--- title: "COMBO Notation Guide - Two-stage Misclassification Model" author: "Kim Hochstedler" output: rmarkdown::html_vignette header-includes: - \usepackage[fontsize=13pt]{scrextend} - \usepackage{kableExtra} - \usepackage{knitr} vignette: > %\VignetteIndexEntry{COMBO Notation Guide - Two-stage Misclassification Model} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include=FALSE} knitr::opts_chunk$set(echo = FALSE) ``` ```{r} Term <- c("$X$", "$Z^{(1)}$", "$Z^{(2)}$", "$Y$", "$y_{ij}$", "$Y^{*(1)}$", "$y^{*(1)}_{ik}$", "$Y^{*(2)}$", "$y^{*(2)}_{i \\ell}$", "True Outcome Mechanism", "First-Stage Observation Mechanism", "Second-Stage Observation Mechanism", "$\\pi_{ij}$","$\\pi^{*(1)}_{ikj}$", "$\\pi^{*(2)}_{i \\ell kj}$", "$\\pi^{*(1)}_{ik}$", "$\\pi^{*(1)}_{jj}$", "$\\pi^{*(2)}_{jjj}$", "First-Stage Sensitivity", "First-Stage Specificity", "$\\beta_X$", "$\\gamma^{(1)}_{11Z^{(1)}}$", "$\\gamma^{(1)}_{12Z^{(1)}}$", "$\\gamma^{(2)}_{111Z^{(2)}}$", "$\\gamma^{(2)}_{121Z^{(2)}}$", "$\\gamma^{(2)}_{112Z^{(2)}}$", "$\\gamma^{(2)}_{122Z^{(2)}}$") Definition <- c("--", "--", "--", "$Y \\in \\{1, 2\\}$", "$\\mathbb{I}\\{Y_i = j\\}$", "$Y^{*(1)} \\in \\{1, 2\\}$", "$\\mathbb{I}\\{Y^{*(1)}_i = k\\}$", "$Y^{*(2)} \\in \\{1, 2\\}$", "$\\mathbb{I}\\{Y^{*(2)}_i = \\ell \\}$", "$\\text{logit} \\{ P(Y = j | X ; \\beta) \\} = \\beta_{j0} + \\beta_{jX} X$", "$\\text{logit}\\{ P(Y^{*(1)} = k | Y = j, Z^{(1)} ; \\gamma^{(1)}) \\} = \\gamma^{(1)}_{kj0} + \\gamma^{(1)}_{kjZ^{(1)}} Z^{(1)}$", "$\\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)}$", "$P(Y_i = j | X ; \\beta) = \\frac{\\text{exp}\\{\\beta_{j0} + \\beta_{jX} X_i\\}}{1 + \\text{exp}\\{\\beta_{j0} + \\beta_{jX} X_i\\}}$", "$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)}\\}}$", "$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)}\\}}$", "$P(Y^{*(1)}_i = k | X, Z^{(1)} ; \\gamma^{(1)}) = \\sum_{j = 1}^2 \\pi^{*(1)}_{ikj} \\pi_{ij}$", "$P(Y^{*(1)} = j | Y = j, Z^{(1)} ; \\gamma^{(1)}) = \\sum_{i = 1}^N \\pi^{*(1)}_{ijj}$", "$P(Y^{*(2)} = j | Y^{*(1)}_i = j, Y = j, Z^{(2)} ; \\gamma^{(2)}) = \\sum_{i = 1}^N \\pi^{*(2)}_{ijjj}$", "$P(Y^{*(1)} = 1 | Y = 1, Z^{(1)} ; \\gamma^{(1)}) = \\sum_{i = 1}^N \\pi^{*(1)}_{i11}$", "$P(Y^{*(1)} = 2 | Y = 2, Z^{(1)} ; \\gamma^{(1)}) = \\sum_{i = 1}^N \\pi^{*(1)}_{i22}$", "--", "--", "--", "--", "--", "--", "--") Description <- c("Predictor matrix for the true outcome.", "Predictor matrix for the first-stage observed outcome, conditional on the true outcome.", "Predictor matrix for the second-stage observed outcome, conditional on the true outcome and first-stage observed outcome.", "True binary outcome. Reference category is 2.", "Indicator for the true binary outcome.", "First-stage observed binary outcome. Reference category is 2.", "Indicator for the first-stage observed binary outcome.", "Second-stage observed binary outcome. Reference category is 2.", "Indicator for the second-stage observed binary outcome.", "Relationship between $X$ and the true outcome, $Y$.", "Relationship between $Z^{(1)}$ and the first-stage observed outcome, $Y^{*(1)}$, given the true outcome $Y$.", "Relationship between $Z^{(2)}$ and the second-stage observed outcome, $Y^{*(2)}$, given the first-stage observed outcome, $Y^{*(1)}$, and the true outcome $Y$.", "Response probability for individual $i$'s true outcome category.", "Response probability for individual $i$'s first-stage observed outcome category, conditional on the true outcome.", "Response probability for individual $i$'s second-stage observed outcome category, conditional on the first-stage observed outcome and the true outcome.", "Response probability for individual $i$'s first-stage observed outcome cateogry.", "Average probability of first-stage correct classification for category $j$.", "Average probability of first-stage and second-stage correct classification for category $j$.", "True positive rate. Average probability of observing first-stage outcome $k = 1$, given the true outcome $j = 1$.", "True negative rate. Average probability of observing first-stage outcome $k = 2$, given the true outcome $j = 2$.", "Association parameter of interest in the true outcome mechanism.", "Association parameter of interest in the first-stage observation mechanism, given $j=1$.", "Association parameter of interest in the first-stage observation mechanism, given $j=2$.", "Association parameter of interest in the second-stage observation mechanism, given $k = 1$ and $j = 1$.", "Association parameter of interest in the second-stage observation mechanism, given $k = 2$ and $j = 1$.", "Association parameter of interest in the second-stage observation mechanism, given $k = 1$ and $j = 2$.", "Association parameter of interest in the second-stage observation mechanism, given $k = 2$ and $j = 2$.") notation_table <- data.frame(Term, Definition, Description) ``` ## Notation This guide is designed to summarize key notation and quantities used the *COMBO* R Package and associated publications. ```{r, results = "asis"} library(xtable) xtable_notation <- xtable(notation_table) print(xtable(notation_table), type = "html", include.rownames=FALSE) ```