mixmeta {mixmeta}

Fitting Standard and Extended Meta-Analysis and Meta-Regression Models

Description

The function mixmeta performs various meta-analytical models under a common mixed-effects framework, including standard univariate fixed and random-effects meta-analysis and meta-regression, and non-standard extensions such as multivariate, multilevel, longitudinal, and dose-response models. The function mixmeta.fit is a wrapper for actual fitting functions based on different estimation methods, usually called internally. See mixmeta-package for an overview.

Usage

mixmeta(formula, S, data, random, method="reml", bscov="unstr", offset, subset,
  contrasts=NULL, na.action, model=TRUE, control=list())

mixmeta.fit(X, Z, y, S, groups, method, bscov, control)

Arguments

Assuming a meta-analysis or meta-regression based on n units aggregated within m (outer-level) groups, k outcomes, p fixed-effects predictors, and q random-effects predictors:

formula

an object of class "formula" (or one that can be coerced to that class) offering a symbolic description of the linear predictor for the fixed-effects part of the model. Alternatively, for meta-analysis with no fixed-effects predictors, a single vector (for univariate models) or matrix-type object (for multivariate models). Terms in formula must be vector or matrix-type objects, optionally provided in the data argument below. See mixmetaFormula.

S

series of within-unit variances (or (co)variance matrices for multivariate models) of the estimated outcome(s). For univariate models, this is usually a n-dimensional vector. For multivariate models, it can be provided as: a m-dimensional list of k \times k matrices; a tri-dimensional k \times k \times m array; a matrix or data frame with n rows and k(k+1)/2 or k columns, depending on the availability of the within-unit correlations. mixmeta.fit accepts only the last option. Optionally, for more complex error structures, this argument can be omitted and passed through addSlist in control. See Details below.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in formula and random. If not found in data, the variables are taken from environment(formula), typically the environment from which mixmeta is called.

random

a one-sided formula (or a list of formulae for multilevel models) offering a symbolic description of the linear predictor(s) and grouping structure for the random-effects part of the model. The usual form is ~ z1 + ... + zq | g, with the grouping factor separated from the linear predictor by the symbol '|'. Multiple levels with the same linear predictor can be defined by separating multiple grouping factors using the symbol '/'. Alternatively, in a list form the grouping factors can also be provided as list names. In both cases, the levels are considered nested (from outer to inner following the order). See mixmetaFormula and Details below.

method

estimation method: "fixed" for fixed-effects models, "ml" or "reml" for random-effects models fitted through (restricted) maximum likelihood, "mm" for random-effects models fitted through method of moments, and "vc" for random-effects models fitted through variance components. See Details below. If "model.frame", the model frame is returned, as in lm or glm.

bscov

a character vector defining the structure of the random-effects (co)variance matrices. Default to "unstr" (unstructured). Names corresponding to grouping factors (see random above) can be used to refer to specific random-effects levels for non-default values. If unnamed, the values can be recycled. Among various (co)variance structures, the user can select "diag" (diagonal), "cs" (compound symmetry), "hcs" (heterogeneous compound symmetry), "ar1" (autoregressive of first order), or "fixed" (fixed). See also Details.

offset

optionally, a n-dimensional numeric vector used to specify an a priori known component in the linear predictor. One or more offset terms can be included in the formula instead or as well. See model.offset.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

contrasts

an optional list. See the contrasts.arg of model.matrix.

na.action

a function which indicates what should happen when the data contain NAs. Default to na.action setting of options, usually na.omit. na.exclude can be useful. See details on missing values in mixmeta.

model

a logical value indicating whether the model frame should be included as a component of the returned value. See the model.frame method function.

control

list of parameters for controlling the fitting process. These are passed to mixmeta.control to replace otherwise selected default values.

X

a n \times p design matrix containing the p fixed-effects predictors, appropriately ordered by groups. Usually produced internally by mixmeta from formula above.

Z

a n \times q design matrix (or a list of design matrices for multilevel models) containing the q random-effects predictors, appropriately ordered by groups. Usually produced internally by mixmeta from random above.

y

a n-dimensional vector (for univariate models) or m \times k matrix (for multivariate models) of outcomes, appropriately ordered by groups. Usually produced internally by mixmeta from formula above.

groups

matrix with n rows, with each column identifying the groups for each level of random-effects, appropriately ordered. Usually produced internally by mixmeta from random above.

Details

The function mixmeta resembles standard regression functions in R. See lme in particular, or lm or glm, for information on most of the arguments. Internally, this function assembles the data components, defines the (potentially multiple) grouping levels and re-order the data accordingly, and then pass them to mixmeta.fit. This is a wrapper for actual fitting functions that are automatically selected. Functions other than mixmeta are not expected to be called directly for model fitting.

