What is a non-formal $Ainfty$ algebra?

For an $A_infty$ algebra (say I do everything over $mathbb{Q}$) modeled by dg-algebras I can understand the definition of formality (that its homology groups is equivalent in a $A_infty$ way to the original algebra). I imagine that this definition also extends to the Lurie/spectra setting? As in take homotopy groups and ask whether the algebra I get is isomorphic to the original one as an $E_1$-algebra. So with all this in mind, let me ask a very basic question. When is this false?