metaprop: Meta-analysis of single proportions

Description

Calculation of an overall proportion from studies reporting a single proportion. Inverse variance method and generalised linear mixed model (GLMM) are available for pooling. For GLMMs, the rma.glmm function from R package metafor (Viechtbauer 2010) is called internally.

Usage

metaprop(
  event,
  n,
  studlab,
  data = NULL,
  subset = NULL,
  exclude = NULL,
  cluster = NULL,
  method,
  sm = gs("smprop"),
  incr = gs("incr"),
  method.incr = gs("method.incr"),
  method.ci = gs("method.ci.prop"),
  level = gs("level"),
  common = gs("common"),
  random = gs("random") | !is.null(tau.preset),
  overall = common | random,
  overall.hetstat = common | random,
  prediction = gs("prediction") | !missing(method.predict),
  method.tau = ifelse(!is.na(charmatch(tolower(method), "glmm", nomatch = NA)), "ML",
    gs("method.tau")),
  method.tau.ci = gs("method.tau.ci"),
  tau.preset = NULL,
  TE.tau = NULL,
  tau.common = gs("tau.common"),
  level.ma = gs("level.ma"),
  method.random.ci = gs("method.random.ci"),
  adhoc.hakn.ci = gs("adhoc.hakn.ci"),
  level.predict = gs("level.predict"),
  method.predict = gs("method.predict"),
  adhoc.hakn.pi = gs("adhoc.hakn.pi"),
  null.effect = NA,
  method.bias = gs("method.bias"),
  backtransf = gs("backtransf"),
  pscale = 1,
  text.common = gs("text.common"),
  text.random = gs("text.random"),
  text.predict = gs("text.predict"),
  text.w.common = gs("text.w.common"),
  text.w.random = gs("text.w.random"),
  title = gs("title"),
  complab = gs("complab"),
  outclab = "",
  subgroup,
  subgroup.name = NULL,
  print.subgroup.name = gs("print.subgroup.name"),
  sep.subgroup = gs("sep.subgroup"),
  test.subgroup = gs("test.subgroup"),
  prediction.subgroup = gs("prediction.subgroup"),
  byvar,
  hakn,
  adhoc.hakn,
  keepdata = gs("keepdata"),
  warn = gs("warn"),
  warn.deprecated = gs("warn.deprecated"),
  control = NULL,
  ...
)

Value

An object of class c("metaprop", "meta") with corresponding generic functions (see meta-object).

