# Simple Data Structures

Here, we’ll use an example dataset taken from the metafor website, which also hosts the very famous metafor package.1

library(metafor)
#> and introduction to the package please type: help(metafor).
library(concurve)
#> Please see the documentation on https://data.lesslikely.com/concurve/ or by typing help(concurve)
library(ggplot2)
dat.hine1989
#>   study         source n1i n2i ai ci
#> 1     1  Chopra et al.  39  43  2  1
#> 2     2       Mogensen  44  44  4  4
#> 3     3    Pitt et al. 107 110  6  4
#> 4     4   Darby et al. 103 100  7  5
#> 5     5 Bennett et al. 110 106  7  3
#> 6     6 O'Brien et al. 154 146 11  4

I will quote Wolfgang (the creator of the metafor package) here, since he explains it best,

"As described under help(dat.hine1989), variables n1i and n2i are the number of patients in the lidocaine and control group, respectively, and ai and ci are the corresponding number of deaths in the two groups. Since these are 2×2 table data, a variety of different outcome measures could be used for the meta-analysis, including the risk difference, the risk ratio (relative risk), and the odds ratio (see Table III). Normand (1999) uses risk differences for the meta-analysis, so we will proceed accordingly. We can calculate the risk differences and corresponding sampling variances with:

dat <- escalc(measure = "RD", n1i = n1i, n2i = n2i, ai = ai, ci = ci, data = dat.hine1989)
dat
#>   study         source n1i n2i ai ci     yi     vi
#> 1     1  Chopra et al.  39  43  2  1 0.0280 0.0018
#> 2     2       Mogensen  44  44  4  4 0.0000 0.0038
#> 3     3    Pitt et al. 107 110  6  4 0.0197 0.0008
#> 4     4   Darby et al. 103 100  7  5 0.0180 0.0011
#> 5     5 Bennett et al. 110 106  7  3 0.0353 0.0008
#> 6     6 O'Brien et al. 154 146 11  4 0.0440 0.0006

"Note that the yi values are the risk differences in terms of proportions. Since Normand (1999) provides the results in terms of percentages, we can make the results directly comparable by multiplying the risk differences by 100 (and the sampling variances by $$100^{2}$$):

dat$yi <- dat$yi * 100
dat$vi <- dat$vi * 100^2

We can fit a fixed-effects model with the following

fe <- rma(yi, vi, data = dat, method = "FE")

Now that we have our metafor object, we can compute the consonance function using the curve_meta() function.

fecurve <- curve_meta(fe, cores = 1L)

Now we can graph our function.

ggcurve(fecurve[], nullvalue = TRUE) We used a fixed-effects model here, but if we wanted to use a random-effects model, we could do so with the following, where we can specify a restricted maximum likelihood estimator (REML) for the random-effects model

re <- rma(yi, vi, data = dat, method = "REML")

And then we could use curve_meta() to get the relevant list

recurve <- curve_meta(re, cores = 1L)

Now we can plot our object.

ggcurve(recurve[], nullvalue = TRUE) We could also compare our two models to see how much consonance/overlap there is

curve_compare(fecurve[], recurve[], plot = TRUE)
#>  "AUC = Area Under the Curve"
#> []
#>
#>
#> | AUC 1| AUC 2| Shared AUC| AUC Overlap (%)| Overlap:Non-Overlap AUC Ratio|
#> |-----:|-----:|----------:|---------------:|-----------------------------:|
#> | 2.084| 2.084|      2.084|             100|                           Inf|
#>
#> [] The results are practically the same and we cannot actually see any difference, and the AUC % overlap also is consisten with this.

# Complex Data Structures

concurve can also handle complex data structures from metafor that are clustered.


