Use a residual plot to assess a transformed model

Which calculation gives the vertical coordinate in a residual plot?

Reuse the five-point residual experiment from 3D. At s=0, note the curved pattern even though the residual sum is zero. Shift the intercept and decide whether moving the plot vertically removes its curvature

What does a zero residual sum establish here?

Analysing a residual plot

A residual plot uses the transformed explanatory coordinate horizontally and the signed difference between observed and predicted transformed responses vertically. Residuals scattered around zero without a clear structure support the linearity assumption

1. 1Use the fitted explanatory coordinate Place the transformed explanatory coordinate on the horizontal axis of the residual plot \(\displaystyle \text{horizontal coordinate}=x'\)
2. 2Subtract predicted from observed At the same transformed explanatory value, subtract the fitted response from the observed transformed response \(\displaystyle e_i=y'_i-\widehat{y'}_i\)
3. 3Keep the residual sign A point above the fitted line has a positive residual and a point below it has a negative residual \(\displaystyle e_i>0\Rightarrow\text{above},\quad e_i<0\Rightarrow\text{below}\)
4. 4Inspect the pattern around zero A usable linear model has residuals scattered around zero without a clear curve, trend or changing spread \(\displaystyle \text{no clear structure around }e=0\Rightarrow\text{linear fit is supported}\)

Data or fit

\(\displaystyle \widehat{y^{\prime}}=2+3x^{\prime},\quad x^{\prime}=4\)

Calculate the predicted transformed response

Data or fit

\(\displaystyle y^{\prime}=13,\quad\widehat{y^{\prime}}=14\)

Calculate the signed residual on the fitted response scale

The observed transformed response is 13 and the predicted transformed response is 14. Which line gives the signed residual?

Residuals belong to the fitted coordinates

Write down the actual X and Y used by regression. Calculate e=Y−Ŷ at the same X, then plot (X,e). An input transformation changes the horizontal coordinate; a response transformation changes the residual scale

1. 1Preserve the scale If X=x² and Y=y, residuals remain in original y units. If X=x and Y=log₁₀(y), residuals are logarithmic differences. If Y=1/y or Y=y², use reciprocal or squared-response differences respectively. Do not inverse-transform individual residuals
2. 2Keep precision Use full coefficients and transformed coordinates until the final requested rounding. A displayed zero slope can hide a small non-zero fitted slope. The stated full equation or a recomputed full fit is needed for accurate residuals
3. 3Choose a useful plot Use an XY scatterplot with numerical spacing, an explicitly labelled horizontal coordinate and residual scale, and a visible zero reference. Include every observation, including repeated inputs. Choose a vertical window that makes structure visible without clipping points; do not connect the dots

Produce the squared-input residual analysis

Reuse the original 4B observations (x,y)=(1,3),(2,6),(3,10),(4,17). Fit y on u=x², with full equation ŷ=179/86+(119/129)u. Build all four transformed inputs, predictions and signed residuals. Independently draw the residual plot against u, recording your formulas, chosen tool and window. Report the residual at x=3 to three decimal places

1. 1Verify and interpret Check the signed sum and u-weighted sum of the unrounded residuals. Explain why these checks do not by themselves establish suitability. State what changes if you use original x rather than u on the horizontal axis, and distinguish a small-sample judgement from evidence of future accuracy
2. 2Save your complete work first Save your work before opening the model. Complete plot construction, reports and actual external-tool use are not automatically marked; review them yourself or with a teacher

Model response: Produce the squared-input residual analysis

Compare your saved construction and explanation with each step. Other suitable windows and equivalent exact expressions are acceptable

1. 1Complete table Inputs u are 1,4,9,16. Predictions and signed residuals are listed in row order below; the third residual rounds to −0.384. The unrounded signed sum and u-weighted sum are both zero \(\displaystyle \text{Predictions: }\frac{775}{258},\frac{1489}{258},\frac{893}{86},\frac{4345}{258};\quad\text{residuals: }-\frac{1}{258},\frac{59}{258},-\frac{99}{258},\frac{41}{258};\quad\mathrm{SSE}=\frac{29}{129}\)
2. 2Plot and scale Use u=0..17 and e=−0.5..0.5, with labels “Squared input u=x²” and “Residual (response units)”. Plot the four exact residual coordinates and the horizontal zero line, without joins. The original-x plot has different spacing; the fitted-coordinate plot must use u. Its residual values do not change merely because you change the horizontal display
3. 3Qualified judgement The signs alternate, with no decisive smooth curve in these four points. This small plot supports checking the model further, not a claim of random errors or a proven predictive relationship. Algebraic balance verifies the least-squares calculation; inspect residual structure and gather relevant additional observations before relying on extrapolation

Data or fit

\(\displaystyle y=10,\quad u=9,\quad\hat y=179/86+(119/129)u\)

Report the signed residual for x=3 to three decimal places

Which horizontal coordinate belongs to this fitted-coordinate residual plot?