Arguments

event: Number of events.
n: Number of observations.
studlab: An optional vector with study labels.
data: An optional data frame containing the study information, i.e., event and n.
subset: An optional vector specifying a subset of studies to be used.
exclude: An optional vector specifying studies to exclude from meta-analysis, however, to include in printouts and forest plots.
cluster: An optional vector specifying which estimates come from the same cluster resulting in the use of a three-level meta-analysis model.
method: A character string indicating which method is to be used for pooling of studies. One of "Inverse" and "GLMM", can be abbreviated.
sm: A character string indicating which summary measure ("PLOGIT", "PAS", "PFT", "PLN", or "PRAW") is to be used for pooling of studies, see Details.
incr: A numeric which is added to event number and sample size of studies with zero or all events, i.e., studies with an event probability of either 0 or 1.
method.incr: A character string indicating which continuity correction method should be used ("only0", "if0all", or "all"), see Details.
method.ci: A character string indicating which method is used to calculate confidence intervals for individual studies, see Details.
level: The level used to calculate confidence intervals for individual studies.
common: A logical indicating whether a common effect meta-analysis should be conducted.
random: A logical indicating whether a random effects meta-analysis should be conducted.
overall: A logical indicating whether overall summaries should be reported. This argument is useful in a meta-analysis with subgroups if overall results should not be reported.
overall.hetstat: A logical value indicating whether to print heterogeneity measures for overall treatment comparisons. This argument is useful in a meta-analysis with subgroups if heterogeneity statistics should only be printed on subgroup level.
prediction: A logical indicating whether a prediction interval should be printed.
method.tau: A character string indicating which method is used to estimate the between-study variance \(\tau^2\) and its square root \(\tau\) (see meta-package).
method.tau.ci: A character string indicating which method is used to estimate the confidence interval of \(\tau^2\) and \(\tau\) (see meta-package).
tau.preset: Prespecified value for the square root of the between-study variance \(\tau^2\).
TE.tau: Overall treatment effect used to estimate the between-study variance tau-squared.
tau.common: A logical indicating whether tau-squared should be the same across subgroups.
level.ma: The level used to calculate confidence intervals for meta-analysis estimates.
method.random.ci: A character string indicating which method is used to calculate confidence interval and test statistic for random effects estimate (see meta-package).
adhoc.hakn.ci: A character string indicating whether an ad hoc variance correction should be applied in the case of an arbitrarily small Hartung-Knapp variance estimate (see meta-package).
level.predict: The level used to calculate prediction interval for a new study.
method.predict: A character string indicating which method is used to calculate a prediction interval (see meta-package).
adhoc.hakn.pi: A character string indicating whether an ad hoc variance correction should be applied for prediction interval (see meta-package).
null.effect: A numeric value specifying the effect under the null hypothesis.
method.bias: A character string indicating which test is to be used. Either "Begg", "Egger", or "Thompson", can be abbreviated. See function metabias.
backtransf: A logical indicating whether results for transformed proportions (argument sm != "PRAW") should be back transformed in printouts and plots. If TRUE (default), results will be presented as proportions; otherwise transformed proportions will be shown. See Details for presentation of confidence intervals.
pscale: A numeric defining a scaling factor for printing of single event probabilities.
text.common: A character string used in printouts and forest plot to label the pooled common effect estimate.
text.random: A character string used in printouts and forest plot to label the pooled random effects estimate.
text.predict: A character string used in printouts and forest plot to label the prediction interval.
text.w.common: A character string used to label weights of common effect model.
text.w.random: A character string used to label weights of random effects model.
title: Title of meta-analysis / systematic review.
complab: Comparison label.
outclab: Outcome label.
subgroup: An optional vector to conduct a meta-analysis with subgroups.
subgroup.name: A character string with a name for the subgroup variable.
print.subgroup.name: A logical indicating whether the name of the subgroup variable should be printed in front of the group labels.
sep.subgroup: A character string defining the separator between name of subgroup variable and subgroup label.
test.subgroup: A logical value indicating whether to print results of test for subgroup differences.
prediction.subgroup: A logical indicating whether prediction intervals should be printed for subgroups.
byvar: Deprecated argument (replaced by 'subgroup').
hakn: Deprecated argument (replaced by 'method.random.ci').
adhoc.hakn: Deprecated argument (replaced by 'adhoc.hakn.ci').
keepdata: A logical indicating whether original data (set) should be kept in meta object.
warn: A logical indicating whether the addition of incr to studies with zero or all events should result in a warning.
warn.deprecated: A logical indicating whether warnings should be printed if deprecated arguments are used.
control: An optional list to control the iterative process to estimate the between-study variance \(\tau^2\). This argument is passed on to rma.uni or rma.glmm, respectively.
...: Additional arguments passed on to rma.glmm function and to catch deprecated arguments.

Author

Guido Schwarzer guido.schwarzer@uniklinik-freiburg.de

Details

This function provides methods for common effect and random effects meta-analysis of single proportions to calculate an overall proportion. Note, you should use R function metabin to compare proportions of pairwise comparisons instead of using metaprop for each treatment arm separately which will break randomisation in randomised controlled trials.

The following transformations of proportions are implemented to calculate an overall proportion:

Logit transformation (sm = "PLOGIT", default)
Arcsine transformation (sm = "PAS")
Freeman-Tukey Double arcsine transformation (sm = "PFT")
Log transformation (sm = "PLN")
No transformation (sm = "PRAW")

List elements TE, TE.common, TE.random, etc., contain the transformed proportions. In printouts and plots these values are back transformed if argument backtransf = TRUE (default).

A generalised linear mixed model (GLMM) - more specific, a random intercept logistic regression model - can be utilised for the meta-analysis of proportions (Stijnen et al., 2010). This is the default method for the logit transformation (argument sm = "PLOGIT"). Internally, the rma.glmm function from R package metafor is called to fit a GLMM.

