-up visit due to missing data. Thus, if VF_OU is used as the summary measure, then one may obtain a biased estimate of the rate of change for some subjects. Therefore, it seems more appropriate to use the eye as the unit of analysis.

With the eye as the unit of analysis, there are a number of challenging issues in analyzing these data. First, there is correlation between 2 eyes of the same person at a given visit (i.e., cross-sectional correlation). Second, there is correlation between VF in the same eye followed over time (i.e., longitudinal correlation). Third, there is cross correlation between VF_OD at time t₁ and VF_OS at time t₂. Fourth, there is possible non-normality of the distribution of VF.

Thus, we used the following mixed effects regression model:

$$y_{ijt}=\alpha +u_i+{\beta }_{1}t+{\beta }_{2}tr{t}_i+{\beta }_{3}ey{e}_i+{\beta }_4{t}^{\ast }tr{t}_i+{\beta }_5{t}^{\ast }ey{e}_j+e_{ijt},$$

where y_ijt is the visual field in the jth eye of the ith patient at time t, u_i is a random effect for the intercept which is assumed to follow a normal distribution with mean 0 and variance σ², trt_i is the treatment assigned to the ith patient (1 if lutein, 0 if placebo), eye_j =1 if OD, =0 if OS, t =time from baseline in years taking values 0, 1,…, 4, and e_ijt is the error term. Thus, β₁ is an estimate of the rate of decline of visual field in the control group, β₁ + β₄ is an estimate of the rate of decline of visual field in the lutein group, β₂ is an estimate of the average difference in visual field at baseline between the lutein and control groups, β₃ is an estimate of the average difference in visual field at baseline between right and left eyes, and β₅ is an estimate of the difference in rate of decline of visual field between right and left eyes. This provides a specification of the effects of covariates on the baseline level and rate of change of visual field over time (i.e. the “mean model”).

Another important element of mixed effects models is their covariance structure. This is complex in this setting because there is correlation between visual field in right and left eyes at each time, correlation over time between repeated measures of visual field in a specific eye, and cross-correlation between visual field measured at one time in the right eye and visual field measured at another time in the left eye of the same person. Furthermore, the variance of visual field measurements may vary between right and left eyes, as well as over time. To accommodate these effects, we used the UN@UN correlation structure available in SAS Proc MIXED, the features of which are summarized as follows:

$$\begin{array}{cc}\hfill \text{var}\left(e_{ijt}\right)& ={\sigma }_{jt}^2\hfill \\ \hfill corr(y_{i{j}_1{t}_1},y_{i{j}_2{t}_2})& ={{\Sigma }_{time}}^{\ast }{\Sigma }_{eye}\hfill \\ \hfill & =\left(\begin{array}{ccc}\hfill \hfill & \hfill OD\hfill & \hfill OS\hfill \\ \hfill OD\hfill & \hfill {\Sigma }_{time}\hfill & \hfill {\rho }_{12}{\Sigma }_{time}\hfill \\ \hfill OS\hfill & \hfill {\rho }_{12}{\Sigma }_{time}\hfill & \hfill {\Sigma }_{time}\hfill \end{array}\right).\hfill \end{array}$$

where the * denotes elementwise multiplication.

In this context, ${\sigma }_{jt}^2$ is the residual variation for the jth eye at time t, which is allowed to vary for both j and t, ${\Sigma }_{time}$ represents the longitudinal correlation for the same eye over time and is specified as an unstructured correlation structure, whereby ${\Sigma }_{time,t_1{t}_2}=a_{t_1{t}_2},t_1,t_2=0.1\dots ,4$ and Σ_eye represents the cross-sectional correlation between fellow eyes at the same time, whereby

$${\Sigma }_{eye,jj}=1and{\Sigma }_{eye,12}={\Sigma }_{eye,21}={\rho }_{12}.$$

${\rho }_{12}{\Sigma }_{time}$ represents the cross-correlation between visual field in the right eye at one time and in the left eye at another.

There are also other correlation structures available in SAS for longitudinal clustered data. Specifically, the UN@CS correlation structure would replace the Σ_time matrix by a compound symmetry correlation structure, where correlation between repeated measures over time in a given eye are all the same (i.e. a_t1t2 ≡ a ). Similarly, the UN@AR correlation structure would assume an autoregressive correlation structure over time, whereby ${\Sigma }_{time,t_1{t}_2}={\rho }^{\mid t_1-t_2\mid }$. We did not feel comfortable with imposing either of these more specific longitudinal correlation structures; hence we used an unstructured longitudinal correlation which allowed correlation over time to be arbitrary. However, as an alternative simple model, we used generalized estimating equations to fit a linear model with the eye and time as repeated measures, and a working independence covariance structure.

