The setting

Consider the simple regression model:

$$$$y_i=\hat{\beta}_0 + \hat{\beta}_1 x_i + \hat{\epsilon}_i$$$$

Now, suppose I want to apply a linear transformation to the response, say:

$$$$u_i = b_0 + b_1 y_i$$$$

The model with the transformed response would be

$$$$u_i = \hat{\gamma}_0 + \hat{\gamma}_1x_i + \hat{\epsilon}_i, \textrm{with} \\ \hat{\gamma}_0 = b_0 + b_1 \hat{\beta}_0 \\ \hat{\gamma}_1 = b_1\hat{\beta}_1$$$$

The problem

So far, so good. Now let's say I have a response in liters, and I want to transform to milliliters. $$$$1ml = 0.001 l, \textrm{ i.e.} \\ u_i = 0.001 y_i$$$$

So I think, "Ah, that's a linear transformation. I can just apply the formula from above." But when I do, I end up with

$$$$\hat{\gamma}_0 = 0.001\hat{\beta}_0, \textrm{and} \\ \hat{\gamma}_1 = 0.001\hat{\beta}_1$$$$ which is obviously wrong. The right transformation would be:

$$$$\hat{\gamma}_0 = 1000\hat{\beta}_0, \textrm{and} \\ \hat{\gamma}_1 = 1000\hat{\beta}_1$$$$

So I suppose, the transformation formula would have to go: $$$$u_i = 1000y_i$$$$

The question

How can I systematically get it right, when starting from something like $$1ml = 0.001l$$?

• – whuber Jan 26 '19 at 15:17