Choose a regression direction from what is being predicted and how the data were obtained. The labels x and y do not tell you whether an input was controlled.
| Context | Appropriate reasoning |
|---|---|
| Two ordinary measured variables; predict y from x | Use the regression of y on x. |
| Two ordinary measured variables; predict x from y | Use the regression of x on y; rearranging the y-on-x line generally gives a different fit. |
| Experimenter controls x and observes response y | Fit response y on controlled input x. A justified target-response calculation can solve this same response model for x. |
Bridge the idea
Height and mass are both measured. To estimate height from mass, use height on mass. In a different experiment, an operator sets a heater input and measures temperature. Fit temperature on the controlled heater input; solving that calibration model for a desired temperature can be appropriate.
For a controlled calibration \(y=12+3x\), a desired response \(y=30\) gives \(x=6\), subject to the fitted range and unchanged experimental conditions. This inversion uses the controlled response model; it does not claim to produce the reverse least-squares line for two ordinary measured variables.
Exam wording: “Use response on the controlled input because the input was fixed by the experimenter.” For ordinary measured pairs: “Use the variable to be estimated regressed on the given variable.” Check reliability separately.
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