Results

Continuous outcome in a cross-sectional study

Table 1 summarizes the cross-sectional relationships of patient characteristics with percent of normal visual field in 197 patients seen in a glaucoma consultation clinic. Estimates in the first column, based on ordinary linear regression without consideration of the correlation between paired eyes, can be unbiased under reasonable assumptions, but their standard errors are generally too small so confidence intervals and hypothesis tests based on this approach are generally invalid. Separate analyses of right and left eyes (columns 2 and 3) are valid but vary somewhat between eyes so the best overall estimate of the relationship of a variable with the outcome is unclear, and it can occur that a variable that is not significantly related to the outcome in each eye separately (e.g. age), is found to be significant in a more efficient analysis that appropriately includes data from both eyes. The random effects and estimating equation approaches agreed in their estimated effects, but the former approach had smaller standard errors for the eye-specific covariates while the latter approach had smaller standard errors for the person-specific covariates. An advantage of the mixed model approach is that it allows for estimation of the intraclass correlation coefficient to quantify the agreement between eyes in the same person in the outcome variable, here the percent of normal visual field. The estimating equation approach with exchangeable correlation structure also obtains a measure of association between outcomes in the two eyes of a person which was similar to the estimated intraclass correlation coefficient in this case.

Table 1.

Estimated linear regression coefficients from alternative models predicting percent of normal visual field in 197 patients followed up for glaucoma

	Standard linear regression
Variable	394 eyes	197 right eyes	197 left eyes	Mixed effects model	Estimating equation
Age (SE)	−0.53 (0.20)†	−0.52 (0.28)	−0.51 (0.29)	−0.57 (0.25)*	−0.57 (0.23)*
Female sex (SE)	0.31 (1.9)	2.7 (2.6)	−2.1 (2.6)	0.15 (2.3)	0.15 (2.2)
Hypertensive (SE)	−3.1 (1.9)	−1.7 (2.7)	−4.2 (2.7)	−3.0 (2.3)	−3.0 (2.3)
Aphakic (SE)	−18.4 (3.8)	−23.4 (5.6)‡	−14.0 (5.3)†	−18.5 (3.5)‡	−18.5 (5.8)†
% Acuity loss	−0.48 (0.08)‡	−0.42 (0.11)‡	−0.56 (0.12)‡	−0.41 (0.07)‡	−0.41 (0.09)‡
Correlation between eyes				0.55	0.56

^*P<0.05;

^†P<0.01;

^‡P<0.001

Dichotomous outcome in a cross-sectional study

A logistic regression analysis that includes data from two eyes of some study subjects but ignores the correlation between outcomes in these eyes will generally yield estimated standard errors and P-values that are biased towards smaller (more significant) values (Table 2, column 1). Separate logistic regression analyses of right and left eyes is generally an inefficient approach that does not yield a single summary estimate of the effect of each variable on the outcome. The polytomous logistic and estimating equation approaches yield similar and principled P-values for evaluation of covariate effects. The effect estimates from the polytomous model have a conditional interpretation, i.e. they reflect the impact of covariates conditional upon the outcome status of the other eye, and are thus generally closer to the null than estimates from the estimating equation approach. Note, however, that standard errors are also smaller in the polytomous model. An advantage of the polytomous model is that it allows for explicit prediction of the probability that an individual is bilaterally affected, which may be a question of clinical interest.

Table 2.

Estimated logistic regression coefficients from alternative models predicting visual acuity deficit (20/50 or worse) in 444 patients with retinitis pigmentosa

