Measures of Similarity / Association

Flowgorithm file: FL-EStetrachoricBrown.fprg.

Cole C₁:

Cole C₂, see Yule's (= Pearson's Phi) (1920)

Cole C₃, see McEwen and Michael (1920)

Cole C₄, see Yule's Q (= Pearson Q ₂) (1900)

Cole C₅:

Cole C₆, see Pearson Q ₃ (1900)

Cole C₇:

And helper functions:

Flowgorithm file: FL-EStetrachoricKirk.fprg.

same as Cole C₃:

Pearson Q₁:

Pearson Q₂, see Yule's Q (1900)

Pearson Q₃ (= Yule's r, Cole's C₆):

Pearson Q₄:

Pearson Q₅:

Yule's Q (= Pearson's Q₂, Cole's C₄:

Yule's Phi = Pearson's Phi:

Yule's r see Pearson Q₃ (1900)

Alroy adjusts the Forbes coefficient by setting (Alroy, 2015):

\(F' = \frac{a\times\left(n' + \sqrt{n'}\right)}{a\times\left(n' + \sqrt{n'}\right) + \frac{3}{2}\times b\times c}\)

With:

\(n' = a + b + c\)

Alroy refers to the Forbes coefficient as a measure of similarity. Alroy then sets out to improve the measure by disregarding the lower left value (d).

Equation 76 found in Choi et al. (2010):

\(\left|\frac{a\times R_2}{c\times R_1}\right|\)

Equation 70 found in Choi et al. (2010):

\(\frac{\sigma - \sigma'}{2\times n}\)

With:

\(\sigma = \max{\left(a, b\right)} + \max{\left(c, d\right)} + \max{\left(a, c\right)} + \max{\left(b, d\right)}\)

\(\sigma' = \max{\left(R_1, R_2\right)} + \max{\left(C_1, C_2\right)}\)

Supposedly from Anderberg (1973):

Austin and Colwell (1977, p. 205):

\(S = \frac{2}{\pi} \text{arcsin} \sqrt{\frac{a + d}{n}}\)

Equation 21 found in Hubálek (1982)

Baroni-Urbani and Buser S (1976, p. 258):

\(S_{**} = \frac{\sqrt{a\times d} + a}{\sqrt{a\times d} + a + b + c}\)

\(S_{*} = \frac{a - b - c + \sqrt{a\times d}}{a + b + c\sqrt{a\times d}}\)

Equations 71 and 72 from Choi et al. (2010):, equation 32 and 33 in Hubálek (1982) and equation 38a and 38b from Warrens (2008)

Becker and Clogg (1988, pp. 410-412)

\( \rho^* = \frac{g-1}{g+1} \)

\( \rho^{**} = \frac{OR^{13.3/\Delta} - 1}{OR^{13.3/\Delta} + 1} \)

with:

\(g=e^{12.4\times\phi - 24.6\times\phi^3}\)

\(\phi = \frac{\ln\left(OR\right)}{\Delta}\)

\(OR=\frac{\left(\frac{a}{c}\right)}{\left(\frac{b}{d}\right)} = \frac{a\times d}{b\times c}\)

\(\Delta = \left(\mu_{R1} - \mu_{R2}\right) \times \left(v_{C1} - v_{C2}\right)\)

\(\mu_{R1} = \frac{-e^{-\frac{t_r^2}{2}}}{p_{R1}}, \mu_{R2} = \frac{e^{-\frac{t_r^2}{2}}}{p_{R2}} \)

\(v_{C1} = \frac{-e^{-\frac{t_c^2}{2}}}{p_{C1}}, v_{C2} = \frac{e^{-\frac{t_c^2}{2}}}{p_{C2}} \)

\(t_r = \Phi^{-1}\left(p_{R1}\right), t_c = \Phi^{-1}\left(p_{C1}\right)\)

\(p_{x} = \frac{x}{n}\)

\(\Phi^{-1}\left(x\right)\) is the inverse standard normal cumulative distribution function

\(OR\) is the Odds Ratio

Bonett and Price (2007, pp. 433-434)

\(Y* = \frac{\hat{\omega}^x-1}{\hat{\omega}^x+1}\)

With:

\(x = \frac{1}{2}-\left(\frac{1}{2}-p_{min}\right)^2\)

\(p_{min} = \frac{\text{MIN}\left(R_1, R_2, C_1, C_2\right)}{n}\)

\(\hat{\omega} = \frac{\left(a+0.1\right)\times\left(d+0.1\right)}{\left(b+0.1\right)\times\left(c+0.1\right)}\)

Note that \(\hat{\omega}\) is a biased corrected version of the Odds Ratio.