Classic meta-analysis (Borenstein et al., 2010) utilising the (un)transformed proportions and corresponding standard errors in the inverse variance method is conducted by calling the metagen function internally. This is the only available method for all transformations but the logit transformation. The classic meta-analysis model with logit transformed proportions is used by setting argument method = "Inverse".

A three-level random effects meta-analysis model (Van den Noortgate et al., 2013) is utilized if argument cluster is used and at least one cluster provides more than one estimate. Internally, rma.mv is called to conduct the analysis and weights.rma.mv with argument type = "rowsum" is used to calculate random effects weights.

Default settings are utilised for several arguments (assignments using gs function). These defaults can be changed for the current R session using the settings.meta function.

Furthermore, R function update.meta can be used to rerun a meta-analysis with different settings.

Choice of transformation / meta-analysis method

Contradictory recommendations on the use of transformations of proportions have been published in the literature. For example, Barendregt et al. (2013) recommend the use of the Freeman-Tukey double arcsine transformation instead of the logit transformation whereas Warton & Hui (2011) strongly advise to use generalised linear mixed models with the logit transformation instead of the arcsine transformation.

Schwarzer et al. (2019) describe seriously misleading results in a meta-analysis with very different sample sizes due to problems with the back-transformation of the Freeman-Tukey transformation which requires a single sample size (Miller, 1978). Accordingly, Schwarzer et al. (2019) also recommend to use GLMMs for the meta-analysis of single proportions, however, admit that individual study weights are not available with this method. Meta-analysts which require individual study weights should consider the inverse variance method with the arcsine or logit transformation.

In order to prevent misleading conclusions for the Freeman-Tukey double arcsine transformation, sensitivity analyses using other transformations or using a range of sample sizes should be conducted (Schwarzer et al., 2019).

Continuity correction

Three approaches are available to apply a continuity correction:

Only studies with a zero cell count (method.incr = "only0")
All studies if at least one study has a zero cell count (method.incr = "if0all")
All studies irrespective of zero cell counts (method.incr = "all")

If the summary measure is equal to "PLOGIT", "PLN", or "PRAW", the continuity correction is applied if a study has either zero or all events, i.e., an event probability of either 0 or 1.

By default, 0.5 is used as continuity correction (argument incr). This continuity correction is used both to calculate individual study results with confidence limits and to conduct meta-analysis based on the inverse variance method. For GLMMs no continuity correction is used.

Confidence intervals for individual studies

Various methods are available to calculate confidence intervals for individual study results (see Agresti & Coull 1998 and Newcombe 1988):

Clopper-Pearson interval also called 'exact' binomial interval (method.ci = "CP", default)
Wilson Score interval (method.ci = "WS")
Wilson Score interval with continuity correction (method.ci = "WSCC")
Agresti-Coull interval (method.ci = "AC")
Simple approximation interval (method.ci = "SA")
Simple approximation interval with continuity correction (method.ci = "SACC")
Normal approximation interval based on summary measure, i.e. defined by argument sm (method.ci = "NAsm")

Note, with exception of the normal approximation based on the summary measure, i.e. method.ci = "NAsm", the same confidence interval is calculated for individual studies for any summary measure (argument sm) as only number of events and observations are used in the calculation disregarding the chosen transformation.

Results will be presented for transformed proportions if argument backtransf = FALSE. In this case, argument method.ci = "NAsm" is used, i.e. confidence intervals based on the normal approximation based on the summary measure.

Subgroup analysis

Argument subgroup can be used to conduct subgroup analysis for a categorical covariate. The metareg function can be used instead for more than one categorical covariate or continuous covariates.

Specify the null hypothesis of test for an overall proportion

Argument null.effect can be used to specify the proportion used under the null hypothesis in a test for an overall effect.

By default (null.effect = NA), no hypothesis test is conducted as it is unclear which value is a sensible choice for the data at hand. An overall proportion of 50%, for example, could be tested by setting argument null.effect = 0.5.

Note, all tests for an overall effect are two-sided with the alternative hypothesis that the effect is unequal to null.effect.

