This guide is designed to summarize key notation and quantities used the COMMA R Package and associated publications.
| Term | Definition | Description |
|---|---|---|
| X | – | Predictor matrix for the true mediator and outcome. |
| C | – | Covariate matrix for the true mediator and outcome. |
| Z | – | Predictor matrix for the observed mediator, conditional on the true mediator |
| Y | – | Outcome variable. |
| M | M ∈ {1, 2} | True binary mediator. Reference category is 2. |
| mij | 𝕀{Mi = j} | Indicator for the true binary mediator. |
| M* | M* ∈ {1, 2} | Observed binary mediator. Reference category is 2. |
| miℓ* | 𝕀{Mi* = ℓ} | Indicator for the observed binary mediator. |
| True Mediator Mechanism | logit{P(M = 1|X, C; β)} = β0 + βXX + βCC | Relationship between X and C and the true mediator, M. |
| Observed Mediator Mechanism | logit{P(M* = 1|M = m, Z; γ)} = γ1m0 + γ1mZZ | Relationship between Z and the observed mediator, M*, given the true mediator M. |
| Outcome Mechanism | E(Y|X, C, M; θ)} = θ0 + θXX + θCCθMM + θXMXM | Relationship between X, C, and M and the outcome of interest Y. |
| πij | $P(M_i = j | X, C ; \beta) = \frac{\text{exp}\{\beta_{j0} + \beta_{jX} X_i + \beta_{jC} C_i\}}{1 + \text{exp}\{\beta_{j0} + \beta_{jX} X_i + \beta_{jC} C_i\}}$ | Response probability for individual i’s true mediator category. |
| πiℓj* | $P(M^*_i = \ell | M_i = j, Z ; \gamma) = \frac{\text{exp}\{\gamma_{\ell j 0} + \gamma_{ \ell jZ} Z_i\}}{1 + \text{exp}\{\gamma_{\ell j0} + \gamma_{kjZ} Z_i\}}$ | Response probability for individual i’s observed mediator category, conditional on the true mediator. |
| πiℓ* | $P(M^*_i = \ell | M_i, X, Z ; \gamma) = \sum_{j = 1}^2 \pi^*_{i \ell j} \pi_{ij}$ | Response probability for individual i’s observed mediator cateogry. |
| πjj* | $P(M^* = j | M = j, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{ijj}$ | Average probability of correct classification for category j. |
| Sensitivity | $P(M^* = 1 | M = 1, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i11}$ | True positive rate. Average probability of observing mediator k = 1, given the true mediator j = 1. |
| Specificity | $P(M^* = 2 | M = 2, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i22}$ | True negative rate. Average probability of observing mediator k = 2, given the true mediator j = 2. |
| βX | – | Association parameter of interest in the true mediator mechanism. |
| γ11Z | – | Association parameter of interest in the observed mediator mechanism, given j = 1. |
| γ12Z | – | Association parameter of interest in the observed mediator mechanism, given j = 2. |
| θX | – | Association parameter of interest in the outcome mechanism. |
| θM | – | Association parameter relating the true mediator to the outcome. |
| θXM | – | Association parameter for the interaction between X and M in the outcome mechanism. |