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introduction

2025-01-09

Source: vignettes/introduction.Rmd
introduction.Rmd

MetaROC Overview

The MetaROC package provides a suite of models for the meta-analysis of receiver operating characteristic (ROC) curves. It includes generalized linear mixed models (GLMM), linear mixed models (LMM), copula models, and Weibull models. While the latter three have already been established in the field, the MetaROC package introduces a novel GLMM approach to the meta-analysis of ROC curves.

A brief overview of this new approach will be provided. However, for a deeper understanding, we recommend reading the publication: ‘A Discrete Time-to-Event Model for the Meta-Analysis of Full ROC Curves’, by Ferdinand V. Stoye, Claudia Tschammler, Oliver Kuss, and Annika Hoyer (via).

The MetaROC package provides functions to preprocess real event data, simulate data for any of the mentioned models, fit the models to the data, evaluate the models, and create a comprehensive wrapper to perform these steps seamlessly. Finally some basic plotting functions will be provided to plot dataset as well as the result of a model evaluation.

GLMM Specification

The newly implemented Generalized Linear Mixed Model (GLMM) approach utilizes a discrete proportional hazard model to handle different diagnostic thresholds. These thresholds, used to estimate sensitivity and specificity as reported by individual studies, are treated as categorical variables within the GLMM framework.

This approach employs both the logit and the asymmetric complementary log-log (cloglog) link functions. For instance, using the cloglog link, the GLMM modeling the discrete hazard as the event probability of a study for data from a given threshold can be formulated as follows:

For studies ii, i=1,…,Ii=1,\dots,I, individuals jj, j=1,…,Nij=1,\dots,N_i, thresholds ckc_k, k=1,…,Kik=1,\dots,K_i:

yijk∼𝐵𝑒𝑟𝑛(λijck),λijck=h(α0ik+𝐱1ij⊤⋅𝛃𝟏+𝐱1ij⊤⋅𝛃2⋅x2ij+γ1⋅x2ij+γ2⋅(1−x2ij)),h(x)=1−exp(−exp(x)). \begin{align*} y_{ijk} &\sim \textit{Bern}(\lambda_{ijc_k}),\nonumber\\ \lambda_{ijc_k} &= h\left(\alpha_{0ik} + \boldsymbol{x}_{1ij}^\top\cdot\boldsymbol{\beta_1} + \boldsymbol{x}_{1ij}^\top\cdot\boldsymbol{\beta}_2\cdot x_{2ij} + \gamma_1\cdot x_{2ij} + \gamma_2\cdot (1-x_{2ij})\right),\nonumber\\ h(x) &= 1-\exp(-\exp(x)). \end{align*}

GLMM estimation

The sensitivity of our model, given a threshold value, is the probability of diseased individuals having a test value larger than the threshold, thus being detected as true positives. Correspondingly, the specificity is the probability of non-diseased individuals having a smaller test value than the given threshold. When modeling the discrete hazard h(x)h(x) by applying the inverse complementary log-log (cloglog) link to the linear term, this results in:

sens(ck∣𝐱)=∏t=1k(1−λct)=∏t=1k1−(1−exp(−exp(β1t)))=∏t=1kexp(−exp(β1t)) \text{sens}(c_k\mid \boldsymbol{x}) = \prod_{t=1}^{k} (1-\lambda_{c_t}) = \prod_{t=1}^{k} 1-(1-\exp(-\exp(\beta_{1t}))) = \prod_{t=1}^{k} \exp(-\exp(\beta_{1t}))

spec(ck∣𝐱)=1−∏t=1k(1−λct)=1−∏t=1k1−(1−exp(−exp(β1t+β2t)))=1−∏t=1kexp(−exp(β1t+β2t)) \text{spec}(c_k\mid \boldsymbol{x}) = 1-\prod_{t=1}^{k} (1-\lambda_{c_t}) = 1-\prod_{t=1}^{k} 1-(1-\exp(-\exp(\beta_{1t}+\beta_{2t}))) = 1-\prod_{t=1}^{k} \exp(-\exp(\beta_{1t}+\beta_{2t}))

Here, β1t\beta_{1t} and β2t\beta_{2t} are the fixed effects of the linear term. By plugging the estimated values of β1t\beta_{1t} and β2t\beta_{2t} into these equations, we obtain the estimated sensitivity and specificity for each threshold.