rcalc: Calculate the Variance-Covariance of Dependent Correlation...
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rcalc R Documentation
Calculate the Variance-Covariance of Dependent Correlation Coefficients
Description
Function to calculate the variance-covariance matrix of correlation coefficients computed based on the same sample of subjects. \loadmathjax
Usage
rcalc(x, ni, data, rtoz=FALSE, nfun="min", sparse=FALSE, ...)
Arguments
x
a formula of the form ri ~ var1 + var2 | study. Can also be a correlation matrix or list thereof. See ‘Details’.
ni
vector to specify the sample sizes based on which the correlations were computed.
data
data frame containing the variables specified via the formula (and the sample sizes).
rtoz
logical to specify whether to transform the correlations via Fisher's r-to-z transformation (the default is FALSE).
nfun
a character string to specify how the ‘common’ sample size within each study should be computed. Possible options are "min" (for the minimum), "harmonic" (for the harmonic mean), or "mean" (for the arithmetic mean). Can also be a function. See ‘Details’.
sparse
logical to specify whether the variance-covariance matrix should be returned as a sparse matrix (the default is FALSE).
...
other arguments.
Details
A meta-analysis of correlation coefficients may involve multiple correlation coefficients extracted from the same study. When these correlations are computed based on the same sample of subjects, then they are typically not independent. The rcalc function can be used to create a dataset with the correlation coefficients (possibly transformed with Fisher's r-to-z transformation) and the corresponding variance-covariance matrix. The dataset and variance-covariance matrix can then be further meta-analyzed using the rma.mv function.
When computing the covariance between two correlation coefficients, we can distinguish two cases:
In the first case, one of the variables involved in the two correlation coefficients is the same. For example, in \mjseqnr_12 and \mjseqnr_13, variable 1 is common to both correlation coefficients. This is sometimes called the (partially) ‘overlapping’ case. The covariance between the two correlation coefficients, \mjeqn\mboxCov[r_12, r_13]Cov[r_12, r_13], then depends on the degree of correlation between variables 2 and 3 (i.e., \mjseqnr_23).
In the second case, none of the variables are common to both correlation coefficients. For example, this would be the case if we have correlations \mjseqnr_12 and \mjseqnr_34 based on 4 variables. This is sometimes called the ‘non-overlapping’ case. The covariance between the two correlation coefficients, \mjeqn\mboxCov[r_12, r_34]Cov[r_12, r_34], then depends on \mjseqnr_13, \mjseqnr_14, \mjseqnr_23, and \mjseqnr_24.
Equations to compute these covariances can be found, for example, in Steiger (1980) and Olkin and Finn (1990).
To use the rcalc function, one needs to construct a data frame that contains a study identifier (say study), two variable identifiers (say var1 and var2), the corresponding correlation coefficients (say ri), and the sample sizes based on which the correlation coefficients were computed (say ni). Then the first argument should be a formula of the form ri ~ var1 + var2 | study, argument ni is set equal to the variable name containing the sample sizes, and the data frame containing these variables is specified via the data argument. When using the function for a single study, one can leave out the study identifier from the formula.
When argument rtoz is set to TRUE, then the correlations are transformed with Fisher's r-to-z transformation (Fisher, 1921) and the variance-covariance matrix is computed for the transformed values.
In some cases, the sample size may not be identical within a study (e.g., \mjseqnr_12 may have been computed based on 120 subjects while \mjseqnr_13 was computed based on 118 subjects due to 2 missing values in variable 3). For constructing the variance-covariance matrix, we need to assume a ‘common’ sample size for all correlation coefficients within the study. Argument nfun provides some options for how the common sample size should be computed. Possible options are "min" (for using the minimum sample size within a study as the common sample size), "harmonic" (for using the harmonic mean), or "mean" (for using the arithmetic mean). The default is "min", which is a conservative choice (i.e., it will overestimate the sampling variances of coefficients that were computed based on a sample size that was actually larger than the minimum sample size). One can also specify a function via the nfun argument (which should take a numeric vector as input and return a single value).