Bonett and Price ρ^* (2005, p. 216)

\(\rho^* = \cos\left(\frac{\pi}{1+\omega^c}\right)\)

With:

\(\omega = OR = \frac{a\times d}{b\times c}\)

\(c = \frac{1-\frac{\left|R_1-C_1\right|}{5\times n} - \left(\frac{1}{2}-p_{min}\right)^2}{2}\)

\(p_{min} = \frac{\text{MIN}\left(R_1, R_2, C_1, C_2\right)}{n}\)

Bonett and Price ρ_hat^* (2005, p. 216)

\(\hat{\rho}^* = \cos\left(\frac{\pi}{1+\hat{\omega}^\hat{c}}\right)\)

With:

\(\hat{\omega} = \frac{\left(a+\frac{1}{2}\right)\times \left(d+\frac{1}{2}\right)}{\left(b+\frac{1}{2}\right)\times \left(c+\frac{1}{2}\right)}\)

\(\hat{c} = \frac{1-\frac{\left|R_1-C_1\right|}{5\times \left(n+2\right)} - \left(\frac{1}{2}-\hat{p}_{min}\right)^2}{2}\)

\(\hat{p}_{min} = \frac{\text{MIN}\left(R_1, R_2, C_1, C_2\right)+1}{n+2}\)

Braun and Blanquet supposedly from (1932):

\(\frac{a}{\max{\left(R_1, C_1\right)}}\)

\(S_{*} = \frac{a - b - c + \sqrt{a\times d}}{a + b + c\sqrt{a\times d}}\)

Equation 46 from Choi et al. (2010), equation 1 in Hubálek (1982) and equation 12 from Warrens (2008)

Camp (1934, pp. 309) describes the following steps for the calculation:

Step 1: If total of column 1 (\(C_1\)) is less than column 2 (\(C_2\)), swop the two columns

Step 2: Calculate \(p = \frac{C_1}{n}\), \(p_1 = \frac{a}{n}\), and \(p_2 = \frac{c}{C_2}\)

Step 3: Determine \(z_1\), \(z_2\) as the normal deviate corresponding to the area \(p_1\), \(p_2\) resp. (inverse standard normal cumulative distribution), i.e. \(z_1 = \Phi^{-1}\left(p_1\right), z_2 = \Phi^{-1}\left(p_2\right)\)

Step 4: Determine y the normal ordinate corresponding to \(p\) (the height of the normal distribution)

Step 5: Calculate \(m = \frac{p\times\left(1-p\right)\times\left(z_1 + z_2\right)}{y}\)

Step 6: Find phi in a table of phi values

Camp suggested for a very basic approximation to simply use \(\phi=1\).

For a better approximation Camp made the following table:

Cureton (1968, p. 241) expanded on this table and produced:

Step 7: Calculate \(r_t = \frac{m}{\sqrt{1+\phi\times m^2}}\)

Cureton (1968) describes quite a few shortcomings with this approximation, and circumstances when it might be appropriate.

Chen and Popovich (2002, pp. 37-38):

\(\lambda_x = \Phi^{-1}\left(\frac{R_1}{n}\right)\)

\(\lambda_y = \Phi^{-1}\left(\frac{C_1}{n}\right)\)

Clement Inter-observer Agreement (1976, p. 258):

\(IAR = \frac{a \times R_2}{n \times R_1} + \frac{d \times R_1}{n \times R_2}\)

Equation 37 from Warrens (2008)

Cohen Kappa (1960, p. 40):

\(\kappa = \frac{n \times P - Q}{n^2 - Q} = \frac{2\times\left(a\times d - b\times c \right)}{R_1\times C_2 + R_2\times C_1}\)

With:

\(P = \sum{i=1}^r F_{i,i}\)

\(Q = \sum{i=1}^r R_i\times C_i\)

Equation 24 from Warrens (2008)

Cohen w (1988, p. 216):

\(w = \sqrt{\frac{\chi^2}{n}}\)

With:

\(\chi^2 = \frac{n\times\left(a\times d - b\times c\right)^2}{R_1\times R_2\times C_1\times C_2}\)

See Cohen W for more information and rules-of-thumb.

