Module stikpetP.effect_sizes.eff_size_bin_bin

Functions

def es_bin_bin(field1, field2, categories1=None, categories2=None, method='odds-ratio')

Effect Sizes for Binary vs. Binary

Various measures of association/agreement/similarity for binary vs. binary cases.

Parameters

field1 : pandas series

data with categories for the rows

field2 : pandas series

data with categories for the columns

categories1 : list or dictionary, optional

the two categories to use from field1. If not set the first two found will be used

categories2 : list or dictionary, optional

the two categories to use from field2. If not set the first two found will be used

method : string, optional

the method to use. Default is odds-ratio

Returns

the effect size measure value

 

Notes

The method can be set to any of the following:

alroy	ample	anderberg	austin-colwell
baroni-urbani-buser-1	baroni-urbani-buser-2	becker-clogg-1	becker-clogg-2
bonett-price-1	bonett-price-2	bonett-price-3	braun-blanquet
camp-1	camp-2	camp-3	chen-popovich
clement	cohen-kappa	cohen-w	cole-c1
cole-c2	cole-c3	cole-c4	cole-c5
cole-c6	cole-c7	cole-c8	contingency
czekanowski	dennis	dice-1	dice-2
dice-3	digby	doolittle	driver-kroeber-1
driver-kroeber-2	edward	eyraud	fager-mcgowan-1
fager-mcgowan-2	faith	fleiss	forbes-1
forbes-2	fossum-kaskey	gilbert	gilbert-wells
gk-lambda-1	gk-lambda-2	gleason	gower
gower-legendre	hamann	harris-lahey	hawkins-dotson
hurlbert	jaccard	johnson	kent-foster-1
kent-foster-2	kuder-richardson	kulczynski-1	kulczynski-2
loevinger	matching	maxwell-pilliner	mcconnaughey
mcewen-michael	mountford	nei-li	ochiai-1
ochiai-2	odds-ratio	otsuka	pearson
pearson-heron	pearson-q1	pearson-q2	pearson-q3
pearson-q4	pearson-q5	peirce-1	peirce-2
peirce-3	phi	rogers-tanimoto	rogot-goldberg
russell-rao	scott	simpson	sokal-michener
sokal-sneath-1	sokal-sneath-2	sokal-sneath-3	sokal-sneath-4
sokal-sneath-5	sorgenfrei	stiles	tanimoto
tarantula	tarwid	tulloss	yule-q
yule-r	yule-y

If we have a 2x2 table with the following values:

	Column 1	Column 2	total
row 1	$a$	$b$	$R_1$
row 2	$c$	$d$	$R_2$
total	$C_1$	$C_2$	$n$

then the following formula are used:

Nr	Label	Formula
1	Russell-Rao	$\frac{a}{n}$
2	Dice-1	$\frac{a}{R_1}$
3	Dice-2	$\frac{a}{C_1}$
4	Braun-Blanquet	$\frac{a}{\max\left(R_1, C_1\right)}$
5	Simpson Similarity	$\frac{a}{\min\left(R_1, C_1\right)}$
6	Kulczynski-1	$\frac{a}{b+c}$
7	Jaccard	$\frac{a}{a+b+c}$
8	Sokal-Sneath-1	$\frac{a}{a+2b+2c}$
9	Gleasson	$\frac{2a}{2a+b+c}$
10	Mountford	$\frac{2a}{a\left(b+c\right)+2bc}$
11	Driver-Kroeber	$\frac{a}{\sqrt{R_1 C_1}}$
12	Sorgenfrei	$\frac{a^2}{R_1 C_1}$
13	Johnson	$\frac{a}{R_1}+\frac{a}{C_1}$
14	Kulczynski-2	$\frac{1}{2}\left(\frac{a}{R_1}+\frac{a}{C_1}\right)$
15	Fager-McGowan-1	$\frac{a}{\sqrt{R_1 C_1}}-\frac{1}{2\sqrt{\max\left(R_1, C_1\right)}}$
16	Fager-McGowan-2	$\frac{a}{\sqrt{R_1 C_1}}-\frac{\sqrt{\max\left(R_1, C_1\right)}}{2}$
17	tarantula	$\frac{aR_2}{cR_1}$
18	Ample	$\frac{aR_2}{cR_1}$
19	Gilbert	$\frac{an-R_1 C_1}{C_1 n + R_1 n - an - R_1 C_1}$
20	Fossum-Kaskey	$\frac{n\left(a - \frac{1}{2}\right)^2}{R_1 C_1}$
21	Forbes - 1	$\frac{na}{R_1 C_1}$
22	Eyraud	$\frac{a-R_1 C_1}{R_1 R_2 C_1 C_2}$
23	Sokal-Michener	$\frac{a+d}{n}$
24	Faith	$\frac{a+\frac{1}{2}}{n}$
25	Sokal-Sneath-5	$\frac{a+d}{b+c}$
26	Rogers-Tanimoto	$\frac{a+d}{a+2\left(b+c\right)+d}$
27	Sokal-Sneath-2	$\frac{2a+2d}{2a+b+c+2d}$
28	Gower	$\frac{a+d}{\sqrt{R_1 R_2 C_1 C_2}}$
29	Sokal-Sneath-4	$\frac{ad}{\sqrt{R_1 R_2 C_1 C_2}}$
30	Rogot-Goldberg	$\frac{a}{R_1 + C_1}+\frac{d}{R_2 + C_2}$
31	Sokal-Sneath-3	$\frac{1}{4}\left(\frac{a}{R_1}+\frac{a}{C_1}+\frac{d}{R_2}+\frac{d}{C_2}\right)$
32	Hawkin-Dotson	$\frac{1}{2}\left(\frac{a}{a+b+c}+\frac{d}{b+c+d}\right)$
33	Clement	$\frac{aR_2}{nR_1}+\frac{dR_1}{nR_2}$
34	Harris-Lahey	$\frac{a\left(R_2+C_2\right)}{2n\left(a+b+c\right)}+\frac{d\left(R_1+C_1\right)}{2n\left(b+c+d\right)}$
35	Austin-Colwell	$\frac{2}{\pi}\text{arcsin}\sqrt{\frac{a+d}{n}}$
36	Baroni-Urbani-Buser-1	$\frac{\sqrt{ad}+a}{\sqrt{ad}+a+b+c}$
37	Peirce-1	$\frac{ad-bc}{R_1 R_2}$
38	Peirce-2	$\frac{ad-bc}{C_1 C_2}$
39	Cole C1	$\frac{ad-bc}{R_1 C_1}$
40	Loevinger	$\frac{ad-bc}{\min\left(R_1 C_2, R_2 C_1\right)}$
41	Cole C7	$\begin{cases} \frac{ad-bc}{R_1 C_2} & \text{ if } ad \geq bc \\ \frac{ad-bc}{R_1 C_1} & \text{ if } ad < bc \text{ and } a \leq d \\ \frac{ad-bc}{R_2 C_2} & \text{ if } ad < bc \text{ and } a > d\end{cases}$
42	Dennis	$\frac{ad-bc}{\sqrt{nR_1 C_1}}$
43	Phi	$\frac{ad-bc}{\sqrt{R_1 R_2 C_1 C_2}}$
44	Doolittle	$\frac{\left(ad-bc\right)^2}{R_1 R_2 C_1 C_2}$
45	Peirce-3	$\frac{ad-bc}{ab+2bc+cd}$
46	Cohen-kappa	$\frac{2\left(ad-bc\right)}{R_1 C_2 + R_2 C_1}$
47	McEwen-Michael	$\frac{4\left(ad-bc\right)}{\left(a+d\right)^2 + \left(b+c\right)^2}$
48	Kuder-Richardson	$\frac{4\left(ad-bc\right)}{R_1 R_2 + C_1 C_2 + 2ad-2bc}$
49	Scott	$\frac{4ad-\left(b+c\right)^2}{\left(R_1 + C_1\right)\left(R_2 + C_2\right)}$
50	Maxwell-Pilliner	$\frac{2\left(ad-bc\right)}{R_1 R_2 + C_1 C_2}$
51	Cole C5	$\frac{\sqrt{2}\left(ad-bc\right)}{\sqrt{\left(ad-bc\right)^2+R_1 R_2 C_1 C_2}}$
52	Hamann	$\frac{\left(a+d\right)-\left(b+c\right)}{n}$
53	Fleiss	$\frac{\left(ad-bc\right)\left(R_1 C_2 + R_2 C_1\right)}{2R_1 R_2 C_1 C_2}$
54	Yule Q	$\frac{ad-bc}{ad+bc}$
55	Yule Y	$\frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}$
56	Digby H	$\frac{\left(ad\right)^{3/4}-\left(bc\right)^{3/4}}{\left(ad\right)^{3/4}+\left(bc\right)^{3/4}}$
57	Edward Q	$\frac{OR^{\pi/4}-1}{OR^{\pi/4}+1}$
58	Tarwid	$\frac{na-R_1 C_1}{na+R_1 C_1}$
59	Bonett-Price Y*	$\frac{\hat{w}^x-1}{\hat{w}^x+1}$
60	Contingency coefficient	$\sqrt{\frac{\chi^2}{n+\chi^2}}$
61	Cohen w	$\sqrt{\frac{\chi^2}{\chi^2}}$
62	Pearson	$\sqrt{\frac{\phi^2}{n+\phi^2}}$
63	Hurlbert	$\frac{ad-bc}{\left\lvert ad-bc\right\rvert}\sqrt{\frac{\chi^2-\chi_{min}^2}{\chi_{max}^2-\chi_{min}^2}}$
64	Stiles	$\log_{10}\left(\frac{n\left(\left\lvert ad-bc \right\rvert -\frac{n}{2}\right)^2}{R_1 R_2 C_1 C_2}\right)$
65	McConnaughey	$\frac{a^2-bc}{R_1 C_1}$
66	Baroni-Urbani-Buser-2	$\frac{a-b-c+\sqrt{ad}}{a+b+c+\sqrt{ad}}$
67	Kent-Foster-1	$\frac{-bc}{bR_1 + cC_1 + bc}$
68	Kent-Foster-2	$\frac{-bc}{bR_2 + cC_2 + bc}$
69	Tulloss	$\sqrt{U\times S\times R}$
70	Gilbert-Wells	$\ln\left(a\right)-\ln\left(n\right)-\ln\left(\frac{R_1}{n}\right)-\ln\left(\frac{C_1}{n}\right)$
71	Yule r	$\cos\left(\frac{\pi\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}\right)$
72	Anderberg	$\frac{\sigma-\sigma'}{2n}$
73	Alroy F	$\frac{a\left(n' + \sqrt{n'}\right)}{a\left(n' + \sqrt{n'}\right)+\frac{3}{2}bc}$
74	Pearson Q1	$\sin\left(\frac{\pi}{2}\times\frac{R_2 C_1}{R_1 C_2}\right)$
75	Goodman-Kruskal Lambda-1	$\frac{\sigma-\sigma'}{2n-\sigma'}$
76	Goodman-Kruskal Lambda-2	$\frac{2\min\left(a,d\right)-b-c}{2\min\left(a,d\right)+b+c}$
77	Odds Ratio	$\frac{ad}{bc}$
78	Pearson Q4	$\sin\frac{\pi}{2}\times\frac{1}{1+\frac{2bcn}{\left(ad-bc\right)\left(b+c\right)}}$
79	Pearson Q5	$\sin\frac{\pi}{2}\times\frac{1}{\sqrt{1+\frac{4abcdn^2}{\left(ad-bc\right)^2\left(a+d\right)\left(b+c\right)}}}$
80	Camp (3 ver.)	$\frac{m}{\sqrt{1+\Theta\times m^2}}$
81	Becker-Clogg-1	$\frac{g-1}{g+1}$
82	Becker-Clogg-2	$\frac{OR^{13.3/\Delta}-1}{OR^{13.3/\Delta}+1}$
83	Bonett-Price-r	$\cos\left(\frac{\pi}{1+\omega^c}\right)$
84	Bonett-Price-rhat	$\cos\left(\frac{\pi}{1+\hat{\omega}^{\hat{c}}}\right)$
85	Chen-Popovich	$\frac{ad-bc}{\lambda_x \lambda_y n^2}$

Equation 57

$$OR = \frac{ad}{bc}$$

Equation 59

$$x = \frac{1}{2}-\left(\frac{1}{2}-p_{min}\right)^2$$ $$p_{min} = \frac{\min\left(R_1, R_2, C_1, C_2\right)}{n}$$ $$\hat{\omega} = \frac{\left(a+0.1\right)\times\left(d+0.1\right)}{\left(b+0.1\right)\times\left(c+0.1\right)}$$

Equations 60, 61, and 63

$$\chi^2 = \frac{n\left(ad-bc\right)^2}{R_1 R_2 C_1 C_2}$$

Equation 62

$$\Phi = \frac{\left\lvert ad-bc\right\rvert}{\sqrt{R_1 R_2 C_1 C_2}}$$ Note that Choi et. al ommit the absolute value, but this would create problems with taking the square root if bc>ad.

