When and have a measurement error component, we wish to write
where and are assumed to be independently distributed error components. Therefore,
So, the angle representing the slope is expressed as
In order to remove the dependence of the distribution of on , we create a new angle which we call where
To form an intuitive notion of this transformation, let us suppose that e = 1 and . If the change in x, is large, the error in the angle will be small. However, if is small, the error in the angle will be large. When we weight by , we remove the dependence of the error on . Of course, e is an unobserved variable, but even so, we can remove this bias from . The adequacy of this bias correction factor requires the independence of and .
Thus, we may estimate with
the standard regression equation.