Exclusion of studies from meta-analysis

Arguments subset and exclude can be used to exclude studies from the meta-analysis. Studies are removed completely from the meta-analysis using argument subset, while excluded studies are shown in printouts and forest plots using argument exclude (see Examples in metagen). Meta-analysis results are the same for both arguments.

Presentation of meta-analysis results

Internally, both common effect and random effects models are calculated regardless of values choosen for arguments common and random. Accordingly, the estimate for the random effects model can be extracted from component TE.random of an object of class "meta" even if argument random = FALSE. However, all functions in R package meta will adequately consider the values for common and random. E.g. function print.meta will not print results for the random effects model if random = FALSE.

Argument pscale can be used to rescale proportions, e.g. pscale = 1000 means that proportions are expressed as events per 1000 observations. This is useful in situations with (very) low event probabilities.

A prediction interval will only be shown if prediction = TRUE.

References

Agresti A & Coull BA (1998): Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52, 119--26

Barendregt JJ, Doi SA, Lee YY, Norman RE, Vos T (2013): Meta-analysis of prevalence. Journal of Epidemiology and Community Health, 67, 974--8

Borenstein M, Hedges LV, Higgins JP, Rothstein HR (2010): A basic introduction to fixed-effect and random-effects models for meta-analysis. Research Synthesis Methods, 1, 97--111

Freeman MF & Tukey JW (1950): Transformations related to the angular and the square root. Annals of Mathematical Statistics, 21, 607--11

Miller JJ (1978): The inverse of the Freeman-Tukey double arcsine transformation. The American Statistician, 32, 138

Newcombe RG (1998): Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine, 17, 857--72

Pettigrew HM, Gart JJ, Thomas DG (1986): The bias and higher cumulants of the logarithm of a binomial variate. Biometrika, 73, 425--35

Schwarzer G, Chemaitelly H, Abu-Raddad LJ, Rücker G (2019): Seriously misleading results using inverse of Freeman-Tukey double arcsine transformation in meta-analysis of single proportions. Research Synthesis Methods, 10, 476--83

Stijnen T, Hamza TH, Ozdemir P (2010): Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29, 3046--67

Van den Noortgate W, López-López JA, Marín-Martínez F, Sánchez-Meca J (2013): Three-level meta-analysis of dependent effect sizes. Behavior Research Methods, 45, 576--94

Viechtbauer W (2010): Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36, 1--48

Warton DI, Hui FKC (2011): The arcsine is asinine: the analysis of proportions in ecology. Ecology, 92, 3--10

Examples

Run this code

# Meta-analysis using generalised linear mixed model
#
metaprop(4:1, 10 * 1:4)

# Apply various classic meta-analysis methods to estimate
# proportions
#
m1 <- metaprop(4:1, 10 * 1:4, method = "Inverse")
m2 <- update(m1, sm = "PAS")
m3 <- update(m1, sm = "PRAW")
m4 <- update(m1, sm = "PLN")
m5 <- update(m1, sm = "PFT")
#
m1
m2
m3
m4
m5
#
forest(m1)
if (FALSE) {
forest(m2)
forest(m3)
forest(m3, pscale = 100)
forest(m4)
forest(m5)
}

# Do not back transform results, e.g. print logit transformed
# proportions if sm = "PLOGIT" and store old settings
#
oldset <- settings.meta(backtransf = FALSE)
#
m6  <- metaprop(4:1, c(10, 20, 30, 40), method = "Inverse")
m7  <- update(m6, sm = "PAS")
m8  <- update(m6, sm = "PRAW")
m9  <- update(m6, sm = "PLN")
m10 <- update(m6, sm = "PFT")
#
forest(m6)
if (FALSE) {
forest(m7)
forest(m8)
forest(m8, pscale = 100)
forest(m9)
forest(m10)
}

# Use old settings
#
settings.meta(oldset)

# Examples with zero events
#
m1 <- metaprop(c(0, 0, 10, 10), rep(100, 4), method = "Inverse")
m2 <- metaprop(c(0, 0, 10, 10), rep(100, 4), incr = 0.1, method = "Inverse")
#
m1
m2
#
if (FALSE) {
forest(m1)
forest(m2)
}

