# Objective Prediction of Human Visual Acuity Using Image Quality Metrics
## 3 Results and Discussion
The critical value was 2.01 with an alpha value equaling 0.05 (probability of the null hypothesis). As the obtained values were higher than the critical value, it can be stated that fittings were statistically significant with a probability over 95%. All the metrics also provided high coefficients of determination, which means that the logistic Function (5) allowed the studied variables to be correlated as a high percentage. The normality of the distribution of residuals was studied by applying three types of tests: Lilliefors, Anderson-Darling and Jarque-Bera with a significance level \(\alpha=0.05\). The three tests indicated that the distributions of residuals could be accepted as normal. Only the obtained residuals using the _nGMSD_ metric did not follow normal distribution. Regarding the independence of residuals, Table 4 shows the computed DW statistics following (7) for the considered metrics. If the value of this statistic was two, the \begin{table}
\begin{tabular}{l l l l l l l l} \hline  & **a** & **b** & **c** & **d** & **e** & & \\ \hline _nMSE_ & \(-4.000\) & \(8.992\) & \(0.000\) & \(0.078\) & \(2.016\) & & \\ _nPSNR_ & \(-4.364\) & \(-8.983\) & \(0.775\) & \(-4.522\) & \(4.211\) & & \\ _SSIM_ & \(10.29\) & \(2.228\) & \(0.746\) & \(-3.054\) & \(3.255\) & \(\beta_{\mathbf{1}}\) & \(\beta_{\mathbf{2}}\) \\ _nGMSD_ & \(0.002\) & \(0.000\) & \(-0.352\) & \(-2.501\) & \(2.580\) & & \\ _MSSSIM_ & \(24.57\) & \(3.075\) & \(0.712\) & \(-15.19\) & \(11.55\) & & \\ _FSIM_ & \(21.24\) & \(3.134\) & \(0.864\) & \(-12.47\) & \(11.78\) & & \\ _nPSNR-HVS_ & \(-23.08\) & \(-3.773\) & \(0.729\) & \(-17.55\) & \(13.51\) & & \\ \hline _nMSE-nPSNR_ & \(64.78\) & \(0.021\) & \(45.16\) & \(-0.335\) & \(15.18\) & \(72.10\) & \(-12.75\) \\ _nMSE-SSIM_ & \(0.173\) & \(155,1\) & \(-5.512\) & \(-0.223\) & \(-1.093\) & \(-0.772\) & \(-12.21\) \\ _nMSE-nGMSD_ & \(-0.708\) & \(4.516\) & \(-4.870\) & \(0.213\) & \(2.418\) & \(-0.026\) & \(-12.78\) \\ _nMSE-MSSSIM_ & \(37.89\) & \(17.24\) & \(-0.873\) & \(-1.326\) & \(17.27\) & \(0.111\) & \(-2.583\) \\ _nMSE-FSIM_ & \(7.478\) & \(8.858\) & \(-3.770\) & \(-0.632\) & \(1.011\) & \(-0.944\) & \(-6.714\) \\ _nMSE-nPSNR-HVS_ & \(41.54\) & \(0.282\) & \(3.670\) & \(-2.878\) & \(10.63\) & \(5.839\) & \(-1.391\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters obtained for the fittings of the normalized metrics and the linear combinations of _nMSE_ with the other metrics to the logistic function (5). \begin{table}
\begin{tabular}{l l l l l} \hline  & \(\mathbf{R^{2}}\) & \(\mathbf{\sigma_{est}}\) & \(\mathbf{F}\)-Number & \(\mathbf{t}\) Statistic \\ \hline _nMSE_ & \(0.9141\) & \(0.1563\) & \(532\) & \(23.1\) \\ _nPSNR_ & \(0.9114\) & \(0.1588\) & \(514\) & \(22.7\) \\ _SSIM_ & \(0.9218\) & \(0.1495\) & \(590\) & \(24.3\) \\ _nGNSD_ & \(0.8891\) & \(0.1777\) & \(401\) & \(20.0\) \\ _MSSSIM_ & \(0.9170\) & \(0.1540\) & \(553\) & \(23.5\) \\ _FSIM_ & \(0.9266\) & \(0.1448\) & \(631\) & \(25.1\) \\ _nPSNR-HVS_ & \(0.9152\) & \(0.1554\) & \(539\) & \(23.2\) \\ _nMSE-nPSNR_ & \(0.9105\) & \(0.1600\) & \(509\) & \(22.6\) \\ _nMSE-SSIM_ & \(0.9261\) & \(0.1454\) & \(626\) & \(25.0\) \\ _nMSE-nGMSD_ & \(0.9178\) & \(0.1533\) & \(558\) & \(23.6\) \\ _nMSE-MSSSIM_ & \(0.9228\) & \(0.1486\) & \(597\) & \(24.4\) \\ _nMSE-ESIM_ & \(0.9309\) & \(0.1406\) & \(673\) & \(25.9\) \\ _nMSE-nPSNR-HVS_ & \(0.9116\) & \(0.1589\) & \(516\) & \(22.7\) \\ \hline \end{tabular}
\end{table} 
Table 3: ANOVA parameters for the above fittings. Coefficient of determination (\(\mathbf{R^{2}}\)), standard error of estimate (\(\mathbf{\sigma_{est}}\)), F-number, and t statistic. residuals were completely independent. For a sample size of 52 eyes and a significance level of \(\alpha=0.05\), the critical values of the statistic were approximately \(DW_{(L,A)}=1.49\) and \(DW_{(U,A)}=1.60\). The obtained \(DW\) values indicated that no performed fit led to a negative self-correlation in residuals.