Title: | Comparing Correlations |
---|---|
Description: | Statistical tests for the comparison between two correlations based on either independent or dependent groups. Dependent correlations can either be overlapping or nonoverlapping. A web interface is available on the website <http://comparingcorrelations.org>. A plugin for the R GUI and IDE RKWard is included. Please install RKWard from <https://rkward.kde.org> to use this feature. The respective R package 'rkward' cannot be installed directly from a repository, as it is a part of RKWard. |
Authors: | Birk Diedenhofen [aut, cre] |
Maintainer: | Birk Diedenhofen <[email protected]> |
License: | GPL (>= 3) |
Version: | 1.1-4 |
Built: | 2025-03-12 03:58:49 UTC |
Source: | https://github.com/cran/cocor |
Statistical tests for the comparison between two correlations based on either independent or dependent groups. Dependent correlations can either be overlapping or nonoverlapping. A web interface is available on the website <http://comparingcorrelations.org>. A plugin for the R GUI and IDE RKWard is included. Please install RKWard from <https://rkward.kde.org> to use this feature. The respective R package 'rkward' cannot be installed directly from a repository, as it is a part of RKWard.
The DESCRIPTION file:
Package: | cocor |
Type: | Package |
Version: | 1.1-4 |
Date: | 2022-06-28 |
Depends: | methods |
Enhances: | rkward |
Encoding: | UTF-8 |
License: | GPL (>= 3) |
LazyLoad: | yes |
URL: | http://comparingcorrelations.org |
Birk Diedenhofen [aut, cre]
Maintainer: Birk Diedenhofen <[email protected]>
Useful links:
Data of two samples of testees who completed an aptitude test consisting of general knowledge questions, logic tasks, and two measures of intelligence.
data("aptitude")
data("aptitude")
A list that contains two data.frames holding 291 and 334 observations on the following 4 variables.
knowledge
Score achieved on the general knowledge questions (numeric vector)
logic
Score achieved on the logic tasks (numeric vector)
intelligence.a
Intelligence measure A (numeric vector)
intelligence.b
Intelligence measure B (numeric vector)
data("aptitude")
data("aptitude")
Convert a cocor result object of class 'cocor.indep.groups', 'cocor.dep.groups.overlap', or 'cocor.dep.groups.nonoverlap' to a list of class 'htest'.
as.htest(result.object) ## S4 method for signature 'cocor' as.htest(result.object)
as.htest(result.object) ## S4 method for signature 'cocor' as.htest(result.object)
result.object |
A cocor result object of class 'cocor.indep.groups', 'cocor.dep.groups.overlap', or 'cocor.dep.groups.nonoverlap'. |
Returns a list containing a list of class 'htest' for the result of each test with the following elements:
data.name |
A character string giving the names of the data. |
estimate |
The two correlations that have been compared and the related correlations. |
method |
A character string indicating the performed test. |
null.value |
The specified hypothesized value of the difference between the two correlations. |
alternative |
A character string describing the alternative hypothesis. |
parameter |
The degrees of freedom of the distribution of the test statistic. |
statistic |
The value of the test statistic. |
p.value |
The p-value of the test. |
conf.int |
The confidence interval of the difference between the two correlations. |
cocor, cocor.indep.groups, cocor.dep.groups.overlap, cocor.dep.groups.nonoverlap
data("aptitude") cocor.result <- cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) as.htest(cocor.result)
data("aptitude") cocor.result <- cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) as.htest(cocor.result)
Performs a test of significance for the difference between two correlations based on either dependent or independent groups. Dependent correlations can be either overlapping (they share a variable) or nonoverlapping (they have no variable in common). The function expects raw data input from which the correlations are calculated.