Cole's C₁(1949, p. 415) is an adjusted Forbes Coefficient to:

\( C_1=\frac{a\times d - b\times c}{\left(a+b\right)\times\left(a+c\right)} \)

Cole's C₅ (1949, p. 416) is an adjustment of the Contingency Coefficient for 2x2 tables:

\( C_5 = \frac{\sqrt{2}\times\left(a\times d-b\times c\right)} {\sqrt{\left(a\times d-b\times c\right)^2 + R_1\times R_2\times C_1\times C_2}} \)

Cole's C₇ (1949, pp. 420-421) is the measure Cole derived himself and calls it 'coefficient of interspecific association" given as:

\(\frac{a\times d - b\times c}{\left(a+b\right) \times \left(b + d\right)}\)

Contingency Coefficient (Pearson, 1904, p. 9):

\(C = \sqrt{\frac{\chi^2}{n + \chi^2}}\)

With:

\(\chi^2 = \frac{n\times\left(a\times d - b\times c\right)^2}{R_1\times R_2\times C_1\times C_2}\)

Dennis (1965, p. 69):

\(R^2 = \frac{a \times d - b \times c}{sqrt{n \times R_1 \times C_1}}\)

Equation 44 from Choi et al. (2010)

Dice (1945, p. 302):

\(\text{Coefficient of Association} = \frac{a}{a + b}\)

\(\text{Coefficient of Association} = \frac{a}{a + c}\)

\(\text{coincidence index} = \frac{2 \times a}{2 \times a + b + c}\)

The Coincidence Index is the same as Gleason.

Equation 17a and 17b from Warrens (2008)

Digby's H (1983, p. 754)

\(H = \frac{\left(a\times d\right)^{3/4} - \left(b\times c\right)^{3/4}}{\left(a\times d\right)^{3/4} + \left(b\times c\right)^{3/4}}\)

Doolittle i (1885, p. 123):

\(\frac{\left(a \times d - b \times c\right)^2}{R_1 \times R_2 \times C_1 \times C_2}\)

Equation 31 in Hubálek (1982) and equation 2 from Warrens (2008)

Driver and Kroeber (1932, p. 219):

\(G = \frac{a}{\sqrt{R_1 \times C_1}}\)

\(A = \frac{1}{2} \times \left(\frac{a}{R_1} + \frac{a}{C_1}\right)\)

G is listed as equation 31, 33 and 38 from Choi et al. (2010) but attributes it as Ochiai-I refering to Ochaiai (1957) in equation 33, while the same equation is used for 38 but then attributed to Otsuka (1936), equation 11 in Hubálek (1982) and equation 13 from Warrens (2008)

A is listed as equation 41 and 42 from Choi et al. (2010) , equation 7 and 8 in Hubálek (1982) and equation 11a from Warrens (2008). The original source is most likely Kulczynski (1927)

Edwards Q (1957, p. ???) as a variation on Yule's Q:

\( Q_E = \frac{OR^{\pi/4} - 1}{OR^{\pi/4} + 1} \)

\(OR=\frac{\left(\frac{a}{c}\right)}{\left(\frac{b}{d}\right)} = \frac{a\times d}{b\times c}\)

Eyraud supposedly from (1936):

\(\frac{a - R_1 \times C_1}{R_1 \times R_2 \times C_1 \times C_2}\)

Equation 74 from Choi et al. (2010) but with an additonal n factor in denominator, equation 17 in Hubálek (1982) and equation 12 from Warrens (2008)

Faith C (1983, p. 290):

\(C = \frac{a + \frac{1}{2}\times d}{n}\)

\(S_{*} = \frac{a - b - c + \sqrt{a\times d}}{a + b + c\sqrt{a\times d}}\)

Equation 10 from Choi et al. (2010).

