copas: Copas selection model analysis

Description

Perform a Copas selection model analysis for selection bias in meta-analysis.

Usage

copas(
  x,
  level.ma = x$level.ma,
  gamma0.range = NULL,
  gamma1.range = NULL,
  ngrid = 20,
  nlevels = 10,
  levels = NULL,
  slope = NULL,
  left = NULL,
  rho.bound = 0.9999,
  sign.rsb = 0.1,
  backtransf = x$backtransf,
  title = x$title,
  complab = x$complab,
  outclab = x$outclab,
  silent = TRUE,
  warn = options()$warn
)

Value

An object of class copas with corresponding

print, summary, and plot function. The object is a list containing the following components:

TE: Vector of treatment effects plotted in treatment effect plot
seTE: Vector of standard error of TE
TE.random: Usual random effects estimate of treatment effect
seTE.random: Standard error of TE.random
lower.random: Lower confidence limit of usual random effects estimate
upper.random: Upper confidence limit of usual random effects estimate
statistic.random: Test statistic of an overall effect (usual random effects model)
pval.random: P-value of test of overall effect (usual random effects model)
TE.adjust: Adjusted random effects estimate from Copas selection model
seTE.adjust: Standard error of TE.adjust
lower.adjust: Lower confidence limit of adjusted treatment estimate
upper.adjust: Upper confidence limit of adjusted treatment estimate
statistic.adjust: Test statistic of an overall effect (Copas selection model)
pval.adjust: P-value of test of overall effect (Copas selection model)
left: Whether selection bias expected on left or right
rho.bound: Bound on rho
gamma0.range: Range of gamma0 (see help on copas arguments above)
gamma1.range: Range of gamma1 (see help on copas arguments above)
slope: Slope of line approximately orthogonal to contours in contour plot
regr: A list containing information on regression lines fitted to contours in contour plot
ngrid: Square root of grid size
nlevels: Number of contour lines
gamma0: Vector of gamma0 values at which model fitted (determined by gamma0.range and grid). x-axis values for contour plot
gamma1: vector of gamma1 values at which model fitted (determined by gamma1.range and grid). y-axis values for contour plot
TE.contour: Treatment values (ie z-axis values) used to draw contour plot.
x.slope: x coordinates for 'orthogonal line' in contour plot
y.slope: y coordinates for 'orthogonal line' in contour plot
TE.slope: Vector of treatment values plotted in treatment effect plot
seTE.slope: Standard error of TE.slope
rho.slope: Vector of estimated rho values corresponding to treatment estimates in TE.slope
tau.slope: Vector of estimated heterogeneity values corresponding to treatment estimates in TE.slope
loglik1: Vector of log-likelihood values corresponding to treatment estimates in TE.slope
conv1: Numerical vector indicating convergence status for each treatment estimate in TE.slope - see parameter convergence in function optim
message1: Character vector - translation of conv1
loglik2: Vector of log-likelihoods from fitting model to evaluate presence of residual selection bias
conv2: Numerical vector indicating convergence status for models to evaluate presence of residual selection bias - see parameter convergence in function optim
message2: Character vector - translation of conv2
publprob: Vector of probabilities of publishing the smallest study, used in x-axis of bottom two panels in function plot.copas
pval.rsb: P-values for tests on presence of residual selection bias, plotted in bottom right panel in plot.copas
sign.rsb: The significance level for the test of residual selection bias
N.unpubl: Approximate number of studies the model suggests remain unpublished
sm: Effect measure (e.g., for binary data, OR - odds ratio, RR - risk ratio, RD - risk difference, AS - arcsin difference)
title: Title of meta-analysis / systematic review.
complab: Comparison label.
outclab: Outcome label.
call: Call to copas function
version: Version of R package metasens used to create object.
x: Details of meta-analysis object used as input into copas function

Arguments

x

An object of class meta, obtained from one of the functions metabin, metacont and metagen in the package meta.

level.ma

The level used to calculate confidence intervals for pooled estimates.

gamma0.range

(Advanced users only) A numerical vector of length two specifying the range of gamma0 values the program will explore.

The parameter gamma0 is the constant in the probit selection model for study publication. Thus, the cumulative normal of gamma0 is approximately the probability that a small study is published (in non-technical terms gamma0 relates to the probability of publishing a small study, although its values are not restricted to the range [0,1]; larger values correspond to higher probabilities of publishing a small study). Most users will not need to specify a range for this parameter. When no argument is specified, the program uses an algorithm to determine a suitable range. This is based on the range of treatment effect standard errors in the meta-analysis, and is described in more detail below.

gamma1.range

(Advanced users only) A numerical vector of length two specifying the range of gamma1 values the program will explore.