Produce the logged-response residual analysis

Reuse the original 4B observations (x,y)=(1,3),(2,30),(3,30),(4,300), with positive dimensionless response index y. Let c=log₁₀(3); the full line is v̂=c−1/2+(3/5)x, where v=log₁₀(y). Build the transformed responses, predictions and residuals for all four observations. Draw a labelled residual plot and verify its two least-squares balance sums without rounding c.

1. 1Test the meaning At x=2, compare the signed log residual with the observed/predicted original-response ratio. Explain why taking 10 to the power of the residual produces a ratio, not an original-unit residual. Multiply every original y by 10 and predict what happens to this residual plot before checking
2. 2Save your complete work first Save the table, formulas and tool settings, labelled residual plots, numerical checks and written judgement before opening the model. Complete plot construction, reports and actual external-tool use are not automatically marked; review them yourself or with a teacher

Model response: Produce the logged-response residual analysis

Compare your saved construction and explanation with each step. Other suitable windows and equivalent exact expressions are acceptable

1. 1Exact table and checks Transformed responses are c,c+1,c+1,c+2. Predicted log-responses are c+1/10,c+7/10,c+13/10,c+19/10. Residuals are −1/10,3/10,−3/10,1/10. Their sum and x-weighted sum are zero; SSE=1/5 on the squared log-response scale
2. 2Plot Use x=0..5,e=−0.4..0.4 and label the vertical axis “Residual on log₁₀(y) scale”. The horizontal coordinates remain 1,2,3,4. Both signs and zero sum do not prove randomness; with four observations, describe the alternating pattern and retain the sample-size limitation
3. 3Ratio and changed units At x=2 the log residual is 0.3. Therefore observed y divided by inverse-fitted y is 10^0.3≈1.995, not a difference of 1.995 response units. Multiplying all original y by 10 raises every log-response and the fitted intercept by 1, leaving every log residual and this plot unchanged

Data or fit

\(\displaystyle v=c+1,\quad\hat v=c+7/10\)

Calculate the exact signed residual at x=2 on the log-response scale

Data or fit

\(\displaystyle \log_{10}(y)-\widehat{\log_{10}(y)}=0.3\)

The signed log-response residual is 0.3. Find observed y divided by inverse-fitted y, to three decimal places

What is the correct interpretation of the positive 0.3 log residual?

Repeat the modelling process after curved residuals

Reuse the original 3D synthetic observations (x,y)=(1,1),(2,4),(3,9),(4,16),(5,25). First reproduce the residual plot for the raw least-squares line ŷ=6x−7. Propose and justify an input transformation, build the new columns, refit and construct a second residual plot. Compare the results and make a qualified recommendation

1. 1Complete the cycle Record raw/transformed scatterplots, full fitted coefficients, all predictions and residuals, and tool formulas. Explain whether changing only the raw fitted intercept could remove the curve. Then change the middle observed response from 9 to 10 and check whether the exact transformed fit survives
2. 2Save your complete work first Save the table, formulas and tool settings, labelled residual plots, numerical checks and written judgement before opening the model. Complete plot construction, reports and actual external-tool use are not automatically marked; review them yourself or with a teacher

Model response: Repeat the modelling process after curved residuals

Compare your saved construction and explanation with each step. Other suitable windows and equivalent exact expressions are acceptable

1. 1Initial evidence and transformation The raw line has residuals 2,−1,−2,−1,2, with zero sum and a clear curve. Changing only the intercept shifts all residuals by the same amount and cannot remove curvature. Try u=x²: the transformed pairs are (1,1),(4,4),(9,9),(16,16),(25,25), giving exact fitted line ŷ=u.
2. 2Refit and recommendation Every transformed residual is zero. A useful window is u=0..26,e=−1..1; explicitly state that all points lie on zero. Prefer this simple transformed rule for the supplied exact data. Perfect fit to five constructed observations does not guarantee future accuracy or identify a cause
3. 3Test a changed observation After changing the middle y to 10, the exact rule y=u no longer passes through all observations. Refit rather than hiding the change. The correct line, transformed mean point and residuals in row order are shown below. The fit is no longer exact; review the new residual evidence before retaining a recommendation \(\displaystyle \hat y=\frac{22}{85}+\frac{186}{187}u;\quad(\bar u,\bar y)=(11,11.2);\quad e=-\frac{237}{935},-\frac{222}{935},\frac{738}{935},-\frac{162}{935},-\frac{117}{935}\)

Data or fit

\(\displaystyle u=x^2,\quad y=16,\quad\hat y=u,\quad x=4\)

For the unchanged exact transformed dataset, find the residual at original x=4.

The paired residuals widen from ±1 to ±8 as input increases. What should the report say?