Accounting for the Correlation Between Regression Analysis Robert J.
Eyes
in
Glynn, ScD, Bernard Rosner, PhD
\s=b\ Regression techniques that appropriately use all available eyes have infrequently been applied in the ophthalmologic literature, despite advances both in the development of statistical models and in the availability of computer soft-
ware
Fellow
to fit these models. We considered
the
general linear model and polychotomous logistic regression approaches of Rosner and the estimating equation approach of Liang and Zeger, applied to both linear and logistic regression. Meth-
ods were illustrated with the use of two real data sets: (1) impairment of visual acuity in patients with retinitis pigmentosa and (2) overall visual field impairment in elderly patients evaluated for glaucoma. We discuss the interpretation of coefficients from these models and the advantages of these approaches compared with alternative approaches, such as treating individuals rather than eyes as the unit of analysis, separate regression analyses of right and left eyes, or utilization of ordinary regression techniques without accounting for the correlation between fellow eyes. Specific advantages include enhanced statistical power, more interpretable regression coefficients, greater precision of estimation, and less sensitivity to missing data for some eyes. We concluded that these models should be used more frequently in ophthalmologic research, and we provide guidelines for choosing between alternative models.
(Arch Ophthalmol. 1992;110:381-387) Accepted for publication July 23, 1991. From the Department of Medicine, Brigham and Women's Hospital and the Channing Laboratory, Harvard Medical School (Drs Glynn and Rosner) and the Department of Biostatistics, Harvard School of Public Health (Dr Rosner), Boston, Mass.
Reprint requests to Brigham Hospital, 55 Pond Ave, Brookline, Glynn).
and Women's MA 02146 (Dr
common statistical problem that faces clinical investigators in oph¬ thalmology is appropriate methodology for addressing the correlation between pairs of eyes in the same individual. The problem arises because measure¬ ments or diagnoses in the fundamental unit of observation, ie, the eye, tend to have a greater correspondence with the fellow eye than with the eye of another individual.1'" This problem af¬ fects the analysis of ophthalmologic data that arise from different study designs (eg, clinical trials, case-con¬ trol, cross-sectional, and cohort stud¬ ies) and impacts on study design inso¬ far as efficient analysis improves the power of a study and, hence, may reduce the sample size needed to achieve a prespecified power. We focus here on appropriate meth¬ ods for fitting both linear and logistic regression models when data are col¬ lected from both eyes of at least some study participants. Regression tech¬ niques are being used with increasing frequency in ophthalmologic research. Straightforward application of regres¬ sion techniques with the use of data from all assessed eyes is generally invalid because the correlation be¬ tween eyes in the same individual vio¬ lates the assumption of statistical inde¬ pendence among all sampled eyes. Three classes of strategies are em¬ ployed for addressing the problem of correlation between fellow eyes when performing regression analyses. The first approach is to analyze the data on an eye-specific basis and ignore the correlation between eyes. Invalid in¬ ferences may result from this approach since values will usually be underes¬ timated, confidence intervals (CIs) will generally be too narrow, and some
Downloaded From: http://archopht.jamanetwork.com/ by a Yale University User on 05/16/2015
'
reportedly significant results may, in fact, be nonsignificant. The second ap¬ proach is to discard some information to satisfy the assumption of indepen¬ dence among eyes that are being ex¬ amined. Alternative implementations of this approach include the following types of analyses: analysis of one eye chosen at random; analysis of only right (or left) eyes, or both right and left eyes separately; and analysis on a person-specific basis with individuals classified
on
the basis of their
(or better) eye
or
worse
by averaging
re¬
sponses across eyes. Limitations of this strategy are as follows: unless pairs of eyes in an individual are per¬ fectly correlated, discarding some data is inefficient; separate analyses of right and left eyes will be problematic if results are discordant; and if analyses are on a person-specific basis with the worse (or better) eye characterizing that person, then difficulties arise in the interpretation of regression coeffi¬ cients associated with eye-specific oph¬ thalmologic predictor variables. For example, if the relationship of elevated intraocular pressure to visual field de¬ fects is being examined, then it is most appropriate to link directly pressure in an eye to defects in that eye. Averag¬ ing outcomes across pairs of eyes will tend to dilute the value of the more impaired eye and also produces diffi¬ culties for the interpretation of eyespecific predictor variables. The third approach is to analyze the data on an eye-specific basis, but to use a regres¬ sion technique that formally accounts for the correlation between eyes. Whether the outcome is a continuous
discrete characteristic, a variety of appropriate techniques exist.4*11 This third approach offers distinct ador a