Fixed or random-effects models for meta-analysis are simply defined using y ~ 1 in formula, where y is a response vector optionally stored in data. In meta-regression models, other terms are added in the right-hand side of the formula as y ~ x1 + ... + xp, defining the linear meta-predictor. Factors, variable transformations and interactions are allowed, following the usual formula specification (see mixmetaFormula).

In this standard usage, each of the n rows is assumed to represent a single estimate of an outcome from a set of independent studies. In random-effects models, the grouping structure is automatically derived by assigning a group to each row of data (with m=n). Extensions to multivariate models (k>1) are straightforward, and only require using a matrix in the left-hand side, where each of the k columns represents a different outcome, or the form cbind(y1, ..., yk) ~ 1. See mixmetaFormula.

Non-standard random-effects models can be specified through the optional argument random. This is commonly represented by a one-sided formula, whose basic random-intercept form is ~ 1 | g, where g is a grouping factor. A more complex linear meta-predictor for the random-effects part can be also specified by ~ z1 + ... + zq | g. The argument random also accepts a list of one-sided formulae referring to multiple random-effects levels (see mixmetaFormula). The use of random extends the standard meta-analytical setting by relaxing the assumption of independence between units, allowing multiple estimates from the same group (with m<n) and multiple nested grouping levels. This provides the possibility to fit longitudinal, multilevel, and dose-response meta-analysis, among other extensions. See the examples below.

The argument bscov allows the definition of specific structures for the random-effects (co)variance matrices corresponding the each level. The default unstructured form requires kq(kq+1)/2 parameters for a single-level meta-analysis. The choice of other structures reduces the number of parameters, although requiring stronger assumptions. More information and complete list of options is available at a specific help page (see mixmetaCovStruct).

The within-unit (co)variances are provided through the argument S, usually as a matrix. If the correlations are available, each of the m row represents the k(k+1)/2 vectorized entries of the lower triangle of the related (co)variance matrix, taken by column (see xpndMat). If correlations are not available, each row represents the k variances, and the correlations are inputted internally through the argument Scor of the control list (see inputcov). For more complex error structures that span multiple units, the argument S can be omitted and passed through addSlist in control, although requiring the observations to be re-ordered accordingly to groups (see mixmeta.control).

Different estimator are available in the package mixmeta and chosen through the argument method, with related fitting functions called internally. In the current version, the options are:

• method="fixed": Fixed-effects estimator

• method="ml": Maximum likelihood (ML) estimator

• method="reml": Restricted maximum likelihood (REML) estimator

• method="mm": Method of moments estimator

• method="vc": Variance components estimator

Note that non-standard random-effects models and the use of structured (co)variance matrices are only available for "ml" and "reml" methods. See their help pages for further details on the estimation procedures, following the links above.

Missing values are allowed in both sides of formula. In the case of missing predictors (right-hand side of formula), the related unit is entirely excluded from estimation. In contrast, a unit still contributes to estimation if at least outcome is non-missing. This behaviour is different from standard regression functions such as lm or glm. Before the call to mixmeta.fit, units matching such stricter missing definition are removed from the the model frame. The missing pattern in S must be consistent with that in y. See further details on handling missing values in mixmeta.

The fitting procedure can be controlled through the additional terms specified in control, which are passed to the function mixmeta.control.

Value

The mixmeta function typically returns a list object of class "mixmeta" representing the meta-analytical model fit, as described in mixmetaObject. When method="data.frame", the model is not fitted and the model frame is returned, namely a data frame with special attributes (see the default method model.frame) and, in this case, the additional class "data.frame.mixmeta".

The wrapper function mixmeta.fit is usually called internally in mixmeta, and returns an intermediate list object with some of the components expected in the "mixmeta" class.

Several method functions for regression objects are available, either default or specifically written for the "mixmeta" class. See mixmetaObject for a complete list.

Author(s)

Antonio Gasparrini <[e-mail]> and Francesco Sera <[e-mail]>

References

Sera F, Gasparrini A. (2019). An extended mixed-effects framework for meta-analysis.Statistics in Medicine. 2019;38(29):5429-5444. [Freely available here].

Gasparrini A, Armstrong B, Kenward MG (2012). Multivariate meta-analysis for non-linear and other multi-parameter associations. Statistics in Medicine. 31(29):3821–3839. [Freely available here].

See Also

See additional info on the estimation procedures at the related page of the fitting functions See mixmetaFormula for the use of formulae to define the fixed and random parts of the model. See alternative (co)variance structures for likelihood-based estimation methods. See handling of missing values in mixmeta. See lme, lm or glm for standard regression functions. See mixmeta-package for an overview of this modelling framework.

