Correlation Coefficients: $r_p$ vs $r_s$

This question compares the product moment correlation coefficient ($r_p$) and the rank correlation coefficient ($r_s$) for bivariate data. These coefficients measure different types of relationships between variables.

Product Moment Correlation ($r_p$)

The product moment correlation coefficient ($r_p$) quantifies the linear relationship between two continuous variables. A value of $r_p = 1$ indicates a perfect positive linear relationship, meaning all data points lie exactly on a straight line with a positive slope.

Rank Correlation ($r_s$)

The rank correlation coefficient ($r_s$), such as Spearman's rank correlation, measures the monotonic relationship between two variables. A value of $r_s = 1$ indicates a perfect positive monotonic relationship, where as one variable increases, the other consistently increases, but not necessarily in a linear fashion.

Correct Statement: $r_p = 1$ implies $r_s = 1$

If the product moment correlation coefficient ($r_p$) is exactly 1, it signifies that all $n$ observations $(X_i, Y_i)$ fall perfectly on a straight line $Y = aX + b$ with a positive slope ($a > 0$).

This perfect linear association is a specific and stricter form of a perfect monotonic association. As $X$ increases linearly and perfectly, the ranks of $X$ will perfectly correspond to the ranks of $Y$ in the same order.

Therefore, the rank correlation coefficient ($r_s$) must also be 1 in this scenario. This statement is correct.

Why Other Statements Are Incorrect

$r_s = 1$ implies $r_p = 1$

A perfect monotonic relationship ($r_s = 1$) means ranks are perfectly ordered, but the underlying relationship between the original variables might be curved, not strictly linear. Thus, $r_p$ may not be 1.

This statement is incorrect.

$r_p \ge 0$ implies $r_s \ge 0$

A positive linear relationship ($r_p \ge 0$) implies a positive monotonic relationship ($r_s \ge 0$) because linearity is a subset of monotonicity. While generally true, the implication involving equality ($r_p=1 \implies r_s=1$) is a more precise and universally certain statement concerning the relationship between the two coefficients.

$r_s \ge 0$ implies $r_p \ge 0$

A positive monotonic relationship ($r_s \ge 0$) does not guarantee a positive linear relationship ($r_p \ge 0$). The relationship could be monotonic but non-linear (e.g., curved).

This statement is incorrect.