Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them with iterated Brownian integrals and the concept of "simple bracket extension", developed in the PhD thesis of David Kelly. As a consequence of these results, we have a sufficient criterion to show for any $γ\in (0,1)$ and any sufficiently smooth function $\varphi \colon \mathbb{R}^d\to \mathbb{R}$ a rough change of variable formula on any $γ$-Hölder continuous path $x\colon [0, T]\to \mathbb{R}^d$, i.e. an explicit expression of $\varphi(x_t)$ in terms of rough integrals.