cocor( formula, data, alternative = "two.sided", test = "all", na.action = getOption("na.action"), alpha = 0.05, conf.level = 0.95, null.value = 0, return.htest = FALSE )
cocor( formula, data, alternative = "two.sided", test = "all", na.action = getOption("na.action"), alpha = 0.05, conf.level = 0.95, null.value = 0, return.htest = FALSE )
formula |
A formula specifying the correlations and their underlying variables (See details). |
data |
A list holding two data.frames/matrices for independent groups or a single data.frame/matrix for dependent groups that contain the variables specified in |
alternative |
A character string specifying whether the alternative hypothesis is two-sided (" |
test |
For the tests available, see cocor.indep.groups,
cocor.dep.groups.overlap,
and cocor.dep.groups.nonoverlap. Use |
na.action |
A function which handles missing data. Defaults to |
alpha |
A number defining the alpha level for the hypothesis test. The default value is |
conf.level |
A number defining the level of confidence for the confidence interval (if a test is used that calculates confidence intervals). The default value is |
null.value |
A number defining the hypothesized difference between the two correlations used for testing the null hypothesis. The default value is |
return.htest |
A logical indicating whether the result should be returned as a list containing a list of class 'htest' for each test. The default value is |
The formula
parameter for the comparison of two correlations based on independent groups can either be ~a + b | a + b
,
~a + b | a + c
,
or ~a + b | c + d
. The variables of the first correlation – a
and b
before the "|
" character – must refer to columns in the data.frame/matrix of the first element in the list of the data
object,
whereas the variables of the second correlation – a
, b
, c
,
and d
after the "|
" character – must refer to columns in the data.frame/matrix of the second element in the list.
The formula
parameterfor correlations based on dependent groups with overlapping variables must follow the pattern ~a + b | a + c
. The variables of the two correlation – a
,
b
,
and c
– must refer to columns in the data.frame/matrix of the data
object.
The formula
for correlations based on dependent groups with nonoverlapping variables must have the form ~a + b | c + d
. The variables of the two correlation – a
,
b
, c
,
and d
– must refer to columns in the data.frame/matrix of the data
object.
Returns an object of class 'cocor.indep.groups', 'cocor.dep.groups.overlap', or 'cocor.dep.groups.nonoverlap' depending on the invoked comparison function.
cocor.indep.groups, cocor.dep.groups.overlap, cocor.dep.groups.nonoverlap, as.htest
data("aptitude") # Compare two correlations based on two independet groups cocor(~logic + intelligence.a | logic + intelligence.a, aptitude) # Compare two correlations based on two depenendent groups # The correlations are overlapping cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample2) # The correlations are nonoverlapping cocor(~logic + intelligence.b | knowledge + intelligence.a, aptitude$sample1) cocor(~logic + intelligence.b | knowledge + intelligence.a, aptitude$sample2) # Return result as a list of class 'htest' cocor(~knowledge + intelligence.b | logic + intelligence.a, aptitude$sample1, return.htest=TRUE)
data("aptitude") # Compare two correlations based on two independet groups cocor(~logic + intelligence.a | logic + intelligence.a, aptitude) # Compare two correlations based on two depenendent groups # The correlations are overlapping cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample2) # The correlations are nonoverlapping cocor(~logic + intelligence.b | knowledge + intelligence.a, aptitude$sample1) cocor(~logic + intelligence.b | knowledge + intelligence.a, aptitude$sample2) # Return result as a list of class 'htest' cocor(~knowledge + intelligence.b | logic + intelligence.a, aptitude$sample1, return.htest=TRUE)
Performs a test of significance for the difference between two correlations based on dependent groups (e.g.,
the same group). The two correlations are nonoverlapping, i.e.,
they have no variable in common. The function tests whether the correlations between j
and k
(r.jk
) and between h
and m
(r.hm
) differ in magnitude. Because the significance depends on the pairwise intercorrelations between all of the variables involved (j
,
k
, h
, and m
),
these intercorrelations have to be provided as additional parameters. The function expects correlation coefficients as input.