Fager-McGowan (Index of Affinity) (1963, p. 454) as interpreted by Warrens (2008):

\(\frac{a}{\sqrt{R_1 \times C_1}} - \frac{1}{2 \times \sqrt{\max{\left(R_1, C_1\right)}}}\)

as interpreted by Choi et al. (2010) and Hubálek (1982):

\(\frac{a}{\sqrt{R_1 \times C_1}} - \frac{\sqrt{\max{\left(R_1, C_1\right)}}}{2}\)

Equation 47 from Choi et al. (2010), equation 13 in Hubálek (1982) and equation 29 from Warrens (2008)

Fleiss M (1975, p. 656):

\(M = \frac{\left(a \times d - b \times c\right) \times \left(R_1 \times C_2 + R_2 \times C_1\right)}{2 \times R_1 \times R_2 \times C_1 \times C_2}\)

Equation 36 from Warrens (2008)

Forbes Coefficient (1907, p. 279)

\( F=\frac{n\times a}{\left(a+b\right)\times\left(a+c\right)} \)

Cole (1949) later adjusted this to range from -1 to 1 and dubbed it C₁.

Fossum-Kaskey Chi-Square (1966, p.65):

\(\chi_{FK}^2 = \frac{n \times \left(a - \frac{1}{2}\right)^2}{\left(a + b\right) \times \left(a + c\right)}\)

Equation 35 from Choi et al. (2010).

Gilbert i' (1884, p. 171):

\(i' = \frac{a \times n - R_1 \times C_1}{C_1 \times n + R_1 \times n - a \times n - R_1 \times C_1}\)

Equation 46 from Choi et al. (2010), equation 1 in Hubálek (1982) and equation 12 from Warrens (2008)

Gilbert-Wells supposedly from (1966):

\(\ln{\left(a\right)} - \ln{\left(n\right)} - \ln{\left(\frac{R_1}{n}\right)} - \ln{\left(\frac{C_1}{n}\right)}\)

Equation 55 from Choi et al. (2010), equation 38 in Hubálek (1982).

Gleason (1920, p. 31):

\(\frac{2 \times a}{2 \times a + b + c}\)

Choi et al. (2010) has this labelled as Dice (eq. 2) and Czekanowski (eq. 3) and Nei-Li (eq 5), equation 5 in Hubálek (1982) and equation 9 from Warrens (2008)

Goodman-Kruskal Lambda (1954, p. 743):

\(\frac{\sigma-\sigma'}{2n-\sigma'}\)

With:

\(\sigma = \max\left(a,b\right)+\max\left(c,d\right)+\max\left(a,c\right)+\max\left(b,d\right)\)

\(\sigma' = \max\left(R_1, R_2\right)+\max\left(C_1, C_2\right)\)

Warrens (2008, p. 220) however, describes a different version:

\(\frac{2\min\left(a,d\right)-b-c}{2\min\left(a,d\right)+b+c}\)

Equation 20 from Warrens (2008)

Gower supposedly from (1971):

\(\frac{a + d}{\sqrt{R_1 \times R_2 \times C_1 \times C_2}}\)

Equation 50 from Choi et al. (2010).

Hamann supposedly from (1961):

\(\frac{\left(a + d\right) - \left(b + c\right)}{n}\)

Equation 67 from Choi et al. (2010), equation 24 in Hubálek (1982) and equation 27 from Warrens (2008)

Harris-Lahey Weighted Agreement (1978) is an adjustment to Clement formula:

\(WA = \frac{a \times \left(R_2 + C_2\right)}{2 \times n \ times \left(a + b + c\right)} + \frac{d \times \left(R_1 + C_1\right)}{2 \times n \ times \left(b + c + d\right)}\)

Equation 40 from Warrens (2008) although Warrens ommits the factor n in each denominator.

Hawkins-Dotson (1975):

\(\frac{1}{2} \times \left(\frac{a}{a + b + c} + \frac{d}{b + c + d}\right)\)

Equation 34 from Warrens (2008)

Hurlbert (1969, p3):

\(C_8 = \frac{ad-bc}{\left\lvert ad-bc\right\rvert}\sqrt{\frac{\chi^2-\chi_{min}^2}{\chi_{max}^2-\chi_{min}^2}}\)

With:

\(\chi_{max}^2 = \begin{cases} \frac{nR_1 C_2}{R_2 C_1} & \text{ if } ad \geq bc \\ \frac{nR_1 C_1}{R_2 C_2} & \text{ if } ad < bc \text{ and } a \leq d \\ \frac{nR_2 C_2}{R_1 C_1} & \text{ if } ad < bc \text{ and } a > d \end{cases}\)

\(\chi_{min}^2 = \frac{n^3\left(\hat{a} - g\left(\hat{a}\right)\right)^2}{R_1 R_2 C_1 C_2}\)

\(\hat{a}=\frac{R_1 C_1}{n}\)

\(g\left(\hat{a}\right) = \begin{cases} \lfloor \hat{a} \rfloor & \text{ if } ad < bc \\ \lceil \hat{a}\rceil & \text{ if } ad \geq bc \end{cases}\)

Note that Hurlbert showed that Cole’s C’s can be rewritten to use the chi-square value, and then introduced a new one and labelled it C8.

