conv.2x2 {metafor}

Reconstruct Cell Frequencies of \(2 \times 2\) Tables

Description

Function to reconstruct the cell frequencies of \(2 \times 2\) tables based on other summary statistics.

Usage

conv.2x2(ori, ri, x2i, ni, n1i, n2i, correct=TRUE, data, include,
         var.names=c("ai","bi","ci","di"), append=TRUE, replace="ifna")

Arguments

ori

optional vector with the odds ratios corresponding to the tables.

ri

optional vector with the phi coefficients corresponding to the tables.

x2i

optional vector with the (signed) chi-square statistics corresponding to the tables.

ni

vector with the total sample sizes.

n1i

vector with the marginal counts for the outcome of interest on the first variable.

n2i

vector with the marginal counts for the outcome of interest on the second variable.

correct

optional logical (or vector thereof) to indicate whether chi-square statistics were computed using Yates's correction for continuity (the default is TRUE).

data

optional data frame containing the variables given to the arguments above.

include

optional (logical or numeric) vector to specify the subset of studies for which the cell frequencies should be reconstructed.

var.names

character vector with four elements to specify the names of the variables for the reconstructed cell frequencies (the default is c("ai","bi","ci","di")).

append

logical to specify whether the data frame provided via the data argument should be returned together with the reconstructed values (the default is TRUE).

replace

character string or logical to specify how values in var.names should be replaced (only relevant when using the data argument and if variables in var.names already exist in the data frame). See the ‘Value’ section for more details.

Details

For meta-analyses based on \(2 \times 2\) table data, the problem often arises that some studies do not directly report the cell frequencies. The present function allows the reconstruction of such tables based on other summary statistics.

In particular, assume that the data of interest for a particular study are of the form:

		variable 2, outcome +		variable 2, outcome -		total
variable 1, outcome +		ai		bi		n1i
variable 1, outcome -		ci		di
total		n2i				ni

where ai, bi, ci, and di denote the cell frequencies (i.e., the number of individuals falling into a particular category), n1i (i.e., ai+bi) and n2i (i.e., ai+ci) are the marginal totals for the outcome of interest on the first and second variable, respectively, and ni is the total sample size (i.e., ai+bi+ci+di) of the study.

For example, if variable 1 denotes two different groups (e.g., treated versus control) and variable 2 indicates whether a particular outcome of interest has occurred or not (e.g., death, complications, failure to improve under the treatment), then n1i denotes the number of individuals in the treatment group, but n2i is not the number of individuals in the control group, but the total number of individuals who experienced the outcome of interest on variable 2. Note that the meaning of n2i is therefore different here compared to the escalc function (where n2i denotes ci+di).

If a study does not report the cell frequencies, but it reports the total sample size (which can be specified via the ni argument), the two marginal counts (which can be specified via the n1i and n2i arguments), and some other statistic corresponding to the table, then it may be possible to reconstruct the cell frequencies. The present function currently allows this for three different cases:

1. If the odds ratio \[OR = \frac{a_i d_i}{b_i c_i}\] is known, then the cell frequencies can be reconstructed (Bonett, 2007). Odds ratios can be specified via the ori argument.

2. If the phi coefficient \[\phi = \frac{a_i d_i - b_i c_i}{\sqrt{n_{1i}(n_i-n_{1i})n_{2i}(n_i-n_{2i})}}\] is known, then the cell frequencies can again be reconstructed (own derivation). Phi coefficients can be specified via the ri argument.

3. If the chi-square statistic from Pearson's chi-square test of independence is known (which can be specified via the x2i argument), then it can be used to recalculate the phi coefficient and hence again the cell frequencies can be reconstructed. However, the chi-square statistic does not carry information about the sign of the phi coefficient. Therefore, values specified via the x2i argument can be positive or negative, which allows the specification of the correct sign. Also, when using a chi-square statistic as input, it is assumed that it was computed using Yates's correction for continuity (unless correct=FALSE). If the chi-square statistic is not known, but its p-value, one can first back-calculate the chi-square statistic using qchisq(<p-value>, df=1, lower.tail=FALSE).

Typically, the odds ratio, phi coefficient, or chi-square statistic (or its p-value) that can be extracted from a study will be rounded to a certain degree. The calculations underlying the function are exact only for unrounded values. Rounding can therefore introduce some discrepancies.

If a marginal total is unknown, then external information needs to be used to ‘guestimate’ the number of individuals that experienced the outcome of interest on this variable. Depending on the accuracy of such an estimate, the reconstructed cell frequencies will be more or less accurate and need to be treated with due caution.

The true marginal counts also put constraints on the possible values for the odds ratio, phi coefficient, and chi-square statistic. If a marginal count is replaced by a guestimate which is not compatible with the given statistic, one or more reconstructed cell frequencies may be negative. The function issues a warning if this happens and sets the cell frequencies to NA for such a study.

If only one of the two marginal counts is unknown but a 95% CI for the odds ratio is also available, then the estimraw package can also be used to reconstruct the corresponding cell frequencies (Di Pietrantonj, 2006; but see Veroniki et al., 2013, for some cautions).

Value

If the data argument was not specified or append=FALSE, a data frame with four variables called var.names with the reconstructed cell frequencies.

If data was specified and append=TRUE, then the original data frame is returned. If var.names[j] (for \(\textrm{j} \in \{1, \ldots, 4\}\)) is a variable in data and replace="ifna" (or replace=FALSE), then only missing values in this variable are replaced with the estimated frequencies (where possible) and otherwise a new variable called var.names[j] is added to the data frame.

