PMC:4157149 / 9786-11469 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/4157149","sourcedb":"PMC","sourceid":"4157149","source_url":"https://www.ncbi.nlm.nih.gov/pmc/4157149","text":"The conditional distribution Pr(D|M,X) accounts for both the LD between marker and causal variant and the dependence between exposure and causal variant. Equations 1 and 2 allow the marker interaction term to be nonzero even when the interaction term for the causal variant is zero. Some conditions under which the marker interaction term is in fact zero are given in the following lemma.Lemma\nIf βDX∗=0, then βMX = 0 if any of the following conditions hold\n(1) there is no main effect of the causal variant on the outcome, βD∗=0\n(2) the marker is perfectly correlated with the causal variant, D = M\n(3) the causal variant is independent of the marker, conditional on the exposure, Pr(D|M,X) = Pr(D|X)\nFurthermore, under linear (h(x) = x) or log-linear (h(x) = log(x)) regression, βMX = 0 if\n(4) the causal variant is independent of the exposure, conditional on the marker, Pr(D|M,X) = Pr(D|M)\nProof\nIf βDX∗=0 then the terms in βMX are explicitly\nh−1(η1,1)=∑dh−1(β0∗+βD∗d+βX∗)Pr(d|M=1,X=1)\nh−1(η1,0)=∑dh−1(β0∗+βD∗d)Pr(d|M=1,X=0)\nh−1(η0,1)=∑dh−1(β0∗+βD∗d+βX∗)Pr(d|M=0,X=1)\nh−1(η0,0)=∑dh−1(β0∗+βD∗d)Pr(d|M=0,X=0)\nIf βD∗=0 then\nh−1(η1,1)=∑dh−1(β0∗+βX∗)Pr(d|M=1,X=1)=h−1(β0∗+βX∗)η1,1=β0∗+βX∗\nSimilarly, η0,1=β0∗+βX∗ and η1,0=η0,0=β0∗, so βMX = 0 proving (1).\nIf marker and causal variant are perfectly correlated, then trivially βMX=βDX∗=0, which proves (2).\nIf Pr(D|M,X) = Pr(D|X), then η1,1 = η0,1 and η1,0 = η0,0, which proves (3).\nFinally, if Pr(D|M,X) = Pr(D|M) and either h(x) = x or h(x) = log(x), then\nη1,1=β0∗+βX∗+∑dh−1(βD∗d)Pr(d|M=1)η1,0=β0∗+∑dh−1(βD∗d)Pr(d|M=1)η1,1−η1,0=βX∗\nSimilarly, η0,1−η0,0=βX∗ under either link function, so βMX = 0 as required, which proves (4). Q.E.D.","divisions":[{"label":"p","span":{"begin":388,"end":393}},{"label":"p","span":{"begin":394,"end":457}},{"label":"label","span":{"begin":458,"end":461}},{"label":"p","span":{"begin":462,"end":529}},{"label":"label","span":{"begin":530,"end":533}},{"label":"p","span":{"begin":534,"end":599}},{"label":"label","span":{"begin":600,"end":603}},{"label":"p","span":{"begin":604,"end":701}},{"label":"p","span":{"begin":702,"end":791}},{"label":"label","span":{"begin":792,"end":795}},{"label":"p","span":{"begin":796,"end":893}},{"label":"p","span":{"begin":894,"end":899}},{"label":"p","span":{"begin":900,"end":946}},{"label":"p","span":{"begin":947,"end":989}},{"label":"p","span":{"begin":990,"end":1028}},{"label":"p","span":{"begin":1029,"end":1071}},{"label":"p","span":{"begin":1072,"end":1110}},{"label":"p","span":{"begin":1111,"end":1124}},{"label":"p","span":{"begin":1125,"end":1187}},{"label":"p","span":{"begin":1188,"end":1254}},{"label":"p","span":{"begin":1255,"end":1354}},{"label":"p","span":{"begin":1355,"end":1430}},{"label":"p","span":{"begin":1431,"end":1505}},{"label":"p","span":{"begin":1506,"end":1581}}],"tracks":[]}