Instead of specifying a formula, one can also pass a correlation matrix to the function via argument x. Argument ni then specifies the (common) sample size based on which the elements in the correlation matrix were computed. One can also pass a list of correlation matrices via argument x, in which case argument ni should be a vector of sample sizes of the same length as x.
Value
A list containing the following components:
dat
a data frame with the study identifier, the two variable identifiers, a variable pair identifier, the correlation coefficients (possibly transformed with Fisher's r-to-z transformation), and the (common) sample sizes.
V
corresponding variance-covariance matrix (given as a sparse matrix when sparse=TRUE).
Note that a particular covariance can only be computed when all of the correlation coefficients involved in the covariance equation are included in the dataset. If one or more coefficients needed for the computation are missing, then the resulting covariance will also be missing (i.e., NA).
Note
For raw correlation coefficients, the variance-covariance matrix is computed with \mjseqnn-1 in the denominator (instead of \mjseqnn as suggested in Steiger, 1980, and Olkin & Finn, 1990). This is more consistent with the usual equation for computing the sampling variance of a correlation coefficient (which also typically uses \mjseqnn-1 in the denominator).
For raw and r-to-z transformed coefficients, the variance-covariance matrix will only be computed when the (common) sample size for a study is at least 5.
Author(s)
Wolfgang Viechtbauer wvb@metafor-project.org https://www.metafor-project.org
References
Fisher, R. A. (1921). On the “probable error” of a coefficient of correlation deduced from a small sample. Metron, 1, 1–32. http://hdl.handle.net/2440/15169
Olkin, I., & Finn, J. D. (1990). Testing correlated correlations. Psychological Bulletin, 108(2), 330–333. https://doi.org/10.1037/0033-2909.108.2.330
Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological Bulletin, 87(2), 245–251. https://doi.org/10.1037/0033-2909.87.2.245
See Also
rma.mv for a model fitting function that can be used to meta-analyze dependent correlation coefficients.
dat.craft2003 for an illustrative example.
Examples
############################################################################
### copy data into 'dat' and examine the first 12 rows
dat <- dat.craft2003
head(dat, 12)
### construct dataset and var-cov matrix of the correlations
tmp <- rcalc(ri ~ var1 + var2 | study, ni=ni, data=dat)
V <- tmp$V
dat <- tmp$dat
### examine data and var-cov matrix for study 1
dat[dat$study == 1,]
blsplit(V, dat$study, round, 4)$`1`
### examine data and var-cov matrix for study 6
dat[dat$study == 6,]
blsplit(V, dat$study, round, 4)$`6`
### examine data and var-cov matrix for study 17
dat[dat$study == 17,]
blsplit(V, dat$study, round, 4)$`17`
############################################################################
### copy data into 'dat' and examine the first 12 rows
dat <- dat.craft2003
head(dat, 12)
### restructure data from study 1 into a correlation matrix
R1 <- diag(4)
R1[lower.tri(R1)] <- dat$ri[dat$study == 1]
R1[upper.tri(R1)] <- t(R1)[upper.tri(R1)]
rownames(R1) <- colnames(R1) <- c("perf", "acog", "asom", "conf")
R1
### restructure data from study 3 into a correlation matrix
R3 <- diag(4)
R3[lower.tri(R3)] <- dat$ri[dat$study == 3]
R3[upper.tri(R3)] <- t(R3)[upper.tri(R3)]
rownames(R3) <- colnames(R3) <- c("perf", "acog", "asom", "conf")
R3
### an example where a correlation matrix is passed to rcalc()
rcalc(R1, ni=142)
### an example where a list of correlation matrices is passed to rcalc()
tmp <- rcalc(list("1"=R1,"3"=R3), ni=c(142,37))
V <- tmp$V
dat <- tmp$dat
### examine data and var-cov matrix for study 1
dat[dat$id == 1,]
blsplit(V, dat$id, round, 4)$`1`
### examine data and var-cov matrix for study 3
dat[dat$id == 3,]
blsplit(V, dat$id, round, 4)$`3`
############################################################################
metafor documentation built on May 29, 2024, 7:44 a.m.
