Select plausible transformations from scatterplot shape

What should be checked first when selecting transformations from a scatterplot?

Reuse the log-input experiment. Compare equal steps in the transformed coordinate with multiplicative changes in the original input. This shows why changing scale can straighten a relation

When t grows by one, what happens to original x?

Selecting candidate transformations

The shape guide uses the direction and curvature of a scatterplot to identify transformations worth trying. It gives a candidate set rather than a final conclusion, so each candidate must be fitted and checked

1. 1Check the direction Use the shape guide only when the association is consistently increasing or consistently decreasing \(\displaystyle \text{all secant slopes}>0\Rightarrow\text{increasing}\)
2. 2Use the curvature Decide whether the high end needs stretching or compressing before selecting the candidate axes \(\displaystyle \text{increasing, concave down}\Rightarrow\left\{\log_{10}(x),\frac{1}{x},y^{2}\right\}\)
3. 3Respect the domains A logarithm needs a positive coordinate and a reciprocal needs a non-zero coordinate on the axis being changed \(\displaystyle x>0\Rightarrow\log_{10}(x)\text{ is defined}\)
4. 4Treat candidates as trials Fit each suggested transformation and inspect the transformed data or residual plot before accepting a model \(\displaystyle \text{candidate}\Rightarrow\text{fit and check residuals}\)

A positive-domain association is increasing and concave down. Which complete candidate set is plausible for trial?

After a shape guide suggests a candidate set, what should happen next?

Treating a candidate as guaranteed. One of these lines is right. Which one?

A positive-scale trial guide

For positive original x and y, read the raw scatter from left to right. Increasing or decreasing describes slope sign; concave up or down describes how successive slopes change, including negative slopes. Keep the requested complete trial set distinct from a single candidate and from a final fitted model

1. 1Increasing, concave up Try x², log₁₀(y), or 1/y separately. Stretch the high positive input values or compress the high positive response values
2. 2Increasing, concave down Try log₁₀(x), 1/x, or y² separately. Compress positive input values or stretch high positive response values
3. 3Decreasing, concave up Try each transformation separately. Reciprocal input is a valid trial: for the example below it gives an exact line \(\displaystyle log_{10}(x),quadrac{1}{x},quadlog_{10}(y),quadrac{1}{y};qquad y=rac{12}{x},quad t=rac{1}{x}Rightarrow y=12t\)
4. 4Decreasing, concave down Try x² or y² separately. Check the transformed scatter and residuals before deciding whether either trial is useful
5. 5Limits of the guide These positive-scale directions suggest trials, not a uniquely best transform or all possible mathematical transformations. A turn, unclear direction or nonpositive original value falls outside this particular guide. Algebraic domains still apply; a square remains defined at negative values even when this guide is unsuitable

Data or rule

\(\displaystyle (x,y)=(1,12),(2,6),(3,4),(4,3)\)

Compute the first raw secant slope

Data or rule

\(\displaystyle (x,y)=(1,12),(2,6),(3,4),(4,3)\)

Compute the last raw secant slope for the same observations

How is the raw association described?

Which complete guide set applies to this positive decreasing, concave-up association?

Data or rule

\(\displaystyle (t,y)=(1,12),(1/2,6),(1/3,4),(1/4,3)\)

Check the slope after replacing only x by its reciprocal

A different dataset includes y=0. Can log₁₀(y) or 1/y be tried on the full dataset?

Select, test and challenge a candidate

Use the four positive synthetic observations (x,y)=(1,2),(4,4),(9,6),(16,8). Independently plot the raw data, calculate all raw slopes, select the complete positive-scale trial set and choose one candidate to test. Produce its full transformed table and plot, state a useful window and verify every transformed slope. Explain why choosing a candidate is different from accepting a model

1. 1Generate a counterexample Change only the final response from 8 to 9. Predict whether the raw direction/curvature and candidate guide change, then recalculate. Check whether your previously exact transformed line still fits. Explain the distinction between an exact-data test and a noisy measured-data fit

Model response: Select, test and challenge a candidate

Compare the complete construction and reasoning with your saved work. Other justified windows or tools may also be suitable. The counterexample changes only the final response from eight to nine

1. 1Raw evidence and trial selection Raw slopes are 2/3,2/5,2/7: positive and decreasing, so the data are increasing and concave down. The guide trials are log₁₀(x),1/x,y². Use x=0..17,y=0..10 for the original scatter. The guide does not guarantee that all three or any particular one will linearise these points
2. 2Test a response-square candidate Taking v=y² gives (1,4),(4,16),(9,36),(16,64). All consecutive slopes are four, so every original point satisfies v=4x. Use x=0..17,v=0..70 with axes x and y²; preserve x and verify the first/last square by hand. The responses are positive, so their sign is known even though the square transform itself loses sign
3. 3Counterexample and conclusion Changing the final response to 9 gives raw slopes 2/3,2/5,3/7. They are not monotonically decreasing, so this exact four-point example no longer has one decisive concavity throughout. The squared-response final slope becomes (81−36)/(16−9)=45/7, not four; the previous exact line fails. A trend or the first few correct slopes cannot prove a complete fit. For noisy measurements, inspect the full transformed scatter, fitted residuals and context rather than demanding identical successive slopes

In measured data, two consecutive transformed slopes differ slightly. What follows?