vantages over the first two approaches insofar as it efficiently uses all avail¬ able data in a single model. However, it has received little attention in the ophthalmologic literature. This article compares these three approaches in a regression context where the outcome is either a continu¬ ous characteristic of an eye and a lin¬ ear regression is appropriate or a dichotomous characteristic and a logistic regression model is appropriate. We focus on analytic strategies where computer software is readily available and discuss the interpretability of pa¬ rameters obtained from alternative an¬ alyses. Two separate data sets, from patients with retinitis pigmentosa and patients evaluated for glaucoma, re¬ spectively, are used to illustrate these concepts. Our goal is to make appropri¬
regression techniques more acces¬ interpretable to ophthalmo¬ logic investigators. ate
sible and
METHODS Data
Two data sets were used to illustrate alternative approaches to fitting regression models to ophthalmologic data. The first data set consisted of measurements from 888 eyes in 444 individuals with retinitis pigmentosa (age range, 6 to 80 years) who were seen at the Berman-Gund Laboratory of the Massachusetts Eye and Ear Infirma¬ ry, Boston, Mass, from 1970 to 1979. Pa¬ tients were classified on the basis of a detailed family history into the genetic types of autosomal dominant, autosomal recessive, sex-linked, and isolate retinitis pigmentosa to study differences among the four groups. For simplicity, only one ran¬ domly selected person was chosen from each family if multiple affected family mem¬ bers were present. Details of the study design have been published.1" The second data set consisted of measure¬ ments from 394 eyes in 197 individuals (age range, s=65 years) who were patients with glaucoma or patients suspected of having glaucoma and who were seen in the Glauco¬ ma Consultation Service of the Massachu¬ setts Eye and Ear Infirmary in 1987 and 1988. Patients were all participants in a study of falls and functional limitations as¬ sociated with glaucoma who underwent bi¬ lateral automated perimetry testing with the use of program 32 on the Octopus perimeter (Interzeag AG, Schlieren, Swit¬ zerland). The percentage of normal visual field in an eye was the average threshold in the central 30° for that eye divided by the normal average value for a 65-year-old per¬ son, then multiplied by 100 to convert to a percentage. Distance visual acuity with the use of regular spectacles was assessed with and without a pinhole; the better of the two measures in each eye was recorded as the Snellen acuity for that eye. This Snellen acuity was transformed to a measure of percent impairment in each eye, by using a standard approach.13 Other measures that
assessed at glaucoma examination in¬ cluded intraocular pressure and phakic sta¬ tus of each eye and systemic hypertension and diabetes as indicated by current use of medications for these conditions. Additional details of the study design have been de¬ scribed elsewhere.1 were
Analyses We considered fitting linear or logistic regression models to predict continuous or
discrete characteristics of an eye. The basic linear regression model has the following form: y a + l,ßixi + e, where y is the outcome of interest (eg, some transformation of visual acuity or percent¬ age of normal visual field); the predictors x¡ may include characteristics of people, such as genetic types of retinitis pigmentosa, as well as of eyes, such as phakic status; and the errors, e, are assumed to be indepen¬ dent and normally distributed for different eyes. It is the assumption of independent errors that tends to be violated because of the correlation between eyes in the same =
individual. The regression coefficient ß, can be interpreted as the predicted rate of change in the outcome y per unit change in the predictor *„ adjusting for the effects of the other predictors in the model. We used five alternative approaches to obtain estimated linear regression coeffi¬ cients. First, estimated coefficients were obtained by using data from both eyes of each participant and by fitting ordinary regression models as implemented by the GLM procedure (PROC GLM) in the SAS ' statistical package.lo
Eyes
in
Glynn, ScD, Bernard Rosner, PhD
\s=b\ Regression techniques that appropriately use all available eyes have infrequently been applied in the ophthalmologic literature, despite advances both in the development of statistical models and in the availability of computer soft-
ware
Fellow
to fit these models. We considered
the
general linear model and polychotomous logistic regression approaches of Rosner and the estimating equation approach of Liang and Zeger, applied to both linear and logistic regression. Meth-
ods were illustrated with the use of two real data sets: (1) impairment of visual acuity in patients with retinitis pigmentosa and (2) overall visual field impairment in elderly patients evaluated for glaucoma. We discuss the interpretation of coefficients from these models and the advantages of these approaches compared with alternative approaches, such as treating individuals rather than eyes as the unit of analysis, separate regression analyses of right and left eyes, or utilization of ordinary regression techniques without accounting for the correlation between fellow eyes. Specific advantages include enhanced statistical power, more interpretable regression coefficients, greater precision of estimation, and less sensitivity to missing data for some eyes. We concluded that these models should be used more frequently in ophthalmologic research, and we provide guidelines for choosing between alternative models.