cocor.dep.groups.nonoverlap( r.jk, r.hm, r.jh, r.jm, r.kh, r.km, n, alternative = "two.sided", test = "all", alpha = 0.05, conf.level = 0.95, null.value = 0, data.name = NULL, var.labels = NULL, return.htest = FALSE )
cocor.dep.groups.nonoverlap( r.jk, r.hm, r.jh, r.jm, r.kh, r.km, n, alternative = "two.sided", test = "all", alpha = 0.05, conf.level = 0.95, null.value = 0, data.name = NULL, var.labels = NULL, return.htest = FALSE )
r.jk |
A number specifying the correlation between |
r.hm |
A number specifying the correlation between |
r.jh |
A number specifying the correlation between |
r.jm |
A number specifying the correlation between |
r.kh |
A number specifying the correlation between |
r.km |
A number specifying the correlation between |
n |
An integer defining the size of the group |
alternative |
A character string specifying whether the alternative hypothesis is two-sided (" |
test |
A vector of character strings specifying the tests to be used ( |
alpha |
A number defining the alpha level for the hypothesis test. The default value is |
conf.level |
A number defining the level of confidence for the confidence interval (if test |
null.value |
A number defining the hypothesized difference between the two correlations used for testing the null hypothesis. The default value is |
data.name |
A character string giving the name of the data/group. |
var.labels |
A vector of four character strings specifying the labels for j, k, h, and m (in this order). |
return.htest |
A logical indicating whether the result should be returned as a list containing a list of class 'htest' for each test. The default value is |
Returns an S4 object of class 'cocor.dep.groups.nonoverlap' with the following slots:
r.jk |
Input parameter |
r.hm |
Input parameter |
r.jh |
Input parameter |
r.jm |
Input parameter |
r.kh |
Input parameter |
r.km |
Input parameter |
n |
Input parameter |
alternative |
Input parameter |
alpha |
Input parameter |
conf.level |
Input parameter |
null.value |
Input parameter |
data.name |
Input parameter |
var.labels |
Input parameter |
diff |
Difference between the two correlations, r.jk and r.hm, that were compared |
For each test a slot of the same name exists with a list containing the following elements:
statistic |
The value of the test statistic (unless test |
distribution |
The distribution of the test statistic (unless test |
p.value |
The p-value of the test (unless test |
conf.int |
The confidence interval of the difference between the two correlations (if test |
In the following,
and
are the two correlations that are being compared;
and
are their
transformed equivalents.
,
,
,
and
are the related correlations that are also required.
specifies the size of the group the two correlations are based on.
Some tests make use of Fisher's
-to-
transformation (1921, p. 26):
Pearson and Filon's (1898) z
This test was proposed by Pearson and Filon (1898, p. 262, formula xl).
The formula for the test statistic is computed as
(Raghunathan, Rosenthal, and Rubin, 1996, p. 179, formula 1), where
(Raghunathan et al. (1996, p. 179, formula 2). The two formulae can also be found in Steiger (1980, p. 245, formula 2 and p. 246, formula 5).
Dunn and Clark's (1969) z
The test statistic of this test is calculated as
(Dunn and Clark, 1969, p. 370, formula 15), where
(Dunn and Clark, 1969, p. 368, formula 9).
Steiger's (1980) modification of Dunn and Clark's (1969) z using average correlations
This test was proposed by Steiger (1980) and is a modification of Dunn and Clark's (1969) .
Instead of
and
the mean of the two is being used.
The test statistic
is given by
(Steiger, 1980, p. 247, formula 15), where
(Steiger, 1980, p. 247) and
(Steiger, 1980, p. 247, formula 11; in the original article, there are brackets missing around the divisor).
Raghunathan, Rosenthal, and Rubin's (1996) modification of Pearson and Filon's (1898) z
This test of Raghunathan, Rosenthal,
and Rubin (1996) is based on Pearson and Filon's (1898) .
Unlike Pearson and Filon (1898),
Raghunathan et al. (1996) use
transformed correlation coefficients.
