Is there a way to use Artificial Neural Network to identify the primary cause(s) for the resultant value?

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Suppose I have a set of data ( attached ) and one column is viewed as consequence ( e.g. wage per hour ) while otheres are considered as potential factors for now. My task is to find out what assumed factor(s) are the most important one(s) and which one(s) are less imporant or even irrelevant to the final result. An ANN can be built and trained using data provided. But the depth and complexity in ANN makes it difficult to tell the primary factor(s) just from weights in each layer.

So the question is : given a trained ANN, how to identify the dimension that has the most significant influence on the predictions and BTW those has little? ( to be note : there are one-hot coding in the input )

Thank you !

Example data to be analyzed