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| rcalc | R Documentation |
Calculate the Variance-Covariance of Dependent Correlation Coefficients
Description
Function to calculate the variance-covariance matrix of correlation coefficients computed based on the same sample of subjects. \loadmathjax
Usage
rcalc(x, ni, data, rtoz=FALSE, nfun="min", sparse=FALSE, ...)
Arguments
x |
a formula of the form |
ni |
vector to specify the sample sizes based on which the correlations were computed. |
data |
data frame containing the variables specified via the formula (and the sample sizes). |
rtoz |
logical to specify whether to transform the correlations via Fisher's r-to-z transformation (the default is |
nfun |
a character string to specify how the ‘common’ sample size within each study should be computed. Possible options are |
sparse |
logical to specify whether the variance-covariance matrix should be returned as a sparse matrix (the default is |
... |
other arguments. |
Details
A meta-analysis of correlation coefficients may involve multiple correlation coefficients extracted from the same study. When these correlations are computed based on the same sample of subjects, then they are typically not independent. The rcalc function can be used to create a dataset with the correlation coefficients (possibly transformed with Fisher's r-to-z transformation) and the corresponding variance-covariance matrix. The dataset and variance-covariance matrix can then be further meta-analyzed using the rma.mv function.
When computing the covariance between two correlation coefficients, we can distinguish two cases:
In the first case, one of the variables involved in the two correlation coefficients is the same. For example, in \mjseqnr_12 and \mjseqnr_13, variable 1 is common to both correlation coefficients. This is sometimes called the (partially) ‘overlapping’ case. The covariance between the two correlation coefficients, \mjeqn\mboxCov[r_12, r_13]Cov[r_12, r_13], then depends on the degree of correlation between variables 2 and 3 (i.e., \mjseqnr_23).
In the second case, none of the variables are common to both correlation coefficients. For example, this would be the case if we have correlations \mjseqnr_12 and \mjseqnr_34 based on 4 variables. This is sometimes called the ‘non-overlapping’ case. The covariance between the two correlation coefficients, \mjeqn\mboxCov[r_12, r_34]Cov[r_12, r_34], then depends on \mjseqnr_13, \mjseqnr_14, \mjseqnr_23, and \mjseqnr_24.
Equations to compute these covariances can be found, for example, in Steiger (1980) and Olkin and Finn (1990).
To use the rcalc function, one needs to construct a data frame that contains a study identifier (say study), two variable identifiers (say var1 and var2), the corresponding correlation coefficients (say ri), and the sample sizes based on which the correlation coefficients were computed (say ni). Then the first argument should be a formula of the form ri ~ var1 + var2 | study, argument ni is set equal to the variable name containing the sample sizes, and the data frame containing these variables is specified via the data argument. When using the function for a single study, one can leave out the study identifier from the formula.
When argument rtoz is set to TRUE, then the correlations are transformed with Fisher's r-to-z transformation (Fisher, 1921) and the variance-covariance matrix is computed for the transformed values.
In some cases, the sample size may not be identical within a study (e.g., \mjseqnr_12 may have been computed based on 120 subjects while \mjseqnr_13 was computed based on 118 subjects due to 2 missing values in variable 3). For constructing the variance-covariance matrix, we need to assume a ‘common’ sample size for all correlation coefficients within the study. Argument nfun provides some options for how the common sample size should be computed. Possible options are "min" (for using the minimum sample size within a study as the common sample size), "harmonic" (for using the harmonic mean), or "mean" (for using the arithmetic mean). The default is "min", which is a conservative choice (i.e., it will overestimate the sampling variances of coefficients that were computed based on a sample size that was actually larger than the minimum sample size). One can also specify a function via the nfun argument (which should take a numeric vector as input and return a single value).
