---
title: "Introduction to dtametaTMB"
format: 
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vignette: >
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---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

This vignette introduces the main functionality of the `dtametaTMB` package for meta-analysis of diagnostic test accuracy (DTA) studies.
We demonstrate how to fit three commonly used models:

- the Reitsma bivariate model
- the Rutter and Gatsonis (HSROC) model
- the Hoyer model for multiple thresholds

using real datasets included in the package.

Subgroup analyses, meta-regression, and latent class analysis allowing for an imperfect reference standard are covered in separate vignettes.

*Note:* Rendering forest plots via `forest` may take longer in an interactive R session due to the underlying grid graphics.
Performance is typically faster when knitting the vignette (e.g. via RMarkdown or Quarto).

## How do I load the package?

```{r setup}
#| echo: TRUE
#install.packages("dtametaTMB")
library(dtametaTMB)
```

## Which model should I use?

- **Reitsma model:** single threshold per study: use when studies apply similar thresholds.
- **Rutter and Gatsonis model:** single threshold per study: use when thresholds vary between studies
- **Hoyer model:** multiple thresholds per study

The Reitsma and Rutter-Gatsonis model classes support parameter constraints for simplified analyses and sparse datasets.

## Validation

The implementations reproduce benchmark results from the Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy (including parameter estimates, standard errors, and graphical outputs) for standard datasets such as the Anti‑CCP and RF examples.

## How do I use the Reitsma model?

Unlike the `mada` package, `dtametaTMB` uses the binomial likelihood to model the within study distribution as recommended by The Cochrane Handbook.

For sensitivity in study $i$, we have the number of diseased individuals testing positive: $y_{Ai} \sim \mathcal{B}(n_{Ai},\pi_{Ai})$.

Similarly for specificity, we have the number of non-diseased individuals testing negative: $y_{Bi} \sim \mathcal{B}(n_{Bi},\pi_{Bi})$.

At the study level, we have

$$ 
\begin{pmatrix}
\operatorname{logit}(\pi_{Ai}) \\
\operatorname{logit}(\pi_{Bi})
\end{pmatrix}
=
\begin{pmatrix}
\mu_{Ai} \\
\mu_{Bi}
\end{pmatrix}
\sim \mathcal{N}
\left(
\begin{pmatrix}
\mu_A \\
\mu_B
\end{pmatrix},
\;
\Sigma
\right),
\quad \text{with} \quad
\Sigma =
\begin{pmatrix}
\sigma_A^2 & \sigma_{AB} \\
\sigma_{AB} & \sigma_B^2
\end{pmatrix}.$$

```{r}
#| echo: TRUE
data("anticcp")
reitsma <- fitReitsma(data=anticcp,
                      TP=TP,
                      FP=FP,
                      FN=FN,
                      TN=TN,
                      study=study)
reitsma
summary(reitsma)
```

## How do I get a summary plot of the Reitsma model?

The confidence region reflects uncertainty in the estimated summary point, whereas the prediction region reflects the likely location of a future study.

```{r}
#| fig.height: 8
#| fig.width: 8
#| echo: TRUE
plot(reitsma)
plot(reitsma,scale=0.01)
```

## How do I get a coupled forest plot?

*Note:* Rendering forest plots may take longer in an interactive R session due to the underlying grid graphics.
Performance is typically faster when knitting the vignette (e.g. via RMarkdown or Quarto).

```{r}
#| fig.height: 11
#| fig.width: 13
#| echo: TRUE
forest(reitsma)
```

## How do I export results to RevMan?

```{r}
#| echo: TRUE
as_revman(reitsma)
```

## How do I use the Rutter and Gatsonis model?

The number of diseased individuals from study $i$ who test positive is denoted by $y_{i1} \sim \mathcal{B}(n_{i1},\pi_{i1})$.

Similarly, the number of non-diseased individuals who test positive is $y_{i2} \sim \mathcal{B}(n_{i2},\pi_{i2})$.