Equation 63

$$\chi_{max}^2 = \begin{cases} \frac{nR_1 C_2}{R_2 C_1} & \text{ if } ad \geq bc \\ \frac{nR_1 C_1}{R_2 C_2} & \text{ if } ad < bc \text{ and } a \leq d \\ \frac{nR_2 C_2}{R_1 C_1} & \text{ if } ad < bc \text{ and } a > d \end{cases}$$ $$\chi_{min}^2 = \frac{n^3\left(\hat{a} - g\left(\hat{a}\right)\right)^2}{R_1 R_2 C_1 C_2}$$ $$\hat{a}=\frac{R_1 C_1}{n}$$ $$g\left(\hat{a}\right) = \begin{cases} \lfloor \hat{a} \rfloor & \text{ if } ad < bc \\ \lceil \hat{a}\rceil & \text{ if } ad \geq bc \end{cases}$$

Equation 69

$$U = \log_2\left(1+\frac{\min\left(b,c\right)+a}{\max\left(b,c\right)+a}\right)$$ $$S = \frac{1}{\sqrt{\log_2\left(2+\frac{\min\left(b,c\right)}{a+1}\right)}}$$ $$R = \log_2\left(1+\frac{a}{R_1}\right)\log_2\left(1+\frac{a}{RC1}\right)$$

Equation 71 and 75

$$\sigma = \max\left(a,b\right)+\max\left(c,d\right)+\max\left(a,c\right)+\max\left(b,d\right)$$ $$\sigma' = \max\left(R_1, R_2\right)+\max\left(C_1, C_2\right)$$

Equation 73

$$n' = a+b+c$$

Equation 80 (Camp)

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

Step 1: If total of column 1 (C1) is less than column 2 (C2), swop the two columns

Step 2: Calculate $p = \frac{C1}{n}$, $p_1 = \frac{a}{n}$, and $p_2 = \frac{c}{C2}$

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)

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 \eqn{\phi=1}.

For a better approximation Camp made the following table:

p	0.5	0.6	0.7	0.8	0.9
phi	0.637	0.63	0.62	0.60	0.56

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

p	0	1	2	3	4	5	6	7	8	9	10
0.5	0.637	0.636	0.636	0.635	0.635	0.634	0.634	0.633	0.633	0.632	0.631
0.6	0.631	0.631	0.630	0.629	0.628	0.627	0.626	0.625	0.624	0.622	0.621
0.7	0.621	0.620	0.618	0.616	0.614	0.612	0.610	0.608	0.606	0.603	0.600
0.8	0.600	0.597	0.594	0.591	0.587	0.583	0.579	0.574	0.569	0.564	0.559

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.

Equations 81 and 82

Version 1 will calculate: $$\rho^* = \frac{g-1}{g+1}$$

Version 2 will calculate: $$\rho^{**} = \frac{OR^{13.3/\Delta} - 1}{OR^{13.3/\Delta} + 1}$$

With: $$g=e^{12.4\times\phi - 24.6\times\phi^3}$$ $$\phi = \frac{\ln\left(OR\right)}{\Delta}$$ $$OR=\frac{\left(\frac{a}{c}\right)}{\left(\frac{b}{d}\right)} = \frac{a\times d}{b\times c}$$ $$\Delta = \left(\mu_{R1} - \mu_{R2}\right) \times \left(v_{C1} - v_{C2}\right)$$ $$\mu_{R1} = \frac{-e^{-\frac{t_r^2}{2}}}{p_{R1}}, \mu_{R2} = \frac{e^{-\frac{t_r^2}{2}}}{p_{R2}}$$ $$v_{C1} = \frac{-e^{-\frac{t_c^2}{2}}}{p_{C1}}, v_{C2} = \frac{e^{-\frac{t_c^2}{2}}}{p_{C2}}$$ $$t_r = \Phi^{-1}\left(p_{R1}\right), t_c = \Phi^{-1}\left(p_{C1}\right)$$ $$p_{x} = \frac{x}{n}$$

Equations 83 and 84

Formula for version 1 is (Bonett & Price, 2005, p. 216): $$\rho^* = \cos\left(\frac{\pi}{1+\omega^c}\right)$$

With: $$\omega = OR = \frac{a\times d}{b\times c}$$ $$c = \frac{1-\frac{\left|R_1-C_1\right|}{5\times n} - \left(\frac{1}{2}-p_{min}\right)^2}{2}$$ $$p_{min} = \frac{\text{MIN}\left(R_1, R_2, C_1, C_2\right)}{n}$$

Formula for version 2 is (Bonett & Price, 2005, p. 216): $$\hat{\rho}^* = \cos\left(\frac{\pi}{1+\hat{\omega}^{\hat{c}}}\right)$$

with: $$\hat{\omega} = \frac{\left(a+\frac{1}{2}\right)\times \left(d+\frac{1}{2}\right)}{\left(b+\frac{1}{2}\right)\times \left(c+\frac{1}{2}\right)}$$ $$\hat{c} = \frac{1-\frac{\left|R_1-C_1\right|}{5\times \left(n+2\right)} - \left(\frac{1}{2}-\hat{p}_{min}\right)^2}{2}$$ $$\hat{p}_{min} = \frac{\text{MIN}\left(R_1, R_2, C_1, C_2\right)+1}{n+2}$$

Equation 85

$$\lambda_x = \Phi^{-1}\left(\frac{R_1}{n}\right)$$ $$\lambda_y = \Phi^{-1}\left(\frac{C_1}{n}\right)$$

with $\Phi^{-1}\left(\dots\right)$ being the inverse standard normal cumulative distribution

Sources for formulas

The formulas were obtained from the following sources. The columns W-C-H show which equation corresponds to my label in:

• W: Warrens (2008, pp. 219–222). Equation 4 from Warrens is the chi-square value and not added.
• C: Choi et al. (2010, pp. 44–45). Equations not added from this source are: Eq. 4 is a ‘3w Jaccard’, could not find a source for this and not added. Equation 12 is just the intersection (a), eq. 13 the innerproduct (a+d), and 66 the dispersion. Equation 51 is the chi-square value and measures 15 to 30 and 62 are just distance measures.
• H: Hubálek (Hubálek, 1982, pp. 671–673)

If no page number is listed in the original source, the formula was taken from Warrens, Choi et al. and/or Hubálek.

numb	Label	Original	Wa	Ch	Har
1	Russell-Rao	(Russell & Rao, 1940)	15	14	14
2	Dice-1	(Dice, 1945, p. 302)	17a
3	Dice-2	(Dice, 1945, p. 302)	17b
4	Braun-Blanquet	(Braun-Blanquet, 1932)	12	46	1
5	Simpson Similarity	(Simpson, 1943, p. 20, 1960, p. 301)	16	45	2
6	Kulczynski-1	(Kulczynski, 1927)	11b	64	3
7	Jaccard =	(Jaccard, 1901, 1912, p. 39)	6	1	4
	Tanimoto	(Tanimoto, 1958, p. 5)		65
8	Sokal-Sneath-1	(Sokal & Sneath, 1963, p. 129)	30a	6	6
9	Gleasson =	(Gleason, 1920, p. 31)	9		5
	Dice-3 =	(Dice, 1945, p. 302)		2
	Nei-Li =	(Nei & Li, 1979, p. 5270)		5
	Czekanowski			3
10	Mountford	(Mountford, 1962, p. 45)	28	37	15
11	Driver-Kroeber =	(Driver & Kroeber, 1932, p. 219)	13	31	11
	Ochiai-1 =	(Ochiai, 1957)		33
	Otsuka	(Otsuka, 1936)		38
12	Sorgenfrei	(Sorgenfrei, 1958)	23	36	12
13	Johnson	(Johnson, 1967)	33	43	9
14	Driver-Kroeber-2 =	(Driver & Kroeber, 1932, p. 219)	11a	42	8
	Kulczynski-2	(Kulczynski, 1927)		41	7
15	Fager-McGowan-1	(Fager & McGowan, 1963, p. 454)	29
16	Fager-McGowan-2	(Fager & McGowan, 1963, p. 454)		47	13
17	tarantula	(Jones & Harrold, 2005)		75
18	ample			76
19	Gilbert	(Gilbert, 1884, p. 171)
20	Fossum-Kaskey	(Fossum & Kaskey, 1966, p. 65)		35
21	Forbes - 1	(Forbes, 1907, p. 279)	5	34	40
22	Eyraud	(Eyraud, 1936)		74	17
23	Sokal-Michener	(Sokal & Michener, 1958, p. 1417)	22	7	20
24	Faith	(Faith, 1983, p. 290)		10
25	Sokal-Sneath-5	(Sokal & Sneath, 1963, p. 129)	30e	56	19
26	Rogers-Tanimoto	(Rogers & Tanimoto, 1960)	25	9	23
27	Sokal-Sneath-2 =	(Sokal & Sneath, 1963, p. 129)	30b	8	22
	Gower-Legendre	(Gower & Legendre, 1986)		11
28	Gower	(Gower, 1971)		50
29	Sokal-Sneath-4 =	(Sokal & Sneath, 1963, p. 130)	30d	57	25
	Ochiai-2	(Ochiai, 1957)		60
30	Rogot-Goldberg	(Rogot & Goldberg, 1966, p. 997)	32
31	Sokal-Sneath-3	(Sokal & Sneath, 1963, p. 130)	30c	49	18
32	Hawkin-Dotson	(Hawkins & Dotson, 1975, pp. 372–373)	34
33	Clement	(Clement, 1976, p. 258)	37
34	Harris-Lahey	(Harris & Lahey, 1978, p. 526)	40
35	Austin-Colwell	(Austin & Colwell, 1977, p. 205)			21
36	Baroni-Urbani-Buser-1	(Baroni-Urbani & Buser, 1976, p. 258)	38a	71	32
37	Peirce-1	(Peirce, 1884, p. 453)	1a
38	Peirce-2	(Peirce, 1884, p. 453)	1b		26
39	Cole C1	(Cole, 1949, p. 415)
40	Loevinger =	(Loevinger, 1947, p. 30)	18
	Forbes 2	(Forbes, 1925)		48	42
41	Cole C7	(Cole, 1949, p. 420)	19		34
42	Dennis	(Dennis, 1965, p. 69)		44
43	Pearson Phi =	(Pearson, 1900a, p. 12)	7	54	30
	Yule Phi =	(Yule, 1912, p. 596)
	Cole C2	(Cole, 1949, p. 415)
44	Doolittle	(Doolittle, 1885, p. 123)	2		31
45	Peirce-3	(Peirce, 1884)		73	16
46	Cohen-kappa	(Cohen, 1960, p. 40)	24
47	McEwen-Michael =	(Michael, 1920, p. 57)	10	68	39
	Cole C3	(Cole, 1949, p. 415)
48	Kuder-Richardson	(Kuder & Richardson, 1937)	14
49	Scott	(Scott, 1955, p. 324)	21
50	Maxwell-Pilliner	(Maxwell & Pilliner, 1968)	35
51	Cole C5	(Cole, 1949, p. 416)		58	29
52	Hamann	(Hamann, 1961)	27	67	24
53	Fleiss	(Fleiss, 1975, p. 656)	36
54	Yule Q =	(Yule, 1900, p. 272)	3	61	36
	Cole C4 =	(Cole, 1949, p. 415)
	Pearson Q2	(Pearson, 1900, p. 15)
55	Yule Y	(Yule, 1912, p. 592)	8	63	37
56	Digby H	(Digby, 1983, p. 754)	41
57	Edward Q	(Edwards, 1957; Becker & Clogg, 1988, p. 409)
58	Tarwid	(Tarwid, 1960, p. 117)		40	43
59	Bonett-Price-1	(Bonett & Price, 2007, p. 433)
60	Contingency	(Pearson, 1904, p. 9)		52	28
61	Cohen w	(Cohen, 1988, p. 216)
62	Pearson	(Pearson, 1904)		53
63	Hurlbert / Cole C8	(Hurlbert, 1969, p. 1)			35
64	Stiles	(Stiles, 1961, p. 272)	26	59
65	McConnaughey	(McConnaughey, 1964)	31	39	10
66	Baroni-Urbani-Buser-2	(Baroni-Urbani & Buser, 1976, p. 258)	38b	72	33
67	Kent-Foster-1	(Kent & Foster, 1977, p. 311)	39a
68	Kent-Foster-2	(Kent & Foster, 1977, p. 311)	39b
69	Tulloss	(Tulloss, 1997, p. 133)
70	Gilbert-Wells	(Gilbert & Wells, 1966)		32	41
71	Yule r	(Yule, 1900, p. 276)
	Pearson-Q3	(Pearson, 1900a, p. 16)
	Cole C6	(Cole, 1949, p. 416)
	Pearson-Heron	(Pearson & Heron, 1913)		55	38
72	Anderberg	(Anderberg, 1973)		70
73	Alroy F	(Alroy, 2015, eq. 6)
74	Pearson Q1	(Pearson, 1900a, p. 15)
75	Goodman-Kruskal Lambda-1	(Goodman & Kruskal, 1954, p. 743)		69
76	Goodman-Kruskal Lambda-2	(Goodman & Kruskal, 1954)	20
77	Odds Ratio	(Fisher, 1935, p. 50)
78	Pearson Q4	(Pearson, 1900, p. 16)
79	Pearson Q5	(Pearson, 1900, p. 16)
80	Camp	(Camp, 1934, p. 309)
81	Becker-Clogg-1	(Becker & Clogg, 1988, pp. 410–412)
82	Becker-Clogg-2	(Becker & Clogg, 1988, pp. 410–412)
83	Bonett-Price-2	(Bonett & Price, 2005, p. 216)
84	Bonett-Price-3	(Bonett & Price, 2005, p. 216)
85	Ched-Popovich	(Chen & Popovich, 2002, p. 37)

Before, After and Alternatives

Before determining the effect size measure, you might want to use: * tab_cross to get a cross table and/or * ts_fisher to perform a Fisher exact test

Some of the measures in this function are approximations of the so-called tetrachoric correlation. A separate function for this is also available via r_tetrachoric

Rules of thumb for the classification for some of the measures can be found using: * Odds Ratio -> th_odds_ratio * Yule Q -> th_yule_q, or according to Lloyd et al. (2013, p. 484) the rules-of-thumb from the Odds Ratio from Rosenthal could be used. * Cohen Kappa -> th_cohen_kappa

References

Alroy, J. (2015). A new twist on a very old binary similarity coefficient. Ecology, 96(2), 575–586. doi:10.1890/14-0471.1

Anderberg, M. R. (1973). Cluster analysis for applications. New York, NY, Academic Press.