	Standard logistic regression
Variable	888 eyes	444 right eyes	444 left eyes	Estimating equation	Polytomous logistic
Age (SE)	0.017 (0.0047)‡	0.017 (0.0067)‡	0.016 (0.0067)*	0.018 (0.0059)‡	0.0098 (0.0037)†
Male sex (SE)	0.069 (0.14)	0.050 (0.20)	0.089 (0.20)	0.071 (0.18)	0.041 (0.11)
Genetic type  DOM§ (SE)	−0.55 (0.24)*	−0.67 (0.34)*	−0.43 (0.34)	−0.54 (0.30)	−0.33 (0.19)
AR§ (SE)	0.35 (0.20)	0.42 (0.29)	0.27 (0.29)	0.34 (0.26)	0.21 (0.16)
SL§ (SE)	1.00 (0.32)†	0.83 (0.44)	1.20 (0.47)	1.02 (0.42)*	0.60 (0.25)*
PSC§ cataract (SE)	0.44 (0.15)†	0.41 (0.21)*	0.47 (0.21)*	0.36 (0.17)*	0.28 (0.12)*
Correlation between eyes				0.66

^*P<0.05;

^†P<0.01;

^‡P<0.001

^§Abbreviations: DOM, autosomal dominant; AR, autosomal regressive; SL, sex-linked; PSC, posterior subcapsular

Time to event analysis

A Cox proportional hazards analysis which considers time until progression of disease separately in right and left eyes of at least some study subjects will generally under-estimate the standard errors of covariate effects if the correlation between disease progression in right and left eyes is not considered (Table 3, column 1). Separate analyses of right and left eyes yields valid estimates (Table 3, columns 2 and 3), but any differences between estimated effects between right and left eyes complicate interpretation. One way to resolve these issues is to define both progression and covariate values at the level of the person (Table 3, column 4). Enhanced statistical power and generally more meaningful estimates of eye-specific covariates are obtained in analyses that consider the eye as the unit of observation (Table 3, columns 5 and 6). For example, evaluation of the impact of time-varying progression in the other eye requires analysis at the level of the eye. This could be of clinical interest to guide the frequency of monitoring or the consideration of interventions in individuals at high risk of bilateral progression. If the status of the contralateral eye is considered to be a time-varying covariate, one obtains a conditional interpretation of covariate effects, and these effects are generally attenuated, relative to the marginal model that ignores the status of the other eye. The model without time-varying covariates is particularly relevant for evaluation of the long-term relationship of a set of covariates measured at a single time with retinopathy progression. One could argue that progression in the other eye is on the causal pathway and its inclusion in the model could therefore distort the effect of the baseline covariates on progression.

Table 3.

Estimated coefficients from alternative proportional hazards models predicting progression in diabetic retinopathy (2-step progression) in 478 patients with insulin-dependent diabetes mellitus

	Standard proportional hazards model
Variable	956 eyes	478 right eyes	478 left eyes	478 people	Estimating equation	Estimating equation
Glycosylated hemoglobin (SE)	0.24 (0.035)‡	0.27 (0.048)‡	0.22 (0.052)‡	0.23 (0.040)‡	0.24 (0.039)‡	0.23 (0.038)‡
Duration of diabetes (SE)	0.077 (0.024)†	0.11 (0.032)‡	0.039 (0.037)	0.077 (0.027)†	0.077 (0.025)†	0.076 (0.023)†
Diastolic blood pressure (SE)	0.032 (0.0088)‡	0.024 (0.012)*	0.039 (0.013)†	0.028 (0.010)†	0.032 (0.0098)†	0.031 (0.0091)‡
Total cholesterol (SE)	0.0038 (0.0018)*	0.0002 (0.0026)	0.0078 (0.0026)	0.0024 (0.0020)	0.0038 (0.0021)	0.0035 (0.0020)
Sorbinil assignment (SE)	−0.045 (0.16)	−.31 (0.21)	0.26 (0.23)	−0.024 (0.18)	−0.045 (0.18)	−0.040 (0.17)
Progression in contralateral eye						0.63 (0.23)†

^*P<0.05;

^†P<0.01;

^‡P<0.001

Longitudinal data with a continuous outcome

With repeated measures over time of a continuous variable such as total point score on the 60-4 test from the Humphrey Field Analyzer, interest focuses on the pattern of change and how it might differ according to treatment (Table 4). Analyses of these changes must account for both the correlations between repeated measures of the same eye (Table 5), as well as the correlations between the two eyes of the same person.

Table 4.