At the study level, we have

$$\operatorname{logit}(\pi_{ij}) =
\left( \theta_i + \alpha_i x_{ij} \right)
\exp(-\beta x_{ij}), \quad \quad \quad
\alpha_i \sim \mathcal{N}(\Lambda, \sigma_\alpha^2),
\qquad
\theta_i \sim \mathcal{N}(\Theta, \sigma_\theta^2),$$

where $$x_{ij} =
\begin{cases}
-0.5 & \text{for non-diseased individuals}, \\
\phantom{-}0.5 & \text{for diseased individuals}.
\end{cases}$$

In this parameterization $\Lambda$ corresponds to accuracy, $\Theta$ to threshold, and $\beta$ controls the shape of the HSROC curve.

```{r}
#| echo: TRUE
#| results: 'hide'
data("RF")
ruttergatsonis <- fitRutterGatsonis(data=RF,
                                    TP=TP,
                                    FP=FP,
                                    FN=FN,
                                    TN=TN,
                                    study=study)
```

```{r}
#| echo: TRUE
ruttergatsonis
summary(ruttergatsonis)
```

## How do I get a summary plot of the Rutter and Gatsonis model?

```{r}
#| fig.height: 8
#| fig.width: 8
#| echo: TRUE
plot(ruttergatsonis)
plot(ruttergatsonis,size="se",specrange=c(0.3,0.995),scale=0.01)
```

## How do I get a coupled forest plot?

*Note:* Rendering forest plots may take longer in an interactive R session due to the underlying grid graphics.
Performance is typically faster when knitting the vignette (e.g. via RMarkdown or Quarto).

```{r}
#| fig.height: 14
#| fig.width: 13
#| echo: TRUE
forest(ruttergatsonis)
```

## How do I simplify the models via parameter constraints?

The package allows model parameters to be constrained to zero during estimation.
This can be useful in meta-analysis of sparse data, when convergence is difficult, or when fitting simplified models as described in the Cochrane Handbook.
For example,

```{r}
#| echo: TRUE
reitsmacons <- fitReitsma(data = anticcp, 
                          TP = TP,
                          FP = FP,
                          FN = FN,
                          TN = TN,
                          study = study,
                          constrain = "sigma_AB")
summary(reitsmacons)$estimates
```

fixed the between-study random-effects covariance at zero.
Other options include `constrain="sigma2_A"`, `"constrain=sigma2_B"`, or `"constrain=all"`.

The Rutter and Gatsonis model allows for the constraint $\sigma_\alpha^2=0$, $\sigma_\theta^2=0$ and/or $\beta=0$ (shape parameter).
For example

```{r}
#| echo: TRUE
ruttergatsoniscons <- fitRutterGatsonis(data = RF, 
                                        TP = TP,
                                        FP = FP,
                                        FN = FN,
                                        TN = TN,
                                        study = study,
                                        constrain = c("sigma2_theta","shape"))
summary(ruttergatsoniscons)$estimates
```

Note that we can also compare the constrained model with the full model using likelihood ratio tests.

```{r}
anova(ruttergatsoniscons,ruttergatsonis)
```

## How do I use the Hoyer model?

The Hoyer model assumes that the diagnostic test measurement arises from a latent continuous variable $Y_{j}$, with $j=0$ for the non-diseased individuals and $j=1$ for the diseased, which follows a parametric survival distribution.
The observed test results correspond to interval-censored observations of this latent variable due to the use of thresholds.