Equation 35 in Hubálek (1982).

Jaccard Coefficient of Cummunity (1901, 1912, p. 39) and Tanimoto (1958, p. 5):

\(\frac{a}{a + b + c}\)

Equation 1+65 from Choi et al. (2010), equation 4 in Hubálek (1982) and equation 6 from Warrens (2008)

Johnson supposedly from (1967):

\(\frac{a}{a + b} + \frac{a}{a + c}\)

Equation 43 from Choi et al. (2010), equation 9 in Hubálek (1982) and equation 33 from Warrens (2008)

Kent and Foster (1977, p. 311):

\(K_{occ} = \frac{-b \times c}{b \times R_1 + c \times C_1 + b \times c}\)

\(K_{non occ} = \frac{-b \times c}{b \times R_2 + c \times C_2 + b \times c}\)

Equation 39a+b from Warrens (2008)

Kuder-Richardson supposedly from (1937):

\(\frac{4 \times \left(a \times d - b \times c\right)}{R_1 \times R_2 \times C_1 \times C_2 + 2 \times a \times d - 2 \times b \times c}\)

Equation 14 from Warrens (2008)

A first version would be:

\(\frac{a}{b + c}\)

Equation 64 from Choi et al. (2010), equation 3 in Hubálek (1982) and equation 11b from Warrens (2008)

A second version would be, supposedly from Kulczynski (1927):

\(A = \frac{1}{2} \times \left(\frac{a}{R_1} + \frac{a}{C_1}\right)\)

Equation 41 and 42 from Choi et al. (2010), equation 7 and 8 in Hubálek (1982) and equation 11a from Warrens (2008)

Forbes supposedly from (1925), but also found in Loevinger (1947, p. 30):

\(\frac{a \times d - b \times c}{\min{\left(R_1 \times R_2 \times C_1 \times C_2\right)}}\)

Equation 48 from Choi et al. (2010), equation 42 in Hubálek (1982) and equation 18 from Warrens (2008)

Maxwell-Pilliner supposedly from (1968):

\(\frac{2 \times \left(a \times d - b \times c \right)}{\left(a + b\right) \times \left(c + d\right) + \left(a + c\right) \times \left(b + d\right)}\)

Equation 35 from Warrens (2008)

McConnaughey supposedly from (1964):

\(\frac{a^2 - b \times c}{R_1 \times C_1}\)

Equation 39 from Choi et al. (2010), equation 10 in Hubálek (1982) and equation 31 from Warrens (2008)

Michael (1920, p. 55) worked together with McEwen and named the following 'McEwen and Michael coefficient':

\(\frac{a\times d - b\times c}{\left(\frac{a+ d}{2}\right)^2 + \left(\frac{b+ c}{2}\right)^2}= \frac{4\times\left(a\times d - b\times c\right)}{\left(a+d\right)^2+\left(b+c\right)^2}\)

Later Cole (1949, p. 415) rewrote this equation for his C₃ into:

\(C_3 = \frac{4\times\left(a\times d - b\times c\right)}{\left(a+d\right)^2+\left(b+c\right)^2}\)

Equation 68 from Choi et al. (2010), equation 39 in Hubálek (1982) and equation 10 from Warrens (2008)

Mountford (1962, p. 45):

\(\frac{2 \times a}{a \times \left(b + c\right) + 2 \times b \times c}\)

Equation 37 from Choi et al. (2010), equation 15 in Hubálek (1982) and equation 28 from Warrens (2008)

A measure labelled ‘Pearson’ from Choi et al. (2010, p. 45):

\(\sqrt{\frac{\phi^2}{n + \phi^2}}\)

Equation 53 from Choi et al. (2010).