The parameter gamma1 is the coefficient of study precision (1/standard error) in the probit selection model for study publication (in non-technical terms gamma1 relates to the rate at which the probability of publishing a study increases as the standard error of the treatment effect it reports decreases; larger values correspond to higher probabilities of publishing a small study). Most users will not need to specify a range for this parameter. When no argument is specified, the program uses an algorithm to determine a suitable range. This is based on the range of treatment effect standard errors in the meta-analysis, and is described in more detail below.

ngrid

The program fits the Copas selection model over a grid defined by the range of values of gamma0 and gamma1 specified in the previous two arguments. This parameter fixes the square-root of the number of points in the grid.

nlevels

(Advanced users only). Fitting the Copas model over the grid specified by the previous three arguments results in a treatment estimate at every point in the grid. These can then be displayed on a contour plot where contours of treatment effect (z-axis) are shown by gamma0 (x-axis) and gamma1 (y-axis). This argument specifies the number of contour lines that will be drawn.

Note

(i) Calculations for the contour plot are performed by the function copas, so this argument has no effect in the plot function.

(ii) If a large number of contour lines are desired, then you may wish to consider increasing the grid size (argument ngrid above).

Leave this option unspecified if you are using the option levels below.

levels

A numerical vector of treatment values for which contour lines will be drawn. In more detail, fitting the Copas model over the grid specified by the arguments gamma0.range, gamma1.range and ngrid results in a treatment estimate at every point in the grid. These are then displayed on a contour plot where contours of treatment effect (z-axis) are shown by gamma0 (x-axis) and gamma1 (y-axis). This argument is a numerical vector which specifies the treatment effects for which contour lines will be drawn.

It is usually not a good idea to set this argument for initial runs, as one does not know the range of treatment values that the contour plot will cover, and treatment values which do not correspond to values in the contour plot (defined by the range of gamma0 and gamma1) will not be plotted.

Note

(i) Calculations for the contour plot are performed by the function copas, so this argument has no effect in the plot function.

(ii) Contours will not be drawn if a large number of contour lines are desired, then you may wish to consider increasing the grid size (argument ngrid above).

Leave this option unspecified if you are using the option nlevels above.

slope

A numeric providing the slope of the line approximately orthogonal to contours in the contour plot. If the argument slope is NULL (default) the program seeks to estimate the slope of the contours in the region of the maximum, which are usually approximately parallel. Most users will leave the argument slope unspecified, at least for the first analysis of a data set, but in certain cases setting it manually can improve the results.

left

A logical indicating whether the cause of any selection bias is due to missing studies on the left or right of the funnel plot: left hand side if left=TRUE, right hand side if left=FALSE. This information is needed in order to be sure the test for presence of residual selection bias is calculated correctly. If not set, the linear regression test for funnel plot asymmetry (i.e., function metabias(..., meth="linreg")) is used to determine whether studies are missing on the left or right hand side. In the majority of cases this will work correctly.

rho.bound

(Advanced users only) A number giving the upper bound for the correlation parameter rho (see details below). This must be < 1, and usually > 0.95. The lower bound is calculated as -(the upper bound).

sign.rsb

The significance level for the test of residual selection bias (between 0 and 1).

backtransf

A logical indicating whether results should be back transformed in printouts and plots. If backtransf=TRUE (default), results for sm="OR" are printed as odds ratios rather than log odds ratio, for example.

title

Title of meta-analysis / systematic review.

complab

Comparison label.

outclab

Outcome label.

silent

A logical indicating whether information on progress in fitting the Copas selection model should be printed: silent=TRUE, do not print information (the default); silent=FALSE, print information.

warn

A number setting the handling of warning messages. It is not uncommon for numerical problems to be encountered during estimation over the grid of (gamma0, gamma1) values. Usually this does not indicate a serious problem. This option specifies what to do with warning messages. warn=-1: ignore all warnings; warn=0 (the default): store warnings till function finishes; if there are less than 10, print them, otherwise print a message saying warning messages were generated; warn=1: print warnings as they occur; warn=2: stop the function when the first warning is generated. For further details see help(options).

Author

James Carpenter James.Carpenter@lshtm.ac.uk, Guido Schwarzer guido.schwarzer@uniklinik-freiburg.de

Details

The program takes an object of class meta, which is most easily created by an analysis using one of the functions metabin, metacont and metagen in the package meta, performs a 'Copas selection model analysis' and presents a graphical and tabular summary of the results. An object of class copas is created and this can be used to recreate the results table and graphs subsequently, without re-running the analysis, using the print, summary and plot function.

Conduct a Copas selection model analysis to investigate, and attempt to correct for, selection / publication bias in a meta-analysis.