### Which distributions does `dtametaTMB` support?

The latent variable $Y_{j}$ is assumed to follow a location–scale family:

$$Y_{j} \sim F(\mu_{j}, \phi_j),$$

where:

- $\mu_j$ is a study-specific location parameter
- $\phi_j$ is a scale parameter

Common choices for $F$ include:

#### Weibull distribution

$$f(y; \mu_j, \phi_j) = \mu_j \phi_j y^{\phi_j - 1} \exp(-\mu_j y^{\phi_j})$$

#### Log-normal distribution

$$f(y; \mu_j, \phi_j) =
\frac{1}{\phi_j y}
\varphi\!\left(\frac{\log(y) - \mu_j}{\phi_j}\right)$$

#### Log-logistic distribution

$$f(y; \mu_j, \phi_j) =
\frac{\mu_j \phi_j y^{\phi_j - 1}}{\left(1 + \mu_j y^{\phi_j}\right)^2}$$

### Random effects structure

To account for between-study heterogeneity and the correlation between diseased and non-diseased groups, we introduce study-specific random effects:

$$\log(y_{0i}) = \beta_{0} + \varepsilon_{0i} + u_{0i}, \quad
\log(y_{1i}) = \beta_{1} + \varepsilon_{1i} + u_{1i},$$

with

$$\begin{pmatrix}
u_{0i} \\
u_{1i}
\end{pmatrix}
\sim
\mathcal{N}
\left(
\begin{pmatrix}
0 \\
0
\end{pmatrix},
\begin{pmatrix}
\sigma_0^2 & \rho \sigma_0 \sigma_1 \\
\rho \sigma_0 \sigma_1 & \sigma_1^2
\end{pmatrix}
\right).$$

## Censoring mechanism

Because only thresholded outcomes (TN,TP,FN,FP) are observed, the latent variable $Y_{j}$ is subject to censoring.

- Left censoring: $(-\infty,y_k^R]$
- Interval censoring between adjacent thresholds
- Right censoring: $(y_k^L, +\infty)$

where $y_k^L$ and $y_k^R$ denote finite threshold values.

### How do I compute sensitivities and specificities?

We can model the ROC curve by predicting sensitivities and specificities (1 - false positive rates) at several threshold values $y$.

#### Weibull distribution

$$S_1(y) = \exp\left(-\exp\left(\frac{\log(y) - \beta_1}{\lambda_1}\right)\right),
\quad \text{with} \quad \beta_1=-\log{(\mu_1}), \quad \lambda_1 = 1/\phi_1,$$ $$S_0(y) = \exp\left(-\exp\left(\frac{\log(y) - \beta_0}{\lambda_0}\right)\right),
\quad \text{with} \quad \beta_0=-\log{(\mu_0}), \quad \lambda_0 = 1/\phi_0.$$

#### Log-normal distribution

$$S_1(y) = 1-\Phi\left(\frac{  \log(y) - \beta_1 }{\lambda_1}  \right), \quad \text{with} \quad \beta_1=\mu_1, \quad \lambda_1 = \phi_1,$$

$$S_0(y) = 1-\Phi\left(\frac{ \log(y) - \beta_0 }{\lambda_0}  \right), \quad \text{with} \quad \beta_0=\mu_0, \quad \lambda_0 = \phi_0.$$

#### Log-logistic distribution

$$S_1(y) = \left(1 +\exp\left(\frac{\log(y) - \beta_1}{\lambda_1}\right) \right)^{-1}, \quad \text{with} \quad \beta_1=-\phi^{-1}_1\log{(\mu_1}), \quad \lambda_1 = 1/\phi_1,$$ $$S_0(y) = \left(1 +\exp\left(\frac{\log(y) - \beta_0}{\lambda_0}\right) \right)^{-1}, \quad \text{with} \quad \beta_0=-\phi^{-1}_0\log{(\mu_0}), \quad \lambda_0 = 1/\phi_0.$$

#### How do I interpret test direction?

For tests where larger values indicate disease (`testdirection = "greater"`):

- $S_1(y)$ corresponds to **sensitivity** at threshold y
- $S_0(y)$ corresponds to the **false positive rate (FPR) = 1 - specificity**

For tests where smaller values indicate disease (`testdirection = "less"`):

- $S_1(y)$ corresponds to the **false negative rate (FNR) = 1 - sensitivity**
- $S_0(y)$ corresponds to **specificity**

