2×2 tables of binary markers with random margins can be considered as tetranomial distributions with a symmetry structure. Symmetry of 2×2 tables can be described by the dihedral group D₄ generated by the transposition of the binary markers (matrix transposition) and transposition of their values (transposition of columns or rows).

We consider the manifold of all non-degenerate tetranominal probability models which we write in two by two lay-out: consists of all two by two matrices t with entries p_ij , (i, j {0, 1}) subject to the constraints p_ij > 0, Σ_i,jp_ij = 1. The p_ij denote the probabilities of the corresponding combination of the states of two binary markers i and j. In the following, we abbreviate Σ¹_{i = 0} Σ¹_{j = 0} = Σ_i,j , p_i. = p_io + p_i1 and p._j = p_oj + p_1j . The margins p_i. and p._j give the marginal distributions of the marker i and j respectively.

In we have several relevant submanifolds. There is a marked point m₀ , namely the midpoint . There is the 1-dimensional submanifold of all tables with diagonal symmetry of the form . And there is the 2-dimensional submanifold of independent tables with p_ij = p_i. • p_.j i, j.

By we denote the closure of . The border ∂ = ‒ consists of tables with at least one zero: four two dimensional sides {p_ij = 0} for any i, j, six one dimensional edges of vanishing rows {p._j = 0}, vanishing columns {p_i = 0} and two vanishing diagonals {p₀₀ = p₁₁ = 0}, {p₀₁ = p₁₀ = 0} as well as four triple zero vertices {p_ij = 1}.

Manipulating the margins defines an additional structure on . We can multiply rows or columns with positive numbers and renormalise: Formally, consider the group G = ( ⁺ × ⁺, •) with component-wise multiplication.

For every (µ, ν) ⁺ × ⁺ we define a map: g(µ, ν) : →

(1)

Since g(µ, ν) ◦ g(µ', ν') = g(µ · µ', ν · ν') and g(1, 1) = Id this defines a G-group action on .

Lying in the same group orbit defines an equivalence relation on : We say two elements t₁, t₂ are equivalentt₁ ~ t₂ if and only if there are (µ, ν) ⁺ × ⁺ with g(µ, ν)(t₁) = t₂. G-Orbits are diffeomorph to ⁺ × ⁺.

A real function η : → is G-invariant if η(t) = η(g(µ, ν)(t)) for all (µ, ν) ⁺ × ⁺.

Proposition 1 (odds-ratio):

a) The odds-ratio is G-invariant.

b) The odds-ratio classifies the G-orbits. Let be the quotient space of by the equivalence relation induced by G. λ induces a bijective map λ̃ : → ⁺.

c) The inverse mapping λ̃^-1 : ⁺ → can be described by

d) Every G-invariant function η : → can be written as a function of λ, namely η = (η̃ ◦ λ̃^-1) ◦ λ.

Proof: a) is easily verified. b) Every equivalence class [t] in has a representant with margins ½ , namely which has the form given in c). d) is trivial.

We next define new coordinates on to make use of this insight.

Proposition 2 (Margin transformation coordinates on T highlighting the G-action and its invariant):

The map Θ : → ³

(2)

is a diffeomorphism.

The inverse Ψ = Θ^‒1 : ³ → is given by

In these new coordinates, x corresponds to the logarithmized odds-ratio [2], while y and z determine the G-transformation that maps the table to diagonal symmetry. In addition, the midpoint m₀ corresponds to the origin (0, 0, 0). G-orbits (odds-ratio = constant) correspond to planes {a} × ². In particular, the submanifold of independent tables maps to {0} × ². The tables with diagonal symmetry form the line R×{0}×{0}. Transposing rows and columns of a table is equivalent to transformations y → ‒y and z → ‒z, while matrix transposition is equivalent to the transformation y ↔ z.

Let := {‒∞, +∞} be the two point compactification of . is a compactification of ³ as a cube. We use a short hand notation to describe the boundaries abbreviating +∞ as ”+”, ‒∞ as ”‒” and any finite real number as ”*”. The eight vertices V = {(± ± ±)} split into two sets of four: V_g = {(+ + +), (+ ‒ ‒), (‒ + ‒), (‒ ‒ +)} and V_b = {(‒ ‒ ‒), (‒ + +), (+ ‒ +), (+ + ‒)}.