The test statistic
is computed as
(Raghunathan et al., 1996, p. 179, formula 3), where
(Raghunathan et al., 1996, p. 179, formula 2).
Silver, Hittner, and May's (2004) modification of Dunn and Clark's (1969) z using a backtransformed average Fisher's (1921) Z procedure
The approach to backtransform averaged Fisher's (1921) s was first proposed in Silver and Dunlap (1987) and was applied to the comparison of nonoverlapping correlations by Silver et al. (2004).
The test is based on Steiger's (1980) approach.
The formula of the test statistic
is given by
(Silver et al., 2004, p. 55, formula 5), where
(Silver et al., 2004, p. 56),
(Silver and Dunlap, 1987, p. 146, formula 4), and
(Silver et al., 2004, p. 55).
Zou's (2007) confidence interval
This test calculates the confidence interval of the difference between the two correlations and
.
If the confidence interval includes zero,
the null hypothesis that the two correlations are equal must be retained.
If the confidence interval does not include zero, the null hypothesis has to be rejected.
A lower and upper bound for the interval (
and
, respectively) is given by
and
(Zou, 2007, pp. 409-410), where
(Zou, 2007, p. 406),
(Zou, 2007, p. 409), and
(Zou, 2007, p. 406).
denotes the desired alpha level of the confidence interval.
Dunn, O. J., & Clark, V. A. (1969). Correlation coefficients measured on the same individuals. Journal of the American Statistical Association, 64, 366-377. doi:10.2307/2283746
Pearson, K., & Filon, L. N. G. (1898). Mathematical contributions to theory of evolution: IV. On the probable errors of frequency constants and on the influence of random selection and correlation. Philosophical Transactions of the Royal Society of London, Series A, 191, 229-311. doi:10.1098/rsta.1898.0007
Raghunathan, T. E., Rosenthal, R., & Rubin, D. B., (1996). Comparing correlated but nonoverlapping correlations. Psychological Methods, 1, 178-183. doi:10.1037//1082-989X.1.2.178
Silver, N. C., & Dunlap, W. P. (1987). Averaging correlation coefficients: Should Fisher's Z transformation be used? Journal of Applied Psychology, 72, 146-148. doi:10.1037//0021-9010.72.1.146
Silver, N. C., Hittner, J. B., & May, K. (2004). Testing dependent correlations with nonoverlapping variables: A Monte Carlo simulation. Journal of Experimental Education, 73, 53-69. doi:10.3200/JEXE.71.1.53-70
Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological Bulletin, 87, 245-251. doi:10.1037//0033-2909.87.2.245
Zou, G. Y. (2007). Toward using confidence intervals to compare correlations. Psychological Methods, 12, 399-413. doi:10.1037/1082-989X.12.4.399
cocor, cocor.indep.groups, cocor.dep.groups.overlap, as.htest
# Compare the difference between the correlations (age, intelligence) and # body mass (index, shoe size) measured in the same group (all values are fictional): r.jk <- .2 # Correlation (age, intelligence) r.hm <- .7 # Correlation (body mass index, shoe size) r.jh <- .4 # Correlation (age, body mass index) r.jm <- .5 # Correlation (age, shoe size) r.kh <- .1 # Correlation (intelligence, body mass index) r.km <- .3 # Correlation (intelligence, shoe size) n <- 232 # Size of the group cocor.dep.groups.nonoverlap(r.jk, r.hm, r.jh, r.jm, r.kh, r.km, n, var.labels=c("age", "intelligence", "body mass index", "shoe size"))
# Compare the difference between the correlations (age, intelligence) and # body mass (index, shoe size) measured in the same group (all values are fictional): r.jk <- .2 # Correlation (age, intelligence) r.hm <- .7 # Correlation (body mass index, shoe size) r.jh <- .4 # Correlation (age, body mass index) r.jm <- .5 # Correlation (age, shoe size) r.kh <- .1 # Correlation (intelligence, body mass index) r.km <- .3 # Correlation (intelligence, shoe size) n <- 232 # Size of the group cocor.dep.groups.nonoverlap(r.jk, r.hm, r.jh, r.jm, r.kh, r.km, n, var.labels=c("age", "intelligence", "body mass index", "shoe size"))
Performs a test of significance for the difference between two correlations based on dependent groups (e.g.,
the same group). The two correlations are overlapping, i.e.,
they have one variable in common. The comparison is made between r.jk
and r.jh
. The function tests whether the correlations between j
and k
(r.jk
) and between j
and h
(r.jh
) differ in magnitude. Because the significance depends on the intercorrelation between k
and h
(r.kh),
this intercorrelation has to be provided as an additional parameter. The function expects correlation coefficients as input.
