Assuming a single, well-mixed, homogeneous fluid and a single acceleration due to gravity (both are good assumptions in natural rivers, and the second is a good assumption for processes on Earth, or any planetary body with a dominant influence on the local gravitational field), the only two variables that determine the boundary shear stress are the depth and the slope. This is the significance of the name of the formula.
For natural streams, in the mks or SI system (units of pascals for shear stress), a typical useful relationship to remember is that:
for water with a density of 1000 kg/m and approximating the acceleration due to gravity as 10 m/s (the error in this assumption is typically much smaller than the error from measurements).