Proposition 3 (Extension to the borders): Ψ and Θ considered as set valued functions can be extended to respectively . They remain inverse to each other. The mappings of the borders can be characterized as follows:

• The vertices V_g together with their respective adjacent edges map to the vertices in .
• The faces of correspond to the vertices V_b.
• The faces (± * *) of the cube map to the diagonal edges p₀₀ = p₁₁ = 0 and p₀₁ = p₁₀ = 0 in .
• The faces (* ± *), (* * ±) correspond to tables with vanishing rows {p._j = 0} or vanishing columns {p_i. = 0} in respectively.

This behaviour is illustrated in Fig. (1). These different compactifications will later be used to characterise the limit behaviour of association measures. It will turn out that the limit behaviour can be easier described using the margin transformation coordinates.

We will now investigate various measures of associations between two binary markers. First we define the objects of interest.

Definition (Measures of association): A measure of association between binary markers is a continuous function η : → with the following properties:

a) η is zero on independent tables.

b) η is a strictly increasing function of the odds-ratio when restricted to tables with fixed margins.

c) η respects the symmetry group D₄, namely:

 c1) η is symmetric in the markers, i.e. invariant to matrix transposition.

 c2) η changes sign when states of a marker are transposed (row or column transposition).

d) The range of the function is restricted to (−1, 1).

The first two conditions are equivalent to basic properties proposed by Piatetsky-Shapiro [3]. Condition c) is added to acknowledge that associations between interchangeable markers are of interest here. Finally, condition d) is added to define a unique scale for all measures of association which is often referred as standardization.

Fig. (1). Illustration of the maps θ and Ψ on the boundaries of and : ”~” represents positive numbers adding up to 1.

The odds-ratio Odds-ratio λ:

can be used to define measures of association. As λ is G-invariant, monotone transformations automatically fulfill condition b) of the definition.

Measures of association derived from the odds-ratio include Yule’s Q [4]:

and Yule’s Y [4]:

Obviously, both Q and Y are measures of association in our sense. Similar to the odds-ratio, both are extremal if one of the p_ij tends to zero.

Fixing margins results is a one dimensional submanifold of tables that can be additively parametrised by a parameter D.

All such tables have the form:

D = p₀₀p₁₁ ‒ p₀₁p₁₀ = p₀₀ ‒ p₀ .p.₀ describes the additive deviation from the independent table with the given margins. This measure is zero in case of independence of the markers but extremal values depend on the margins.

Lewontin’s D^′ [5]: The measure D^′ is a standardisation of the original measure D:

Lewontin’s D^′ ranges from −1 to 1 and tends to these values if at least one of the p_ij tends to zero.

D^′ is widely used in genetics to measure linkage disequilibrium. When a new SNP emerges in a population by a single mutation event, the new allele is exclusively found in conjunction with only one of the two alleles of already existing SNPs. As long as no recombination events occurs, the new SNP remains in complete linkage disequilibrium with the other SNPs. The corresponding 2×2 tables feature a single zero cell. Thus in this context a measure is needed that is extremal whenever a single entry tends to zero.

Since D_max is constant for tables with fixed margins and D increases with increasing odds-ratio, D^′ is a monotone function of the odds-ratio for constant margins. Symmetry is obvious. Hence, D^′ is a measure of association in our sense.

Correlation coefficient r [6]: The correlation coefficient applied to binary data has similar popularity in genetics as D^′. It ranges also from −1 to 1, but, in contrast to D^′, the absolute value 1 is obtained when a diagonal of t tends to zero:

With reasoning similar as for D^′, r is a measure of association.

Proposition 4 (Equality of r, D^′ and Y on diagonal tables): The measures r, D^′ and Y coincide on the set of diagonal tables, i.e. tables with pair-wise equal diagonal elements.

Proof: This follows directly after calculating these measures for the tables .

The mutual information [7] is defined as the difference between the information of the given table and the independent table with the same margins.

MutInf takes values only in (0, 1). In order to make it a measure of association according to our definition, we define a signed version:

Proposition 5: sMutInf is a measure of association.