Examples

### STANDARD MODELS

# RANDOM-EFFECTS META-ANALYSIS, ESTIMATED WITH REML
model <- mixmeta(logor, logorvar, data=bcg)
summary(model)

# RANDOM-EFFECTS META-REGRESSION, ESTIMATED WITH ML
model <- mixmeta(logor~ablat, logorvar, data=bcg, method="ml")
summary(model)


### MAIN METHOD FUNCTIONS

# COEFFICIENTS AND (CO)VARIANCE MATRIX
coef(model)
vcov(model)

# RESIDUALS AND FITTED VALUES
residuals(model)
fitted(model)

# MODEL FRAME AND MODEL MATRIX
model.frame(model)
model.matrix(model)

# LOG-LIKELIHOOD AND AIC VALUE
logLik(model)
AIC(model)

# COCHRAN Q TEST FOR RESIDUAL HETEROGENEITY
qtest(model)


### PREDICTIONS

# PREDICTED EFFECTS
predict(model)
predict(model, se=TRUE)
predict(model, newdata=data.frame(ablat=2:5*10), ci=TRUE)

# BEST LINEAR UNBIASED PREDICTION
blup(model)
blup(model, pi=TRUE)

# SEE help(predict.mixmeta) AND help(BLUP.mixmeta) FOR MORE INFO


### MULTIVARIATE MODELS

### BIVARIATE MODELS
model <- mixmeta(cbind(PD,AL) ~ pubyear, S=berkey98[5:7], data=berkey98)
summary(model)
residuals(model)

### MULTIVARIATE META-ANALYSIS WITH MORE THAN 2 OUTCOMES
y <- as.matrix(fibrinogen[2:5])
S <- as.matrix(fibrinogen[6:15])
model <- mixmeta(y, S)
summary(model)
predict(model, se=TRUE)
predict(model, se=TRUE, aggregate="outcome")


### OTHER EXTENSIONS

# MULTILEVEL META-ANALYSIS
model <- mixmeta(effect, var, random= ~ 1|district/study, data=school)
summary(model)
# SEE help(school) AND help(thrombolytic) FOR MORE EXAMPLES

# DOSE-RESPONSE META-ANALYSIS (SIMPLIFIED)
model <- mixmeta(logrr ~ 0 + dose, S=se^2, random= ~ 0 + dose|id, data=alcohol,
 subset=!is.na(se))
summary(model)
# SEE help(alcohol) FOR MORE EXAMPLES

# LONGITUDINAL META-ANALYSIS
model <- mixmeta(logOR~time, S=logORvar, random=~I(time-15)|study, data=gliomas)
summary(model)
# SEE help(gliomas) AND help(dbs) FOR MORE EXAMPLES


### FIXED-EFFECTS MODELS AND ALTERNATIVE ESTIMATORS

# FIXED-EFFECTS MODEL
model <- mixmeta(sbp~ish, S=sbp_se^2, data=hyp, method="fixed")
summary(model)

# METHOD OF MOMENTS
S <- as.matrix(hsls[5:10])
model <- mixmeta(cbind(b1,b2,b3), S, data=hsls, method="mm")
summary(model)

# VARIANCE COMPONENTS ESTIMATOR
model <- mixmeta(cbind(PD,AL)~pubyear, S=berkey98[5:7], data=berkey98,
  method="vc")
summary(model)


### IN THE PRESENCE OF MISSING VALUES

# RUN THE MODEL
y <- as.matrix(smoking[11:13])
S <- as.matrix(smoking[14:19])
model <- mixmeta(y, S)
summary(model)
model.frame(model)

# SEE help(na.omit.data.frame.mixmeta) FOR MORE EXAMPLES


### WHEN WITHIN-STUDY COVIARIANCES ARE NOT AVAILABLE AND/OR NEED TO BE INPUTTED

# GENERATE S
(S <- inputcov(hyp[c("sbp_se","dbp_se")], cor=hyp$rho))

# RUN THE MODEL
model <- mixmeta(cbind(sbp,dbp), S=S, data=hyp)

# INPUTTING THE CORRELATION DIRECTLY IN THE MODEL
model <- mixmeta(cbind(y1,y2), cbind(V1,V2), data=p53, control=list(Scor=0.95))
summary(model)

# SEE help(hyp) AND help(p53) FOR MORE EXAMPLES


### STRUCTURING THE BETWEEN-STUDY (CO)VARIANCE

# DIAGONAL
S <- as.matrix(hsls[5:10])
model <- mixmeta(cbind(b1,b2,b3), S, data=hsls, bscov="diag")
summary(model)
model$Psi

# COMPOUND SYMMETRY
model <- mixmeta(cbind(b1,b2,b3), S, data=hsls, bscov="cs")
summary(model)
model$Psi

# SEE help(mixmetaCovStruct) FOR DETAILS AND ADDITIONAL EXAMPLES


### USE OF THE CONTROL LIST

# PRINT THE ITERATIONS AND CHANGE THE DEFAULT FOR STARTING VALUES
y <- as.matrix(smoking[11:13])
S <- as.matrix(smoking[14:19])
model <- mixmeta(y, S, control=list(showiter=TRUE, igls.inititer=20))

# SEE help(mixmeta.control) FOR FURTHER DETAILS