Question

1. suppose that the scatterplot of (in(x), in(y)) shows a strong positive correlation. which of the following must be true?

i. the variables x and y also have a correlation close to 1.
ii. a scatterplot of (x, y) shows a strong nonlinear pattern.
iii. the residual plot of the variables x and y shows a random pattern.
(a) i only
(b) ii only
(c) iii only
(d) i and ii
(e) i, ii, and iii

Explanation:

• For statement I:
• The transformation \(u = \ln(x)\) and \(v=\ln(y)\) having a strong positive correlation implies a power - law relationship \(y = ax^{b}\) (by taking the exponential of \(v = b u+\ln(a)\)). A power - law relationship is a non - linear relationship. But the existence of a non - linear relationship between \(\ln(x)\) and \(\ln(y)\) does not imply a linear relationship (which is what correlation measures) between \(x\) and \(y\). So, statement I is false.
• For statement II:
• Correlation (\(r\)) measures the strength of a linear relationship. Since the relationship between \(x\) and \(y\) is non - linear (because of the non - linear transformation \(\ln(x)\) and \(\ln(y)\) having a strong relationship), the correlation coefficient \(r\) between \(x\) and \(y\) is not close to \(1\). So, statement II is false.
• For statement III:
• When we fit a regression model (even a non - linear regression model, after transformation). A random residual plot is a sign that the model is a good fit. If the scatterplot of \((\ln(x),\ln(y))\) shows a strong pattern (which can be modeled by a linear regression of \(\ln(y)\) on \(\ln(x)\)), then when we go back to the original variables \(x\) and \(y\) (by exponentiating the regression equation \(y = ax^{b}\)), the residual plot of \(y\) vs \(x\) (after fitting the non - linear model \(y = ax^{b}\)) will show a random pattern.

Answer:

C. III only