Proof: The symmetry of this measure is clear. To show that sMutInf is a monotone function of the odds-ratio, we consider the tables for a sufficiently small ε > 0. These tables have the same margins as the table but higher odds-ratios. Assume that λ > 1, we see that sMutInf (t_ε ) = log₂ λ > 0. Hence sMutInf is monotone, and thus, a measure of association.

MutInf approaches 1 only if tables approach ; while r approaches 1 if tables approach the form .

Kappa coefficient [8]: The Kappa coefficient which is useful in quantifying the agreement between two raters is defined as:

Kappa is not a measure of association in our sense. Although it fulfils the condition of monotonicity, it is not symmetric.

We use the coordinates introduced in Proposition 2 in order to describe and visualise how measures of association depend on the margins. In particular we study measures of association η restricted to x=const i.e. for fixed odds-ratios. The restricted functions will be denoted η_x and called margin weighting functions. We characterise the shape of the margin weighting functions and study their limiting behaviours and extensibility to the compactification in comparison to .

The association measure r expressed in margin transformation coordinates reads:

(3)

The margin weighting function of r for odds-ratio λ = 40 is shown in Fig. (2).

Fig. (2). Margin-weighting function of r.

Proposition 6 (Margin weighting function for r): For all x \ {0}:

a) r_x has exactly one extremum at the origin (y, z) = (0, 0), corresponding to the diagonal symmetric table with the fixed odds-ratio.

b) lim_{‖(y,z)‖→∞}r_x = 0.

c) lim_x→±∞r_x = ±1

d) r can be extended to except for the lines (±,±, *) and (±, *,±) and the vertices V.

e) r can be extended to except for the vertices.

Proof: see Supplement Material.

The measure r down-weights tables with skewed margins.

The association measure D' expressed in margin transformation coordinates reads:

(4)

The margin weighting function of D^′ for odds-ratio λ = 40 is shown in Fig. (3).

Fig. (3). Margin-weighting function of D′.

Proposition 7 (Margin weighting function for D^′): For all x {0}:

a) D_x^′ has a non-differentiable edge along the diagonal y = zfor D^′ > 0 and along the diagonal y = −zfor D^′ < 0. There is a non-smooth saddle point in the origin.

b)

Thus, limit functions have a range of (0, 1 ‒ e^‒2x) for x > 0 and (e^2x ‒ 1, 0) for x < 0, where 0 is obtained for y → ±∞, z → ±∞, x > 0 and y → ∞, z → ±∞, x < 0.

c) lim_x→±∞D_x′ = ±1

d) D′ can be extended to except for the vertices V_g.

e) D′ can be extended to except for the edges and vertices.

Proof: see Supplement Material.

D^′ gives higher weights to certain tables without diagonal symmetry. The measure up-weights or down-weights tables with skewed margins depending on the position of zeros which occur in the limiting tables (see Fig. 3). Comparing d) and e) one recognizes that the introduction of the odds-ratio as coordinate allows extending D^′ to limit tables with vanishing colums or rows.

The association measure sMutInf can also be written in margin transformation coordinates but this is skipped due to the lengthy formula. The margin weighting function of sMutInf for odds-ratio λ = 40 is shown in Fig. (4).

Fig. (4). Margin-weighting function of sMutInf.

Proposition 8 (Margin weighting function for sMutInf): For all x \ {0}:

a) sMutInf_xhas exactly one maximum at the origin (y, z) = (0, 0).

b) lim_{‖(y,z)‖→∞} sMutInf_x = 0.

c) lim_x→±∞ sMutInf_x =

Thus sMutInf_x → ±1 for y = zand x→ ±∞ respectively.

d) sMutInf can be extended to except for the vertices V_b.

e) sMutInf can be extended completely to .

Proof: see Supplement Material.

Thus, similarly to r, sMutInf down-weights tables with skewed margins (see Fig. 4).

The association measure Y in margin transformation coordinates can be simply written as:

(5)

Proposition 9 (Margin weighting function for Y ): For all x :

a) Y_x is constant.

b) lim_{‖(y,z)‖→∞}Y_x = tanh

c) lim_x→±∞Y_x = ±1

d) Y can be extended completely to .

e) Y can be extended to except for edges and vertices corresponding to vanishing rows or columns.

Proof: is trivial.

