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Fig. 1 | BMC Neurology

Fig. 1

From: The nature of genetic susceptibility to multiple sclerosis: constraining the possibilities

Fig. 1

Plots of the distribution of MS-probabilities in the (W) set − i.e., from the combined set (G T ) − are shown in two hypothetical circumstances, chosen to match the conditions derived from the Model. Because, as noted in the text, {PMS|G) > 12 * P(MS)}, the minimum asymmetry of the distribution is far too great to be explained by a log-normal distribution (see Additional file 1). For illustrative purposes only, the distriburtion of the MS-probabilities in both of the (G T ) subsets are plotted as normal. However, because of the large difference in sub-subset means (see text and below), this arbitrary choice makes no difference to the final conclusions [29]. The means (μ) and standard deviations (σ) for the two distributions have been chosen to fit with the following four derived or defined relationships (see text and Additional file 1): 1. E(X) = x = P(MS|G) ≥ 0.059; 2. 0 < y = P(MS|G min) < P(MS) ≤ 0.005; 3. P(MS) = P(G)x + P(G min)y and: 4. σ 2 X  = E(xi − x)2 ≤ 0.0013 ;  σ X  ≤ 0.036. Thus, for the distribution surrounding P(MS|G), these values have been taken to be (μ1 = 0.055 ; σ1 = 0.036) and for the distribution surrounding P(MS|G min), they have been taken to be (μ2 = 0.0025 ; σ2 = 0.001). The total area underneath the entire (W) distribution (depicted) is equal to P(G T ). The areas underneath each sub-distribution {i.e., splitting the two distributions at P(MS)}, are equal to P(G) and P(G min), respectively. For the first distribution, (μ 1) is slightly less than {x = P(MS|G)} because a small portion of the left-hand tail of this distribution belongs to the (G min) subset. The first plot (panel A), considers the distribution of (W) in a circumstance where the distribution of MS probabilities in the combined (G T -) set is unlikely to be bimodal {i.e., where P(G-) = 0}. In this circumstance, the area underneath (and, thus, the height of) the curve representing the (X) distribution (i.e., for the “genetically susceptible” subset) depends upon the value of P(MS,G min). Thus: 5. when: P(MS, Gmin) ≈ 0; then: P(G) ≈ P(MS)/x; 6. when: P(MS, Gmin) ≈ 0.5 * P(MS); then: P(G) ≈ 0.5 * P(MS)/x; 7. when: P(MS, Gmin) ≈ P(MS); then: P(G) ≈ 0. However, because {P(MS, Gmin) ≤ 0.56 * P(MS)}, circumstance #7 is impossible (see Additional file 1). Panel A represents circumstance #6. By contrast, the second plot (panel B), considers the distribution of (W) in a circumstance where the distribution of (Z) is definitely bimodal. Specifically this figure considers the circumstsnce, in which: P(G−) > 0.83; and: P(Gmin) ≤ P(G). In the particular case illustrated − i.e., where P(G min) ≈ P(G) ≈ 0.0425 − the distribution is actually trimodal although the zero probability of MS for the subset (G-) subset is not depicted in the graph, despite the fact that this subset constitutes the large majority of the population. If the distribution for the (G min) subset were assumed to have a uniform distribution, on the plot in Panel B, the plateau of the (G min) distribution would be above the population frequency of (0.0008) and, thus, would still be clearly bimodal. This same pattern persists regradless of the actual value chosen for P(G) ≈ P(Gmin). Both panels A and B demonstrate the severly bimodal character for the (W) distribution that results under any circumstance. Indeed, such severe bimodality will exist, regardless of the acutal shape of the distribution of MS probabilities for members of each of the two subsets of (G T ). This is both because of the extreme separation of the subset means and because of the very restricted range for the MS probabilities within the (G min) subset [29]

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