Function to create funnel plots.

funnel(x, …)

# S3 method for rma
funnel(x, yaxis="sei",
       xlim, ylim, xlab, ylab,
       steps=5, at, atransf, targs, digits, level=x$level,
       addtau2=FALSE, type="rstandard",
       back="lightgray", shade="white", hlines="white",
       refline, pch=19, pch.fill=21, col, bg, legend=FALSE, ci.res=1000, …)

# S3 method for default
funnel(x, vi, sei, ni, subset, yaxis="sei",
       xlim, ylim, xlab, ylab,
       steps=5, at, atransf, targs, digits, level=95,
       back="lightgray", shade="white", hlines="white",
       refline=0, pch=19, col, bg, legend=FALSE, ci.res=1000, …)

Arguments

x

an object of class "rma" or a vector with the observed effect sizes or outcomes.

vi

vector with the corresponding sampling variances.

sei

vector with the corresponding standard errors.

ni

vector with the corresponding sample sizes.

subset

optional vector indicating the subset of studies that should be included in the plot. This can be a logical vector of the same length as x or a numeric vector indicating the indices of the observations to include. Only relevant when passing a vector via x.

yaxis

either "sei", "vi", "seinv", "vinv", "ni", "ninv", "sqrtni", "sqrtninv", "lni", or "wi" to indicate what values should be placed on the y-axis. See ‘Details’.

xlim

x-axis limits. If unspecified, the function tries to set the x-axis limits to some sensible values.

ylim

y-axis limits. If unspecified, the function tries to set the y-axis limits to some sensible values.

xlab

title for the x-axis. If unspecified, the function tries to set an appropriate axis title.

ylab

title for the y-axis. If unspecified, the function tries to set an appropriate axis title.

steps

the number of tick marks for the y-axis (the default is 5).

at

position of the x-axis tick marks and corresponding labels. If unspecified, the function tries to set the tick mark positions/labels to some sensible values.

atransf

optional argument specifying the name of a function that should be used to transform the x-axis labels (e.g., atransf=exp; see also transf). If unspecified, no transformation is used.

targs

optional arguments needed by the function specified via atransf.

digits

integer specifying the number of decimal places to which the tick mark labels of the x- and y-axis should be rounded. Can also be a vector of two integers, the first specifying the number of decimal places for the x-axis, the second for the y-axis labels (e.g., c(2,3)). If unspecified, the function tries to set the argument to some sensible values.

level

numerical value between 0 and 100 specifying the level of the pseudo confidence interval region (for "rma" objects, the default is to take the value from the object). May also be a vector of values to obtain multiple regions. See ‘Examples’.

addtau2

logical to indicate whether the amount of heterogeneity should be accounted for when drawing the pseudo confidence interval region (the default is FALSE). Ignored when the model includes moderators and residuals are plotted. See ‘Details’.

type

either "rstandard" (default) or "rstudent" indicating whether the usual or deleted residuals should be used in creating the funnel plot when the model involves moderators. See ‘Details’.

back

color to use for the background of the plotting region.

shade

color to use for shading the pseudo confidence interval region. When level is a vector of values, different shading colors can be specified for each region.

hlines

color of the horizontal reference lines.

refline

value at which to draw the vertical reference line and, if drawn, where the pseudo confidence interval should be centered. If unspecified, the reference line is drawn at the fixed- or random-effects model estimate when the model does not include moderators and at zero when moderators are included (and therefore residuals are plotted) or when directly plotting observed effect sizes or outcomes.

pch

plotting symbol to use for the observed effect sizes or outcomes. By default, a solid circle is used. Can also be a vector of values. See points for other options.

pch.fill

plotting symbol to use for the effect sizes or outcomes filled in by the trim and fill method. By default, a circle is used. Only relevant when plotting an object created by the trimfill function.

col

optional character string specifying the name of a color to use for the points ("black" is used by default if not specified). Can also be a vector of color names.

bg

optional character string specifying the name of a background color for open plot symbols ("white" is used by default if not specified). Can also be a vector of color names.

legend

logical to indicate whether a legend should be added to the plot (can also be a keyword to indicate the position of the legend; see as in legend).

ci.res

integer specifying the number of y-axis values at which to calculate the bounds of the pseudo confidence interval. The default is 1000, which usually provides a sufficient resolution for the plotting.

other arguments.

