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COMBO Notation Guide

Kim Hochstedler

Notation

This guide is designed to summarize key notation and quantities used the COMBO R Package and associated publications.
Term Definition Description
X Predictor matrix for the true outcome.
Z Predictor matrix for the observed outcome, conditional on the true outcome.
Y Y{1,2} True binary outcome. Reference category is 2.
yij I{Yi=j} Indicator for the true binary outcome.
Y Y{1,2} Observed binary outcome. Reference category is 2.
yik I{Yi=k} Indicator for the observed binary outcome.
True Outcome Mechanism logit{P(Y=j|X;β)}=βj0+βjXX Relationship between X and the true outcome, Y.
Observation Mechanism logit{P(Y=k|Y=j,Z;γ)}=γkj0+γkjZZ Relationship between Z and the observed outcome, Y, given the true outcome Y.
πij P(Yi=j|X;β)=exp{βj0+βjXXi}1+exp{βj0+βjXXi} Response probability for individual i’s true outcome category.
πikj P(Yi=k|Yi=j,Z;γ)=exp{γkj0+γkjZZi}1+exp{γkj0+γkjZZi} Response probability for individual i’s observed outcome category, conditional on the true outcome.
πik P(Yi=k|Yi,X,Z;γ)=2j=1πikjπij Response probability for individual i’s observed outcome cateogry.
πjj P(Y=j|Y=j,Z;γ)=Ni=1πijj Average probability of correct classification for category j.
Sensitivity P(Y=1|Y=1,Z;γ)=Ni=1πi11 True positive rate. Average probability of observing outcome k=1, given the true outcome j=1.
Specificity P(Y=2|Y=2,Z;γ)=Ni=1πi22 True negative rate. Average probability of observing outcome k=2, given the true outcome j=2.
βX Association parameter of interest in the true outcome mechanism.
γ11Z Association parameter of interest in the observation mechanism, given j=1.
γ12Z Association parameter of interest in the observation mechanism, given j=2.