# dosresmeta-package: Multivariate dose-response meta-analysis In dosresmeta: Multivariate Dose-Response Meta-Analysis

## Description

It consists of a collection of functions to estimate dose-response relations from summarized dose-response data for both continuous and binary outcomes, and to combine them according to principles of (multivariate) random-effects model.

## Modeling framework

Dose-response meta-analysis represents a specific type of meta-analysis. Aim of such analysis is to reconstruct and combine study-specific curves from summarized dose-response data. Greenland and Longnecker originally developed the methodology in 1992 for pooling associations from epidemiological studies of binary outcomes. Extensions are currently proposed for other types of outcomes (e.g. continuous) from others study design, such as clinical trials.

The summarized dose-response data are most often presented in a tabular way, reporting the levels of the exposure (doses) and the corresponding outcome variable. The latter is usually expressed as contrast to the unexposed or baseline category (referent level). Examples are (log) relative risks, (log) odds ratios, (log) incidence rate ratios, mean differences, and standardized mean differences. Thus the outcome cannot be regarded as independent and a (co)variance matrix needs to be provided or approximated from the available data.See `covar.smd` and `covar.logrr` for more details.

## Estimation procedure

The pooled dose-response association can be estimated using two different approaches. The former consists of a two-stage procedure, where the study-specific trend are first estimated and then pooled across studies. Assuming yj is the vector of non-referent outcome values in each of i = 1, …, m studies, and Xi the related matrix of p transformations of the exposure (typically p = 1, 2), the dose-response model can be written as

yi = Xiβi + εi

with Si = (co)variance of εi known (available or reconstructed from the available data). The βi are then combined according to principles of (multivariate) random-effects meta-analytical models

βi ~ N( β, Vi + Ψ )

where Vi and Ψ indicate, respectively, the within study (co)variance (obtained in the first stage analysis) and the between study (co)variance.

The alternative approach, instead, consists of a one-stage (also known as pool-first) procedure. The data are pooled by concatenating the vector yi and vectors (or matrices) Xi. The (multivariate) random effects-model can be written as

yi = X_iβi + Z_iηi + εi

where β represents the fixed-effects parameter, η_i the vector (or matrix) of unobserved random-effects for thei-th study, and Zi coincides with Xi. The marginal model has a co(variance) matrix equal to Σ + ZiΨZi', where Σ is the block diagonal (co)variance with i-th diagonal block Si.

The two approaches provide similar results, despite the two-stage procedure may be more stable and faster in terms of convergence. In both the procedures the aim is to estimate the coefficients β and, for random-effects models, the components of the between-study (co)variance matrix Psi. Different estimators are implemented in the package. The estimation options available are

• Fixed-effects

• Maximum likelihood (ML)

• Restricted maximum likelihood (REML)

• Method of moments (currently available only for the two-stage procedure)

The fixed-effects model is fitted through generalized least squares (GLS), assuming the (co)variance structure, composed by the within-study error only, as completely known. Among random-effects models, ML and REML approaches provides fit criteria and inferential test derived from likelihood theory, such as AIC and likelihood ratio test, purticularly useful in a one-stage procedure. Further details on estimation methods are given in the related help pages.

## Functions and data included in the package

The structure of the package and the internal functions resemble those of the `mvmeta` package. See `mvmeta-package` for a general overview. The main function is `dosresmeta`, which performs the various models illustrated above. The function returns a list object of class "`dosresmeta`" (see `dosresmetaObject`).

The estimation is carried out internally through `dosresmeta.fit`, a wrapper which prepares the data and calls specific estimation functions for fitting the models, depending on the chosen procedure. For the two-stage procedure, the second part of the analysis is performed using the function `mvmeta.fit` while estimators for random-effects models are implemented in the functions `dosresmeta.ml` and `dosresmeta.reml` for (restricted) maximum likelihood. For likelihood-based methods, iterative optimizations algorithms are used for maximizing the (restricted) likelihood. Fitting parameter options are set by `dosresmeta.control`.

Method functions are available for objects of class "`dosremeta`" (see `dosresmetaObject` for a complete list). The method `summary` produces a list of class "`summary.dosremeta`" for summarizing the fit of the model and providing additional results. The method function `predict` computes predicted values, optionally for a set of new values of the predictors. `blup` gives the (empirical) best linear unbiased predictions for the unobserved random-effects. Other default or specific method functions for regression can be used on objects of class "`dosremeta`", such as `logLik`, `AIC` and `BIC`, among others. The method function `qtest.dosresmeta` (producing an object with class of the same name) performs the Cochran Q test for (residual) heterogeneity currently appropriate only for the two-stage approach.

Printing functions for the objects of classes defined above are also provided. Other functions are used internally in the source code, and not exported in the namespace. For users interested in getting into details of the package structure, these functions can be displayed using the triple colon (':::') operator. For instance, dosresmeta:::glsfit displays the code of the function glsfit.

The package includes the datasets `alcohol_crc`, `alcohol_cvd`, `ari`, and `cc_ex` as data frames, which are used in the examples.

Use citation("dosresmeta") to cite this package.

## Author(s)

Alessio Crippa, [email protected]

## References

Alessio Crippa, Nicola Orsini (2016). Multivariate Dose-Response Meta-Analysis: The dosresmeta R Package. Journal of Statistical Software, Code Snippets, 72(1), 1-15.doi:10.18637/jss.v072.c01

Greenland, S., Longnecker, M. P. (1992). Methods for trend estimation from summarized dose-response data, with applications to meta-analysis. American journal of epidemiology, 135(11), 1301-1309.

Orsini, N., Bellocco, R., Greenland, S. (2006). Generalized least squares for trend estimation of summarized dose-response data. Stata Journal, 6(1), 40.

Orsini, N., Li, R., Wolk, A., Khudyakov, P., Spiegelman, D. (2012). Meta-analysis for linear and nonlinear dose-response relations: examples, an evaluation of approximations, and software. American journal of epidemiology, 175(1), 66-73.

Gasparrini, A., Armstrong, B., Kenward, M. G. (2012). Multivariate meta-analysis for non-linear and other multi-parameter associations. Statistics in Medicine, 31(29), 3821-3839.

`dosresmeta` `mvmeta`