Today, a large number of glaucoma cases remain undetected, resulting in irreversible blindness . In a quest for cost-effective screening, deep learning-based methods are being evaluated to detect glaucoma from color fundus images . Although unprecedented sensitivity and specificity values are reported, recent glaucoma detection deep learning models lack in decision transparency . Here, we propose a methodology that advances explainable deep learning in the field of glaucoma detection and vertical cup-disc ratio (VCDR), an important risk factor . We trained and evaluated a total of 64 deep learning models using fundus images that undergo a certain cropping policy . We defined the circular crop radius as a percentage of image size, centered on the optic nerve head (ONH), with an equidistant spaced range from 10% -60% (ONH crop policy). The inverse of the cropping mask was also applied to quantify the performance of models trained on ONH information exclusively (periphery crop policy). The performance of the models evaluated on original images resulted in an area under the curve (AUC) of 0.94 [95% CI : 0.92-0.96] for glaucoma detection, and a coefficient of determination (R^2) equal to 77% [95% CI : 0.77-0.79] for VCDR estimation . Models that were trained on images with absence of the ONH are still able to obtain significant performance (0.88 [95% CI : 0.85-0.90] AUC for glaucoma detection and 37% [95% CI : 0.35-0.40] R^2 score for VCDR estimation in the most extreme setup of 60% ONH crop). We validated our glaucoma detection models on a recent public data set (REFUGE) that contains images captured with a different camera, still achieving an AUC of 0.80 [95% CI : 0.76-0.84] when ONH crop policy of 60% image size was applied . Our findings provide the first irrefutable evidence that deep learning can detect glaucoma from fundus image regions outside the ONH.