(Arch Ophthalmol. 1992;110:381-387) Accepted for publication July 23, 1991. From the Department of Medicine, Brigham and Women's Hospital and the Channing Laboratory, Harvard Medical School (Drs Glynn and Rosner) and the Department of Biostatistics, Harvard School of Public Health (Dr Rosner), Boston, Mass.
Reprint requests to Brigham Hospital, 55 Pond Ave, Brookline, Glynn).
and Women's MA 02146 (Dr
common statistical problem that faces clinical investigators in oph¬ thalmology is appropriate methodology for addressing the correlation between pairs of eyes in the same individual. The problem arises because measure¬ ments or diagnoses in the fundamental unit of observation, ie, the eye, tend to have a greater correspondence with the fellow eye than with the eye of another individual.1'" This problem af¬ fects the analysis of ophthalmologic data that arise from different study designs (eg, clinical trials, case-con¬ trol, cross-sectional, and cohort stud¬ ies) and impacts on study design inso¬ far as efficient analysis improves the power of a study and, hence, may reduce the sample size needed to achieve a prespecified power. We focus here on appropriate meth¬ ods for fitting both linear and logistic regression models when data are col¬ lected from both eyes of at least some study participants. Regression tech¬ niques are being used with increasing frequency in ophthalmologic research. Straightforward application of regres¬ sion techniques with the use of data from all assessed eyes is generally invalid because the correlation be¬ tween eyes in the same individual vio¬ lates the assumption of statistical inde¬ pendence among all sampled eyes. Three classes of strategies are em¬ ployed for addressing the problem of correlation between fellow eyes when performing regression analyses. The first approach is to analyze the data on an eye-specific basis and ignore the correlation between eyes. Invalid in¬ ferences may result from this approach since values will usually be underes¬ timated, confidence intervals (CIs) will generally be too narrow, and some
Downloaded From: http://archopht.jamanetwork.com/ by a Yale University User on 05/16/2015
'
reportedly significant results may, in fact, be nonsignificant. The second ap¬ proach is to discard some information to satisfy the assumption of indepen¬ dence among eyes that are being ex¬ amined. Alternative implementations of this approach include the following types of analyses: analysis of one eye chosen at random; analysis of only right (or left) eyes, or both right and left eyes separately; and analysis on a person-specific basis with individuals classified
on
the basis of their
(or better) eye
or
worse
by averaging
re¬
sponses across eyes. Limitations of this strategy are as follows: unless pairs of eyes in an individual are per¬ fectly correlated, discarding some data is inefficient; separate analyses of right and left eyes will be problematic if results are discordant; and if analyses are on a person-specific basis with the worse (or better) eye characterizing that person, then difficulties arise in the interpretation of regression coeffi¬ cients associated with eye-specific oph¬ thalmologic predictor variables. For example, if the relationship of elevated intraocular pressure to visual field de¬ fects is being examined, then it is most appropriate to link directly pressure in an eye to defects in that eye. Averag¬ ing outcomes across pairs of eyes will tend to dilute the value of the more impaired eye and also produces diffi¬ culties for the interpretation of eyespecific predictor variables. The third approach is to analyze the data on an eye-specific basis, but to use a regres¬ sion technique that formally accounts for the correlation between eyes. Whether the outcome is a continuous
discrete characteristic, a variety of appropriate techniques exist.4*11 This third approach offers distinct ador a