Details

For fixed- and random-effects models (i.e., models not involving moderators), the plot shows the individual observed effect sizes or outcomes on the x-axis against the corresponding standard errors (i.e., the square root of the sampling variances) on the y-axis. A vertical line indicates the estimate based on the model. A pseudo confidence interval region is drawn around this value with bounds equal to \(± 1.96 SE\), where \(SE\) is the standard error value from the y-axis (assuming level=95). If addtau2=TRUE (only for models of class "rma.uni"), then the bounds of the pseudo confidence interval region are equal to \(± 1.96 \sqrt{SE^2 + \tau^2}\), where \(\tau^2\) is the amount of heterogeneity as estimated by the model.

For models involving moderators, the plot shows the residuals on the x-axis against their corresponding standard errors. Either the usual or deleted residuals can be used for that purpose (set via the type argument). See residuals.rma for more details on the different types of residuals.

With the atransf argument, the labels of the observed effect sizes or outcomes on the x-axis can be transformed with some suitable function. For example, when plotting log odds ratios, one could use transf=exp to obtain a funnel plot with the values on the x-axis corresponding to the odds ratios. See also transf for some transformation functions useful for meta-analyses.

Instead of placing the standard error value on the y-axis, several other options are available by setting the yaxis argument to:

  • yaxis="vi" for the sampling variance,

  • yaxis="seinv" for the inverse of the standard error,

  • yaxis="vinv" for the inverse of the sampling variance,

  • yaxis="ni" for the sample size,

  • yaxis="ninv" for the inverse of the sample size,

  • yaxis="sqrtni" for the square root sample size,

  • yaxis="sqrtninv" for the inverse of the square root sample size,

  • yaxis="lni" for the log of the sample size,

  • yaxis="wi" for the weights.

However, only when yaxis="sei" (the default) will the pseudo confidence region have the expected (upside-down) funnel shape with straight lines. Also, when placing (a function of) the sample size on the y-axis or the weights, then the pseudo confidence region cannot be drawn. See Sterne and Egger (2001) for more details on the choice of the y-axis.

If the object passed to the function comes from the trimfill function, the effect sizes or outcomes that are filled in by the trim and fill method are also added to the funnel plot. The symbol to use for plotting the filled in values can then be specified via the pch.fill argument.

One can also directly pass a vector of observed effect sizes or outcomes (via x) and the corresponding sampling variances (via vi), standard errors (via sei), and/or sample sizes (via ni) to the function. By default, the vertical reference line is then drawn at zero.

The arguments back, shade, and hlines can be set to NULL to suppress the shading and the horizontal reference lines.

Note

Placing (a function of) the sample size on the y-axis (i.e., using yaxis="ni", yaxis="ninv", yaxis="sqrtni", yaxis="sqrtninv", or yaxis="lni") is only possible when information about the sample sizes is actually stored within the object passed to the funnel function. That should automatically be the case when the observed outcomes were computed with the escalc function or when the observed outcomes were computed within the model fitting function. On the other hand, this will not automatically be the case when rma.uni was used together with the yi and vi arguments and the yi and vi values were not computed with escalc. In that case, it is still possible to pass information about the sample sizes to the rma.uni function (i.e., use rma.uni(yi, vi, ni)).

When using unweighted estimation, using yaxis="wi" will place all points on a horizontal line. When directly passing a vector of observed effect sizes or outcomes to the function, yaxis="wi" is equivalent to yaxis="vinv", except that the weights are expressed in percent.

