There is a specific method to measure this, called Partial Least Squares (PLS). PLS computes the correlation of the input variables (X) with the output variable (Y). Basically, the method finds a rotation matrix, w, w.r.t. arg max(Cov(Xw, Y)). In particular, the weights of w represent the maximum correlation between X and Y. For more details, please consider to examine the following works:
"Latent hypernet: Exploring all Layers from Convolutional Neural Networks"
"Partial least squares regression and projection on latent structure regression (PLS Regression)"
Question is not explicit. PLS is a good method to understand relations among variables but do not provide a "significancy". Interpretation must be made in a subjective manner.