Model-Fitting Adequacy and Clinical Rationality in Multivariate Linear Regression Analysis

I read with great interest the article by Wong et al. ¹ reporting the distribution and determinants of ocular biometric parameters in a Singapore Malaysian population. This population-based, cross-sectional study was well designed, and the use of partial coherence interferometry ensured the superior accuracy of the ocular biometric measurements. Two-step multivariate linear regression was used for evaluation of the relationship between ocular parameters and other independent variables. An age- and sex-adjusted model was used for each variable first, and those that were significant during the first step were recruited for second-step multivariate regression with backward selection. However, I was puzzled about Table 5, which showed the results of multivariate linear regression models. For example, the standardized β statistics for “Height, cm” are 0.162, 0.075, and 0.250 for AL, ACD, and CC, respectively. These statistics mean that when body height increases 10 cm, AL and ACD will increase 1.62 and 0.75 mm, respectively, which are quite large numbers for AL, averaging 23.55 mm, and ACD, averaging 3.1 mm. Also, for the model of ACD, the standardized β for “Age, y” is −0.358, which means an average of 0.358 mm decrease of ACD per year, meaning that one's ACD would decrease to 0 when he or she is 9 years older! 

It is possible that the unit of height in regression models was the number of quartiles, not centimeters, or that there was a typographical error in Table 5. If not, perhaps the adequacy of the regression models should be examined. Plots of residuals against questionable variables can be used to check for model-fitting adequacy. For the model of ACD, at least two parameter estimates are extreme (age and height), in which overparameterization or collinearity among independent variables may result. The former should not happen, because there are only five parameters to be estimated for 2788 samples. As for the problem of collinearity, it is difficult to handle because it does exist. Wong et al. had considered variables that were closely related to one another (which was supposed to be evaluated by Pearson's correlation coefficients or by clinical judgment) and chose only the most significant one. However, correlations among selected variables cannot be totally avoided. The variance inflation factor can be used to evaluate the extent of collinearity, but selection of independent variables should not depend only on that factor. The criterion of P < 0.05 may be too strict for backward selection in this study, since the adjusted R ² statistics were all less than 0.2. It is suggested that Mallows' Cp statistic be used to choose several acceptable models. If the estimates of parameters vary a lot among different models, clinical rationality as well as the adjusted R ² should be considered when choosing the best-fitting one.