If replace="all" (or replace=TRUE), then all values in var.names[j] where a reconstructed cell frequency can be computed are replaced, even for cases where the value in var.names[j] is not missing.

Author(s)

Wolfgang Viechtbauer [e-mail] https://www.metafor-project.org

References

Bonett, D. G. (2007). Transforming odds ratios into correlations for meta-analytic research. American Psychologist, 62(3), 254–255. ⁠https://doi.org/10.1037/0003-066x.62.3.254⁠

Di Pietrantonj, C. (2006). Four-fold table cell frequencies imputation in meta analysis. Statistics in Medicine, 25(13), 2299–2322. ⁠https://doi.org/10.1002/sim.2287⁠

Veroniki, A. A., Pavlides, M., Patsopoulos, N. A., & Salanti, G. (2013). Reconstructing 2 x 2 contingency tables from odds ratios using the Di Pietrantonj method: Difficulties, constraints and impact in meta-analysis results. Research Synthesis Methods, 4(1), 78–94. ⁠https://doi.org/10.1002/jrsm.1061⁠

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

See Also

escalc for a function to compute various effect size measures based on \(2 \times 2\) table data.

Examples

### demonstration that the reconstruction of the 2x2 table works
### (note: the values in rows 2, 3, and 4 correspond to those in row 1)
dat <- data.frame(ai=c(36,NA,NA,NA), bi=c(86,NA,NA,NA), ci=c(20,NA,NA,NA), di=c(98,NA,NA,NA),
                  oddsratio=NA, phi=NA, chisq=NA, ni=NA, n1i=NA, n2i=NA)
dat$oddsratio[2] <- round(exp(escalc(measure="OR", ai=ai, bi=bi, ci=ci, di=di, data=dat)$yi[1]), 2)
dat$phi[3] <- round(escalc(measure="PHI", ai=ai, bi=bi, ci=ci, di=di, data=dat)$yi[1], 2)
dat$chisq[4] <- round(chisq.test(matrix(c(t(dat[1,1:4])), nrow=2, byrow=TRUE))$statistic, 2)
dat$ni[2:4]  <- with(dat, ai[1] + bi[1] + ci[1] + di[1])
dat$n1i[2:4] <- with(dat, ai[1] + bi[1])
dat$n2i[2:4] <- with(dat, ai[1] + ci[1])
dat

### reconstruct cell frequencies for rows 2, 3, and 4
dat <- conv.2x2(ri=phi, ori=oddsratio, x2i=chisq, ni=ni, n1i=n1i, n2i=n2i, data=dat)
dat

### same example but with cell frequencies that are 10 times as large
dat <- data.frame(ai=c(360,NA,NA,NA), bi=c(860,NA,NA,NA), ci=c(200,NA,NA,NA), di=c(980,NA,NA,NA),
                  oddsratio=NA, phi=NA, chisq=NA, ni=NA, n1i=NA, n2i=NA)
dat$oddsratio[2] <- round(exp(escalc(measure="OR", ai=ai, bi=bi, ci=ci, di=di, data=dat)$yi[1]), 2)
dat$phi[3] <- round(escalc(measure="PHI", ai=ai, bi=bi, ci=ci, di=di, data=dat)$yi[1], 2)
dat$chisq[4] <- round(chisq.test(matrix(c(t(dat[1,1:4])), nrow=2, byrow=TRUE))$statistic, 2)
dat$ni[2:4]  <- with(dat, ai[1] + bi[1] + ci[1] + di[1])
dat$n1i[2:4] <- with(dat, ai[1] + bi[1])
dat$n2i[2:4] <- with(dat, ai[1] + ci[1])
dat <- conv.2x2(ri=phi, ori=oddsratio, x2i=chisq, ni=ni, n1i=n1i, n2i=n2i, data=dat)
dat # slight inaccuracy in row 3 due to rounding

### demonstrate what happens when a true marginal count is guestimated
escalc(measure="PHI", ai=176, bi=24, ci=72, di=128)
conv.2x2(ri=0.54, ni=400, n1i=200, n2i=248) # using the true marginal counts
conv.2x2(ri=0.54, ni=400, n1i=200, n2i=200) # marginal count for variable 2 is guestimated
conv.2x2(ri=0.54, ni=400, n1i=200, n2i=50)  # marginal count for variable 2 is incompatible

### demonstrate that using the correct sign for the chi-square statistic is important
chisq <- round(chisq.test(matrix(c(40,60,60,40), nrow=2, byrow=TRUE))$statistic, 2)
conv.2x2(x2i=-chisq, ni=200, n1i=100, n2i=100) # correct reconstruction
conv.2x2(x2i=chisq, ni=200, n1i=100, n2i=100) # incorrect reconstruction

### demonstrate use of the 'correct' argument
tab <- matrix(c(28,14,12,18), nrow=2, byrow=TRUE)
chisq <- round(chisq.test(tab)$statistic, 2) # chi-square test with Yates' continuity correction
conv.2x2(x2i=chisq, ni=72, n1i=42, n2i=40) # correct reconstruction
chisq <- round(chisq.test(tab, correct=FALSE)$statistic, 2) # without Yates' continuity correction
conv.2x2(x2i=chisq, ni=72, n1i=42, n2i=40) # incorrect reconstruction
conv.2x2(x2i=chisq, ni=72, n1i=42, n2i=40, correct=FALSE) # correct reconstruction

### recalculate chi-square statistic based on p-value
pval <- round(chisq.test(tab)$p.value, 2)
chisq <- qchisq(pval, df=1, lower.tail=FALSE)
conv.2x2(x2i=chisq, ni=72, n1i=42, n2i=40)