Comparison of Treatment groups^* with Respect to Ophthalmological Findings for Subjects Seen at all Visits Humphrey Field Analyzer

Mean(SR&BS†) Mean SE (n)	1 Year Mean SE (n)	2 Year Mean SE (n)	3 Year Mean SE (n)	4 Year Mean SE (n)
60-4 Condition Total Point Score
Treatment Group 1*
OD	400 47 (69)	361 47 (69)	341 46 (69)	326 47 (69)	286 44 (69)
OS	384 49 (59)	341 50 (59)	309 48 (59)	310 49 (59)	260 43 (59)
Treatment Group 2*
OD	434 39 (74)	389 40 (74)	355 38 (74)	323 37 (74)	289 35 (74)
OS	423 41 (67)	386 42 (67)	344 41 (67)	310 39 (67)	283 38 (67)

^*Treatment group 1 is the lutein group; group 2 is the placebo group

^†SR and BS are the screening and baseline visits

Table 5.

Estimated longitudinal correlation for the Humphrey visual field over time in the same eye

Year
	0	1	2	3	4
60-4
0	1.0	0.87	0.78	0.79	0.73
1		1.0	0.79	0.79	0.76
2			1.0	0.82	0.83
3				1.0	0.86

Separate analyses of right and left eyes constitute a valid analytic strategy in this design. However, as in our previous examples, this approach yields two estimates of each covariate effect, and can be inefficient relative to a single integrated approach (Table 6). One can simplify the problem by averaging outcomes in a person’s two eyes at one visit. However, this is inefficient, especially if some individuals contribute one eye while others contribute two. In particular, note that while effect estimates from this approach are generally similar to those obtained from analyses that treat the eye as the unit of analysis, estimated standard errors are larger. In particular, the mixed effects model that treats the eye as the unit of analysis and appropriately accounts for the correlation between paired eyes in a person finds a significant relationship between randomized lutein treatment and a slower rate of loss of visual field (time*group effect), as measured by the 60-4 test on the Humphrey Field Analyzer. Compared to this approach, the simpler generalized estimating equation approach that also treats the eye as the unit of analysis finds similar estimated effects, but estimated standard errors are slightly larger.

Table 6.

Estimated regression coefficient in a longitudinal analysis in change in visual field among retinitis pigmentosa patients in a randomized trial

	60-4 Humphrey Field
	715 Right eyes	630 Left eyes	1345 Eyes ††	1345 Eyes†††	815 Visits ††††
Intercept	431.6 (43.9)	422.7 (45.3)	419.3 (37.8)	418.0 (40.4)	416.4 (40.7)
Time (yrs)	−35.5 (5.5)‡	−35.2 (5.9)‡	−34.3 (3.0)‡	−36.0 (3.9)‡	−34.2 (5.3)‡
Group	−36.1 (63.2)	−46.4 (66.2)	−48.1 (52.6)	−38.7 (59.1)	−48.5 (58.1)
Eye	---	---	−1.8 (11.7)	13.3 (21.7)	---
Time * Group	8.7 (7.9)	7.2 (8.6)	7.8 (3.7)*	8.6 (4.4)	7.7 (7.6)
Time * Eye	---	---	−0.4 (1.9)	0.9 (2.1)	---

^*P<0.05;

^†P<0.01;

^‡P<0.001

^††Estimates from the mixed effects model accounting for correlations within persons and over time

^†††Estimates from a linear model accounting for correlations within persons and over time via a generalized estimating equation approach with independent working correlation

^††††Uses the person as the unit of analysis, but accounts for correlations over time, based on the average over available eyes for a subject

Discussion

Our four examples, drawn from different study designs and with varying outcome scales, demonstrate several advantages of regression approaches that treat the eye as the unit of analysis and appropriately account for the correlation between the two eyes in a person. First, if model forms are correct, they yield valid P-values and have confidence intervals with nominal coverage, which is not the case for models that assume data from the two eyes of the same person are independent observations. Second, these models that appropriately consider the eye as the unit of analysis often have greater statistical power than analyses based on average or other composites of the outcomes in the two eyes of a person. Moreover, these latter approaches are problematic if eye-specific covariates are considered, or if some subjects contribute data from one eye while others contribute data from two eyes. Third, models that appropriately consider the eye as the unit of analysis often have greater statistical power, and yield single, optimally weighted overall effect estimates and confidence intervals, relative to results of separate analyses of right and left eyes.

Advances have occurred in development and accessibility of statistical software to fit and evaluate these regression models with the eye as the unit of analysis. We have illustrated these approaches with analyses that used SAS⁸ and Stata¹¹ software, but other software packages, including S-Plus¹⁷ and R¹⁸, can readily fit these models. In many situations, these models will be the best approaches for analysis of ophthalmologic data.