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
\(j=2\).