## In a nutshell

By modelling the diagnostic marker as a latent continuous outcome, the Hoyer model:

- naturally accommodates **multiple thresholds per study**
- provides a unified framework for **interval-censored diagnostic data**
- preserves the correlation between sensitivity and specificity

The following parameters have to be estimated: $\beta_{0},\; \beta_{1},\; \lambda_0,\; \lambda_1,\; \sigma_0^2,\; \sigma_1^2,\; \text{and } \rho$, where $\lambda_0$ and $\lambda_1$ are transformations of $\phi_0$ and $\phi_1$.

To obtain initial values, it is necessary to specify a smallest (\<\< $\min y$) and largest (\>\> $\max y$) value across all studies, essentially replacing negative and positive infinity for the left and right censored intervals.

## How do I fit the Hoyer model?

```{r}
#| echo: TRUE
data("diabetes")
head(diabetes)
summary(diabetes$threshold)
```

The argument `eval_threshold` specifies the values at which sensitivities and specificities are evaluated after model fitting.

```{r}
#| echo: TRUE
#| results: 'hide'
hoyer <- fitHoyer(data=diabetes,
                  TP=TP,
                  FP=FP,
                  FN=FN,
                  TN=TN,
                  threshold=threshold,
                  study=study,
                  dist="loglogistic",
                  testdirection="greater",
                  eval_threshold=c(5.0,5.5,6.0,6.5,7.0,7.5),
                  smallest=2,
                  largest=10)
```

```{r}
#| echo: TRUE
hoyer
summary(hoyer)
```

## How do I get a summary plot of the Hoyer model?

```{r}
#| echo: TRUE
#| fig.height: 8
#| fig.width: 8
plot(hoyer,
     main="Diagnostic Test Accuracy Meta-Analysis\nLoglogistic Model")
plot(hoyer,size="sampsize",scale=0.025,
     main="Diagnostic Test Accuracy Meta-Analysis\nLoglogistic Model")
plot(hoyer,size="se",scale=0.025,
     main="Diagnostic Test Accuracy Meta-Analysis\nLoglogistic Model")
```

## How do I get a coupled forest plot?

*Note:* Rendering forest plots may take longer in an interactive R session due to the underlying grid graphics.
Performance is typically faster when knitting the vignette (e.g. via RMarkdown or Quarto).

```{r}
#| fig.height: 32
#| fig.width: 12
#| echo: TRUE
forest(hoyer)
```

## How do I fit the Hoyer model to the FeNO dataset?

*Note:* The FeNO data are reproduced from the original source and the supplementary material accompanying the Cochrane Handbook.
However, the study 'Schneider 2013' contains inconsistent diseased counts (TP + FN).
To illustrate the Hoyer analysis, we modify the TP counts in rows 118–120 from 39 to 38, restoring a constant total of 154 diseased individuals.
Other corrections restoring internal consistency are also conceivable.

```{r}
#| echo: TRUE
#| fig.height: 8
#| fig.width: 8
data("FENO")
head(FENO)
summary(FENO$cutoff)
# feno <- fitHoyer(data=FENO,
#                  TP=TP,
#                  FP=FP,
#                  FN=FN,
#                  TN=TN,
#                  threshold=cutoff,
#                  study=study,
#                  dist="loglogistic",
#                  testdirection="greater",
#                  eval_threshold=seq(16,40,1),
#                  smallest=2,
#                  largest=120)
#Error in `FUN()`:
#! Inconsistent diseased counts (TP+FN) within study 'Schneider 2013'.
#Rows: 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122
#Counts: 154, 155
#
# Let's check the Schneider 2013 study
check_schneider    <- FENO[110:122,]
check_schneider$n1 <- with(check_schneider,TP+FN)
check_schneider$n0 <- with(check_schneider,TN+FP)
check_schneider
# Inspection suggests that the TP and/or FN counts
# in rows 118, 119, 120 may need to be changed.
# Let's reduce the TP counts from 39 to 38.
# But other corrections are conceivable as well.
FENO[118:120,"TP"] <- 38
### Let's try again
feno <- fitHoyer(data=FENO,
                 TP=TP,
                 FP=FP,
                 FN=FN,
                 TN=TN,
                 threshold=cutoff,
                 study=study,
                 dist="loglogistic",
                 testdirection="greater",
                 eval_threshold=seq(16,40,1),
                 smallest=2,
                 largest=120)
plot(feno,size="se",scale=0.03,
     main="Diagnostic Test Accuracy Meta-Analysis\nLoglogistic Model")
```