### copy data from Berkey et al. (1998) into 'dat'
dat <- dat.berkey1998

### construct list of the variance-covariance matrices of the observed outcomes for the studies
V <- lapply(split(dat[, c("v1i", "v2i")], dat$trial), as.matrix) ### construct block diagonal matrix V <- bldiag(V) ### fit multivariate model res <- rma.mv(yi, V, mods = ~ outcome - 1, random = ~ outcome | trial, struct = "UN", data = dat) ### results based on sandwich method (a <- robust(res, cluster = dat$trial))
#>
#> Number of outcomes:   10
#> Number of clusters:   5
#> Outcomes per cluster: 2
#>
#> Test of Moderators (coefficients 1:2):
#> F(df1 = 2, df2 = 3) = 41.6138, p-val = 0.0065
#>
#> Model Results:
#>
#>            estimate      se     tval    pval    ci.lb    ci.ub
#> outcomeAL   -0.3392  0.1010  -3.3597  0.0437  -0.6605  -0.0179  *
#> outcomePD    0.3534  0.0675   5.2338  0.0136   0.1385   0.5683  *
#>
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

l <- concurve::curve_meta(res, method = "uni", robust = TRUE, cluster = dat\$trial, adjust = TRUE, steps = 1000, cores = 1L)

ggcurve(data = l[], type = "c") +
ggplot2::theme_minimal() The interval function is quite narrow because there were so many iterations that were set by us.

Here, we simulate a more complicated set of trials using the simstudy package.

library(simstudy)
library(parallel)
library(nlme)
library(data.table)
library(metafor)
library(concurve)

defS <- defData(varname = "a.k", formula = 3, variance = 2, id = "study")
defS <- defData(defS, varname = "d.0", formula = 3, dist = "nonrandom")
defS <- defData(defS, varname = "v.k", formula = 0, variance = 6, dist= "normal")
defS <- defData(defS, varname = "s2.k", formula = 16, variance = .2, dist = "gamma")
defS <- defData(defS, varname = "size.study", formula = ".3;.5;.2", dist = "categorical")
defS <- defData(defS, varname = "n.study",
formula = "(size.study==1) * 20 + (size.study==2) * 40 + (size.study==3) * 60",
dist = "poisson")

defI <- defDataAdd(varname = "y", formula = "a.k + x * (d.0 + v.k)", variance = "s2.k")

RNGkind(kind = "L'Ecuyer-CMRG")
set.seed(1031)

ds <- genData(12, defS)
ds
#>     study      a.k d.0        v.k      s2.k size.study n.study
#>  1:     1 4.745069   3 -5.1399796 16.275358          2      38
#>  2:     2 1.983020   3 -3.2693291 13.976232          3      66
#>  3:     3 1.865846   3  1.9742469  8.332148          2      40
#>  4:     4 4.235730   3 -3.2126873 11.100117          2      40
#>  5:     5 4.246219   3 -0.5858786 13.765175          3      57
#>  6:     6 3.899617   3  4.6808278  7.532707          2      41
#>  7:     7 3.059236   3 -5.9924638  7.817771          1      14
#>  8:     8 2.324586   3  5.7588246 10.970013          3      56
#>  9:     9 3.857260   3 -1.1081854 24.039016          3      75
#> 10:    10 4.308413   3 -1.9081253 14.084537          1      24
#> 11:    11 1.607899   3  0.2178001  8.271145          2      48
#> 12:    12 3.717728   3  1.8818612 11.274497          3      67

dc <- genCluster(ds, "study", "n.study", "id", )
dc <- trtAssign(dc, strata = "study", grpName = "x")

d.obs <- dc[, .(study, id,s2.k, x, y)]
d.obs <-  as.data.frame(d.obs)
res <- rma.mv(yi = y, V = s2.k, W = NULL, mods = ~  factor(study) + factor(x),  random = ~ factor(x) | study, struct="CS", data = d.obs, method = "REML")

a <- curve_meta(x = res, measure = "default", method = "mv", parm = "factor(x)1", cores = 1L)
ggcurve(a[], type = "cdf") # Cite R packages

Please remember to cite the packages that you use.

