Redmond and colleagues

^{ 1 }wrote a very interesting manuscript where they used Passing–Bablok regression^{ 2 }(PBR) to test whether or not structural and functional measures are associated in healthy eyes and in eyes with early glaucoma. One of their main conclusions is that even in healthy subjects there is an association between structure and function. This conclusion is in conflict with previous findings.^{ 3,4 }Redmond and colleagues

^{ 1 }decided to use PBR over linear ordinary least-squares (OLS) because both structure and function are subject to measurement errors. This method, which may be valid for other applications, is inappropriate for testing whether structure and function are associated. That is because the slope estimated with PBR depends on the variance ratio.^{ 2 }Therefore, testing whether the slope estimated with PBR is significantly different from zero is not equivalent to testing whether structure and function are associated. This is also true for other bivariate line-fitting methods such as major-axis, standardized major-axis, and Deming regression.^{ 5 }The appropriate regression method for testing independence between variables is the classical OLS regression (see, e.g., Table 1 in Warton et al.^{ 5 }). That is so even though OLS regression is inappropriate for other applications such as methods comparison studies, as was nicely illustrated in the analysts' comparison example.^{ 6 }Simulations were performed here to demonstrate that OLS is the appropriate method to test whether or not structure and function are associated with each other, even if structure and function are both subject to measurement errors. In the Figure, results are shown for OLS regression (solid curves) and PBR (dashed curves) for 1000 simulations with uncorrelated Gaussian data and for different variance ratios. The upper panel shows the average estimated slope, and the lower panel shows the percentage of times that the null hypothesis “slope equals 0” was accepted using each regression method. In more detail, for each of the 1000 simulations, two datasets of 100 values each, one for

*x*and another for*y*, were generated using two Gaussian distributions so that*x*and*y*were uncorrelated with (independent of) each other. The means of the Gaussian distributions were set to zero, the standard deviation of the first dataset was set to 1, and the standard deviation of the second dataset varied from approximately 1/3 to approximately 3 (abscissa). Other simulations were performed, changing the number of values generated from 100 to 50, and similar results were obtained. The estimated slopes depend on the variance ratio, and not on the actual values for mean and standard deviation. Since the significance for the test of hypothesis was set at 0.05, the appropriate method must make correct inferences 95% of the time and fail 5% of the time.**Figure**

**Figure**

It is evident from the Figure that OLS is the appropriate method to assess association between structure and function through the null hypothesis “slope equals 0,” regardless of the variance ratio.

Whether or not structure and function are associated in healthy subjects alone

^{ 1 }remains an unresolved question. More insight would be gained if the analysis were replicated using OLS regression or, alternatively, Spearman correlation, for which linearity is not assumed.References

Redmond
T
Anderson
RS
Russell
RA
Garway-Heath
DF.
Relating retinal nerve fiber layer thickness and functional estimates of ganglion cell sampling density in healthy eyes and in early glaucoma.

*. 2013; 54: 2153–2162. [CrossRef] [PubMed]**Invest Ophthalmol Vis Sci*
Passing
H
Bablok
W.
A new biometrical procedure for testing the equality of measurements from two different analytical methods. Application of linear regression procedures for method comparison studies in clinical chemistry, part I.

*. 1983; 21: 709–720. [PubMed]**J Clin Chem Clin Biochem*
Hood
DC
Anderson
SC
Wall
M
Raza
AS
Kardon
RH.
A test of a linear model of glaucomatous structure-function loss reveals sources of variability in retinal nerve fiber and visual field measurements.

*. 2009; 50: 4254–4266. [CrossRef] [PubMed]**Invest Ophthalmol Vis Sci*
Wollstein
G
Kagemann
L
Bilonick
RA
Retinal nerve fibre layer and visual function loss in glaucoma: the tipping point.

*. 2012; 96: 47–52. [CrossRef] [PubMed]**Br J Ophthalmol*
Warton
DI
Wright
IJ
Falster
DS
Westoby
M.
Bivariate line fitting methods for allometry.

*. 2006; 81: 259–291. [CrossRef] [PubMed]**Biol Rev*
Ripley
BD
Thompson
M.
Regression techniques for the detection of analytical bias.

*. 1987; 112: 377–383. [CrossRef]**Analyst*
R Core Team.

*. Vienna, Austria: R Foundation for Statistical Computing; 2013. ISBN: 3-900051-07-0. Available at: http://www.R-project.org/. Accessed June 28, 2013.**R: A Language and Environment for Statistical Computing*
Carstensen
B
Gurrin
L
Ekstrom
C.

*MethComp: functions for analysis of method comparison studies. R package version 1.15*. 2012. Available at: http://CRAN.R-project.org/package=MethComp. Accessed May 9, 2013.