Austin, B., & Colwell, R. R. (1977). Evaluation of some coefficients for use in numerical taxonomy of microorganisms. International Journal of Systematic Bacteriology, 27(3), 204–210. doi:10.1099/00207713-27-3-204

Baroni-Urbani, C., & Buser, M. W. (1976). Similarity of binary data. Systematic Zoology, 25(3), 251–259. doi:10.2307/2412493

Becker, M. P., & Clogg, C. C. (1988). A note on approximating correlations from Odds Ratios. Sociological Methods & Research, 16(3), 407–424. doi:10.1177/0049124188016003003

Bonett, D. G., & Price, R. M. (2005). Inferential methods for the tetrachoric correlation coefficient. Journal of Educational and Behavioral Statistics, 30(2), 213–225. doi:10.3102/10769986030002213

Bonett, D. G., & Price, R. M. (2007). Statistical inference for generalized yule coefficients in 2 x 2 contingency tables. Sociological Methods & Research, 35(3), 429–446. doi:10.1177/0049124106292358

Braun-Blanquet, J. (1932). Plant sociology: The study of plant communities. McGraw Hill.

Camp, B. H. (1934). Mathematical part of elementary statistics. D.C. Heath and Company, London.

Chen, P. Y., & Popovich, P. M. (2002). Correlation: Parametric and nonparametric measures. Sage Publications.

Choi, S.-S., Cha, S.-H., & Tappert, C. (2010). A survey of binary similarity and distance measures. Journal on Systemics, Cybernetics and Informatics, 8(1), 43–48.

Clement, P. W. (1976). A formula for computing inter-observer agreement. Psychological Reports, 39(1), 257–258. doi:10.2466/pr0.1976.39.1.257

Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20(1), 37–46. doi:10.1177/001316446002000104

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). L. Erlbaum Associates.

Cole, L. C. (1949). The measurement of interspecific associaton. Ecology, 30(4), 411–424. doi:10.2307/1932444

Cureton, E. E. (1968). Tetrachoric correlation by the Camp approximation. Educational and Psychological Measurement, 28(2), 239–244. doi:10.1177/001316446802800202

Dennis, S. F. (1965). The construction of a thesaurus automatically from a sample of text. In M. E. Stevens, V. E. Giuliano, & L. B. Heilprin (Eds.), Statistical Association Methods for Mechanized Documentation (Vol. 14, pp. 61–148). U.S. Government Printing Office.

Dice, L. R. (1945). Measures of the amount of ecologic association between species. Ecology, 26(3), 297–302. doi:10.2307/1932409

Digby, P. G. N. (1983). Approximating the tetrachoric correlation coefficient. Biometrics, 39(3), 753–757. doi:10.2307/2531104

Doolittle, M. H. (1885). The verification of predictions. Bulletin of the Philosophical Society of Washington, 7, 122–127.

Driver, H. E., & Kroeber, A. L. (1932). Quantitative expression of cultural relationships. University Of California Publications in American Archeology and Ethnology, 31(4), 211–256.

Edwards, J. H. (1957). A note on the practical interpretation of 2 x 2 tables. Journal of Epidemiology & Community Health, 11(2), 73–78. doi:10.1136/jech.11.2.73

Edwards, J. H., & Edwards, A. W. F. (1984). Approximating the tetrachoric correlation coefficient. Biometrics, 40(2), 563–563.

Eyraud, H. (1936). Les principes de la mesure des correlations. Ann. Univ. Lyon, III. Ser., Sect. A, 1(30–47), 111.

Fager, E. W., & McGowan, J. A. (1963). Zooplankton species groups in the north pacific: Co-occurrences of species can be used to derive groups whose members react similarly to water-mass types. Science, 140(3566), 453–460. doi:10.1126/science.140.3566.453

Faith, D. P. (1983). Asymmetric binary similarity measures. Oecologia, 57(3), 287–290. doi:10.1007/BF00377169

Fisher, R. A. (1935). The logic of inductive inference. Journal of the Royal Statistical Society, 98(1), 39–82. doi:10.2307/2342435

Fleiss, J. L. (1975). Measuring agreement between two judges on the presence or absence of a trait. Biometrics, 31(3), 651–659. doi:10.2307/2529549

Forbes, S. A. (1907). On the local distribution of certain Illinois fishes: An essay in statistical ecology. Illinois Natural History Survey Bulletin, 7(8), 273–303.

Forbes, S. A. (1925). Method of determining and measuring the associative relations of species. Science, 61, 524.

Fossum, E. G., & Kaskey, G. (1966). Optimization and standardization of information retrieval language and systems (AF49(638)1194; p. 96). Univac Division.

Gilbert, G. K. (1884). Finley’s tornado predictions. American Meteorological Journal, 1(5), 166–172.

Gilbert, N., & Wells, T. C. E. (1966). Analysis of quadrat data. Journal of Ecology, 54(3), 675–685. doi:10.2307/2257810

Gleason, H. A. (1920). Some applications of the quadrat method. Bulletin of the Torrey Botanical Club, 47(1), 21–33. doi:10.2307/2480223

Glen, S. (2017, August 16). Gamma Coefficient (Goodman and Kruskal’s Gamma) & Yule’s Q. Statistics How To. https://www.statisticshowto.com/gamma-coefficient-goodman-kruskal/

Goodman, L. A., & Kruskal, W. H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49(268), 732–764. doi:10.2307/2281536

Gower, J. C. (1971). A general coefficient of similarity and some of its properties. Biometrics, 27(4), 857. doi:10.2307/2528823

Gower, J. C., & Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3(1), 5–48. doi:10.1007/BF01896809

Hamann, U. (1961). Merkmalsbestand und verwandtschaftsbeziehungen der farinosae: Ein beitrag zum system der monokotyledonen. Willdenowia, 2(5), 639–768.

Harris, F. C., & Lahey, B. B. (1978). A method for combining occurrence and nonoccurrence interobserver agreement scores. Journal of Applied Behavior Analysis, 11(4), 523–527. doi:10.1901/jaba.1978.11-523

Hawkins, R. P., & Dotson, V. A. (1975). Reliability scores that delude: An Alice in wonderland trip through the misleading characteristics of inter-observer agreement scores in interval recording. In E. Ramp & G. Semb (Eds.), Behavior analysis: Areas of research and application (pp. 359–376). Prentice Hall.

Hubálek, Z. (1982). Coefficients of association and similarity, based on binary (presence-absence) data: An evaluation. Biological Reviews, 57(4), 669–689. doi:10.1111/j.1469-185X.1982.tb00376.x

Hurlbert, S. H. (1969). A coefficient of interspecific assciation. Ecology, 50(1), 1–9. doi:10.2307/1934657

Jaccard, P. (1901). Étude comparative de la distribution florale dans une portion des Alpes et des Jura. Bulletin Del La Société Vaudoise Des Sciences Naturelles, 37, 547–579.

Jaccard, P. (1912). The distribution of the flora in the alpine zone. The New Phytologist, 11(2), 37–50.

Johnson, S. C. (1967). Hierarchical clustering schemes. Psychometrika, 32(3), 241–254. doi:10.1007/BF02289588

Jones, J. A., & Harrold, M. J. (2005). Empirical evaluation of the tarantula automatic fault-localization technique. Proceedings of the 20th IEEE/ACM International Conference on Automated Software Engineering, 273–282. doi:10.1145/1101908.1101949

Kent, R. N., & Foster, S. L. (1977). Direct observational procedures: Methodological issues in naturalistic settings. In A. R. Ciminero, K. S. Calhoun, & H. E. Adams (Eds.), Handbook of behavioral assessment (pp. 279–328). New York, NY: Wiley.

Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. Psychometrika, 2(3), 151–160. doi:10.1007/BF02288391

Kulczynski, S. (1927). Die Pflanzenassoziationen der Pieninen. Bulletin International de l’Academie Polonaise Des Sciences et Des Lettres, Classe Des Sciences Mathematiques et Naturelles, B (Sciences Naturelles), II, 57–203.

Ling, M. H. T. (2010). COPADS, I: Distance coefficients between two lists or sets. The Python Papers Source Codes, 2(2), 1–31.

Loevinger, J. (1947). A systematic approach to the construction and evaluation of tests of ability. Psychological Monographs, 61(4), i–49. doi:10.1037/h0093565

Maxwell, A. E., & Pilliner, A. E. (1968). Deriving coefficients of reliability and agreement for ratings. The British Journal of Mathematical and Statistical Psychology, 21(1), 105–116. doi:10.1111/j.2044-8317.1968.tb00401.x

McConnaughey, B. H. (1964). The determination and analysis of plankton communities. Marine Research, 7, 1–40.

Michael, E. L. (1920). Marine Ecology and the coefficient of association: A plea in behalf of quantitative biology. Journal of Ecology, 8(1), 54–59. doi:10.2307/2255213

Mountford, M. D. (1962). An index of similarity and its application to classification problems. In P. W. Murphy & D. Phil (Eds.), Progress in soil zoology (pp. 43–50). Butterworths.

Nei, M., & Li, W. H. (1979). Mathematical model for studying genetic variation in terms of restriction endonucleases. Proceedings of the National Academy of Sciences, 76(10), 5269–5273. doi:10.1073/pnas.76.10.5269

Ochiai, A. (1957). Zoogeographical studies on the soleoid fishes found in Japan and its neighbouring regions-I. Nippon Suisan Gakkaishi, 22(9), 522–525. doi:10.2331/suisan.22.522

Otsuka, Y. (1936). The faunal character of the Japanese Pleistocene marine Mollusca, as evidence of the climate having become colder during the Pleistocene in Japan. Bulletin of the Biogeographical Society of Japan, 6(16), 165–170.

Pearson, K. (1900). Contributions to the mathematical theory of evolution. VII. On the correlation of characters not quantitatively measurable. Philosophical Transactions of the Royal Society of London, 195, 1–405. doi:10.1098/rsta.1900.0022

Pearson, K. (1904). Contributions to the Mathematical Theory of Evolution. XIII. On the theory of contingency and its relation to association and normal correlation. Dulau and Co.

Pearson, K., & Heron, D. (1913). On theories of association. Biometrika, 9(1/2), 159–315. doi:10.2307/2331805

Peirce, C. S. (1884). The numerical measure of the success of predictions. Science, 4(93), 453–454. doi:10.1126/science.ns-4.93.453-a

Rogers, D. J., & Tanimoto, T. T. (1960). A computer program for classifying plants. Science, 132(3434), 1115–1118. doi:10.1126/science.132.3434.1115

Rogot, E., & Goldberg, I. D. (1966). A proposed index for measuring agreement in test-retest studies. Journal of Chronic Diseases, 19(9), 991–1006. doi:10.1016/0021-9681(66)90032-4

Russell, P. F., & Rao, T. R. (1940). On habitat and association of species of anopheline larvae in south-eastern Madras. Journal of the Malaria Institute of India, 3(1), 153–178.

Scott, W. A. (1955). Reliability of content analysis: The case of nominal scale coding. The Public Opinion Quarterly, 19(3), 321–325.

Simpson, G. G. (1943). Mammals and the nature of continents. American Journal of Science, 241(1), 1–31. doi:10.2475/ajs.241.1.1

Simpson, G. G. (1960). Notes on the measurement of faunal resemblance. American Journal of Science, 258-A, 300–311.

Sokal, P. H. A., & Sneath, R. R. (1963). Principles of numerical taxonomy. W.H. Freeman and Company.

Sokal, R., & Michener, C. (1958). A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin.

Sorgenfrei, T. (1958). Molluscan assemblages from the marine middle miocene of south Jutland and their Environments. Vol. I. Danmarks Geologiske Undersøgelse II. Række, 79, 1–355. doi:10.34194/raekke2.v79.6868

Stiles, H. E. (1961). The association factor in information retrieval. Journal of the ACM, 8(2), 271–279. doi:10.1145/321062.321074

Tanimoto, T. T. (1958). An elementary mathematical theory of classification and prediction (PB167360). International Business Machines Corp., New York, NY.

Tarwid, K. (1960). Szacowanie zbieinosci nisz ekologicznych gatunkow droga oceny prawdopodobienstwa spotykania sie ich w polowach. Ekologia Polska, 6, 115–130.

Tulloss, R. E. (1997). Assessment of similarity indices for undesirable properties and a new tripartite similarity index based on cost functions. In M. E. Palm & I. H. Chapela (Eds.), Mycology in sustainable development: Expanding concepts, vanishing borders (pp. 122–143). Parkway Pub.

Walker, H. M., & Lev, J. (1953). Statistical inference. Holt.

