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Let’s refer to the case group’s data and the control group’s data collectively as
a database and call the data for an individual a record. We can think of the number
of cases, R, and the number of controls, S as fixed. Recall that we assume the control group’s data are known to the public.
Therefore, for a given genotype table, we assume that s₀, s₁, and s₂ are fixed. Then the ?² statistic can be written as a function of r₀ and r₁. How the value of the ?² statistic changes when we change one record in the database is illustrated in Figure
1. In Figure 1, each dot represent a value of the ?² statistic given r₀ and r₁. When we change one record in the case group, there are 6 possible changes to the
genotype table: (r₀ ? r₀ + 1, r₁ ? r₁), (r₀ ? r₀ + 1, r₁ ? r₁ ? 1), (r₀ ? r₀, r₁ ? r₁ ? 1), (r₀ ? r₀ ? 1, r₁ ? r₁), (r₀ ? r₀ ? 1, r₁ ? r₁ + 1), and (r₀ ? r₀, r₁ ? r₁ + 1); that is, r₀ and r₁ cannot increase or decrease by 1 simultaneously. The possible changes are shown as
arrows in Figure 1. A change in the genotype table results in a change in the allelic table, and we
get a new value for the ?² statistic based on the new allelic table.

Figure 1. Legal moves in the space of genotype tables with fixed R, S, s₀, s₁, and s₂.

Let p^* denote a pre-specified threshold p-value and let c denote the ?² statistic corresponding to p^?, the p-value of the ?² distribution with 1 degree of freedom. Then for a given SNP in the pool of candidate
SNPs, the genotype table of which we denote by D, the shortest Hamming distance is the smallest number of sequential changes made
to D such that the resulting genotype table, D’, satisfies Y_A(D’) ? c if Y_A(D) c and Y_A(D’) c if Y_A(D) ? c; that is, if we call c the significance threshold, then the goal is to make changes to the “insignificant”
(“significant”) table D so that the ?² statistic of D’ goes above (below) the significance threshold c, and D’ becomes a “significant” (“insignificant”) table. The Hamming distance score is defined
as h = (shortest Hamming distance) ? 1 if Y_A(D) ? c and h = ?(shortest Hamming distance) if Y_A(D) c.

Let’s consider the space of genotype tables, , defined by a genotype table D: for all , D’ shares the same values of s₀, s₁, s₂, R, S, and N with D, but D’ does not necessarily have the same values of r₀, r₁, and r₂ as D. Let ₁₀ = 2s₀ + s₁ denote the number of major alleles in the control group, and let x = 2r₀ + r₁ denote the number of major alleles in the control group, then we can write the ?² statistic of a genotype table as a function of x:

where r₀, r₁ and x are derived from D’, and ₁₀, R, S and N are the same for D and D^l. For notational convenience, when r₀ and r₁ are also derived from D, we will simply write the ?² statistic as Y_A(D).

Lemma 1 Y_A is an increasing function of × when xS ? n₁₀R 0, and it is a decreasing function of × when xS ? n₁₀R 0.

Proof. [see Additional file 1].

Additional file 1. Proofs

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To understand the implication of Lemma 1, let’s consider Figure 2. In Figure 2, each dashed line has slope = ?2, representing a value of x, which is defined as x = 2r₀ + r₁. Because we can consider each dot in Figure 2 to be a unique genotype table in the space of genotype tables with fixed control
data and a fixed number of cases, those tables that lie on the same dashed line will
have the same value of ?² statistic. Furthermore, because 0 ? r₀ + r₁ ? R, r₀ ? 0 and r₁ ? 0, the space of genotype tables, represented as dots, fall within a triangle in Figure
2.

Figure 2. An example of a genotype table, D, in the space of genotype tables with fixed R, S, s₀, s₁, and s₂. Each dot represent a genotype table. Each dashed line has slope = fl2, representing the lines x = 2r₀ + r₁. The red line is x = (2s₀ + s₁)R/S = 2r₀ + r₁, and the two black lines correspond to values of (2r₀ + r₁) such that Y_A(r₀, r₁; ) = c, where c is a pre-specified significance threshold value.

