One implication of the results presented above is that different methods of objective refraction that yield similar refractions on average are likely to be statistically correlated. We tested this prediction by computing the correlation coefficient between all possible pairs of methods for predicting
M. The resulting correlation matrix is visualized in
Figure 8. For example, the left-most column of tiles in the matrix represents the Pearson correlation coefficient
r between the first objective refraction method in the list (RMSw) and all other methods in the order specified in
Table 1. Notice that the values of
M predicted by optimizing RMSw are highly correlated with the values returned by methods 3 (RMSs), 8 (Bave), 19 (ENT), and 32 (least-squares fit). As predicted, all of these metrics are grouped at the bottom of the ranking in
Figure 7. To the contrary, refractions using RMSw are poorly correlated with values returned by methods 4 (PFWc), 9 (PFCt), 21 (VSX), 24 (SFcOTF), and 33 (Curvature fit). All of these metrics are grouped at the top of the ranking in
Figure 7, which further supports this connection between accuracy and correlation. A similar analysis of the correlation matrix for astigmatism parameters is not as informative because there was very little difference between the various methods for predicting
J0 and
J45.
Another interesting feature of
Figure 8 is that some refraction methods (e.g. PFCc, VOTF, VNOTF) are very poorly correlated with all other methods. This result for metric PFCc is explained by the fact that PFCc was the only metric to produce hyperopic refractions in the vicinity of
M=+0.25D. However, this argument does not apply to the other two examples that are poorly correlated with most other metrics even though these other metrics produced similar refractions on average (e.g. 20 (NS), 7 (PFSc), and 23 (AreaMTF)). This result suggests that maximizing metrics VOTF and VNOTF optimizes a unique aspect of optical and visual quality that is missed by other metrics. In fact, these two metrics were specifically designed to capture infidelity of spatial phase in the retinal image.