Warrens, M. J. (2008). Similarity coefficients for binary data: Properties of coefficients, coefficient matrices, multi-way metrics and multivariate coefficients [Doctoral dissertation, Universiteit Leiden]. https://hdl.handle.net/1887/12987

Yule, G. U. (1900). On the association of attributes in statistics: With illustrations from the material of the childhood society, &c. Philosophical Transactions of the Royal Society of London, 194, 257–319. doi:10.1098/rsta.1900.0019

Yule, G. U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75(6), 579–652. doi:10.2307/2340126

Author

Made by P. Stikker

Companion website: https://PeterStatistics.com
YouTube channel: https://www.youtube.com/stikpet
Donations: https://www.patreon.com/bePatron?u=19398076

Examples

>>> import pandas as pd
>>> file1 = "https://peterstatistics.com/Packages/ExampleData/GSS2012a.csv"
>>> df1 = pd.read_csv(file1, sep=',', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
>>> es_bin_bin(df1['mar1'], df1['wrksch'], categories1=['MARRIED', 'NEVER MARRIED'], categories2=['STAY HOME', 'WORK FULL-TIME'])
np.float64(2.5555555555555554)

Expand source code

def es_bin_bin(field1, field2, categories1=None, categories2=None, method = "odds-ratio"):
    '''
    Effect Sizes for Binary vs. Binary
    -----------------------------------
    Various measures of association/agreement/similarity for binary vs. binary cases.
    
    Parameters
    ----------
    field1 : pandas series
        data with categories for the rows
    field2 : pandas series
        data with categories for the columns
    categories1 : list or dictionary, optional
        the two categories to use from field1. If not set the first two found will be used
    categories2 : list or dictionary, optional
        the two categories to use from field2. If not set the first two found will be used
    method : string, optional
        the method to use. Default is odds-ratio
    
    Returns
    -------
    the effect size measure value
    
    Notes
    -----
    The method can be set to any of the following:
    
    | | | | |
    |----|----|----|----|
    |alroy|ample|anderberg|austin-colwell|
    |baroni-urbani-buser-1|baroni-urbani-buser-2|becker-clogg-1|becker-clogg-2|
    |bonett-price-1|bonett-price-2|bonett-price-3|braun-blanquet|
    |camp-1|camp-2|camp-3|chen-popovich|
    |clement|cohen-kappa|cohen-w|cole-c1|
    |cole-c2|cole-c3|cole-c4|cole-c5|
    |cole-c6|cole-c7|cole-c8|contingency|
    |czekanowski|dennis|dice-1|dice-2|
    |dice-3|digby|doolittle|driver-kroeber-1|
    |driver-kroeber-2|edward|eyraud|fager-mcgowan-1|
    |fager-mcgowan-2|faith|fleiss|forbes-1|
    |forbes-2|fossum-kaskey|gilbert|gilbert-wells|
    |gk-lambda-1|gk-lambda-2|gleason|gower|
    |gower-legendre|hamann|harris-lahey|hawkins-dotson|
    |hurlbert|jaccard|johnson|kent-foster-1|
    |kent-foster-2|kuder-richardson|kulczynski-1|kulczynski-2|
    |loevinger|matching|maxwell-pilliner|mcconnaughey|
    |mcewen-michael|mountford|nei-li|ochiai-1|
    |ochiai-2|odds-ratio|otsuka|pearson|
    |pearson-heron|pearson-q1|pearson-q2|pearson-q3|
    |pearson-q4|pearson-q5|peirce-1|peirce-2|
    |peirce-3|phi|rogers-tanimoto|rogot-goldberg|
    |russell-rao|scott|simpson|sokal-michener|
    |sokal-sneath-1|sokal-sneath-2|sokal-sneath-3|sokal-sneath-4|
    |sokal-sneath-5|sorgenfrei|stiles|tanimoto|
    |tarantula|tarwid|tulloss|yule-q|
    |yule-r|yule-y|  |  |

    
    
     If we have a 2x2 table with the following values:
    
     |   |Column 1| Column 2| total|
     |---|:-------:|:--------:|:------:|
     |row 1| \\(a\\) | \\(b\\) | \\(R_1\\)|
     |row 2| \\(c\\) | \\(d\\) | \\(R_2\\)|
     |total| \\(C_1\\) | \\(C_2\\) | \\(n\\)|
    
     then the following formula are used:
    
     |Nr|Label|Formula|
     |:---:|-----------|---------|
     | 1|Russell-Rao|\\(\\frac{a}{n}\\)|
     | 2|Dice-1|\\(\\frac{a}{R_1}\\)|
     | 3|Dice-2|\\(\\frac{a}{C_1}\\)|
     | 4|Braun-Blanquet|\\(\\frac{a}{\\max\\left(R_1, C_1\\right)}\\)|
     | 5|Simpson Similarity|\\(\\frac{a}{\\min\\left(R_1, C_1\\right)}\\)|
     | 6|Kulczynski-1|\\(\\frac{a}{b+c}\\)|
     | 7|Jaccard|\\(\\frac{a}{a+b+c}\\)|
     | 8|Sokal-Sneath-1|\\(\\frac{a}{a+2b+2c}\\)|
     | 9|Gleasson|\\(\\frac{2a}{2a+b+c}\\)|
     |10|Mountford|\\(\\frac{2a}{a\\left(b+c\\right)+2bc}\\)|
     |11|Driver-Kroeber|\\(\\frac{a}{\\sqrt{R_1 C_1}}\\)|
     |12|Sorgenfrei|\\(\\frac{a^2}{R_1 C_1}\\)|
     |13|Johnson|\\(\\frac{a}{R_1}+\\frac{a}{C_1}\\)|
     |14|Kulczynski-2|\\(\\frac{1}{2}\\left(\\frac{a}{R_1}+\\frac{a}{C_1}\\right)\\)|
     |15|Fager-McGowan-1|\\(\\frac{a}{\\sqrt{R_1 C_1}}-\\frac{1}{2\\sqrt{\\max\\left(R_1, C_1\\right)}}\\)|
     |16|Fager-McGowan-2|\\(\\frac{a}{\\sqrt{R_1 C_1}}-\\frac{\\sqrt{\\max\\left(R_1, C_1\\right)}}{2}\\)|
     |17|tarantula|\\(\\frac{aR_2}{cR_1}\\)|
     |18|Ample|\\(\\frac{aR_2}{cR_1}\\)|
     |19|Gilbert|\\(\\frac{an-R_1 C_1}{C_1 n + R_1 n - an - R_1 C_1}\\)|
     |20|Fossum-Kaskey|\\(\\frac{n\\left(a - \\frac{1}{2}\\right)^2}{R_1 C_1}\\)|
     |21|Forbes - 1|\\(\\frac{na}{R_1 C_1}\\)|
     |22|Eyraud|\\(\\frac{a-R_1 C_1}{R_1 R_2 C_1 C_2}\\)|
     |23|Sokal-Michener|\\(\\frac{a+d}{n}\\)|
     |24|Faith|\\(\\frac{a+\\frac{1}{2}}{n}\\)|
     |25|Sokal-Sneath-5|\\(\\frac{a+d}{b+c}\\)|
     |26|Rogers-Tanimoto|\\(\\frac{a+d}{a+2\\left(b+c\\right)+d}\\)|
     |27|Sokal-Sneath-2|\\(\\frac{2a+2d}{2a+b+c+2d}\\)|
     |28|Gower|\\(\\frac{a+d}{\\sqrt{R_1 R_2 C_1 C_2}}\\)|
     |29|Sokal-Sneath-4|\\(\\frac{ad}{\\sqrt{R_1 R_2 C_1 C_2}}\\)|
     |30|Rogot-Goldberg|\\(\\frac{a}{R_1 + C_1}+\\frac{d}{R_2 + C_2}\\)|
     |31|Sokal-Sneath-3|\\(\\frac{1}{4}\\left(\\frac{a}{R_1}+\\frac{a}{C_1}+\\frac{d}{R_2}+\\frac{d}{C_2}\\right)\\)|
     |32|Hawkin-Dotson|\\(\\frac{1}{2}\\left(\\frac{a}{a+b+c}+\\frac{d}{b+c+d}\\right)\\)|
     |33|Clement|\\(\\frac{aR_2}{nR_1}+\\frac{dR_1}{nR_2}\\)|
     |34|Harris-Lahey|\\(\\frac{a\\left(R_2+C_2\\right)}{2n\\left(a+b+c\\right)}+\\frac{d\\left(R_1+C_1\\right)}{2n\\left(b+c+d\\right)}\\)|
     |35|Austin-Colwell|\\(\\frac{2}{\\pi}\\text{arcsin}\\sqrt{\\frac{a+d}{n}}\\)|
     |36|Baroni-Urbani-Buser-1|\\(\\frac{\\sqrt{ad}+a}{\\sqrt{ad}+a+b+c}\\)|
     |37|Peirce-1|\\(\\frac{ad-bc}{R_1 R_2}\\)|
     |38|Peirce-2|\\(\\frac{ad-bc}{C_1 C_2}\\)|
     |39|Cole C1|\\(\\frac{ad-bc}{R_1 C_1}\\)|
     |40|Loevinger|\\(\\frac{ad-bc}{\\min\\left(R_1 C_2, R_2 C_1\\right)}\\)|
     |41|Cole C7|\\(\\begin{cases} \\frac{ad-bc}{R_1 C_2} & \\text{ if } ad \\geq bc \\\\ \\frac{ad-bc}{R_1 C_1} & \\text{ if } ad < bc \\text{ and } a \\leq d \\\\ \\frac{ad-bc}{R_2 C_2} & \\text{ if } ad < bc \\text{ and } a > d\\end{cases}\\)|
     |42|Dennis|\\(\\frac{ad-bc}{\\sqrt{nR_1 C_1}}\\)|
     |43|Phi|\\(\\frac{ad-bc}{\\sqrt{R_1 R_2 C_1 C_2}}\\)|
     |44|Doolittle|\\(\\frac{\\left(ad-bc\\right)^2}{R_1 R_2 C_1 C_2}\\)|
     |45|Peirce-3|\\(\\frac{ad-bc}{ab+2bc+cd}\\)|
     |46|Cohen-kappa|\\(\\frac{2\\left(ad-bc\\right)}{R_1 C_2 + R_2 C_1}\\)|
     |47|McEwen-Michael|\\(\\frac{4\\left(ad-bc\\right)}{\\left(a+d\\right)^2 + \\left(b+c\\right)^2}\\)|
     |48|Kuder-Richardson|\\(\\frac{4\\left(ad-bc\\right)}{R_1 R_2 + C_1 C_2 + 2ad-2bc}\\)|
     |49|Scott|\\(\\frac{4ad-\\left(b+c\\right)^2}{\\left(R_1 + C_1\\right)\\left(R_2 + C_2\\right)}\\)|
     |50|Maxwell-Pilliner|\\(\\frac{2\\left(ad-bc\\right)}{R_1 R_2 + C_1 C_2}\\)|
     |51|Cole C5|\\(\\frac{\\sqrt{2}\\left(ad-bc\\right)}{\\sqrt{\\left(ad-bc\\right)^2+R_1 R_2 C_1 C_2}}\\)|
     |52|Hamann|\\(\\frac{\\left(a+d\\right)-\\left(b+c\\right)}{n}\\)|
     |53|Fleiss|\\(\\frac{\\left(ad-bc\\right)\\left(R_1 C_2 + R_2 C_1\\right)}{2R_1 R_2 C_1 C_2}\\)|
     |54|Yule Q|\\(\\frac{ad-bc}{ad+bc}\\)|
     |55|Yule Y|\\(\\frac{\\sqrt{ad}-\\sqrt{bc}}{\\sqrt{ad}+\\sqrt{bc}}\\)|
     |56|Digby H|\\(\\frac{\\left(ad\\right)^{3/4}-\\left(bc\\right)^{3/4}}{\\left(ad\\right)^{3/4}+\\left(bc\\right)^{3/4}}\\)|
     |57|Edward Q|\\(\\frac{OR^{\\pi/4}-1}{OR^{\\pi/4}+1}\\)|
     |58|Tarwid|\\(\\frac{na-R_1 C_1}{na+R_1 C_1}\\)|
     |59|Bonett-Price Y*|\\(\\frac{\\hat{w}^x-1}{\\hat{w}^x+1}\\)|
     |60|Contingency coefficient|\\(\\sqrt{\\frac{\\chi^2}{n+\\chi^2}}\\)|
     |61|Cohen w|\\(\\sqrt{\\frac{\\chi^2}{\\chi^2}}\\)|
     |62|Pearson|\\(\\sqrt{\\frac{\\phi^2}{n+\\phi^2}}\\)|
     |63|Hurlbert|\\(\\frac{ad-bc}{\\left\\lvert ad-bc\\right\\rvert}\\sqrt{\\frac{\\chi^2-\\chi_{min}^2}{\\chi_{max}^2-\\chi_{min}^2}}\\)|
     |64|Stiles|\\(\\log_{10}\\left(\\frac{n\\left(\\left\\lvert ad-bc \\right\\rvert -\\frac{n}{2}\\right)^2}{R_1 R_2 C_1 C_2}\\right)\\)|
     |65|McConnaughey|\\(\\frac{a^2-bc}{R_1 C_1}\\)|
     |66|Baroni-Urbani-Buser-2|\\(\\frac{a-b-c+\\sqrt{ad}}{a+b+c+\\sqrt{ad}}\\)|
     |67|Kent-Foster-1|\\(\\frac{-bc}{bR_1 + cC_1 + bc}\\)|
     |68|Kent-Foster-2|\\(\\frac{-bc}{bR_2 + cC_2 + bc}\\)|
     |69|Tulloss|\\(\\sqrt{U\\times S\\times R}\\)|
     |70|Gilbert-Wells|\\(\\ln\\left(a\\right)-\\ln\\left(n\\right)-\\ln\\left(\\frac{R_1}{n}\\right)-\\ln\\left(\\frac{C_1}{n}\\right)\\)|
     |71|Yule r|\\(\\cos\\left(\\frac{\\pi\\sqrt{bc}}{\\sqrt{ad}+\\sqrt{bc}}\\right)\\)|
     |72|Anderberg|\\(\\frac{\\sigma-\\sigma'}{2n}\\)|
     |73|Alroy F|\\(\\frac{a\\left(n' + \\sqrt{n'}\\right)}{a\\left(n' + \\sqrt{n'}\\right)+\\frac{3}{2}bc}\\)|
     |74|Pearson Q1|\\(\\sin\\left(\\frac{\\pi}{2}\\times\\frac{R_2 C_1}{R_1 C_2}\\right)\\)|
     |75|Goodman-Kruskal Lambda-1|\\(\\frac{\\sigma-\\sigma'}{2n-\\sigma'}\\)|
     |76|Goodman-Kruskal Lambda-2|\\(\\frac{2\\min\\left(a,d\\right)-b-c}{2\\min\\left(a,d\\right)+b+c}\\)|
     |77|Odds Ratio|\\(\\frac{ad}{bc}\\)|
     |78|Pearson Q4|\\(\\sin\\frac{\\pi}{2}\\times\\frac{1}{1+\\frac{2bcn}{\\left(ad-bc\\right)\\left(b+c\\right)}}\\)|
     |79|Pearson Q5|\\(\\sin\\frac{\\pi}{2}\\times\\frac{1}{\\sqrt{1+\\frac{4abcdn^2}{\\left(ad-bc\\right)^2\\left(a+d\\right)\\left(b+c\\right)}}}\\)|
     |80|Camp (3 ver.)|\\(\\frac{m}{\\sqrt{1+\\Theta\\times m^2}}\\)|
     |81|Becker-Clogg-1|\\(\\frac{g-1}{g+1}\\)|
     |82|Becker-Clogg-2|\\(\\frac{OR^{13.3/\\Delta}-1}{OR^{13.3/\\Delta}+1}\\)|
     |83|Bonett-Price-r|\\(\\cos\\left(\\frac{\\pi}{1+\\omega^c}\\right)\\)|
     |84|Bonett-Price-rhat|\\(\\cos\\left(\\frac{\\pi}{1+\\hat{\\omega}^{\\hat{c}}}\\right)\\)|
     |85|Chen-Popovich|\\(\\frac{ad-bc}{\\lambda_x \\lambda_y n^2}\\)|
     