Among tables of a fixed odds-ratio, we look for a principled approach to prefer interesting tables and down-weight obscure ”junk” tables. As a candidate we study the table entropy on . The entropy function H : → is defined as the negative expectation of the loglikelihood of the tables:

Why is entropy a candidate to select among tables? It can be characterised in multiple ways: For general finite discrete distributions the entropy was introduced by Shannon (1948) [9]. Shannon characterised H by a set of postulates to measure the uncertainty in a discrete distribution:

Shannon’s Characterisation of Entropy: If functions H_n(p₁, ..., p_n) with p_i ≥ 0, ∑p_i = 1, n ≥ 2 satisfy the conditions

a) H₂(p, 1 − p) is a continuous positive function of p.

b) H_n(p₁, ..., p_n) is symmetric, i.e. invariant under permutations of the p₁, ..., p_n for all n.

c) H_n(p₁, ..., p_n) = H_{n‒ 1} (p₁ + p₂, p₃, ..., p_n) + (p₁ + p₂)·

then H_n(p₁, ..., p_n) = ‒ K ·Σ p_i log₂ (p_i) for some K > 0.

Tables with high entropy are interesting as they have high uncertainty and ”surprise value”.

Jaynes [10] gives an independent combinatorial characterisation: When we sample sequentially from a table t we obtain a vector of observations of length N , which we summarise as a frequency table . Each frequency table is characterised by the number of sequences which realise . Intuitively, tables that can be realised in multiple ways are more plausible than those that can be realised only by few sequences. We can use Stirlings formula for n! to approximate W ( ). In the limit N → ∞, → t in probability and 1/N · log(W(t^N)) → H(t). Thus the entropy describes the combinatorial plausibility of a table.

Given a set of distributions fulfilling certain constraints, Jaynes [10] proposes to pick the corresponding maximum entropy distribution as the most uncommitted and prototypical distribution. Looking at the margin weighting function of the entropy leads to a surprise (see Fig. 5):

Recall that Lambert’s W-function is defined as the inverse function to x exp x. W is a multi-branch function since y = x exp(x) has two solutions for y (−1/e, 0). We can prove the following:

Fig. (5). Margin-weighting function of entropy: The margin-weighting function of the entropy is shown, conditioned to the odds-ratios λ = 5 which results in a single maximum, λ = 12.896 at which the maximum splits into two and λ = 40 with two maxima and a saddle point.

Theorem 1 (magic odds-ratio): Define the ”magic odds-ratio” by L_magic = W (1/e)⁻² ≈ 12.89. Let L > 1. The entropy H restricted to the submanifold of constant odds-ratio L in

• has a single maximum at the diagonal table of odds-ratio L if 1 < L ≤ L_magic.
• has a saddle point at the diagonal table of odds-ratio L and two ”L-shaped” tables as maxima which transpose with matrix transposition if L_magic < L.

”L-shaped” means that for L → ∞ one of the maxima approaches the table . For the case L < 1 a similar result can be derived by transposing principal and secondary diagonals.

Proof: There are two constraints to be considered, one of them not linear in p_ij :

(6)

(7)

Using Langrange multipliers, critical tables of H restricted to odds-ratio equals L can be expressed in terms of Lambert’s W function. The bifurcation occurs for L_magic < L because Lambert’s W is multibranched. See supplement material for details.

This theorem suggests that the ”magic odds-ratio” is a natural cutpoint between weak and strong association. For weak association L < L_magic, interesting tables are those near . For strong association L_magic < L, particularly interesting tables are those that approach ”L-shape”, i.e. those in which one cell differs in magnitude from the three others.

Using these insights on the entropy of a table, in this section we aim to define a measure of association with similar properties to D^′, Y but better limit behaviour, i.e. the measure should down-weight tables with almost vanishing rows or columns or single entries. These tables are denoted as junk tables in the following. We have seen in the last sections that D^′ and Y could be large for these tables.

We also like to recall that both, D^′ and Y become extremal if the table features a single entry equals zero while r, sMutInf require a vanishing diagonal. We like to retain this property for a new measure to be defined. Another feature to be retained is the agreement of measures for diagonal tables which holds for Y , D^′ and r.

According to our definition, an important property of a measure of association is that it is a monotone function of the odds-ratio when the margins are kept fixed. For the entropy, one can prove the following lemma:

Lemma 1 (Monotony of the entropy difference): Let H be the entropy of t and H_diag be the entropy of the corresponding diagonal table of the same odds-ratio λ. Then, H_diag − H is monotonically decreasing for increasing λ > 1 and constant margins.

