01 · Direction
A model predicts fuel used from journey distance. Which regression direction is relevant if x is distance and y is fuel?
Hint
The requested response is y.
Worked solution
Use y on x: predict fuel from distance.
Understand · explore · practise
Choosing the regression direction
Choose y-on-x or x-on-y regression, distinguish rearranging a given equation from fitting the reverse prediction and understand the perfect-line exception.
Before you startRegression equations, least squares and vertical residuals.
01 / Name the response first
Regression direction follows the prediction question.
To predict y from x, use the regression of y on x. To predict x from y, use the regression of x on y.
These generally minimise different squared errors and produce different fitted lines.
Algebraically rearranging a nonhorizontal equation is valid. It does not, by itself, turn the least-squares fit in one direction into the separately fitted least-squares model for the reverse direction.
Choose what is being predictedExplore
Constructed pairs: (0,1), (1,2), (2,2), (3,5). Separate least-squares fits are ŷ = 0.7 + 1.2x and x̂ = −1/6 + (2/3)y.
For supplied x = 1, the y-on-x fit predicts y = 1.9. Its residuals are vertical.
02 / Predict y from a supplied x
Use vertical response errors.
For the constructed data, the y-on-x line is ŷ = 0.7 + 1.2x.Worked example
At x = 2, predict y = 3.1
Substitute the supplied x.
The observed response at x = 2 is 2
The vertical residual is −1.1.
The fit minimises squared vertical residuals over all four pairs
It was not chosen by minimising errors in x.
02 · Substitution
For ŷ = 0.7 + 1.2x, find the prediction at x = 3.
Hint
Keep the intercept.
Worked solution
ŷ = 4.3. This is a fitted response, not necessarily the observed one.
03 / Fit the reverse response separately
Holding y fixed changes the errors being minimised.
The x-on-y fit for the same data is x̂ = −1/6 + (2/3)y.Worked example
At y = 3, x̂ = −1/6 + 2 = 11/6
Approximately 1.833.
Rearranging the y-on-x fit would give x = (3 − 0.7)/1.2 = 23/12
Approximately 1.917, a different number.
Both calculations are algebraically correct for their equations
Only the first uses the separately fitted x-on-y model.
03 · Reverse prediction
Use x̂ = −1/6 + (2/3)y to predict x at y = 2.
Hint
Use the equation whose response is x.
Worked solution
x̂ = −1/6 + 4/3 = 7/6 ≈ 1.167.
04 · Why different?
Why need the reverse fit differ from an algebraic inverse?
Hint
Vertical and horizontal squared errors are different objectives.
Worked solution
The two fits optimise different response errors. Scatter around a line means the optimisation usually selects different lines.
04 / Separate two legitimate tasks
Solving a stated equation is not refitting data.
05 · Solve a target
A question explicitly asks which x makes the stated model prediction ŷ = 0.7 + 1.2x equal to 3. What is the answer?
Hint
This is an algebraic target on the stated line.
Worked solution
x = (3 − 0.7)/1.2 = 23/12. That solves the given model; do not relabel it as the separately fitted x-on-y regression.
06 · Only one fit supplied
Only a y-on-x equation is supplied. Does it determine the separately fitted x-on-y line in general?
Hint
Information about the scatter and both variables is missing.
Worked solution
No. Extra data or appropriate summary statistics are needed. Follow a question that explicitly asks you to rearrange its given model, but distinguish that task from reverse statistical fitting.
05 / See what is held fixed
Vertical and horizontal gaps tell different stories.
Watch: the response direction changes the residuals
Pause, replay or seek freely. The notes explain the same idea and stay in view.
07 · Mean point
Both fits use x̄ = 1.5 and ȳ = 2.5. Verify that both fitted lines pass through that point.
Hint
Substitute the relevant supplied coordinate in each.
Worked solution
0.7 + 1.2×1.5 = 2.5. Also −1/6 + (2/3)×2.5 = 1.5. Ordinary unweighted fits with intercepts share the mean point.
08 · Distance direction
When x is the response, which way is a residual drawn on the usual x-horizontal, y-vertical scatterplot?
Hint
Compare observed and predicted x at the same y.
Worked solution
Horizontally. The prediction direction can change without swapping the physical axes.
06 / Recognise the exact-line exception
A nonzero perfect relationship can be inverted consistently.
09 · Exact relationship
All pairs satisfy y = 2 + 3x exactly, with varying x. What is the reverse equation?
Hint
Solve the exact relationship.
Worked solution
x = (y − 2)/3. With zero scatter and nonzero gradient, both least-squares directions lie on the same geometric line.
10 · Constant y
All y values are 5 while x varies. Can the y-on-x constant fit be inverted to recover x from y?
Hint
One response value corresponds to several x values.
Worked solution
No. There is no unique inverse, and the predictor y has zero variation for reverse fitting. The perfect nonzero-gradient exception does not apply.
07 / State the intended use
Do not choose the line merely by its appearance.
11 · Equation labels
Two fitted equations are given, one for predicted mass from length and one for predicted length from mass. You know an object’s mass and want its length. Which do you use?
Hint
Choose the equation predicting the unknown from the known.
Worked solution
Use the fit predicting length from mass, then assess whether its range and context apply.
12 · Causal direction
Does choosing y-on-x regression prove x causes y?
Hint
Prediction direction is not causal identification.
Worked solution
No. It specifies which response errors are minimised and what is being predicted, not a causal mechanism.
08 / Choose, substitute and qualify
Keep prediction direction separate from inverse algebra.
Name the known variable and the response you want. Use the relevant fitted direction, or solve a stated model when that is explicitly the task. Check range and context, and remember that the two regression lines generally differ when there is scatter.
Section 1 of 8 · Name the response first