cocor.dep.groups.overlap( r.jk, r.jh, r.kh, n, alternative = "two.sided", test = "all", alpha = 0.05, conf.level = 0.95, null.value = 0, data.name = NULL, var.labels = NULL, return.htest = FALSE )
cocor.dep.groups.overlap( r.jk, r.jh, r.kh, n, alternative = "two.sided", test = "all", alpha = 0.05, conf.level = 0.95, null.value = 0, data.name = NULL, var.labels = NULL, return.htest = FALSE )
r.jk |
A number specifying the correlation between |
r.jh |
A number specifying the correlation between |
r.kh |
A number specifying the correlation between |
n |
An integer defining the size of the group |
alternative |
A character string specifying whether the alternative hypothesis is two-sided (" |
test |
A vector of character strings specifying the tests to be used ( |
alpha |
A number defining the alpha level for the hypothesis test. The default value is |
conf.level |
A number defining the level of confidence for the confidence interval (if test |
null.value |
A number defining the hypothesized difference between the two correlations used for testing the null hypothesis. The default value is |
data.name |
A character string giving the name of the data/group. |
var.labels |
A vector of three character strings specifying the labels for j, k, and h (in this order). |
return.htest |
A logical indicating whether the result should be returned as a list containing a list of class 'htest' for each test. The default value is |
Returns an S4 object of class 'cocor.dep.groups.overlap' with the following slots:
r.jk |
Input parameter |
r.jh |
Input parameter |
r.kh |
Input parameter |
n |
Input parameter |
alternative |
Input parameter |
alpha |
Input parameter |
conf.level |
Input parameter |
null.value |
Input parameter |
data.name |
Input parameter |
var.labels |
Input parameter |
diff |
Difference between the two correlations, r.jk and r.jh, that were compared |
For each test a slot of the same name exists with a list containing the following elements:
statistic |
The value of the test statistic (unless test |
distribution |
The distribution of the test statistic (unless test |
df |
The degrees of freedom of the distribution of the test statistic (if test |
p.value |
The p-value of the test (unless test |
conf.int |
The confidence interval of the difference between the two correlations (if test |
In the following,
and
are the two correlations that are being compared;
and
are their
transformed equivalents.
is the related correlation that is additionally required.
specifies the size of the group the two correlations are based on.
Some tests make use of Fisher's
-to-
transformation (1921, p. 26):
Pearson and Filon's (1898) z
This test was proposed by Pearson and Filon (1898, p. 259, formula xxxvii).
The test statistic is computed as
(Steiger, 1980, p. 246, formula 4), where
(Steiger, 1980, p. 245 formula 3).
Hotelling's (1940) t
The test statistic is given by
(Hotelling, 1940, p. 278, formula 7) with , where
(Hotelling, 1940, p. 278). The test statistic is also given in Steiger (1980, p. 246), Glass and Stanley (1984, p. 311, formula 15.7), and Hittner, May, and Silver (2003, p. 152).
Williams' (1959) t
This test is a modification of Hotelling's (1940) and was suggested by Williams (1959).
Two mathematically different formulae for Williams'
can be found in the literature (Hittner et al.,
2003, p. 152).
