# COMBO Notation Guide

## 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.
$$Y$$ $$Y \in \{1, 2\}$$ True binary outcome. Reference category is 2.
$$y_{ij}$$ $$\mathbb{I}\{Y_i = j\}$$ Indicator for the true binary outcome.
$$Y^*$$ $$Y^* \in \{1, 2\}$$ Observed binary outcome. Reference category is 2.
$$y^*_{ik}$$ $$\mathbb{I}\{Y^*_i = k\}$$ Indicator for the observed binary outcome.
True Outcome Mechanism $$\text{logit} \{ P(Y = j | X ; \beta) \} = \beta_{j0} + \beta_{jX} X$$ Relationship between $$X$$ and the true outcome, $$Y$$.
Observation Mechanism $$\text{logit}\{ P(Y^* = k | Y = j, Z ; \gamma) \} = \gamma_{kj0} + \gamma_{kjZ} Z$$ Relationship between $$Z$$ and the observed outcome, $$Y^*$$, given the true outcome $$Y$$.
$$\pi_{ij}$$ $$P(Y_i = j | X ; \beta) = \frac{\text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}{1 + \text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}$$ Response probability for individual $$i$$’s true outcome category.
$$\pi^*_{ikj}$$ $$P(Y^*_i = k | Y_i = j, Z ; \gamma) = \frac{\text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}{1 + \text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}$$ Response probability for individual $$i$$’s observed outcome category, conditional on the true outcome.
$$\pi^*_{ik}$$ $$P(Y^*_i = k | Y_i, X, Z ; \gamma) = \sum_{j = 1}^2 \pi^*_{ikj} \pi_{ij}$$ Response probability for individual $$i$$’s observed outcome cateogry.
$$\pi^*_{jj}$$ $$P(Y^* = j | Y = j, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{ijj}$$ Average probability of correct classification for category $$j$$.
Sensitivity $$P(Y^* = 1 | Y = 1, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i11}$$ True positive rate. Average probability of observing outcome $$k = 1$$, given the true outcome $$j = 1$$.
Specificity $$P(Y^* = 2 | Y = 2, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i22}$$ True negative rate. Average probability of observing outcome $$k = 2$$, given the true outcome $$j = 2$$.
$$\beta_X$$ Association parameter of interest in the true outcome mechanism.
$$\gamma_{11Z}$$ Association parameter of interest in the observation mechanism, given $$j=1$$.
$$\gamma_{12Z}$$ Association parameter of interest in the observation mechanism, given $$j=2$$.