Weighting Lower and Upper Ranks Simultaneously Through Rank-Order Correlation Coefficients

Abstract

Two new weighted correlation coefficients, that allow to give more weight to the lower and upper ranks simultaneously, are proposed. These indexes were obtained computing the Pearson correlation coefficient with a modified Klotz and modified Mood scores. Under the null hypothesis of independence of the two sets of ranks, the asymptotic distribution of these new coefficients was derived. The exact and approximate quantiles were provided. To illustrate the value of these measures an example, that could mimic several biometrical concerns, is presented. A Monte Carlo simulation study was carried out to compare the performance of these new coefficients with other weighted coefficient, the van der Waerden correlation coefficient, and with two non-weighted indexes, the Spearman and Kendall correlation coefficients. The results show that, if the aim of the study is the detection of correlation or agreement between two sets of ranks, putting emphasis on both lower and upper ranks simultaneously, the use of van der Waerden, signed Klotz and signed Mood rank-order correlation coefficients should be privileged, since they have more power to detect this type of agreement, in particular when the concordance was focused on a lower proportion of extreme ranks. The preference for one of the coefficients should take into account the weight one wants to assign to the extreme ranks.

Appendix

1.1 A1 Mean and Variance of $R_{_S}$

Indeed, under the null hypothesis of independence of the two sets of rankings ($\left( R_{11}, R_{12}, \ldots , R_{1n}\right)$ and $\left( R_{21}, R_{22}, \ldots , R_{2n}\right)$), the expected value of $R_{_S}$ is zero:

$$\begin{aligned} \mathbb {E}\left( R_{_S}\right) \,{=}\, \mathbb {E}\left( \frac{1}{C_{_S}}\textstyle \sum _{j=1}^{n} S_{1j}S_{2j}\right) \! \,{=}\, \frac{1}{C_{_S}} \textstyle \sum _{j=1}^{n} \mathbb {E}\left( S_{1j}S_{2j}\right) \! \,{=}\, \frac{n}{C_{_S}}\, \mathbb {E}\left( S_{1j}\right) \mathbb {E}\left( S_{2j}\right) \! \,{=}\, 0 . \end{aligned}$$

In fact, the expected values of each one of the variables $S_{ij}$, with $i=1, 2$ and $j=1,\ldots ,n$, is zero, since it is an expected value of a function of a discrete uniform variable in n points, $X_{ij}$, with probability function $f_{X_{ij}}(x) = \frac{1}{n}$, i.e.,

$$\begin{aligned} \mathbb {E}\left( S_{ij}\right) = \mathbb {E}\left( g \left( X_{ij}\right) \right) = \textstyle \sum _{j=1}^n g(x) f_{X_{ij}}(x)= \frac{1}{n} \textstyle \sum _{j=1}^n s_{ij} = 0 . \end{aligned}$$

Note that for the van der Waerden and for the signed Klotz correlation coefficients one has $X_{ij} = \frac{R_{ij}}{n+1}$, but while $g\left( X_{ij}\right) = \varPhi ^{-1}\left( X_{ij}\right)$ in van der Waerden case, $g\left( X_{ij}\right) = sign\left( R_{ij} - \frac{n+1}{2}\right) \left( \varPhi ^{-1}\left( X_{ij}\right) \right) ^2$ for the signed Klotz. In the case of signed Mood correlation coefficient, $X_{ij} = R_{ij} - \frac{n+1}{2}$ and $g\left( X_{ij}\right) = sign\left( X_{ij}\right) X_{ij}^2$.