Pearson Phi Coefficient (Pearson, 1900, p. 12) is not a tetrachoric correlation, but rather the 'standard' correlation if you would give each of the two values for each of the two variables a 0 and 1. Yule also derived the same result (Yule, 1912, p. 596). The formula can be written as:

\(\phi = \frac{a\times d - b\times c}{\sqrt{R_1\times R_2\times C_1\times C_2}}\)

The formula can also be re-written to use the Pearson chi-square statistic , Cole's C₂ (1949) uses this:

\(C_2 = \sqrt{\frac{\chi^2}{n}}\)

Cohen (1988, p. 216) refers to this as 'w'. He uses the proportions in the cross table and gives the equation:

\(w = \sqrt{\sum_i\sum_j\frac{\left(p_{ij} - \pi_{ij}\right)^2}{\pi_{ij}}}\)

Where \(\pi_{ij}\) is the expected proportion for cell row i column j, and \(p_{ij}\) the sample proportion.

Cohen also provided some guidelines for the interpretation:

Equation 54 from Choi et al. (2010), equation 30 in Hubálek (1982) and equation 7 from Warrens (2008)

Pearson Q₁ (Pearson 1900, p. 15)

\( Q_1 = \sin\left(\frac{\pi}{2} \times \frac{a\times d - b\times c}{\left(a+b\right)\times\left(b+d\right)}\right) \)

Note that Pearson (1900) stated: "Q₁ was found of little service" (p. 16).

Pearson Q₄ (1900, p. 16)

\( Q_4 = \sin\left(\frac{\pi}{2} \times \frac{1}{1 + \frac{2\times b\times c\times n}{\left(a\times d -b\times c\right) \times \left(b+c\right)}}\right) \)

Pearson Q₅ (1900, p. 16)

\( Q_5 = \sin\left(\frac{\pi}{2} \times \frac{1}{\sqrt{1+ k^2}}\right) \)

with \( k^2 = \frac{4\times a\times b\times c\times d\times n^2}{\left(a\times d-b\times c\right)^2\times\left(a+d\right)\times\left(b+c\right)} \)

Peirce i (1884, p. 453):

\(i = \frac{a \times d - b \times c}{R_1 \times R_2}\)

Equation 1a from Warrens (2008)

a second version is also possible:

\(i = \frac{a \times d - b \times c}{C_1 \times C_2}\)

Equation 1b from Warrens (2008)

Equation 73 in Choi et al. (2010) is different and was unable to find the original source for the variation they use. They use:

\(i = \frac{a \times d - b \times c}{a \times b + 2 \times b \times c + c \times d}\)

Equation 73 from Choi et al. (2010), equation 16 in Hubálek (1982).

Rogers-Tanimoto Similarity Ratio (1960, p. 1117):

\(s = \frac{a + d}{a + 2\left(b + c\right) + d}\)

Equation 9 from Choi et al. (2010), equation 23 in Hubálek (1982) and equation 25 from Warrens (2008)

Rogot-Goldberg Index of Adjusted Agreement (1966, p. 997):

\(s = \frac{a}{R_1 + C_1}+\frac{d}{R_2 + C_2}\)

Equation 32 from Warrens (2008)

Russel-Rao supposedly from (1940):

\(\frac{a}{n}\)

Equation 14 from Choi et al. (2010, p. 44), equation 14 in Hubálek (1982, p. 672) and equation 15from Warrens (2008, p. 220)

Scott Index of Reliability (1955, p. 323):

\(\pi = \frac{4 \times a \times d - \left(b + c\right)^2}{\left(R_1 + C_1\right) \times \left(R_2 + C_2\right)}\)

Equation 21 from Warrens (2008)

Sokal-Michener / Matching Coefficient (1958, p. 1417):

\(\frac{a + d}{n}\)

Equation 7 from Choi et al. (2010), equation 20 in Hubálek (1982) and equation 22 from Warrens (2008)

Simpson Index of Taxonomic Resemblance (1943, p.20; 1960, p. 301):

\(\frac{a}{\min{\left(R_1 \times C_1\right)}}\)

Equation 45 from Choi et al. (2010), equation 2 in Hubálek (1982) and equation 16 from Warrens (2008)

Sokal-Sneath version 1 (1963, p. 129):

\(\frac{a}{a + 2 \times b + 2 \times c}\)

Ling labels equation 7 as ‘Anderberg’ and uses a formula that can be rewritten to the above version.