# Example from Miller (1978):
#
death <- c(3, 6, 10, 1)
animals <- c(11, 17, 21, 6)
#
m3 <- metaprop(death, animals, sm = "PFT")
forest(m3)

# Data examples from Newcombe (1998)
# - apply various methods to estimate confidence intervals for
#   individual studies
#
event <- c(81, 15, 0, 1)
n <- c(263, 148, 20, 29)
#
m1 <- metaprop(event, n, method.ci = "SA", method = "Inverse")
m2 <- update(m1, method.ci = "SACC")
m3 <- update(m1, method.ci = "WS")
m4 <- update(m1, method.ci = "WSCC")
m5 <- update(m1, method.ci = "CP")
#
lower <- round(rbind(NA, m1$lower, m2$lower, NA, m3$lower,
  m4$lower, NA, m5$lower), 4)
upper <- round(rbind(NA, m1$upper, m2$upper, NA, m3$upper,
  m4$upper, NA, m5$upper), 4)
#
tab1 <- data.frame(
  scen1 = meta:::formatCI(lower[, 1], upper[, 1]),
  scen2 = meta:::formatCI(lower[, 2], upper[, 2]),
  scen3 = meta:::formatCI(lower[, 3], upper[, 3]),
  scen4 = meta:::formatCI(lower[, 4], upper[, 4])
  )
names(tab1) <- c("r=81, n=263", "r=15, n=148",
  "r=0, n=20", "r=1, n=29")
row.names(tab1) <- c("Simple", "- SA", "- SACC",
  "Score", "- WS", "- WSCC", "Binomial", "- CP")
tab1[is.na(tab1)] <- ""
# Newcombe (1998), Table I, methods 1-5:
tab1

# Same confidence interval, i.e. unaffected by choice of summary
# measure
#
print(metaprop(event, n, method.ci = "WS", method = "Inverse"), ma = FALSE)
print(metaprop(event, n, sm = "PLN", method.ci = "WS"), ma = FALSE)
print(metaprop(event, n, sm = "PFT", method.ci = "WS"), ma = FALSE)
print(metaprop(event, n, sm = "PAS", method.ci = "WS"), ma = FALSE)
print(metaprop(event, n, sm = "PRAW", method.ci = "WS"), ma = FALSE)

# Different confidence intervals as argument sm = "NAsm"
#
print(metaprop(event, n, method.ci = "NAsm", method = "Inverse"), ma = FALSE)
print(metaprop(event, n, sm = "PLN", method.ci = "NAsm"), ma = FALSE)
print(metaprop(event, n, sm = "PFT", method.ci = "NAsm"), ma = FALSE)
print(metaprop(event, n, sm = "PAS", method.ci = "NAsm"), ma = FALSE)
print(metaprop(event, n, sm = "PRAW", method.ci = "NAsm"), ma = FALSE)

# Different confidence intervals as argument backtransf = FALSE.
# Accordingly, method.ci = "NAsm" used internally.
#
print(metaprop(event, n, method.ci = "WS", method = "Inverse"),
  ma = FALSE, backtransf = FALSE)
print(metaprop(event, n, sm = "PLN", method.ci = "WS"),
  ma = FALSE, backtransf = FALSE)
print(metaprop(event, n, sm = "PFT", method.ci = "WS"),
  ma = FALSE, backtransf = FALSE)
print(metaprop(event, n, sm = "PAS", method.ci = "WS"),
  ma = FALSE, backtransf = FALSE)
print(metaprop(event, n, sm = "PRAW", method.ci = "WS"),
  ma = FALSE, backtransf = FALSE)

# Same results (printed on original and log scale, respectively)
#
print(metaprop(event, n, sm = "PLN", method.ci = "NAsm"), ma = FALSE)
print(metaprop(event, n, sm = "PLN"), ma = FALSE, backtransf = FALSE)
# Results for first study (on log scale)
round(log(c(0.3079848, 0.2569522, 0.3691529)), 4)

# Print results as events per 1000 observations
#
print(metaprop(6:8, c(100, 1200, 1000), method = "Inverse"),
  pscale = 1000, digits = 1)

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