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