     **Equation 57**
     
     $$OR = \\frac{ad}{bc}$$
     
     **Equation 59**
     
     $$x = \\frac{1}{2}-\\left(\\frac{1}{2}-p_{min}\\right)^2$$
    $$p_{min} = \\frac{\\min\\left(R_1, R_2, C_1, C_2\\right)}{n}$$
    $$\\hat{\\omega} = \\frac{\\left(a+0.1\\right)\\times\\left(d+0.1\\right)}{\\left(b+0.1\\right)\\times\\left(c+0.1\\right)}$$
    
    **Equations 60, 61, and 63**
    
    $$\\chi^2 = \\frac{n\\left(ad-bc\\right)^2}{R_1 R_2 C_1 C_2}$$
    
    **Equation 62**
    
    $$\\Phi = \\frac{\\left\\lvert ad-bc\\right\\rvert}{\\sqrt{R_1 R_2 C_1 C_2}}$$
    Note that Choi et. al ommit the absolute value, but this would create problems with taking the square root if bc>ad.
    
    **Equation 63**
    
    $$\\chi_{max}^2 = \\begin{cases} \\frac{nR_1 C_2}{R_2 C_1} & \\text{ if } ad \\geq bc \\\\  \\frac{nR_1 C_1}{R_2 C_2} & \\text{ if } ad < bc \\text{ and } a \\leq d \\\\  \\frac{nR_2 C_2}{R_1 C_1} & \\text{ if } ad < bc \\text{ and } a > d \\end{cases}$$
    $$\\chi_{min}^2 = \\frac{n^3\\left(\\hat{a} - g\\left(\\hat{a}\\right)\\right)^2}{R_1 R_2 C_1 C_2}$$
    $$\\hat{a}=\\frac{R_1 C_1}{n}$$
    $$g\\left(\\hat{a}\\right) = \\begin{cases} \\lfloor \\hat{a} \\rfloor & \\text{ if } ad < bc \\\\  \\lceil \\hat{a}\\rceil & \\text{ if } ad \\geq bc \\end{cases}$$
    
    **Equation 69**
    
    $$U = \\log_2\\left(1+\\frac{\\min\\left(b,c\\right)+a}{\\max\\left(b,c\\right)+a}\\right)$$
    $$S = \\frac{1}{\\sqrt{\\log_2\\left(2+\\frac{\\min\\left(b,c\\right)}{a+1}\\right)}}$$
    $$R = \\log_2\\left(1+\\frac{a}{R_1}\\right)\\log_2\\left(1+\\frac{a}{RC1}\\right)$$
    
    **Equation 71 and 75**
    
    $$\\sigma = \\max\\left(a,b\\right)+\\max\\left(c,d\\right)+\\max\\left(a,c\\right)+\\max\\left(b,d\\right)$$
    $$\\sigma' = \\max\\left(R_1, R_2\\right)+\\max\\left(C_1, C_2\\right)$$
    
    **Equation 73**
    
    $$n' = a+b+c$$
    
    **Equation 80 (Camp)**
    
    Camp (1934, pp. 309) describes the following steps for the calculation:
    
    Step 1: If total of column 1 (C1) is less than column 2 (C2), swop the two columns
    
    Step 2: Calculate \\(p = \\frac{C1}{n}\\), \\(p_1 = \\frac{a}{n}\\), and \\(p_2 = \\frac{c}{C2}\\)
    
    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)
    
    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 \\eqn{\\phi=1}.
    
    For a better approximation Camp made the following table:
    
    | p | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
    |---|-----|-----|-----|-----|-----|
    |phi|0.637|0.63|0.62|0.60|0.56|
    
    Cureton (1968, p. 241) expanded on this table and produced:
    
    | p | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
    |-----|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
    | 0.5 | 0.637 | 0.636 | 0.636 | 0.635 | 0.635 | 0.634 | 0.634 | 0.633 | 0.633 | 0.632 | 0.631 |
    | 0.6 | 0.631 | 0.631 | 0.630 | 0.629 | 0.628 | 0.627 | 0.626 | 0.625 | 0.624 | 0.622 | 0.621 |
    | 0.7 | 0.621 | 0.620 | 0.618 | 0.616 | 0.614 | 0.612 | 0.610 | 0.608 | 0.606 | 0.603 | 0.600 |
    | 0.8 | 0.600 | 0.597 | 0.594 | 0.591 | 0.587 | 0.583 | 0.579 | 0.574 | 0.569 | 0.564 | 0.559 |
    
    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.
    
    **Equations 81 and 82**
    
    Version 1 will calculate:
    $$\\rho^* = \\frac{g-1}{g+1}$$
    
    Version 2 will calculate:
    $$\\rho^{**} = \\frac{OR^{13.3/\\Delta} - 1}{OR^{13.3/\\Delta} + 1}$$
    
    With:
    $$g=e^{12.4\\times\\phi - 24.6\\times\\phi^3}$$
    $$\\phi = \\frac{\\ln\\left(OR\\right)}{\\Delta}$$
    $$OR=\\frac{\\left(\\frac{a}{c}\\right)}{\\left(\\frac{b}{d}\\right)} = \\frac{a\\times d}{b\\times c}$$
    $$\\Delta = \\left(\\mu_{R1} - \\mu_{R2}\\right) \\times \\left(v_{C1} - v_{C2}\\right)$$
    $$\\mu_{R1} = \\frac{-e^{-\\frac{t_r^2}{2}}}{p_{R1}}, \\mu_{R2} = \\frac{e^{-\\frac{t_r^2}{2}}}{p_{R2}}$$
    $$v_{C1} = \\frac{-e^{-\\frac{t_c^2}{2}}}{p_{C1}}, v_{C2} = \\frac{e^{-\\frac{t_c^2}{2}}}{p_{C2}}$$
    $$t_r = \\Phi^{-1}\\left(p_{R1}\\right), t_c = \\Phi^{-1}\\left(p_{C1}\\right)$$
    $$p_{x} = \\frac{x}{n}$$
    
    **Equations 83 and 84**
    
    Formula for version 1 is (Bonett & Price, 2005, p. 216):
    $$\\rho^* = \\cos\\left(\\frac{\\pi}{1+\\omega^c}\\right)$$
    
    With:
    $$\\omega = OR = \\frac{a\\times d}{b\\times c}$$
    $$c = \\frac{1-\\frac{\\left|R_1-C_1\\right|}{5\\times n} - \\left(\\frac{1}{2}-p_{min}\\right)^2}{2}$$
    $$p_{min} = \\frac{\\text{MIN}\\left(R_1, R_2, C_1, C_2\\right)}{n}$$
    
    Formula for version 2 is  (Bonett & Price, 2005, p. 216):
    $$\\hat{\\rho}^* = \\cos\\left(\\frac{\\pi}{1+\\hat{\\omega}^{\\hat{c}}}\\right)$$
    
    with:
    $$\\hat{\\omega} = \\frac{\\left(a+\\frac{1}{2}\\right)\\times \\left(d+\\frac{1}{2}\\right)}{\\left(b+\\frac{1}{2}\\right)\\times \\left(c+\\frac{1}{2}\\right)}$$
    $$\\hat{c} = \\frac{1-\\frac{\\left|R_1-C_1\\right|}{5\\times \\left(n+2\\right)} - \\left(\\frac{1}{2}-\\hat{p}_{min}\\right)^2}{2}$$
    $$\\hat{p}_{min} = \\frac{\\text{MIN}\\left(R_1, R_2, C_1, C_2\\right)+1}{n+2}$$
    
    **Equation 85**
    
    $$\\lambda_x = \\Phi^{-1}\\left(\\frac{R_1}{n}\\right)$$
    $$\\lambda_y = \\Phi^{-1}\\left(\\frac{C_1}{n}\\right)$$
    
    with \\(\\Phi^{-1}\\left(\\dots\\right)\\) being the inverse standard normal cumulative distribution
    
    **Sources for formulas**
    
    The formulas were obtained from the following sources.
    The columns W-C-H show which equation corresponds to my label in:
    
    * W: Warrens (2008, pp. 219–222). Equation 4 from Warrens is the chi-square value and not added.
    * C: Choi et al. (2010, pp. 44–45). Equations not added from this source are: Eq. 4 is a ‘3w Jaccard’, could not find a source for this and not added. Equation 12 is just the intersection (a), eq. 13 the innerproduct (a+d), and 66 the dispersion. Equation 51 is the chi-square value and measures 15 to 30 and 62 are just distance measures.
    * H: Hubálek (Hubálek, 1982, pp. 671–673)
    
    If no page number is listed in the original source, the formula was taken from Warrens, Choi et al. and/or Hubálek.
    
