The authors prove that if \(p_n \geq k^k/(k+1)^{k+1}\) for all large \(n\), then every solution of the first order delay difference equation \(\Delta x_n + p_n x_{n-k}=0\) is oscillatory if and only if every solution of the second order ordinary difference equation \[ \Delta^2 y_{n-1}+ 2 \frac{(k+1)^k}{k^{k+1}}\left[p_n-\frac{k^k}{(k+1)^{k+1}}\right] y_n=0 \] is oscillatory.