Instead of specifying a formula, one can also pass a correlation matrix to the function via argument x. Argument ni then specifies the (common) sample size based on which the elements in the correlation matrix were computed. One can also pass a list of correlation matrices via argument x, in which case argument ni should be a vector of sample sizes of the same length as x.
Value
A list containing the following components:
dat |
a data frame with the study identifier, the two variable identifiers, a variable pair identifier, the correlation coefficients (possibly transformed with Fisher's r-to-z transformation), and the (common) sample sizes. |
V |
corresponding variance-covariance matrix (given as a sparse matrix when |
Note that a particular covariance can only be computed when all of the correlation coefficients involved in the covariance equation are included in the dataset. If one or more coefficients needed for the computation are missing, then the resulting covariance will also be missing (i.e., NA).
Note
For raw correlation coefficients, the variance-covariance matrix is computed with \mjseqnn-1 in the denominator (instead of \mjseqnn as suggested in Steiger, 1980, and Olkin & Finn, 1990). This is more consistent with the usual equation for computing the sampling variance of a correlation coefficient (which also typically uses \mjseqnn-1 in the denominator).
For raw and r-to-z transformed coefficients, the variance-covariance matrix will only be computed when the (common) sample size for a study is at least 5.
Author(s)
Wolfgang Viechtbauer wvb@metafor-project.org https://www.metafor-project.org
References
Fisher, R. A. (1921). On the “probable error” of a coefficient of correlation deduced from a small sample. Metron, 1, 1–32. http://hdl.handle.net/2440/15169
Olkin, I., & Finn, J. D. (1990). Testing correlated correlations. Psychological Bulletin, 108(2), 330–333. https://doi.org/10.1037/0033-2909.108.2.330
Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological Bulletin, 87(2), 245–251. https://doi.org/10.1037/0033-2909.87.2.245
See Also
rma.mv for a model fitting function that can be used to meta-analyze dependent correlation coefficients.
dat.craft2003 for an illustrative example.
Examples
############################################################################
### copy data into 'dat' and examine the first 12 rows
dat <- dat.craft2003
head(dat, 12)
### construct dataset and var-cov matrix of the correlations
tmp <- rcalc(ri ~ var1 + var2 | study, ni=ni, data=dat)
V <- tmp$V
dat <- tmp$dat
### examine data and var-cov matrix for study 1
dat[dat$study == 1,]
blsplit(V, dat$study, round, 4)$`1`
### examine data and var-cov matrix for study 6
dat[dat$study == 6,]
blsplit(V, dat$study, round, 4)$`6`
### examine data and var-cov matrix for study 17
dat[dat$study == 17,]
blsplit(V, dat$study, round, 4)$`17`
############################################################################
### copy data into 'dat' and examine the first 12 rows
dat <- dat.craft2003
head(dat, 12)
### restructure data from study 1 into a correlation matrix
R1 <- diag(4)
R1[lower.tri(R1)] <- dat$ri[dat$study == 1]
R1[upper.tri(R1)] <- t(R1)[upper.tri(R1)]
rownames(R1) <- colnames(R1) <- c("perf", "acog", "asom", "conf")
R1
### restructure data from study 3 into a correlation matrix
R3 <- diag(4)
R3[lower.tri(R3)] <- dat$ri[dat$study == 3]
R3[upper.tri(R3)] <- t(R3)[upper.tri(R3)]
rownames(R3) <- colnames(R3) <- c("perf", "acog", "asom", "conf")
R3
### an example where a correlation matrix is passed to rcalc()
rcalc(R1, ni=142)
### an example where a list of correlation matrices is passed to rcalc()
tmp <- rcalc(list("1"=R1,"3"=R3), ni=c(142,37))
V <- tmp$V
dat <- tmp$dat
### examine data and var-cov matrix for study 1
dat[dat$id == 1,]
blsplit(V, dat$id, round, 4)$`1`
### examine data and var-cov matrix for study 3
dat[dat$id == 3,]
blsplit(V, dat$id, round, 4)$`3`
############################################################################
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