Proof: see supplement material.

As a direct consequence of this lemma, it is easy to see that:

Corollary:

(8)

is a measure of association for arbitrary n ≥ 0.

This newly defined measure fulfils all above mentioned properties: It coincides with Y , D^′, r at diagonal tables, is extremal for tables with a single zero, up-weights L-shaped tables for large odds-ratios in the sense that HS_n > Y and down-weights junk-tables in the sense that HS_n < Y at the margins (proof see below). However, the down-weighting is imperfect as HS_n > 0 for junk-tables.

The parameter n can be chosen in order to define the degree of up- and down-weighting. According to our observations, n = 4 is a reasonable choice resulting in a satisfactory down-weighting of junk tables (see later).

The measure HS_n can be written in margin transformation coordinates using

At Fig. (6) we present the margin weighting functions of HS_n for λ = 5 and λ = 40. These functions can be easily characterised using the results of the previous section:

Proposition 10 (Margin weighting function for HS_n): For all x \ {0}:

a) For x (−1 − W(1/e) , 1 +W (1/e)), HS_nxhas exactly one maximum at the origin (y, z) = (0, 0). If x < −1 − W (1/e) or x > 1 + W (1/e), HSn_x has a saddle-point at the origin and two extrema elsewhere. At these extrema, the elements of one diagonal are equal while at the other diagonal there is one (small) element.

b) HS_nx has the following limit functions

where p = (1 + e^x±z )⁻¹ for y → ±∞ or p = (1 + e^x±y)⁻¹ for z → ±∞ respectively. Thus, the limit functions have an extremum at p = 0.5 that is z = x for y → ±∞ and y = x for z → ±∞ respectively.

c) lim_x→±∞ HS_nx = ±1

d) HS_nx < Y_x at the margins, i.e. HS_n down-weights junk-tables.

e) HS_n can be extended completely to .

f) HS_n can be extended to except for edges and vertices corresponding to vanishing rows or columns.

g) For all x , HS_n coincides with Y , D^′, r at diagonal tables.

Proof: a) follows from Theorem 1. b) is easy to see taking the limit of the tables first. c) is clear since lim_x→±∞ tanh ^x/₂ = ±1 and the exponent is finite. d) holds since H_diag > 1 and H ≤ 1 at the margins of finite x. e) and f) are consequences of b) and c). g) is obvious.

Fig. (6). Margin-weighting function of HS4.

We now study the behaviour of the measures Y , r, D^′ and the newly proposed measure HS₄ for a variety of selected tables (see Table 1). For this purpose, we study the odds-ratios λ {1, 2, 5, 10, 20, 50, 100} and consider the following tables for x = ln

• The diagonal table (y = z = 0).
• An L-shaped table, characterized by y = x, z = ‒x.
• A junk table with y = 10, z = −y corresponding to p₀₁ ≈ 1.
• A junk table with y = 10, z = −x corresponding to p₀₀ ≈ p₀₁ ≈ 0.5.
• A junk table with y = 10, z = y corresponding to p₀₀ ≈ 1.

We also like to remark that the table with three equal entries has maximum entropy if λ → ∞.

Per definition of a measure, for λ = 1 all measures equals zero independent of the concrete realization of the table. Since Y is based on the odds-ratio, Y is constant for all tables of the same odds-ratio. Y , r, D^′ and HS₄ always coincide at diagonal tables. r is maximal at diagonal tables and becomes small for all kinds of junk tables. D^′ is always greater for L-shaped tables than for diagonal tables. D^′ is close to zero in case of p₀₀ ≈ 1 but could become large for p₀₁ ≈ 1 which is highly counter-intuitive. HS₄ also becomes larger for L-shaped tables compared to diagonal tables if λ is large. In contrast to D^′, HS₄ is close to zero for both junk configurations p₀₀ ≈ 1 and p₀₁ ≈ 1 respectively. The limit tables have a maximum of the entropy at p₀₀ = p₀₁ = 0.5. This induces a maximum of HS₄ for limit tables which increases with λ (see Table 1, fourth rows of each odds-ratio).