When specifying vectors for pch, col, and/or bg, the variables specified are assumed to be of the same length as the data passed to the funnel function or the model fitting function (when using funnel on a model object). Any subsetting and removal of studies with missing values is automatically applied to the variables specified via pch, col, and bg.

Value

A data frame with components:

x

the x coordinates of the points that were plotted.

y

the y coordinates of the points that were plotted.

slab

study labels of the points that were plotted.

Note that the data frame is returned invisibly.

References

Light, R. J., & Pillemer, D. B. (1984). Summing up: The science of reviewing research. Cambridge, MA: Harvard University Press.

Peters, J. L., Sutton, A. J., Jones, D. R., Abrams, K. R., & Rushton, L. (2008). Contour-enhanced meta-analysis funnel plots help distinguish publication bias from other causes of asymmetry. Journal of Clinical Epidemiology, 61, 991--996.

Sterne, J. A. C., & Egger, M. (2001). Funnel plots for detecting bias in meta-analysis: Guidelines on choice of axis. Journal of Clinical Epidemiology, 54, 1046--1055.

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1--48. http://www.jstatsoft.org/v36/i03/.

See also

Examples

### copy BCG vaccine data into 'dat' dat <- dat.bcg ### calculate log risk ratios and corresponding sampling variances dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat) ### random-effects model res <- rma(yi, vi, data=dat) ### standard funnel plot funnel(res)
### show risk ratio values on x-axis (log scale) funnel(res, atransf=exp)
### passing log risk ratios and sampling variances directly to the function ### note: essentially the same plot, except that reference line is centered at zero funnel(dat$yi, dat$vi)
funnel(res, refline=0)
### funnel plot with risk ratio values on the x-axis (log scale) funnel(res, atransf=exp, at=log(c(.12, .25, .5, 1, 2)))
### contour-enhanced funnel plot centered at 0 (see Peters et al., 2008) funnel(res, level=c(90, 95, 99), shade=c("white", "gray55", "gray75"), refline=0, legend=TRUE)
### same, but show risk ratio values on the x-axis funnel(res, level=c(90, 95, 99), shade=c("white", "gray55", "gray75"), refline=0, legend=TRUE, atransf=exp, at=log(c(.10, .25, .5, 1, 2, 4, 10)))
### illustrate the use of vectors for 'pch' and 'col' res <- rma(yi, vi, data=dat, subset=2:10) funnel(res, pch=ifelse(dat$yi > -1, 19, 21), col=ifelse(sqrt(dat$vi) > .3, "red", "blue"))
### mixed-effects model with absolute latitude in the model res <- rma(yi, vi, mods = ~ ablat, data=dat) ### funnel plot of the residuals funnel(res)
### simulate a large meta-analytic dataset (correlations with rho = 0.2) ### with no heterogeneity or publication bias; then try out different ### versions of the funnel plot gencor <- function(rhoi, ni) { x1 <- rnorm(ni, mean=0, sd=1) x2 <- rnorm(ni, mean=0, sd=1) x3 <- rhoi*x1 + sqrt(1-rhoi^2)*x2 cor(x1, x3) } set.seed(1234) k <- 200 ### number of studies to simulate ni <- round(rchisq(k, df=2) * 20 + 20) ### simulate sample sizes (skewed distribution) ri <- mapply(gencor, rep(0.2,k), ni) ### simulate correlations res <- rma(measure="ZCOR", ri=ri, ni=ni, method="FE") ### use r-to-z transformed correlations funnel(res, yaxis="sei")
funnel(res, yaxis="vi")
funnel(res, yaxis="seinv")
funnel(res, yaxis="vinv")
funnel(res, yaxis="ni")
funnel(res, yaxis="ninv")
funnel(res, yaxis="sqrtni")
funnel(res, yaxis="sqrtninv")
funnel(res, yaxis="lni")
funnel(res, yaxis="wi")