Equation 6 from Choi et al. (2010), equation 6 in Hubálek (1982) and equation 30a from Warrens (2008)

Sokal-Sneath version 2 (1963, p. 129):

\(\frac{2 \times a + 2 \times d}{2 \times a + b + c + 2 \times d}\)

Choi and also Ling lists also a version of ‘Gower & Legendre’ (Gower & Legendre, 1986)

Equation 8 and 11 from Choi et al. (2010), equation 22 in Hubálek (1982) and equation 30b from Warrens (2008)

Sokal-Sneath version 3 (1963, p. 130):

\(\frac{1}{4} \times \left(\frac{a}{R_1} + \frac{a}{C_1} + \frac{d}{R_2} + \frac{d}{C_2}\right)\)

Equation 49 from Choi et al. (2010), equation 18 in Hubálek (1982) and equation 30c from Warrens (2008)

Sorgenfrei supposedly from (1958):

\(\frac{a^2}{R_1 \times C_1}\)

Equation 36 from Choi et al. (2010), equation 12 in Hubálek (1982) and equation 23 from Warrens (2008)

Stiles Association Factor (1961, p. 272) is the log of chi-square with Yates correction:

\(\log_{10}\left(\frac{n\left(\left\lvert a \times d - b \times c \right\rvert -\frac{n}{2}\right)^2}{R_1 \times R_2 \times C_1 \times C_2}\right)\)

Equation 59 from Choi et al. (2010) and equation 26 from Warrens (2008)

tarantula supposedly from Jones and Harrold (2005):

\(\frac{a \times R_2}{c \times R_1}\)

Equation 75 from Choi et al. (2010).

Tarwid (1960, p. 117):

\(\frac{n \times a - R_1 \times C_1}{n \times a + R_1 \times C_1}\)

Equation 40 from Choi et al. (2010), equation 43 in Hubálek (1982).

Tulloss Tripartite Similarity Index (1997, p. 132):

\(T = \sqrt{U \times S\times R}\)

With:

\(U = \log_2\left(1 + \frac{\min\left(b, c\right) + a}{\max\left(b, c\right) + a}\right)\)

\(S = \frac{1}{\sqrt{\log_2\left(2 + \frac{\min\left(b, c\right)}{a + 1}\right)}}\)

\(R = \log_2\left(1 + \frac{a}{R_1}\right) \times \log_2\left(1 + \frac{a}{C_1}\right)\)

Yule's Q (Yule, 1900, p. 272) can be calculated using:

\(Q = \frac{a\times d - b\times c}{a\times d + b\times c}\)

As for the interpretation for Q, there aren't many rules of thumb I could find. One I did find is from Glen (2017):

Alternative, the Q can be converted to an Odds Ratio and in turn, some proposed to convert an Odds Ratio to Cohen's d. Also Cohen's d can be calculated to a correlation coefficient (r), for which again there are different tables.

\(OR = \frac{1 + Q}{1 - Q}\)

Equation 61 from Choi et al. (2010), equation 36 in Hubálek (1982) and equation 3 from Warrens (2008)

Yule proposed to convert his Q to a correlation coefficient using (Yule, 1900, p. 276):

\(r_Q = \cos\left(\frac{\sqrt{k}}{1+\sqrt{k}}\times\pi\right)\)

\(k = \frac{1-Q}{1+Q}\)

Pearson's Q₃ (1900) will give the same result, although he used the sinus function:

\(Q_2 = \sin\left(\frac{\pi}{2}\times\frac{\sqrt{a\times d} - \sqrt{b\times c}}{\sqrt{a\times d} + \sqrt{b\times c}}\right)\)

Cole (1949) rewrote this to:

\(C_6 = \cos\left(\frac{\pi\times\sqrt{b\times c}}{\sqrt{a\times d} + \sqrt{b\times c}}\right)\)

Equation 55 from Choi et al. (2010) and equation 38 in Hubálek (1982).

Yule's Y (1912, p. 592) is a further adaptation of Yule's Q into:

\(Y = \frac{\sqrt{a\times d} - \sqrt{b\times c}}{\sqrt{a\times d} + \sqrt{b\times c}}\)

Yule referred to this as referred to as the coefficient of colligation.

The Odds Ratio can also be calculated from this using:

\(OR = \left(\frac{Y + 1}{1 - Y}\right)^2\)

Tables for the interpretation of the Odds Ratio can then be used.

Equation 63 from Choi et al. (2010), equation 37 in Hubálek (1982) and equation 8 from Warrens (2008)