The association between two variables y and x, as shown in the scatterplot above, is non-linear. Which of the following transformations is most likely to linearise the relationship?

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Question restatement: The association between two variables y and x, as shown in the scatterplot above, is non-linear. Which of the following transformations is most likely to linearise the relationship?

Option a: a 1/y transformation
- Using a 1/y transformation would map higher y values to smaller numbers and vice versa. This approach is typically useful if the underlying relationship is proportional to 1/y or if the model is of the form y ~ 1/f(x). In the given scatterplot, y values start around 4–5 when x is small and decrease toward zero as x increases, but the pattern does not obviously suggest a 1/y relationship being linear in x. Therefore, this transformation is unlikely to produce a straight line across the plotted points.

Option b: a 1/x transformation
- Transforming x to 1/x and plotting y versus 1/x can linearize relationships of the form y ~ a + b(1/x). This is common when the dependent variable responds inversely to x. However, in the current plot, the spread and curvature do not clearly indicate a simple inverse relationship between y and x alone. Thus, a 1/x transform of the predictor is not the most straightforward route here.

Option c: a y^2 transformation
- Squaring y would distort the scale of the dependent variable and typically does not help linearize a monotonic decreasing pattern with x. Unless the true relationship involved a quadratic dependency in y, simply squaring y is unlikely to straighten the scatter.

Option d: a log y transformation
- A log transformation on y is a common technique when the relationship between y and x is exponential in nature, i.e., y ≈ A·e^(−k·x) or similar. In such cases, log(y) plotted against x tends to be approximately linear. The given scatterplot shows a smooth, monotonic decrease of y with increasing x, which is characteristic of an exponential decline. Therefore, applying log y is the most plausible transformation to linearise this pattern.

In summary, among the provided options, the log transformation of y best matches the typical scenario where an exponential-type decrease with x becomes a straight line when taking the logarithm of y, making option d the most suitable choice for linearisation.

The association between two variables y and x, as shown in the scatterplot above, is non-linear. Which of the following transformations is most likely to linearise the relationship?

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