Function to fit fixed-effects models to \(2 \times 2\) table data via Peto's method. See below and the documentation of the metafor-package for more details on these models.
rma.peto(ai, bi, ci, di, n1i, n2i, data, slab, subset, add=1/2, to="only0", drop00=TRUE, level=95, digits, verbose=FALSE, ...)
vector to specify the \(2 \times 2\) table frequencies (upper left cell). See below and the documentation of the
vector to specify the \(2 \times 2\) table frequencies (upper right cell). See below and the documentation of the
vector to specify the \(2 \times 2\) table frequencies (lower left cell). See below and the documentation of the
vector to specify the \(2 \times 2\) table frequencies (lower right cell). See below and the documentation of the
vector to specify the group sizes or row totals (first group). See below and the documentation of the
vector to specify the group sizes or row totals (second group). See below and the documentation of the
optional data frame containing the data supplied to the function.
optional vector with labels for the \(k\) studies.
optional (logical or numeric) vector to specify the subset of studies that should be used for the analysis.
non-negative number to specify the amount to add to zero cells, counts, or frequencies when calculating the observed effect sizes or outcomes of the individual studies. Can also be a vector of two numbers, where the first number is used in the calculation of the observed effect sizes outcomes and the second number is used when applying Peto's method. See below and the documentation of the
character string to specify when the values under
logical to specify whether studies with no cases (or only cases) in both groups should be dropped when calculating the observed effect sizes or outcomes (the outcomes for such studies are set to
numeric value between 0 and 100 to specify the confidence interval level (the default is 95).
integer to specify the number of decimal places to which the printed results should be rounded (if unspecified, the default is 4).
logical to specify whether output should be generated on the progress of the model fitting (the default is
Specifying the Data
The studies are assumed to provide data in terms of \(2 \times 2\) tables of the form:
|outcome 1||outcome 2||total|
di denote the cell frequencies and
n2i the row totals. For example, in a set of randomized clinical trials (RCTs) or cohort studies, group 1 and group 2 may refer to the treatment (exposed) and placebo/control (not exposed) group, with outcome 1 denoting some event of interest (e.g., death) and outcome 2 its complement. In a set of case-control studies, group 1 and group 2 may refer to the group of cases and the group of controls, with outcome 1 denoting, for example, exposure to some risk factor and outcome 2 non-exposure.
An approach for aggregating \(2 \times 2\) table data of this type was suggested by Peto (see Yusuf et al., 1985). The method provides a weighted estimate of the (log) odds ratio under a fixed-effects model. The method is particularly advantageous when the event of interest is rare, but it should only be used when the group sizes within the individual studies are not too dissimilar and effect sizes are generally small (Greenland & Salvan, 1990; Sweeting et al., 2004; Bradburn et al., 2007). Note that the printed results are given both in terms of the log and the raw units (for easier interpretation).
Observed Effect Sizes or Outcomes of the Individual Studies
Peto's method itself does not require the calculation of the observed (log) odds ratios of the individual studies and directly makes use of the \(2 \times 2\) table counts. Zero cells are not a problem (except in extreme cases, such as when one of the two outcomes never occurs in any of the tables). Therefore, it is unnecessary to add some constant to the cell counts when there are zero cells.
However, for plotting and various other functions, it is necessary to calculate the observed (log) odds ratios for the \(k\) studies. Here, zero cells can be problematic, so adding a constant value to the cell counts ensures that all \(k\) values can be calculated. The
to arguments are used to specify what value should be added to the cell frequencies and under what circumstances when calculating the observed (log) odds ratios and when applying Peto's method. Similarly, the
drop00 argument is used to specify how studies with no cases (or only cases) in both groups should be handled. The documentation of the
escalc function explains how the
drop00 arguments work. If only a single value for these arguments is specified (as per default), then these values are used when calculating the observed (log) odds ratios and no adjustment to the cell counts is made when applying Peto's method. Alternatively, when specifying two values for these arguments, the first value applies when calculating the observed (log) odds ratios and the second value when applying Peto's method.
drop00 is set to
TRUE by default. Therefore, the observed (log) odds ratios for studies where
bi=di=0 are set to
NA. When applying Peto's method, such studies are not explicitly dropped (unless the second value of
drop00 argument is also set to
TRUE), but this is practically not necessary, as they do not actually influence the results (assuming no adjustment to the cell/event counts are made when applying Peto's method).
An object of class
c("rma.peto","rma"). The object is a list containing the following components:
aggregated log odds ratio.
standard error of the aggregated value.
test statistics of the aggregated value.
lower bound of the confidence interval.
upper bound of the confidence interval.
test statistic of the test for heterogeneity.
number of studies included in the analysis.
the vector of individual log odds ratios and corresponding sampling variances.
a list with the log-likelihood, deviance, AIC, BIC, and AICc values under the unrestricted and restricted likelihood.
some additional elements/values.
Forest, funnel, radial, L'Abbé, and Baujat plots can be obtained with
qqnorm.rma.peto function provides normal QQ plots of the standardized residuals. One can also just call
plot.rma.peto on the fitted model object to obtain various plots at once.
A cumulative meta-analysis (i.e., adding one observation at a time) can be obtained with
Bradburn, M. J., Deeks, J. J., Berlin, J. A., & Localio, A. R. (2007). Much ado about nothing: A comparison of the performance of meta-analytical methods with rare events. Statistics in Medicine, 26(1), 53--77. https://doi.org/10.1002/sim.2528
Greenland, S., & Salvan, A. (1990). Bias in the one-step method for pooling study results. Statistics in Medicine, 9(3), 247--252. https://doi.org/10.1002/sim.4780090307
Sweeting, M. J., Sutton, A. J., & Lambert, P. C. (2004). What to add to nothing? Use and avoidance of continuity corrections in meta-analysis of sparse data. Statistics in Medicine, 23(9), 1351--1375. https://doi.org/10.1002/sim.1761
Yusuf, S., Peto, R., Lewis, J., Collins, R., & Sleight, P. (1985). Beta blockade during and after myocardial infarction: An overview of the randomized trials. Progress in Cardiovascular Disease, 27(5), 335--371. https://doi.org/10.1016/s0033-0620(85)80003-7
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1--48. https://doi.org/10.18637/jss.v036.i03
### meta-analysis of the (log) odds ratios using Peto's method rma.peto(ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)#> #> Fixed-Effects Model (k = 13) #> #> I^2 (total heterogeneity / total variability): 92.85% #> H^2 (total variability / sampling variability): 13.98 #> #> Test for Heterogeneity: #> Q(df = 12) = 167.7302, p-val < .0001 #> #> Model Results (log scale): #> #> estimate se zval pval ci.lb ci.ub #> -0.4744 0.0407 -11.6689 <.0001 -0.5541 -0.3948 #> #> Model Results (OR scale): #> #> estimate ci.lb ci.ub #> 0.6222 0.5746 0.6738 #>