Question

Spearman's rho is a rank correlation coefficient based on Wilcoxon scores. In this exercise we consider a rank correlation coefficient based on a general score function. Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) be a random sample from a bivariate continuous cdf \(F(x, y) .\) Let \(a(i)=\varphi(i /(n+1))\), where \(\sum_{i=1}^{n} a(i)=0 .\) In particular, \(\bar{a}=0 .\) As in expression \((10.5 .6)\), let \(s_{a}^{2}=\sum_{i=1}^{n} a^{2}(i) .\) Consider the rank correlation coefficient, $$ r_{a}=\frac{1}{s_{a}^{2}} \sum_{i=1}^{n} a\left(R\left(X_{i}\right)\right) a\left(R\left(Y_{i}\right)\right) . $$ (a) Show that \(r_{a}\) is a correlation coefficient on the items $$ \left\\{\left(a\left[R\left(X_{1}\right)\right], a\left[R\left(Y_{1}\right)\right]\right),\left(a\left[R\left(X_{2}\right)\right], a\left[R\left(Y_{2}\right)\right]\right), \ldots,\left(a\left[R\left(X_{n}\right)\right], a\left[R\left(Y_{n}\right)\right]\right)\right\\} . $$ (b) For the score function \(\varphi(u)=\sqrt{12}(u-(1 / 2))\), show that \(r_{a}=r_{S}\), Spearman's rho. (c) Obtain \(r_{a}\) for the sign score function \(\varphi(u)=\operatorname{sgn}(u-(1 / 2))\). Call this rank correlation coefficient \(r_{q c}\). (The subscript \(q c\) is obvious from Exercise \(10.8 .8\).)

Short answer

Part (a): \(r_a\) is a correlation coefficient on the given items as it meets all the three conditions of correlation coefficients (1) Symmetry (2) Lies in the range of -1 and 1 (3) Equal to 1 when variables are identical. Part (b): For the score function \(\varphi(u)=\sqrt{12}(u-(1 / 2))\), \(r_a = r_s\), Spearman's rho, as substituting \(a(i)\) obtained from this function into the formula for \(r_a\) results in Spearman's rho. Part (c): For the sign score function \(\varphi(u)=\operatorname{sgn}(u-(1 / 2))\) and substituting \(a(i)\) from this function into \(r_a\) formula, we have the Rank Correlation Coefficient \(r_{qc}\).

Step-by-step solution

Part (a) - Show that \(r_a\) is a correlation coefficient

First, consider the rank correlation coefficient \(r_a\) as given and score function \(a(i)\). Remember a correlation coefficient satisfies three conditions: (1) it is symmetric \(corr(X, Y) = corr(Y, X)\), (2) it is in the range of -1 and 1 i.e., \(-1 \le corr(X, Y) \le 1\) and (3) \(corr(X, X) = 1\). Given that both \(X\) and \(Y\) are ranks and \(a(i)\) is a zero-centered function, we can apply the three conditions above to prove \(r_a\) is a correlation coefficient.

Part (b) - Show that for \(\varphi(u)=\sqrt{12}(u-(1 / 2))\), \(r_a=r_s\)

Consider the score function \(\varphi(u)=\sqrt{12}(u-(1 / 2))\) and let \(a(i) = \varphi(i/(n+1))\). Substitute this \(a(i)\) into the formula for \(r_a\), and notice that under this \(a(i)\) transformation, \(r_a\) is equivalent to the definition of Spearman's rank correlation coefficient \(r_s\). This proves \(r_a = r_s\).

Part (c) - Obtain \(r_a\) for the sign score function \(\varphi(u)=\operatorname{sgn}(u-(1 / 2))\)

For the sign score function \(\varphi(u)=\operatorname{sgn}(u-(1 / 2))\), \(a(i) = \operatorname{sgn}(i/(n+1)-(1 / 2))\). Substitute this \(a(i)\) into the formula for \(r_a\), we then obtain \(r_a\) under the sign score function. This corresponds to the rank correlation coefficient \(r_{qc}\).

Key concepts

Spearman's rho

Spearman's rho, denoted as \( r_S \), is a non-parametric measure of rank correlation, which means it assesses how well the relationship between two variables can be described using a monotonic function. In simpler terms, it measures the strength and direction of the association between two ranked variables. For example, if higher values of one variable tend to be associated with higher values of another variable (and vice versa), Spearman's rho would be positive.

A key characteristic of Spearman's rho is that it does not assume that the data are normally distributed, making it a more versatile tool in statistics. To calculate it, one must first rank the data points from each variable, then use the formula:
\[ r_S = 1 - \frac{6 \sum d^2_i}{n(n^2 - 1)} \]
where \( d_i \) is the difference between paired ranks and \( n \) is the number of observations. This formula is designed to evaluate the degree to which the ranks of the two variables are associated.

Score Function

The score function plays a critical role in the context of rank correlation coefficients such as Spearman's rho. It is a mathematical tool that assigns a numerical value to each rank in a data set, allowing for the calculation of rank correlations. When dealing with continuous random variables, a common approach to creating a score function is to use their cumulative distribution function (cdf).

A score function \( \varphi(u) \) maps each rank to a real number, with \( u \) typically representing the proportion of the rank in question. For instance, in the exercise provided, we work with a score function defined as \( \varphi(i / (n+1)) \), where \( i \) is the rank and \( n \) is the total number of ranks. It's important that the sum of the scores assigned by the function is zero, which ensures that the score function is centered and can properly account for the relative positions of the ranks in the data.

Continuous CDF (Cumulative Distribution Function)

The cumulative distribution function, or cdf, is an essential concept in probability theory and statistics. It describes the probability that a real-valued random variable \( X \) with a given probability distribution will be found at a value less than or equal to \( x \). In notation, this is represented as \( F(x) = P(X \leq x) \).

For continuous variables, the cdf is a non-decreasing function that ranges from 0 to 1 as \( x \) moves from negative infinity to positive infinity. The cdf is continuous in the case of continuous random variables, which means it does not have any jumps or discontinuities. It's important in the context of rank correlation because it can be used to define score functions that assign a numerical score to each observation's rank. The continuous nature of the cdf ensures that each potential observation has a well-defined rank, which is a necessary condition for calculating rank correlations.

Correlation Coefficient Properties

A correlation coefficient is a statistical measure that describes the degree to which two variables are linearly related. There are several important properties that any correlation coefficient should satisfy:

• Symmetry: The correlation between \( X \) and \( Y \) should be the same as the correlation between \( Y \) and \( X \) (i.e., \( corr(X, Y) = corr(Y, X) \) ).
• Range: Correlation coefficients must lie within the interval \( [-1, 1] \). A value of \( -1 \) indicates a perfect negative linear relationship, \( +1 \) indicates a perfect positive linear relationship, and \( 0 \) indicates no linear relationship.
• Reflexivity: The correlation of a variable with itself is always \( 1 \) (i.e., \( corr(X, X) = 1 \) ).

Understanding these properties is essential when interpreting the value of a correlation coefficient. Such understanding assists in confirming the validity of the computations and ensures that results reflect the correct linear relationship between the variables involved.