    |numb |Label|Original|Wa|Ch|Har|
    |:----:|-----|-------|:----:|:----:|:----:|
    | 1|Russell-Rao|(Russell & Rao, 1940)| 15 | 14 | 14 |
    | 2|Dice-1|(Dice, 1945, p. 302)|17a|   |   |
    | 3|Dice-2|(Dice, 1945, p. 302)|17b|   |   |
    | 4|Braun-Blanquet|(Braun-Blanquet, 1932)|12|46|1|
    | 5|Simpson Similarity|(Simpson, 1943, p. 20, 1960, p. 301)|16|45|2|
    | 6|Kulczynski-1|(Kulczynski, 1927)|11b|64|3|
    | 7|Jaccard =|(Jaccard, 1901, 1912, p. 39)|6|1|4|
    |  |Tanimoto|(Tanimoto, 1958, p. 5)|   |65|   |
    | 8|Sokal-Sneath-1|(Sokal & Sneath, 1963, p. 129)|30a|6|6|
    | 9|Gleasson =|(Gleason, 1920, p. 31)|9|   |5|
    |  |Dice-3 =|(Dice, 1945, p. 302)|   |2|   |
    |  |Nei-Li =|(Nei & Li, 1979, p. 5270)|   |5|   |
    |  |Czekanowski|   |   |3|   |
    |10|Mountford|(Mountford, 1962, p. 45)|28|37|15|
    |11|Driver-Kroeber =|(Driver & Kroeber, 1932, p. 219)|13|31|11|
    |  |Ochiai-1  =|(Ochiai, 1957)|   |33|   |
    |  |Otsuka|(Otsuka, 1936)|   |38|   |
    |12|Sorgenfrei|(Sorgenfrei, 1958)|23|36|12|
    |13|Johnson|(Johnson, 1967)|33|43|9|
    |14|Driver-Kroeber-2 = |(Driver & Kroeber, 1932, p. 219)|11a|42|8|
    |  |Kulczynski-2|(Kulczynski, 1927)|   |41|7|
    |15|Fager-McGowan-1|(Fager & McGowan, 1963, p. 454)|29|   |   |
    |16|Fager-McGowan-2|(Fager & McGowan, 1963, p. 454)|   |47|13|
    |17|tarantula|(Jones & Harrold, 2005)|   |75|   |
    |18|ample|   |   |76|   |
    |19|Gilbert|(Gilbert, 1884, p. 171)|   |   |   |
    |20|Fossum-Kaskey|(Fossum & Kaskey, 1966, p. 65)|   |35|   |
    |21|Forbes - 1|(Forbes, 1907, p. 279)|5|34|40|
    |22|Eyraud|(Eyraud, 1936)|   |74|17|
    |23|Sokal-Michener|(Sokal & Michener, 1958, p. 1417)|22|7|20|
    |24|Faith|(Faith, 1983, p. 290)|   |10|   |
    |25|Sokal-Sneath-5|(Sokal & Sneath, 1963, p. 129)|30e|56|19|
    |26|Rogers-Tanimoto|(Rogers & Tanimoto, 1960)|25|9|23|
    |27|Sokal-Sneath-2 = |(Sokal & Sneath, 1963, p. 129)|30b|8|22|
    |  |Gower-Legendre|(Gower & Legendre, 1986)|   |11|   |
    |28|Gower|(Gower, 1971)|   |50|   |
    |29|Sokal-Sneath-4 = |(Sokal & Sneath, 1963, p. 130)|30d|57|25|
    |  |Ochiai-2|(Ochiai, 1957)|   |60|   |
    |30|Rogot-Goldberg|(Rogot & Goldberg, 1966, p. 997)|32|   |   |
    |31|Sokal-Sneath-3|(Sokal & Sneath, 1963, p. 130)|30c|49|18|
    |32|Hawkin-Dotson|(Hawkins & Dotson, 1975, pp. 372–373)|34|   |   |
    |33|Clement|(Clement, 1976, p. 258)|37|   |   |
    |34|Harris-Lahey|(Harris & Lahey, 1978, p. 526)|40|   |   |
    |35|Austin-Colwell|(Austin & Colwell, 1977, p. 205)|   |   |21|
    |36|Baroni-Urbani-Buser-1|(Baroni-Urbani & Buser, 1976, p. 258)|38a|71|32|
    |37|Peirce-1|(Peirce, 1884, p. 453)|1a|   |   |
    |38|Peirce-2|(Peirce, 1884, p. 453)|1b|   |26|
    |39|Cole C1|(Cole, 1949, p. 415)|   |   |   |
    |40|Loevinger = |(Loevinger, 1947, p. 30)|18|   |   |
    |  |Forbes 2|(Forbes, 1925)|   |48|42|
    |41|Cole C7|(Cole, 1949, p. 420)|19|   |34|
    |42|Dennis|(Dennis, 1965, p. 69)|   |44|   |
    |43|Pearson Phi =|(Pearson, 1900a, p. 12)|7|54|30|
    |  |Yule Phi =|(Yule, 1912, p. 596)|   |   |   |
    |  |Cole C2|(Cole, 1949, p. 415)|   |   |   |
    |44|Doolittle|(Doolittle, 1885, p. 123)|2|   |31|
    |45|Peirce-3|(Peirce, 1884)|   |73|16|
    |46|Cohen-kappa|(Cohen, 1960, p. 40)|24|   |   |
    |47|McEwen-Michael =|(Michael, 1920, p. 57)|10|68|39|
    |  |Cole C3|(Cole, 1949, p. 415)|   |   |   |
    |48|Kuder-Richardson|(Kuder & Richardson, 1937)|14|   |   |
    |49|Scott|(Scott, 1955, p. 324)|21|   |   |
    |50|Maxwell-Pilliner|(Maxwell & Pilliner, 1968)|35|   |   |
    |51|Cole C5|(Cole, 1949, p. 416)|   |58|29|
    |52|Hamann|(Hamann, 1961)|27|67|24|
    |53|Fleiss|(Fleiss, 1975, p. 656)|36|   |   |
    |54|Yule Q =|(Yule, 1900, p. 272)|3|61|36|
    |  |Cole C4 =|(Cole, 1949, p. 415)|   |   |   |
    |  |Pearson Q2|(Pearson, 1900, p. 15)|   |   |   |
    |55|Yule Y|(Yule, 1912, p. 592)|8|63|37|
    |56|Digby H|(Digby, 1983, p. 754)|41|   |   |
    |57|Edward Q|(Edwards, 1957; Becker & Clogg, 1988, p. 409)|   |   |   |
    |58|Tarwid|(Tarwid, 1960, p. 117)|   |40|43|
    |59|Bonett-Price-1|(Bonett & Price, 2007, p. 433)|   |   |   |
    |60|Contingency|(Pearson, 1904, p. 9)|   |52|28|
    |61|Cohen w|(Cohen, 1988, p. 216)|   |   |   |
    |62|Pearson|(Pearson, 1904)|   |53|   |
    |63|Hurlbert / Cole C8|(Hurlbert, 1969, p. 1)|   |   |35|
    |64|Stiles|(Stiles, 1961, p. 272)|26|59|   |
    |65|McConnaughey|(McConnaughey, 1964)|31|39|10|
    |66|Baroni-Urbani-Buser-2|(Baroni-Urbani & Buser, 1976, p. 258)|38b|72|33|
    |67|Kent-Foster-1|(Kent & Foster, 1977, p. 311)|39a|   |   |
    |68|Kent-Foster-2|(Kent & Foster, 1977, p. 311)|39b|   |   |
    |69|Tulloss|(Tulloss, 1997, p. 133)|   |   |   |
    |70|Gilbert-Wells|(Gilbert & Wells, 1966)|   |32|41|
    |71|Yule r|(Yule, 1900, p. 276)|   |   |   |
    |  |Pearson-Q3|(Pearson, 1900a, p. 16)|   |   |   |
    |  |Cole C6|(Cole, 1949, p. 416)|   |   |   |
    |  |Pearson-Heron|(Pearson & Heron, 1913)|   |55|38|
    |72|Anderberg|(Anderberg, 1973)|   |70|   |
    |73|Alroy F|(Alroy, 2015, eq. 6)|   |   |   |
    |74|Pearson Q1|(Pearson, 1900a, p. 15)|   |   |   |
    |75|Goodman-Kruskal Lambda-1|(Goodman & Kruskal, 1954, p. 743)|   |69|   |
    |76|Goodman-Kruskal Lambda-2|(Goodman & Kruskal, 1954)|20|   |   |
    |77|Odds Ratio|(Fisher, 1935, p. 50)|   |   |   |
    |78|Pearson Q4|(Pearson, 1900, p. 16)|   |   |   |
    |79|Pearson Q5|(Pearson, 1900, p. 16)|   |   |   |
    |80|Camp|(Camp, 1934, p. 309)|   |   |   |
    |81|Becker-Clogg-1|(Becker & Clogg, 1988, pp. 410–412)|   |   |   |
    |82|Becker-Clogg-2|(Becker & Clogg, 1988, pp. 410–412)|   |   |   |
    |83|Bonett-Price-2|(Bonett & Price, 2005, p. 216)|   |   |   |
    |84|Bonett-Price-3|(Bonett & Price, 2005, p. 216)|   |   |   |
    |85|Ched-Popovich|(Chen & Popovich, 2002, p. 37)|   |   |   |
    
    Before, After and Alternatives
    -----------------------
    Before determining the effect size measure, you might want to use:
    * [tab_cross](../other/table_cross.html#tab_cross) to get a cross table and/or 
    * [ts_fisher](../tests/test_fisher.html#ts_fisher) to perform a Fisher exact test

    Some of the measures in this function are approximations of the so-called tetrachoric correlation. A separate function for this is also available via [r_tetrachoric](../correlations/cor_tetrachoric.html)

    Rules of thumb for the classification for some of the measures can be found using:
    * Odds Ratio -> [th_odds_ratio](../other/thumb_odds_ratio.html#th_odds_ratio)
    * Yule Q -> [th_yule_q](../other/thumb_yule_q.html#th_yule_q), or according to Lloyd et al. (2013, p. 484) the rules-of-thumb from the Odds Ratio from Rosenthal could be used.
    * Cohen Kappa -> [th_cohen_kappa](../other/thumb_cohen_kappa.html#th_cohen_kappa)
    
    References
    ----------
    Alroy, J. (2015). A new twist on a very old binary similarity coefficient. *Ecology, 96*(2), 575–586. doi:10.1890/14-0471.1

    Anderberg, M. R. (1973). *Cluster analysis for applications*. New York, NY, Academic Press.

    Austin, B., & Colwell, R. R. (1977). Evaluation of some coefficients for use in numerical taxonomy of microorganisms. *International Journal of Systematic Bacteriology, 27*(3), 204–210. doi:10.1099/00207713-27-3-204

    Baroni-Urbani, C., & Buser, M. W. (1976). Similarity of binary data. *Systematic Zoology, 25*(3), 251–259. doi:10.2307/2412493

    Becker, M. P., & Clogg, C. C. (1988). A note on approximating correlations from Odds Ratios. *Sociological Methods & Research, 16*(3), 407–424. doi:10.1177/0049124188016003003

    Bonett, D. G., & Price, R. M. (2005). Inferential methods for the tetrachoric correlation coefficient. *Journal of Educational and Behavioral Statistics, 30*(2), 213–225. doi:10.3102/10769986030002213

    Bonett, D. G., & Price, R. M. (2007). Statistical inference for generalized yule coefficients in 2 x 2 contingency tables. *Sociological Methods & Research, 35*(3), 429–446. doi:10.1177/0049124106292358

    Braun-Blanquet, J. (1932). *Plant sociology: The study of plant communities*. McGraw Hill.

    Camp, B. H. (1934). *Mathematical part of elementary statistics*. D.C. Heath and Company, London.

    Chen, P. Y., & Popovich, P. M. (2002). *Correlation: Parametric and nonparametric measures*. Sage Publications.

    Choi, S.-S., Cha, S.-H., & Tappert, C. (2010). A survey of binary similarity and distance measures. *Journal on Systemics, Cybernetics and Informatics, 8*(1), 43–48.

    Clement, P. W. (1976). A formula for computing inter-observer agreement. *Psychological Reports, 39*(1), 257–258. doi:10.2466/pr0.1976.39.1.257

    Cohen, J. (1960). A coefficient of agreement for nominal scales. *Educational and Psychological Measurement, 20*(1), 37–46. doi:10.1177/001316446002000104

    Cohen, J. (1988). *Statistical power analysis for the behavioral sciences* (2nd ed.). L. Erlbaum Associates.

    Cole, L. C. (1949). *The measurement of interspecific associaton. Ecology, 30*(4), 411–424. doi:10.2307/1932444

    Cureton, E. E. (1968). Tetrachoric correlation by the Camp approximation. *Educational and Psychological Measurement, 28*(2), 239–244. doi:10.1177/001316446802800202

    Dennis, S. F. (1965). The construction of a thesaurus automatically from a sample of text. In M. E. Stevens, V. E. Giuliano, & L. B. Heilprin (Eds.), *Statistical Association Methods for Mechanized Documentation* (Vol. 14, pp. 61–148). U.S. Government Printing Office.

    Dice, L. R. (1945). Measures of the amount of ecologic association between species. *Ecology, 26*(3), 297–302. doi:10.2307/1932409

    Digby, P. G. N. (1983). Approximating the tetrachoric correlation coefficient. *Biometrics, 39*(3), 753–757. doi:10.2307/2531104

    Doolittle, M. H. (1885). The verification of predictions. *Bulletin of the Philosophical Society of Washington, 7*, 122–127.

    Driver, H. E., & Kroeber, A. L. (1932). Quantitative expression of cultural relationships. *University Of California Publications in American Archeology and Ethnology, 31*(4), 211–256.

    Edwards, J. H. (1957). A note on the practical interpretation of 2 x 2 tables. *Journal of Epidemiology & Community Health, 11*(2), 73–78. doi:10.1136/jech.11.2.73

    Edwards, J. H., & Edwards, A. W. F. (1984). Approximating the tetrachoric correlation coefficient. *Biometrics, 40*(2), 563–563.

    Eyraud, H. (1936). Les principes de la mesure des correlations. *Ann. Univ. Lyon, III. Ser., Sect. A, 1*(30–47), 111.

    Fager, E. W., & McGowan, J. A. (1963). Zooplankton species groups in the north pacific: Co-occurrences of species can be used to derive groups whose members react similarly to water-mass types. *Science, 140*(3566), 453–460. doi:10.1126/science.140.3566.453

    Faith, D. P. (1983). Asymmetric binary similarity measures. *Oecologia, 57*(3), 287–290. doi:10.1007/BF00377169

    Fisher, R. A. (1935). The logic of inductive inference. *Journal of the Royal Statistical Society, 98*(1), 39–82. doi:10.2307/2342435

    Fleiss, J. L. (1975). Measuring agreement between two judges on the presence or absence of a trait. *Biometrics, 31*(3), 651–659. doi:10.2307/2529549

    Forbes, S. A. (1907). On the local distribution of certain Illinois fishes: An essay in statistical ecology. *Illinois Natural History Survey Bulletin, 7*(8), 273–303.

    Forbes, S. A. (1925). Method of determining and measuring the associative relations of species. *Science, 61*, 524.