In what concerns the variance of $R_{_S}$, under the null hypothesis of independence between the two sets of rankings, one has:

$$\begin{aligned} Var\left( R_{_S}\right)= & {} Var\left( \frac{1}{C_{_S}}\textstyle \sum _{j=1}^{n} S_{1j} S_{2j}\right) = \frac{1}{C_{_S}^2} Var\left( \textstyle \sum _{j=1}^{n} S_{1j}S_{2j}\right) . \end{aligned}$$

(1)

As a matter of fact,

$$\begin{aligned}&Var \left( \textstyle \sum _{j=1}^{n} S_{1j} S_{2j}\right) \\= & {} n Var\left( S_{1j}\right) Var\left( S_{2j}\right) + n(n-1) Cov\left( S_{1j},S_{1k}\right) Cov\left( S_{2j},S_{2k}\right) \\= & {} n \left( Var\left( S_{1j}\right) \right) ^2 + n(n-1) \left( Cov\left( S_{1j},S_{1k}\right) \right) ^2 . \end{aligned}$$

Attending to the fact that

$$\begin{aligned} Var\left( S_{1j}\right) = \mathbb {E}\left( S_{1j}^2\right) - \mathbb {E}^2\left( S_{1j}\right) = \mathbb {E}\left( S_{1j}^2\right) = \textstyle \sum _{j=1}^n \left( g(x)\right) ^2 f_{X_{ij}}(x) = \frac{C_{_S}}{n} \end{aligned}$$

and, considering the joint probability function of the random sample $\left( X_{1j},X_{1k}\right)$, $f_{_{\left( X_{1j},X_{1k}\right) }}\left( x_{1j},x_{1k}\right) = \frac{1}{n(n-1)}$, for $j \ne k$ and $j, k = 1, \ldots ,n$, then

$$\begin{aligned} Cov\left( S_{1j},S_{1k}\right)= & {} \! \mathbb {E}\left( S_{1j}S_{1k}\right) \!- \!\mathbb {E}\left( S_{1j}\right) \mathbb {E}\left( S_{1k}\right) \! = \!\mathbb {E}\left( S_{1j}S_{1k}\right) = \mathbb {E}\left( g\left( X_{1j}\right) g\left( X_{1k}\right) \right) \\= & {} \! \textstyle \sum _{j \ne k} g\left( x_{1j}\right) g\left( x_{1k}\right) f_{_{\left( X_{1j},X_{1k}\right) }}\left( x_{1j},x_{1k}\right) \!=\!\frac{1}{n(n-1)}\textstyle \sum _{j \ne k} s_{1j} s_{1k}\\= & {} \frac{1}{n(n-1)}\left[ \left( \textstyle \sum _{j=1}^n s_{1j}\right) ^2 - \textstyle \sum _{j=1}^n s_{1j}^2\right] = -\frac{C_{_S}}{n(n-1)} . \end{aligned}$$

Therefore

$$\begin{aligned} Var\left( \textstyle \sum _{j=1}^{n} S_{1j} S_{2j}\right)= & {} n \left( Var\left( S_{1j}\right) \right) ^2 + n(n-1) \left( Cov\left( S_{1j},S_{1k}\right) \right) ^2\nonumber \\= & {} n \left( \frac{C_{_S}}{n}\right) ^2 + n(n-1) \left( -\frac{C_{_S}}{n(n-1)}\right) ^2 = \frac{C^2_{_S}}{n-1} . \end{aligned}$$

(2)

Finally, from Eqs. (1) and (2), it follows that $Var\left( R_{_S}\right) = \frac{1}{n-1}$.

1.2 A2 Tables

Table 3. Exact quantiles of $R_{_W}$, $R_{_{SK}}$, and $R_{_{SM}}$

Table 4. Approximate quantiles of $R_{_W}$, $R_{_{SK}}$, and $R_{_{SM}}$.

Table 5. Mean (standard deviation) of Spearman, Kendall’s Tau, van der Waerden, signed Klotz, and signed Mood correlation coefficient estimates, for two scenarios: on the left, the concordance was targeted for a higher proportion ($p=0.25$) of extreme ranks, and on the right, the concordance was focused on a lower proportion ($p=0.1$) of extreme ranks.

Table 6. Powers ($\%$) of Spearman, Kendall’s Tau, van der Waerden, signed Klotz, and signed Mood correlation coefficients, for two scenarios: on the left, the concordance was targeted for a higher proportion ($p=0.25$) of extreme ranks, and on the right, the concordance was focused on a lower proportion ($p=0.1$) of extreme ranks.