For the moment, let’s treat r₀ and r₁ as continuous values. In Figure 2, the red solid line represents the line 2r₀ + r₁ = x = ₁₀R/S and the two solid black lines represent lines 2r₀ + r₁ = x such that . There are two black lines because by Lemma 1 Y_A(x) is an increasing function when x n₁₀R/S and it is a decreasing function when x n₁₀R/S; that is, there could be two values of x, say x₁ and x₂, such that and x_{1 10}R/S x₂. Because it is possible that

there could be genotype tables for which only one black line exists or no black line
exists at all; in such cases, we will use the lines 0 = 2r₀ + r₁ or 2R = 2r₀ + r₁ wherever appropriate.

In Figure 2, the genotype table D is insignificant and its ?² statistic is below the threhold value. By Lemma 1, we know that the ?² statistics of genotype tables, as represented by the dots on Figure 2, are greater than c when they are in the shaded area, outside of the area between the two black lines
and they are smaller than Y_A(D^?) when they are inside the area between the two black lines. Therefore, finding the
Hamming distance score for D is to find the shortest Hamming distance from the genotype table D to genotype tables in the shaded areas.

For genotype tables that are significant, they will fall into the shaded areas in
Figure 2. Then finding the Hamming distance score for a significant genotype table is to find
the shortest Hamming distance from the genotype table in one of the shaded areas to
genotype tables in the non-shaded area.

Proposition 2 Given a significance threshold value c and an insignificant genotype table D (i.e.,
Y_A(D) c), if there exists such that , then the shortest Hamming distance is min{H₁, H₂}, where H₁ and H₂ are defined as follows:

(i) H₁ is the number of changes made to D in the following manner: (1) keep decreasing r₀ until the new genotype table, D’, becomes significant (i.e.,); (2) when r₀ is minimized but the new table is still insignificant, keep decreasing r₁ until the new table becomes significant.

(ii) H₂ is the number of changes made to D in the following manner: (1) keep increasing r₀ until the new genotype table becomes significant; (2) if r₀ can no longer be increased without decreasing r₁ and the new table is still insignificant, increase r₀ and decrease r₁ in each change until the new table becomes significant.

If for all , , then we define the shortest Hamming distance as min, where and are defined as follows:

(i) When r₀ and r₁ are both minimized but the new table is still insignificant, set to 1 + d₁, where d₁ is smallest the number of changes needed to minimize r₀ and r₁.

(ii) When r₀ and r₁ are both maximized but the new table is still insignificant, set to 1 + d₂, where d₂ is smallest the number of changes needed to maximize r₀ and r₁.

Proof. [see Additional file 1].

Proposition 3 Given a significance threshold value c and a significant genotype table D (that is,
Y_A(D) ? c), the shortest Hamming distance is min{H₁, H₂}, where H₁ and H₂ are defined as follows:

(i) If 2r₀ + r₁ (2s₀ + s₁)R/S, set H₁ = ?; otherwise, H₁ is the number of changes made to D in the following manner: keep decreasing r₀ until the new genotype table, D’, becomes insignificant (i.e., Y_A(D’, D) c).

(ii) If 2r₀ + r₁ (2s₀ + s₁)R/S, set H₂ = ?; otherwise, H₂ is the number of changes made to D in the following manner: keep decreasing r₀ until the new genotype table becomes insignificant.

Proof The proof is similar to that of Proposition 2.

Definition 6 (The Hamming distance score) Given a threshold ?² statistic value c and a genotye table D, the Hamming distance score of D is

where d^? is found using Proposition 2 and d⁺ is found using Proposition 3.

Corollary 4 The sensitivity of the Hamming distance score as defined in Definition 6 is 1.