    Fossum, E. G., & Kaskey, G. (1966). Optimization and standardization of information retrieval language and systems (AF49(638)1194; p. 96). Univac Division.

    Gilbert, G. K. (1884). Finley’s tornado predictions. *American Meteorological Journal, 1*(5), 166–172.

    Gilbert, N., & Wells, T. C. E. (1966). *Analysis of quadrat data. Journal of Ecology, 54*(3), 675–685. doi:10.2307/2257810

    Gleason, H. A. (1920). Some applications of the quadrat method. *Bulletin of the Torrey Botanical Club, 47*(1), 21–33. doi:10.2307/2480223

    Glen, S. (2017, August 16). Gamma Coefficient (Goodman and Kruskal’s Gamma) & Yule’s Q. Statistics How To. https://www.statisticshowto.com/gamma-coefficient-goodman-kruskal/

    Goodman, L. A., & Kruskal, W. H. (1954). Measures of association for cross classifications. *Journal of the American Statistical Association, 49*(268), 732–764. doi:10.2307/2281536

    Gower, J. C. (1971). A general coefficient of similarity and some of its properties. *Biometrics, 27*(4), 857. doi:10.2307/2528823

    Gower, J. C., & Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. *Journal of Classification, 3*(1), 5–48. doi:10.1007/BF01896809

    Hamann, U. (1961). Merkmalsbestand und verwandtschaftsbeziehungen der farinosae: Ein beitrag zum system der monokotyledonen. *Willdenowia, 2*(5), 639–768.

    Harris, F. C., & Lahey, B. B. (1978). A method for combining occurrence and nonoccurrence interobserver agreement scores. *Journal of Applied Behavior Analysis, 11*(4), 523–527. doi:10.1901/jaba.1978.11-523

    Hawkins, R. P., & Dotson, V. A. (1975). Reliability scores that delude: An Alice in wonderland trip through the misleading characteristics of inter-observer agreement scores in interval recording. In E. Ramp & G. Semb (Eds.), *Behavior  analysis: Areas of research and application *(pp. 359–376). Prentice Hall.

    Hubálek, Z. (1982). Coefficients of association and similarity, based on binary (presence-absence) data: An evaluation. *Biological Reviews, 57*(4), 669–689. doi:10.1111/j.1469-185X.1982.tb00376.x

    Hurlbert, S. H. (1969). A coefficient of interspecific assciation. *Ecology, 50*(1), 1–9. doi:10.2307/1934657

    Jaccard, P. (1901). Étude comparative de la distribution florale dans une portion des Alpes et des Jura. *Bulletin Del La Société Vaudoise Des Sciences Naturelles, 37*, 547–579.

    Jaccard, P. (1912). The distribution of the flora in the alpine zone. *The New Phytologist, 11*(2), 37–50.

    Johnson, S. C. (1967). Hierarchical clustering schemes. *Psychometrika, 32*(3), 241–254. doi:10.1007/BF02289588

    Jones, J. A., & Harrold, M. J. (2005). Empirical evaluation of the tarantula automatic fault-localization technique. Proceedings of the 20th IEEE/ACM International Conference on Automated Software Engineering, 273–282. doi:10.1145/1101908.1101949

    Kent, R. N., & Foster, S. L. (1977). Direct observational procedures: Methodological issues in naturalistic settings. In A. R. Ciminero, K. S. Calhoun, & H. E. Adams (Eds.), *Handbook of behavioral assessment* (pp. 279–328). New York, NY: Wiley.

    Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. *Psychometrika, 2*(3), 151–160. doi:10.1007/BF02288391

    Kulczynski, S. (1927). Die Pflanzenassoziationen der Pieninen. *Bulletin International de l’Academie Polonaise Des Sciences et Des Lettres, Classe Des Sciences Mathematiques et Naturelles, B (Sciences Naturelles)*, II, 57–203.

    Ling, M. H. T. (2010). COPADS, I: Distance coefficients between two lists or sets. *The Python Papers Source Codes, 2*(2), 1–31.

    Loevinger, J. (1947). A systematic approach to the construction and evaluation of tests of ability. *Psychological Monographs, 61*(4), i–49. doi:10.1037/h0093565

    Maxwell, A. E., & Pilliner, A. E. (1968). Deriving coefficients of reliability and agreement for ratings. *The British Journal of Mathematical and Statistical Psychology, 21*(1), 105–116. doi:10.1111/j.2044-8317.1968.tb00401.x

    McConnaughey, B. H. (1964). The determination and analysis of plankton communities. *Marine Research, 7*, 1–40.

    Michael, E. L. (1920). Marine Ecology and the coefficient of association: A plea in behalf of quantitative biology. *Journal of Ecology, 8*(1), 54–59. doi:10.2307/2255213

    Mountford, M. D. (1962). An index of similarity and its application to classification problems. In P. W. Murphy & D. Phil (Eds.), *Progress in soil zoology* (pp. 43–50). Butterworths.

    Nei, M., & Li, W. H. (1979). Mathematical model for studying genetic variation in terms of restriction endonucleases. *Proceedings of the National Academy of Sciences, 76*(10), 5269–5273. doi:10.1073/pnas.76.10.5269

    Ochiai, A. (1957). Zoogeographical studies on the soleoid fishes found in Japan and its neighbouring regions-I. *Nippon Suisan Gakkaishi, 22*(9), 522–525. doi:10.2331/suisan.22.522

    Otsuka, Y. (1936). The faunal character of the Japanese Pleistocene marine Mollusca, as evidence of the climate having become colder during the Pleistocene in Japan. *Bulletin of the Biogeographical Society of Japan, 6*(16), 165–170.

    Pearson, K. (1900). Contributions to the mathematical theory of evolution. VII. On the correlation of characters not quantitatively measurable. *Philosophical Transactions of the Royal Society of London, 195*, 1–405. doi:10.1098/rsta.1900.0022

    Pearson, K. (1904). *Contributions to the Mathematical Theory of Evolution. XIII. On the theory of contingency and its relation to association and normal correlation*. Dulau and Co.

    Pearson, K., & Heron, D. (1913). On theories of association. *Biometrika, 9*(1/2), 159–315. doi:10.2307/2331805

    Peirce, C. S. (1884). The numerical measure of the success of predictions. *Science, 4*(93), 453–454. doi:10.1126/science.ns-4.93.453-a

    Rogers, D. J., & Tanimoto, T. T. (1960). A computer program for classifying plants. *Science, 132*(3434), 1115–1118. doi:10.1126/science.132.3434.1115

    Rogot, E., & Goldberg, I. D. (1966). A proposed index for measuring agreement in test-retest studies. *Journal of Chronic Diseases, 19*(9), 991–1006. doi:10.1016/0021-9681(66)90032-4

    Russell, P. F., & Rao, T. R. (1940). On habitat and association of species of anopheline larvae in south-eastern Madras. *Journal of the Malaria Institute of India, 3*(1), 153–178.

    Scott, W. A. (1955). Reliability of content analysis: The case of nominal scale coding. *The Public Opinion Quarterly, 19*(3), 321–325.

    Simpson, G. G. (1943). Mammals and the nature of continents. *American Journal of Science, 241*(1), 1–31. doi:10.2475/ajs.241.1.1

    Simpson, G. G. (1960). Notes on the measurement of faunal resemblance. *American Journal of Science, 258-A*, 300–311.

    Sokal, P. H. A., & Sneath, R. R. (1963). Principles of numerical taxonomy. W.H. Freeman and Company.

    Sokal, R., & Michener, C. (1958). A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin.

    Sorgenfrei, T. (1958). Molluscan assemblages from the marine middle miocene of south Jutland and their Environments. Vol. I. *Danmarks Geologiske Undersøgelse II. Række*, 79, 1–355. doi:10.34194/raekke2.v79.6868

    Stiles, H. E. (1961). The association factor in information retrieval. *Journal of the ACM, 8*(2), 271–279. doi:10.1145/321062.321074

    Tanimoto, T. T. (1958). An elementary mathematical theory of classification and prediction (PB167360). International Business Machines Corp., New York, NY.

    Tarwid, K. (1960). Szacowanie zbieinosci nisz ekologicznych gatunkow droga oceny prawdopodobienstwa spotykania sie ich w polowach. *Ekologia Polska, 6*, 115–130.

    Tulloss, R. E. (1997). Assessment of similarity indices for undesirable properties and a new tripartite similarity index based on cost functions. In M. E. Palm & I. H. Chapela (Eds.), *Mycology in sustainable development: Expanding concepts, vanishing borders* (pp. 122–143). Parkway Pub.

    Walker, H. M., & Lev, J. (1953). *Statistical inference*. Holt. 

    Warrens, M. J. (2008). Similarity coefficients for binary data: Properties of coefficients, coefficient matrices, multi-way metrics and multivariate coefficients [Doctoral dissertation, Universiteit Leiden]. https://hdl.handle.net/1887/12987

    Yule, G. U. (1900). On the association of attributes in statistics: With illustrations from the material of the childhood society, &c. *Philosophical Transactions of the Royal Society of London, 194*, 257–319. doi:10.1098/rsta.1900.0019

    Yule, G. U. (1912). On the methods of measuring association between two attributes. *Journal of the Royal Statistical Society, 75*(6), 579–652. doi:10.2307/2340126
    
    Author
    ------
    Made by P. Stikker
    
    Companion website: https://PeterStatistics.com  
    YouTube channel: https://www.youtube.com/stikpet  
    Donations: https://www.patreon.com/bePatron?u=19398076

    Examples
    --------
    >>> import pandas as pd
    >>> file1 = "https://peterstatistics.com/Packages/ExampleData/GSS2012a.csv"
    >>> df1 = pd.read_csv(file1, sep=',', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    >>> es_bin_bin(df1['mar1'], df1['wrksch'], categories1=['MARRIED', 'NEVER MARRIED'], categories2=['STAY HOME', 'WORK FULL-TIME'])
    np.float64(2.5555555555555554)
    
    '''
    #get the cross table
    
    tab = tab_cross(field1, field2, order1=categories1, order2=categories2, percent=None, totals="exclude")
    
    # cell values of sample cross table
    a = tab.iloc[0,0]
    b = tab.iloc[0,1]
    c = tab.iloc[1,0]
    d = tab.iloc[1,1]
    r1 = a + b
    r2 = c + d
    c1 = a + c
    c2 = b + d
    n = r1 + r2