vantages over the first two approaches insofar as it efficiently uses all avail¬ able data in a single model. However, it has received little attention in the ophthalmologic literature. This article compares these three approaches in a regression context where the outcome is either a continu¬ ous characteristic of an eye and a lin¬ ear regression is appropriate or a dichotomous characteristic and a logistic regression model is appropriate. We focus on analytic strategies where computer software is readily available and discuss the interpretability of pa¬ rameters obtained from alternative an¬ alyses. Two separate data sets, from patients with retinitis pigmentosa and patients evaluated for glaucoma, re¬ spectively, are used to illustrate these concepts. Our goal is to make appropri¬
regression techniques more acces¬ interpretable to ophthalmo¬ logic investigators. ate
sible and
METHODS Data
Two data sets were used to illustrate alternative approaches to fitting regression models to ophthalmologic data. The first data set consisted of measurements from 888 eyes in 444 individuals with retinitis pigmentosa (age range, 6 to 80 years) who were seen at the Berman-Gund Laboratory of the Massachusetts Eye and Ear Infirma¬ ry, Boston, Mass, from 1970 to 1979. Pa¬ tients were classified on the basis of a detailed family history into the genetic types of autosomal dominant, autosomal recessive, sex-linked, and isolate retinitis pigmentosa to study differences among the four groups. For simplicity, only one ran¬ domly selected person was chosen from each family if multiple affected family mem¬ bers were present. Details of the study design have been published.1" The second data set consisted of measure¬ ments from 394 eyes in 197 individuals (age range, s=65 years) who were patients with glaucoma or patients suspected of having glaucoma and who were seen in the Glauco¬ ma Consultation Service of the Massachu¬ setts Eye and Ear Infirmary in 1987 and 1988. Patients were all participants in a study of falls and functional limitations as¬ sociated with glaucoma who underwent bi¬ lateral automated perimetry testing with the use of program 32 on the Octopus perimeter (Interzeag AG, Schlieren, Swit¬ zerland). The percentage of normal visual field in an eye was the average threshold in the central 30° for that eye divided by the normal average value for a 65-year-old per¬ son, then multiplied by 100 to convert to a percentage. Distance visual acuity with the use of regular spectacles was assessed with and without a pinhole; the better of the two measures in each eye was recorded as the Snellen acuity for that eye. This Snellen acuity was transformed to a measure of percent impairment in each eye, by using a standard approach.13 Other measures that
assessed at glaucoma examination in¬ cluded intraocular pressure and phakic sta¬ tus of each eye and systemic hypertension and diabetes as indicated by current use of medications for these conditions. Additional details of the study design have been de¬ scribed elsewhere.1 were
Analyses We considered fitting linear or logistic regression models to predict continuous or
discrete characteristics of an eye. The basic linear regression model has the following form: y a + l,ßixi + e, where y is the outcome of interest (eg, some transformation of visual acuity or percent¬ age of normal visual field); the predictors x¡ may include characteristics of people, such as genetic types of retinitis pigmentosa, as well as of eyes, such as phakic status; and the errors, e, are assumed to be indepen¬ dent and normally distributed for different eyes. It is the assumption of independent errors that tends to be violated because of the correlation between eyes in the same =
individual. The regression coefficient ß, can be interpreted as the predicted rate of change in the outcome y per unit change in the predictor *„ adjusting for the effects of the other predictors in the model. We used five alternative approaches to obtain estimated linear regression coeffi¬ cients. First, estimated coefficients were obtained by using data from both eyes of each participant and by fitting ordinary regression models as implemented by the GLM procedure (PROC GLM) in the SAS ' statistical package.lo
