Term
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Definition
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Description
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X
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–
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Predictor matrix for the true outcome.
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Z
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–
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Predictor matrix for the observed outcome, conditional on the true
outcome.
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Y
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Y∈{1,2}
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True binary outcome. Reference category is 2.
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yij
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I{Yi=j}
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Indicator for the true binary outcome.
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Y∗
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Y∗∈{1,2}
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Observed binary outcome. Reference category is 2.
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y∗ik
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I{Y∗i=k}
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Indicator for the observed binary outcome.
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True Outcome Mechanism
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logit{P(Y=j|X;β)}=βj0+βjXX
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Relationship between X and the true
outcome, Y.
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Observation Mechanism
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logit{P(Y∗=k|Y=j,Z;γ)}=γkj0+γkjZZ
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Relationship between Z and the
observed outcome, Y∗, given the
true outcome Y.
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πij
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P(Yi=j|X;β)=exp{βj0+βjXXi}1+exp{βj0+βjXXi}
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Response probability for individual i’s true outcome category.
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π∗ikj
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P(Y∗i=k|Yi=j,Z;γ)=exp{γkj0+γkjZZi}1+exp{γkj0+γkjZZi}
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Response probability for individual i’s observed outcome category,
conditional on the true outcome.
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π∗ik
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P(Y∗i=k|Yi,X,Z;γ)=∑2j=1π∗ikjπij
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Response probability for individual i’s observed outcome cateogry.
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π∗jj
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P(Y∗=j|Y=j,Z;γ)=∑Ni=1π∗ijj
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Average probability of correct classification for category j.
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Sensitivity
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P(Y∗=1|Y=1,Z;γ)=∑Ni=1π∗i11
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True positive rate. Average probability of observing outcome k=1, given the true outcome j=1.
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Specificity
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P(Y∗=2|Y=2,Z;γ)=∑Ni=1π∗i22
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True negative rate. Average probability of observing outcome k=2, given the true outcome j=2.
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βX
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–
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Association parameter of interest in the true outcome mechanism.
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γ11Z
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–
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Association parameter of interest in the observation mechanism, given
j=1.
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γ12Z
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–
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Association parameter of interest in the observation mechanism, given
j=2.
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