    if (method == "russell-rao"):
        es = a / n
    elif (method == "dice-1"):
        es = a / r1
    elif (method == "dice-2"):
        es = a / c1
    elif (method == "braun-blanquet"):
        if (r1 > c1):
            es = a / r1 
        else:
            es = a / c1
    elif (method == "simpson"):
        if (r1 < c1):
            es = a / r1
        else:
            es = a / c1
    elif (method == "kulczynski-1"):
        es = a / (b + c)
    elif (method == "jaccard" or method == "tanimoto"):
        es = a / (a + b + c)
    elif (method == "sokal-sneath-1"):
        es = a / (a + 2 * b + 2 * c)
    elif (method == "gleason" or method == "dice-3" or method == "nei-li" or method == "czekanowski"):
        es = 2 * a / (2 * a + b + c)
    elif (method == "mountford"):
        es = 2 * a / (a * (b + c) + 2 * b * c)
    elif (method == "driver-kroeber-1" or method == "ochiai-1" or method == "otsuka"):
        es = a / sqrt((a + b) * (a + c))
    elif (method == "sorgenfrei"):
        es = a**2 / (r1 * c1)
    elif (method == "johnson"):
        es = a / r1 + a / c1
    elif (method == "kulczynski-2" or method == "driver-kroeber-2"):
        es = a * (2 * a + b + c) / (2 * r1 * c1)
    elif (method == "fager-mcgowan-1"):
        es = a / sqrt(r1 * c1) - 1 / (2 * sqrt(max(r1, c1)))
    elif (method == "fager-mcgowan-2"):
        es = a / sqrt(r1 * c1) - sqrt(max(r1, c1)) / 2
    elif (method == "tarantula"):
        es = a * r2 / (c * r1)
    elif (method == "ample"):
        es = abs(a * r2 / (c * r1))
    elif (method == "gilbert"):
        es = (a * n - r1 * c1) / (c1 * n + r1 * n - a * n - r1 * c1)
    elif (method == "fossum-kaskey"):
        es = n * (a - 1 / 2)**2 / (r1 * c1)
    elif (method == "eyraud"):
        es = (a - r1 * c1) / (r1 * r2 * c1 * c2)
    elif (method == "sokal-michener" or method == "matching"):
        es = (a + d) / (a + b + c + d)
    elif (method == "faith"):
        es = (a + 1 / 2 * d) / n
    elif (method == "sokal-sneath-5"):
        es = (a + d) / (b + c)
    elif (method == "rogers-tanimoto"):
        es = (a + d) / (a + 2 * b + 2 * c + d)
    elif (method == "sokal-sneath-2" or method == "gower-legendre"):
        es = (2 * a + 2 * d) / (2 * a + b + c + 2 * d)
    elif (method == "gower"):
        es = (a + d) / sqrt(r1 * r2 * c1 * c2)
    elif (method == "sokal-sneath-4" or method == "ochiai-2"):
        es = a * d / sqrt((a + b) * (a + c) * (c + d) * (b + d))
    elif (method == "rogot-goldberg"):
        es = a / (r1 + c1) + d / (r2 + c2)
    elif (method == "sokal-sneath-3"):
        es = 1 / 4 * (a / r1 + a / c1 + d / r2 + d / c2)
    elif (method == "hawkins-dotson"):
        es = 1 / 2 * (a / (a + b + c) + d / (b + c + d))
    elif (method == "clement"):
        es = a * r2 / (n * r1) + d * r1 / (n * r2)
    elif (method == "harris-lahey"):
        es = a * (r2 + c2) / (2 * n * (a + b + c)) + d * (r1 + c1) / (2 * n * (b + c + d))
    elif (method == "austin-colwell"):
        es = 2 / pi * asin(sqrt((a + d) / n))
    elif (method == "forbes-1"):
        es = n * a / (r1 * c1)
    elif (method == "baroni-urbani-buser-1"):
        es = (sqrt(a * d) + a) / (sqrt(a * d) + a + b + c)
    elif (method == "peirce-1"):
        es = (a * d - b * c) / (r1 * r2)
    elif (method == "peirce-2"):
        es = (a * d - b * c) / (c1 * c2)
    elif (method == "cole-c1"):
        es = (a * d - b * c) / ((a + b) * (a + c))
    elif (method == "loevinger" or method == "forbes-2"):
        es = (a * d - b * c) / min(r1 * c2, r2 * c1)
    elif (method == "cole-c7"):
        if (a * d >= (b * c)):
            C7 = (a * d - b * c) / ((a + b) * (b + d))
        elif (a <= d):
            C7 = (a * d - b * c) / ((a + b) * (a + c))
        else :
            C7 = (a * d - b * c) / ((b + d) * (c + d))
        es = C7
    elif (method == "dennis"):
        es = (a * d - b * c) / sqrt(n * r1 * c1)
    elif (method == "phi" or method == "cole-c2"):
        es = (a * d - b * c) / sqrt(r1 * r2 * c1 * c2)
    elif (method == "doolittle"):
        es = (a * d - b * c)**2 / (r1 * r2 * c1 * c2)
    elif (method == "peirce-3"):
        es = (a * d + b * c) / (a * b + 2 * b * c + c * d)
    elif (method == "cohen-kappa"):
        es = 2 * (a * d - b * c) / (r1 * c2 + r2 * c1)
    elif (method == "mcewen-michael" or method == "cole-c3"):
        es = 4 * (a * d - b * c) / ((a + d)**2 + (b + c)**2)
    elif (method == "kuder-richardson"):
        es = 4 * (a * d - b * c) / (r1 * r2 + c1 * c2 + 2 * a * d - 2 * b * c)
    elif (method == "scott"):
        es = (4 * a * d - (b + c)**2) / ((r1 + c1) * (r2 + c2))
    elif (method == "maxwell-pilliner"):
        es = 2 * (a * d - b * c) / (r1 * r2 + c1 * c2)
    elif (method == "cole-c5"):
        es = sqrt(2) * (a * d - b * c) / sqrt((a * d - b * c)**2 + (a + b) * (c + d) * (a + c) * (b + d))
    elif (method == "hamann"):
        es = (a + d - b - c) / (a + b + c + d)
    elif (method == "fleiss"):
        es = (a * d - b * c) * (r1 * c2 + r2 * c1) / (2 * r1 * r2 * c1 * c2)
    elif (method == "yule-q" or method == "cole-c4" or method == "pearson-q2"):
        es = (a * d - b * c) / (a * d + b * c)
    elif (method == "yule-y"):
        es = (sqrt(a * d) - sqrt(b * c)) / (sqrt(a * d) + sqrt(b * c))
    elif (method == "digby"):
        es = ((a * d)**(3 / 4) - (b * c)**(3 / 4)) / ((a * d)**(3 / 4) + (b * c)**(3 / 4))
    elif (method == "edward"):
        OdRat = a * d / (b * c)
        es = (OdRat**(pi / 4) - 1) / (OdRat**(pi / 4) + 1)
    elif (method == "tarwid"):
        es = (n * a - r1 * c1) / (n * a + r1 * c1)
    elif (method == "mcconnaughey"):
        es = (a**2 - b * c) / ((a + b) * (a + c))
    elif (method == "baroni-urbani-buser-2"):
        es = (sqrt(a * d) + a - b - c) / (sqrt(a * d) + a + b + c)
    elif (method == "kent-foster-1"):
        es = -b * c / (b * r1 + c * c1 + b * c)
    elif (method == "kent-foster-2"):
        es = -b * c / (b * r2 + c * c2 + b * c)
    elif (method == "tulloss"):
        mn = b
        mx = c
        if (c < b):
            mn = c
            mx = b

        f1 = (log(1 + (mn + a) / (mx + a)) / log(10)) / (log(2) / log(10))
        f2 = 1 / sqrt((log(2 + mn / (a + 1)) / log(10)) / (log(2) / log(10)))
        f3 = (log(1 + a / (a + b)) / log(10)) * (log(1 + a / (a + c)) / log(10)) / ((log(2) / log(10))**2)

        es = sqrt(f1 * f2 * f3)
    elif (method == "gilbert-wells"):
        es = log(a) - log(n) - log(r1 / n) - log(c1 / n)
    elif (method == "yule-r" or method == "pearson-q3" or method == "cole-c6" or method == "pearson-heron"):
        es = cos(pi * sqrt(b * c) / (sqrt(a * d) + sqrt(b * c)))
    elif (method == "anderberg"):
        S = max(a, b) + max(c, d) + max(a, c) + max(b, d)
        s2 = max(r1, r2) + max(c1, c2)
        es = (S - s2) / (2 * n)
    elif (method == "alroy"):
        na = a + b + c
        es = a * (na + sqrt(na)) / (a * (na + sqrt(na)) + 3 / 2 * b * c)
    elif (method == "pearson-q1"):
        #Pearson requires for Q1 that ad>bc, a>d, and c>b
        sw = 1
        Runs = 0
        while (Runs < 2):
            #swop rows
            if (a * d < (b * c) or a < d or c < b):
            #swop the rows
                at = a
                a = c
                c = at
                bt = b
                b = d
                d = bt
                sw = -sw

            #swop columns
            if (a * d < (b * c) or a < d or c < b):
            #swop the columns
                at = a
                a = b
                b = at
                ct = c
                c = d
                d = ct
                sw = -sw

            #swop rows again
            if (a * d < (b * c) or a < d or c < b):
            #swop the rows
                at = a
                a = c
                c = at
                bt = b
                b = d
                d = bt
                sw = -sw

            #pivot the table
            if (a * d < (b * c) or a < d or c < b):
                bt = b
                b = c
                c = bt
                sw = -sw

            Runs = Runs + 1

        es = sw * sin(pi / 2 * (c + d) * (a + c) / ((a + b) * (b + d)))

    elif (method == "gk-lambda-1"):
        S = max(a, b) + max(c, d) + max(a, c) + max(b, d)
        s2 = max(r1, r2) + max(c1, c2)
        es = (S - s2) / (2 * n - s2)

    elif (method == "gk-lambda-2"):
        m = min(a, d)
        es = (2 * m - b - c) / (2 * m + b + c)
    elif (method == "contingency"):
        chi = n * (a * d - b * c)**2 / (r1 * r2 * c1 * c2)
        es = sqrt(chi / (n + chi))
    elif (method == "cohen-w"):
        chi = n * (a * d - b * c)**2 / (r1 * r2 * c1 * c2)
        es = sqrt(chi / n)
    elif (method == "pearson"):
        Phi = abs(a * d - b * c) / sqrt(r1 * r2 * c1 * c2)
        es = sqrt(Phi / (n + Phi))
    elif (method == "hurlbert" or method == "cole-c8"):
        chi = n * (a * d - b * c)**2 / (r1 * r2 * c1 * c2)
        ahat = r1 * c1 / n
        gahat = floor(ahat)
        if (a * d >= b * c):
            chimax = n * r1 * c2 / r2 * c1
            gahat = ceil(ahat)
        elif (a * d < b * c and a <= d):
            chimax = n * r1 * c1 / r2 * c2
        else :
            chimax = n * r2 * c2 / r1 * c1

        chimin = n**3 * (ahat - gahat)**2 / (r1 * r2 * c1 * c2)
        es = (a * d - b * c) / abs(a * d - b * c) * sqrt((chi - chimin) / (chimax - chimin))
    elif (method == "stiles"):
        es = log((a + b + c + d) * (abs(a * d - b * c) - (a + b + c + d) / 2)**2 / ((a + b) * (a + c) * (b + d) * (c + d))) / log(10)
    elif (method == "bonett-price-1"):
        w = ((a + 0.1) * (d + 0.1)) / ((b + 0.1) * (c + 0.1))
        pmin = min(r1, r2, c1, c2) / n
        x = 1 / 2 - (1 / 2 - pmin)**2
        es = (w**x - 1) / (w**x + 1)

    elif (method == "odds-ratio"):
        es = a * d / (b * c)
    
    elif (method == "pearson-q4"):
        es = sin(pi / 2 * 1 / (1 + 2 * b * c * n / ((a * d - b * c) * (b + c))))
        
    elif (method == "pearson-q5"):
        k = 4 * a * b * c * d * n**2 / ((a * d - b * c)**2 * (a + d) * (b + c))
        es = sin(pi / 2 * 1 / ((1 + k)**0.5))
    
    elif method == "camp-1" or method == "camp-2" or method == "camp-3":
        
        #swop columns if first column total is less than second
    #"always orienting the table initially so that F1 > 0.5" (Camp, 1934, p. 309)
        sg = 1
        if a+c < b+d:
            at = a
            a = b
            b = at

            ct = c
            c = d
            d = ct

            sg = -1

        #The totals
        R1 = a + b
        R2 = c + d
        C1 = a + c
        C2 = b + d
        n = R1 + R2

        #step 2: determine three proportions
        p1 = a/C1
        p2 = d/C2
        p = C1/n

        #step 3: determine the corresponding z-values
        z1 = NormalDist().inv_cdf(p1)
        z2 = NormalDist().inv_cdf(p2)

        #step 4: determine the height of the normal distribution
        y = NormalDist().pdf(p)

        #step 5: calculate m
        m = p*(1-p)*(z1+z2)/y

        if method=="camp-1":
            phi=1

        elif method=="camp-2":
            thetaTable = {0.5:0.637, 0.6:0.63, 0.7:0.62, 0.8:0.6, 0.9:0.56}
            if p < 0.5:
                phi = thetaTable[trunc((1-p)*10)/10]
            else:
                phi = thetaTable[trunc(p*10)/10]

        else:
            #step 6: find phi in table of phi-values
            thetaTable = {0.5:[0.637, 0.636, 0.636, 0.635, 0.635, 0.634, 0.634, 0.633, 0.633, 0.632, 0.631], 
                           0.6:[0.631, 0.631, 0.63, 0.629, 0.628, 0.627, 0.626, 0.625, 0.624, 0.622, 0.621], 
                           0.7:[0.621, 0.62, 0.618, 0.616, 0.614, 0.612, 0.61, 0.608, 0.606, 0.603, 0.6], 
                           0.8:[0.6, 0.597, 0.594, 0.591, 0.587, 0.583, 0.579, 0.574, 0.569, 0.564, 0.559]}

            if p < 0.5:
                phi = thetaTable[trunc((1-p)*10)/10][int(round(p*100,0))-trunc((1-p)*10)*10]
            else:
                phi = thetaTable[trunc(p*10)/10][int(round(p*100,0))-trunc(p*10)*10]

        #step 7: calculate r
        es = sg * m/(1+phi*m**2)**0.5
        
    elif method=="becker-clogg-1" or method=="becker-clogg-2":
        pR1 = r1/n
        pR2 = r2/n
        pC1 = c1/n
        pC2 = c2/n

        tr = NormalDist().inv_cdf(pR1)
        tc = NormalDist().inv_cdf(pC1)

        mR1 = -exp(-tr**2/2) / pR1
        mR2 = exp(-tr**2/2) / pR2

        vC1 = -exp(-tc**2/2) / pC1
        vC2 = exp(-tc**2/2) / pC2

        delta = (mR1 - mR2)*(vC1 - vC2)

        OR = a*d/(b*c)

        if (method=="becker-clogg-2"):
            es = (OR**(13.3/delta) - 1) / (OR**(13.3/delta) + 1)
        elif (method=="becker-clogg-1"):
            phiBC = log(OR) / delta        
            g = exp(12.4*phiBC - 24.6*phiBC**3)    
            es = (g - 1)/(g + 1)
    
    elif(method=="bonett-price-2"):
        pMin = min(min(r1, r2),  min(c1, c2))/n
        cBP = (1 - abs(r1 - c1)/(5*n) - (0.5 - pMin)**2)/2
        omg = a*d/(b*c)
        es = cos(pi / (1 + omg**cBP))
    elif(method=="bonett-price-3"):
        pMin2 = (min(min(r1, r2),  min(c1, c2)) + 1)/(n+2)
        cBP2 = (1 - abs(r1 - c1)/(5*(n+2)) - (0.5 - pMin2)**2)/2
        omg2 = (a+0.5)*(d+0.5)/((b+0.5)*(c+0.5)) 
        es = cos(pi  / (1 + omg2**cBP2))
        
    elif(method == "chen-popovich"):
        p1 = r1 / n
        p2 = c1 / n
        z1 = NormalDist().inv_cdf(p1)
        z2 = NormalDist().inv_cdf(p2)
        es = (a * d - b * c) / (z1 * z2 * n**2)
    
    return (es)