citation("concurve")
#>
#> Rafi Z, Vigotsky A (2020). _concurve: Computes and Plots Compatibility
#> (Confidence) Intervals, P-Values, S-Values, & Likelihood Intervals to
#> Form Consonance, Surprisal, & Likelihood Functions_. R package version
#> 2.7.7, <URL: https://CRAN.R-project.org/package=concurve>.
#>
#> Rafi Z, Greenland S (2020). "Semantic and Cognitive Tools to Aid
#> Statistical Science: Replace Confidence and Significance by
#> Compatibility and Surprise." _BMC Medical Research Methodology_, *20*,
#> 244. ISSN 1471-2288, doi: 10.1186/s12874-020-01105-9 (URL:
#> https://doi.org/10.1186/s12874-020-01105-9), <URL:
#> https://doi.org/10.1186/s12874-020-01105-9>.
#>
#> To see these entries in BibTeX format, use 'print(<citation>,
#> bibtex=TRUE)', 'toBibtex(.)', or set
#> 'options(citation.bibtex.max=999)'.
citation("ggplot2")
#>
#> To cite ggplot2 in publications, please use:
#>
#>   H. Wickham. ggplot2: Elegant Graphics for Data Analysis.
#>   Springer-Verlag New York, 2016.
#>
#> A BibTeX entry for LaTeX users is
#>
#>   @Book{,
#>     title = {ggplot2: Elegant Graphics for Data Analysis},
#>     publisher = {Springer-Verlag New York},
#>     year = {2016},
#>     isbn = {978-3-319-24277-4},
#>     url = {https://ggplot2.tidyverse.org},
#>   }
citation("metafor")
#>
#> To cite the metafor package in publications, please use:
#>
#>   Viechtbauer, W. (2010). Conducting meta-analyses in R with the
#>   metafor package. Journal of Statistical Software, 36(3), 1-48. URL:
#>   https://www.jstatsoft.org/v36/i03/
#>
#> A BibTeX entry for LaTeX users is
#>
#>   @Article{,
#>     title = {Conducting meta-analyses in {R} with the {metafor} package},
#>     author = {Wolfgang Viechtbauer},
#>     journal = {Journal of Statistical Software},
#>     year = {2010},
#>     volume = {36},
#>     number = {3},
#>     pages = {1--48},
#>     url = {https://www.jstatsoft.org/v36/i03/},
#>   }
citation("simstudy")
#>
#> To cite package 'simstudy' in publications use:
#>
#>   Keith Goldfeld and Jacob Wujciak-Jens (2020). simstudy: Simulation of
#>   Study Data. R package version 0.2.1.
#>   https://CRAN.R-project.org/package=simstudy
#>
#> A BibTeX entry for LaTeX users is
#>
#>   @Manual{,
#>     title = {simstudy: Simulation of Study Data},
#>     author = {Keith Goldfeld and Jacob Wujciak-Jens},
#>     year = {2020},
#>     note = {R package version 0.2.1},
#>     url = {https://CRAN.R-project.org/package=simstudy},
#>   }
citation("nlme")
#>
#> Pinheiro J, Bates D, DebRoy S, Sarkar D, R Core Team (2020). _nlme:
#> Linear and Nonlinear Mixed Effects Models_. R package version 3.1-149,
#> <URL: https://CRAN.R-project.org/package=nlme>.
#>
#> A BibTeX entry for LaTeX users is
#>
#>   @Manual{,
#>     title = {{nlme}: Linear and Nonlinear Mixed Effects Models},
#>     author = {Jose Pinheiro and Douglas Bates and Saikat DebRoy and Deepayan Sarkar and {R Core Team}},
#>     year = {2020},
#>     note = {R package version 3.1-149},
#>     url = {https://CRAN.R-project.org/package=nlme},
#>   }
citation("data.table")
#>
#> To cite package 'data.table' in publications use:
#>
#>   Matt Dowle and Arun Srinivasan (2020). data.table: Extension of
#>   data.frame. R package version 1.13.0.
#>   https://CRAN.R-project.org/package=data.table
#>
#> A BibTeX entry for LaTeX users is
#>
#>   @Manual{,
#>     title = {data.table: Extension of data.frame},
#>     author = {Matt Dowle and Arun Srinivasan},
#>     year = {2020},
#>     note = {R package version 1.13.0},
#>     url = {https://CRAN.R-project.org/package=data.table},
#>   }
citation("parallel")
#>
#> The 'parallel' package is part of R.  To cite R in publications use:
#>
#>   R Core Team (2020). R: A language and environment for statistical
#>   computing. R Foundation for Statistical Computing, Vienna, Austria.
#>   URL https://www.R-project.org/.
#>
#> A BibTeX entry for LaTeX users is
#>
#>   @Manual{,
#>     title = {R: A Language and Environment for Statistical Computing},
#>     author = {{R Core Team}},
#>     organization = {R Foundation for Statistical Computing},
#> citing R packages.