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COMBO Notation Guide

Kim Hochstedler

COMBO_notation_guide.Rmd

Notation

This guide is designed to summarize key notation and quantities used the COMBO R Package and associated publications.
Term Definition Description
XX – Predictor matrix for the true outcome.
ZZ – Predictor matrix for the observed outcome, conditional on the true outcome.
YY Y∈{1,2}Y \in \{1, 2\} True binary outcome. Reference category is 2.
yijy_{ij} 𝕀{Yi=j}\mathbb{I}\{Y_i = j\} Indicator for the true binary outcome.
Y*Y^* Y*∈{1,2}Y^* \in \{1, 2\} Observed binary outcome. Reference category is 2.
yik*y^*_{ik} 𝕀{Yi*=k}\mathbb{I}\{Y^*_i = k\} Indicator for the observed binary outcome.
True Outcome Mechanism logit{P(Y=j|X;β)}=βj0+βjXX\text{logit} \{ P(Y = j | X ; \beta) \} = \beta_{j0} + \beta_{jX} X Relationship between XX and the true outcome, YY.
Observation Mechanism logit{P(Y*=k|Y=j,Z;γ)}=γkj0+γkjZZ\text{logit}\{ P(Y^* = k | Y = j, Z ; \gamma) \} = \gamma_{kj0} + \gamma_{kjZ} Z Relationship between ZZ and the observed outcome, Y*Y^*, given the true outcome YY.
πij\pi_{ij} P(Yi=j|X;β)=exp{βj0+βjXXi}1+exp{βj0+βjXXi}P(Y_i = j | X ; \beta) = \frac{\text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}{1 + \text{exp}\{\beta_{j0} + \beta_{jX} X_i\}} Response probability for individual ii’s true outcome category.
πikj*\pi^*_{ikj} P(Yi*=k|Yi=j,Z;γ)=exp{γkj0+γkjZZi}1+exp{γkj0+γkjZZi}P(Y^*_i = k | Y_i = j, Z ; \gamma) = \frac{\text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}{1 + \text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}} Response probability for individual ii’s observed outcome category, conditional on the true outcome.
πik*\pi^*_{ik} P(Yi*=k|Yi,X,Z;γ)=∑j=12πikj*πijP(Y^*_i = k | Y_i, X, Z ; \gamma) = \sum_{j = 1}^2 \pi^*_{ikj} \pi_{ij} Response probability for individual ii’s observed outcome cateogry.
πjj*\pi^*_{jj} P(Y*=j|Y=j,Z;γ)=∑i=1Nπijj*P(Y^* = j | Y = j, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{ijj} Average probability of correct classification for category jj.
Sensitivity P(Y*=1|Y=1,Z;γ)=∑i=1Nπi11*P(Y^* = 1 | Y = 1, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i11} True positive rate. Average probability of observing outcome k=1k = 1, given the true outcome j=1j = 1.
Specificity P(Y*=2|Y=2,Z;γ)=∑i=1Nπi22*P(Y^* = 2 | Y = 2, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i22} True negative rate. Average probability of observing outcome k=2k = 2, given the true outcome j=2j = 2.
βX\beta_X – Association parameter of interest in the true outcome mechanism.
γ11Z\gamma_{11Z} – Association parameter of interest in the observation mechanism, given j=1j=1.
γ12Z\gamma_{12Z} – Association parameter of interest in the observation mechanism, given j=2j=2.