Author: Kim Hochstedler

# COMBO Notation Guide - Two-stage Misclassification Model

#### 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.
\(Z^{(1)}\) | – | Predictor matrix for the first-stage observed outcome, conditional on the true outcome.
\(Z^{(2)}\) | – | Predictor matrix for the second-stage observed outcome, conditional on the true outcome and first-stage observed 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.
\(Y^{*(1)}\) | \(Y^{*(1)} \in \{1, 2\}\) | First-stage observed binary outcome. Reference category is 2.
\(y^{*(1)}_{ik}\) | \(\mathbb{I}\{Y^{*(1)}_i = k\}\) | Indicator for the first-stage observed binary outcome.
\(Y^{*(2)}\) | \(Y^{*(2)} \in \{1, 2\}\) | Second-stage observed binary outcome. Reference category is 2.
\(y^{*(2)}_{i \ell}\) | \(\mathbb{I}\{Y^{*(2)}_i = \ell \}\) | Indicator for the second-stage 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, \(Y\).
First-Stage Observation Mechanism | \(\text{logit}\{ P(Y^{*(1)} = k | Y = j, Z^{(1)} ; \gamma^{(1)}) \} = \gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z^{(1)}\) | Relationship between \(Z^{(1)}\) and the first-stage observed outcome, \(Y^{*(1)}\), given the true outcome \(Y\).
Second-Stage Observation Mechanism | \(\text{logit}\{ P(Y^{*(2)} = \ell | Y^{*(1)} = k, Y = j, Z^{(2)} ; \gamma^{(2)}) \} = \gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z^{(2)}\) | Relationship between \(Z^{(2)}\) and the second-stage observed outcome, \(Y^{*(2)}\), given the first-stage observed outcome, \(Y^{*(1)}\), and 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.
\(\pi^{*(1)}_{ikj}\) | \(P(Y^{*(1)}_i = k | Y = j, Z^{(1)} ; \gamma^{(1)}) = \frac{\text{exp}\{\gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z_i^{(1)}\}}{1 + \text{exp}\{\gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z_i^{(1)}\}}\) | Response probability for individual \(i\)’s first-stage observed outcome category, conditional on the true outcome.
\(\pi^{*(2)}_{i \ell kj}\) | \(P(Y^{*(2)}_i = \ell | Y^{*(1)} = k, Y = j, Z^{(2)} ; \gamma^{(2)}) = \frac{\text{exp}\{\gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z_i^{(2)}\}}{1 + \text{exp}\{\gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z_i^{(2)}\}}\) | Response probability for individual \(i\)’s second-stage observed outcome category, conditional on the first-stage observed outcome and the true outcome.
\(\pi^{*(1)}_{ik}\) | \(P(Y^{*(1)}_i = k | X, Z^{(1)} ; \gamma^{(1)}) = \sum_{j = 1}^2 \pi^{*(1)}_{ikj} \pi_{ij}\) | Response probability for individual \(i\)’s first-stage observed outcome cateogry.
\(\pi^{*(1)}_{jj}\) | \(P(Y^{*(1)} = j | Y = j, Z^{(1)} ; \gamma^{(1)}) = \sum_{i = 1}^N \pi^{*(1)}_{ijj}\) | Average probability of first-stage correct classification for category \(j\).
\(\pi^{*(2)}_{jjj}\) | \(P(Y^{*(2)} = j | Y^{*(1)}_i = j, Y = j, Z^{(2)} ; \gamma^{(2)}) = \sum_{i = 1}^N \pi^{*(2)}_{ijjj}\) | Average probability of first-stage and second-stage correct classification for category \(j\).
First-Stage Sensitivity | \(P(Y^{*(1)} = 1 | Y = 1, Z^{(1)} ; \gamma^{(1)}) = \sum_{i = 1}^N \pi^{*(1)}_{i11}\) | True positive rate. Average probability of observing first-stage outcome \(k = 1\), given the true outcome \(j = 1\).
First-Stage Specificity | \(P(Y^{*(1)} = 2 | Y = 2, Z^{(1)} ; \gamma^{(1)}) = \sum_{i = 1}^N \pi^{*(1)}_{i22}\) | True negative rate. Average probability of observing first-stage outcome \(k = 2\), given the true outcome \(j = 2\).
\(\beta_X\) | – | Association parameter of interest in the true outcome mechanism.
\(\gamma^{(1)}_{11Z^{(1)}}\) | – | Association parameter of interest in the first-stage observation mechanism, given \(j=1\).
\(\gamma^{(1)}_{12Z^{(1)}}\) | – | Association parameter of interest in the first-stage observation mechanism, given \(j=2\).
\(\gamma^{(2)}_{111Z^{(2)}}\) | – | Association parameter of interest in the second-stage observation mechanism, given \(k = 1\) and \(j = 1\).
\(\gamma^{(2)}_{121Z^{(2)}}\) | – | Association parameter of interest in the second-stage observation mechanism, given \(k = 2\) and \(j = 1\).
\(\gamma^{(2)}_{112Z^{(2)}}\) | – | Association parameter of interest in the second-stage observation mechanism, given \(k = 1\) and \(j = 2\).
\(\gamma^{(2)}_{122Z^{(2)}}\) | – | Association parameter of interest in the second-stage observation mechanism, given \(k = 2\) and \(j = 2\).