## How does `testdirection="less"` work?

We use the synthetic anaemia data set.
Smaller values in haemoglobin indicate disease.

```{r}
#| echo: TRUE
data("anaemia")
summary(anaemia$threshold)
```

```{r}
#| echo: TRUE
#| results: 'hide'
less <-  fitHoyer(data=anaemia,
                  TP=TP,
                  FP=FP,
                  FN=FN,
                  TN=TN,
                  threshold=threshold,
                  study=study,
                  dist="loglogistic",
                  testdirection="less",
                  eval_threshold=c(8,9,10,11,12,13),
                  smallest=5,
                  largest=17)
```

```{r}
#| echo: TRUE
#| fig.height: 8
#| fig.width: 8
less
summary(less)
plot(less,main="Diagnostic Test Accuracy Meta-Analysis\nLoglogistic Model")
```

## How does `fitHoyer` work internally? -- Optional advanced workflow

Most users should use the high-level `fitHoyer` function.
The lower-level functions are provided for advanced use and transparency.

### Step 1: How do we restructure the data set?

```{r}
#| echo: TRUE
res <- restructure_data(
    data = diabetes,
    TP = TP,
    FP = FP,
    FN = FN,
    TN = TN,
    study = study,
    threshold = threshold,
    testdirection = "greater",
    smallest = 2,
    largest = 10)
res
```

### Step 2: How do we get initial values?

```{r}
#| echo: TRUE
init <- initHoyerAFT(res$restructured,
                     dist="loglogistic")
init
```

### Step 3: How do we fit the model?

```{r}
#| echo: TRUE
#| results: 'hide'
hoyerAFT <- fitHoyerAFT(data=res,
                        init=init,
                        eval_threshold=c(5.0,5.5,6.0,6.5,7.0,7.5))
```

```{r}
#| echo: TRUE
hoyerAFT
summary(hoyerAFT)
```

## References

Reitsma, J. B., et al. (2005).
Bivariate analysis of sensitivity and specificity produces informative summary measures in diagnostic reviews.
*Journal of Clinical Epidemiology*, 58(10), 982–990.

Rutter, C. M., & Gatsonis, C. A.
(2001).
A hierarchical regression approach to meta-analysis of diagnostic test accuracy evaluations.
*Statistics in Medicine*, 20(19), 2865–2884.

Harbord, R. M., Deeks, J. J., Egger, M., Whiting, P., & Sterne, J. A. C.
(2007).
A unification of models for meta-analysis of diagnostic accuracy studies.
*Biostatistics*, 8(2), 239–251.

Riley, R. D., Ensor, J., Jackson, D., & Burke, D. L.
(2018).
Deriving percentage study weights in multi-parameter meta-analysis models.
*Statistical Methods in Medical Research*, 27(10), 2885–2905.

Hoyer, A., Hirt, S., Kuss, O.
(2018).
Meta-analysis of full ROC curves using bivariate time-to-event models for interval-censored data.
*Research Synthesis Methods*, 9(1), 62-72.

Deeks, J. J., Bossuyt, P. M., Leeflang, M. M., & Takwoingi, Y.
(editors) (2023).
Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy.
Version 2.0 (updated July 2023).
*Cochrane.*
