**Models in this App**

** 1. Reitsma's model**

Suppose that $N~(i=1, \dots, N)$ diagnostic studies are included in meta-analysis. In other words, there are $N$ rows in the input data.
The observed data are the number of true positives (TP), true negative (TN), false positives (FP), and false negatives (FN), so sensitivity (Sens) and specificity (Spec) can be estimated from data.

In the within-study level, we denote that,

- $y_{1i}$ and $y_{2i}$: the logit-Sens and logit-Spec after continuity correction
- $s_{1i}^2$ and $s_{2i}^2$: the observed variances of $y_{1i}$ and $y_{2i}$ within each study
- $\mu_{1i}$ and $\mu_{2i}$: the logit-transformed true Sens and Spec of the $i$th study

Given $(\mu_{1i}, \mu_{2i})$, it is assumed that:
\begin{align}
\binom{y_{1i}}{y_{2i}}
\sim
N \left (\binom{\mu_{1i}}{\mu_{2i}}, \boldsymbol{\Sigma}_i \right )
\mathrm{~with~}
\boldsymbol{\Sigma}_i
= \begin{pmatrix}
s_{1i}^2 & 0\\
0 & s_{2i}^2
\end{pmatrix},
\label{eq:b2}
\end{align}

In the within-study level, we denote that,

- $\mu_1$ and $\mu_2$: the common means of the logit-transformed sensitivity and specificity
- $\tau_1^2$ and $\tau_2^2$: between-study variances
- $\tau_{12} = \rho\tau_{1}\tau_{2}$ is the covariance between $\mu_{1i}$ and $\mu_{2i}$, $\rho~(-1 \le \rho \le 1)$ is the correlation coefficient

It is assumed that $(\mu_{1i}, \mu_{2i})^T$ is normally distributed:
\begin{align}
\binom{\mu_{1i}}{\mu_{2i}}
\sim
N\left ( \binom{\mu_1}{\mu_2}, \boldsymbol{\Omega} \right )
\mathrm{~with~}
\boldsymbol{\Omega} =
\begin{pmatrix}
\tau_1^2 & \tau_{12} \\
\tau_{12} & \tau_2^2
\end{pmatrix},
\label{eq:b1}
\end{align}

The combination of \eqref{eq:b2} and \eqref{eq:b1} leads to the Reitsma's model:

\begin{align}
\boldsymbol{y}_i | \boldsymbol{\Sigma}_i
\sim N_2
\left (\boldsymbol{\mu}, \boldsymbol{\Omega} + \boldsymbol{\Sigma}_i \right ),
\label{eq:b12}
\end{align}
where,$\boldsymbol{y}_i = (y_{1i},y_{2i})^T$, $\boldsymbol{\mu} = (\mu_1,\mu_2)^T$

Reitsma's model requires the number of diseased and non-diseased to be large.

** 2. Generalized linear mixed (GLM) model**

GLM model differs Reitsma's model in the within-study level.

Suppose,

- $n_{00i}, n_{11i}, n_{01i}, n_{10i}$: TP, TN, FP, FN; $N_{1i}=n_{11i}+n_{10i}, N_{0i}=n_{00i}+n_{01i}$
- $s_{1i}^2$ and $s_{2i}^2$: the observed variances of $y_{1i}$ and $y_{2i}$ within each study
- $\mu_{1i}$ and $\mu_{2i}$: the logit-transformed true Sens and Spec of the $i$th study

In the within-study level, it is assumed that:,

\begin{align}
n_{00i}\sim \text{Binomial}(N_{0i}, \text{Spec}_i); ~~~~ n_{11i}\sim \text{Binomial}(N_{1i}, \text{Sens}_i)
\label{eq:b3}
\end{align}

In the between-study level, we denote that:

- $\mu_{1i}=\text{logit}(\text{Sens}_i)$ and $\mu_{2i}=\text{logit}(\text{Spec}_i)$
- $\mu_1$ and $\mu_2$: the common means of the logit-Sens and logit-Spec
- $\tau_1^2$ and $\tau_2^2$: between-study variances
- $\tau_{12} = \rho\tau_{1}\tau_{2}$ is the covariance between $\mu_{1i}$ and $\mu_{2i}$, $\rho~(-1 \le \rho \le 1)$ is the correlation coefficient

In the between-study level, $\mu_{1i}, \mu_{2i}$ have the same distribution with \ref{eq:b2}

The combination of \eqref{eq:b3} and \eqref{eq:b1} leads to the GLM model.

GLM model does not require continuity correction.

** 3. SROC and SAUC**

The SROC curve:
\begin{align}
SROC(x; \boldsymbol{\mu}, \boldsymbol{\Omega})
= \mathrm{logit}^{-1} \left[ \mu_1 - \dfrac{\tau_{12}}{\tau_2^2}\{\mathrm{logit}(x)+\mu_2\} \right].
\label{eq:sroc}
\end{align}
The HSROC curve is given by the SROC curve \eqref{eq:sroc} with $\rho=-1$.
The SAUC:
\begin{align}
SAUC(\boldsymbol{\mu}, \boldsymbol{\Omega})
= \int_{0}^{1}SROC(x; \boldsymbol{\mu}, \boldsymbol{\Omega})dx.
\label{sauc}
\end{align}

** 4. Likelihood based sensitivity analysis method**

Publication bias is the phenomenon that studies with significant results are more likely to be published or selected for meta-analysis.
In meta-analysis of DTA, we consider to model the selective publication of each study by the selection function
\begin{align*}
P(\text{Publish}|t_i)=\Phi(\alpha+\beta\times t_i\},
\end{align*}
which is the probit model on the $t$-type statistic of the linear combination of the logit-transformed sensitivity and specificity:
$
\boldsymbol{c}^T \boldsymbol{y}_i = c_1y_{1i}+c_2y_{2i},
$
where $\boldsymbol{c} = (c_1, c_2)^T$ is a contrast vector.
The $t$-type statistic of the linear combination is
$
t_i = {\boldsymbol{c}^T \boldsymbol{y}_i}/{\sqrt{\boldsymbol{c}^T\boldsymbol{\Sigma}_i\boldsymbol{c}}}.
$

When $(c_1, c_2) = (1/\sqrt{2}, 1/\sqrt{2})$, it gives the $t$-statistic of the lnDOR.
By taking different contrast vectors, the $t$-type statistic can determine a variety of selective publication mechanisms.
For example,
$(c_1, c_2) = (1, 0)$ and $(c_1, c_2) = (0, 1)$ indicate that the selective publication mechanisms are determined by the significance of sensitivity and specificity, respectively.
The contrast vector $\boldsymbol{c}$ can be regarded as unknown parameters to be estimated or specified by user.

In the sensitivity analysis, we introduce the marginal probability of selective publication, $p=P(\mathrm{select})=E\{P(\text{Publish}|t_i)\}$.
The estimating model based on the joint distribution of observed data involving the selection probabilities can be constructed.
Given various specified values of $p$, the bias-adjusted SROC curves and SAUC can be estimated from the maximum likelihood estimation.

**R packages used in this Application:**

- mada,
DT,
ggplot2,
plotly,
shiny,
shinythemes,
shinyWidgets,
mvmeta,
mada,
shinyAce,
latex2exp,
base,