The Copas selection model consists of two models, which are fitted jointly. The first is the usual random effects meta-analysis model, and the second is a selection model, where study i is selected for publication if Z>0, where

Z = gamma0 + gamma1 / (SE(i)) + delta(i)

The error delta(i) is correlated with the error in the random effects meta-analysis, with correlation rho. If rho=0, the model corresponds to the usual random effects meta-analysis. As rho moves from 0 to 1, studies with larger treatment estimates are more likely to be selected/published.

The software chooses a grid of gamma0 and gamma1 values, corresponding to a range of selection / publication probabilities for the study with the largest treatment effect standard error (often the smallest study). For each value in this grid, the treatment effect is estimated using the function optim. This information is used to produce the contour plot (top right panel of output from plot.copas).

Contours of constant treatment effect are usually locally parallel. The software estimates the slope of these contours, and combines this information with other parameter estimates from the model to explore (i) how the treatment estimate, and its standard error, change with increasing selection (bottom left panel, plot.copas) and (ii) how much selection needs to be accounted for before any remaining asymmetry in the funnel plot is likely to have occurred by chance (bottom right panel, plot.copas).

A table of results can be produced by the function summary.copas. A more detail output is provided by the function print.copas.

For a fuller description of the model, our implementation and specifically our approach to estimating the locally parallel contours, see Carpenter et al. (2009) and Schwarzer et al. (2010).

References

Carpenter JR, Schwarzer G, Rücker G, Künstler R (2009): Empirical evaluation showed that the Copas selection model provided a useful summary in 80% of meta-analyses. Journal of Clinical Epidemiology, 62, 624--31

Copas J (1999): What works?: Selectivity models and meta-analysis. Journal of the Royal Statistical Society, Series A, 162, 95--109

Copas J, Shi JQ (2000): Meta-analysis, funnel plots and sensitivity analysis. Biostatistics, 1, 247--62

Copas JB, Shi JQ (2001): A sensitivity analysis for publication bias in systematic reviews. Statistical Methods in Medical Research, 10, 251--65

Schwarzer G, Carpenter J, Rücker G (2010): Empirical evaluation suggests Copas selection model preferable to trim-and-fill method for selection bias in meta-analysis. Journal of Clinical Epidemiology, 63, 282--8

Examples

Run this code

data(Fleiss1993bin, package = "meta")

# Perform meta-analysis
#  (Note d.asp indicates deaths, n.asp total in aspirin group;
#        d.plac indicates deaths, n.plac total in placebo group)
#
m1 <- metabin(d.asp, n.asp, d.plac, n.plac, data = Fleiss1993bin, sm = "OR")
m1

# Perform a basic Copas selection model analysis
#
cop1 <- copas(m1)
plot(cop1)
cop1
#
# Interpretation: 
#
# a. The initial meta-analysis shows the common and random effects
#    pooled ORs differ; consistent with asymmetry in the funnel
#    plot and possible selection bias. Both common effect and random
#    effects model show a significant treatment effect in this
#    dataset.
#
# b. Plotting the copas analysis shows
#
# (i) funnel plot: asymmetry indicates possible selection bias.
#
# (ii) contour plot treatment effect declines steadily as selection
#      increases (no selection, top right, log OR < -0.12;
#      increasing selection as move to left of plot, log OR rises
#      to -0.03.
#

# (iii) Treatment effect plot suggests that even with no selection,
#       p-value for treatment effect is larger than 0.05 which is
#       different from the result of the usual random effects model
#       (see output of summary(cop1). This difference is due to the
#       use of different methods to estimate the between-study
#       variance: maximum-likelihood in Copas analysis compared to
#       method-of-moments in usual random effects model.  The
#       p-value for treatment effect is increasing with increasing
#       selection.
#
# (iv) P-value for residual selection bias plot: this shows that
#      even with no selection bias, the p-value for residual
#      selection bias is non-significant at the 10% level. As
#      expected, as selection increases the p-value for residual
#      selection bias increases too.


# Repeat the same example, setting several arguments of the copas
# function:
#
cop2 <- copas(m1,
   gamma0.range = c(-0.5, 2.1), # range of gamma0 parameter
   gamma1.range = c(0, 0.08),   # range of gamma1 parameter
   ngrid = 20,                  # specify a 20x20 grid (finer than default)
   levels = c(-0.13, -0.12, -0.1, -0.09,
              -0.07, -0.05, -0.03), # specify contour lines
   slope = 0.2,       # specify slope of 'orthogonal' line in contour plot
   left = FALSE,      # as any selection bias due to missing studies on right
   rho.bound = 0.998, # constrain rho between [-0.998, 0.998]
   silent = FALSE,    # update user on progress
   warn = -1          # suppress warning messages
  )
plot(cop2)
#
# Print table of results used to draw treatment effect plot:
#
cop2

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