This is the version that Hittner et al. (2003,
p. 152) labeled as "standard Williams'
":
with , where
and
An alternative formula for Williams' —termed as "Williams' modified
per Hendrickson,
Stanley, and Hills (1970)" by Hittner et al. (2003,
p. 152)—is implemented in this function as
hendrickson1970
(see below).
The test statistic of williams1959
is also given in Steiger (1980, p. 246,
formula 7) and Neill and Dunn (1975, p. 533).
Results of williams1959
are in accordance with the results of the software DEPCORR by Hittner and May (1998) and DEPCOR by Silver,
Hittner, and May (2006).
However,
we found several typographical errors in formulae that also claim to compute Williams' .
For example, the formula reported by Boyer, Palachek, and Schucany (1983,
p. 76) contains an error because the term
is not being cubed.
There are also typographical errors in the formula described by Hittner et al. (2003,
p. 152). For example,
is divided instead of being multiplied by the square root term, and in the denominator of the fraction in the square root term,
there are additional parentheses so that the whole denominator is multiplied by 2.
These same errors can also be found in Wilcox and Tian (2008, p. 107, formula 1).
Olkin's (1967) z
In the original article by Olkin (1967, p. 112) and in Hendrickson, Stanley,
and Hills (1970, p. 190, formula 2), the reported formula contains a typographical error.
Hendrickson and Collins (1970, p. 639) provide a corrected version.
In the revised version, however, in the enumerator is decreased by 1.
This function implements the corrected formula without the decrement.
The formula implemented in this function is used by Glass and Stanley (1970, p. 313,
formula 14.19), Hittner et al. (2003, p. 152), and May and Hittner (1997a, p. 259; 1997b,
p. 480):
Dunn and Clark's (1969) z
The test statistic of this test is calculated as
(Dunn and Clark, 1969, p. 370, formula 15), where
(Dunn and Clark, 1969, p. 368, formula 8).
Hendrickson, Stanley, and Hills' (1970) modification of Williams' (1959) t
This test is a modification of Hotelling's (1940) and was suggested by Williams (1959).
Two mathematically different formulae of Williams' (1959)
can be found in the literature.
hendrickson1970
is the version that Hittner et al. (2003,
p. 152) labeled as "Williams' modified per Hendrickson, Stanley, and Hills (1970)".
An alternative formula termed as "standard Williams'
" by Hittner et al. (2003,
p. 152) is implemented as
williams1959
(see above).
The hendrickson1970
formula can be found in Hendrickson, Stanley, and Hills (1970,
p. 193), May and Hittner (1997a, p. 259; 1997b, p. 480), and Hittner et al. (2003,
p. 152):
with .
A slightly changed version of this formula was provided by Dunn and Clark (1971, p. 905,
formula 1.2), but seems to be erroneous, due to an error in the denominator.
Steiger's (1980) modification of Dunn and Clark's (1969) z using average correlations
This test was proposed by Steiger (1980) and is a modification of Dunn and Clark's (1969) .
Instead of
and
, the mean of the two is used.
The test statistic
is defined as
(Steiger 1980, p. 247, formula 14), where
(Steiger, 1980, p. 247) and
(Steiger ,1980, p. 247, formula 10; in the original article, there are brackets missing around the divisor).
Meng, Rosenthal, and Rubin's (1992) z
This test is based on the test statistic ,
(Meng et al., 1992, p. 173, formula 1), where
(Meng et al., 1992, p. 173, formula 2),
( must be
; Meng et al., 1992, p. 173, formula 3), and
(Meng et al., 1992, p. 173).
This test also constructs a confidence interval of the difference between the two correlation coefficients and
:
(Meng et al., 1992, p. 173, formula 4).
denotes the desired alpha level of the confidence interval.
If the confidence interval includes zero,
the null hypothesis that the two correlations are equal must be retained.
If zero is outside the confidence interval, the null hypothesis can be rejected.
Hittner, May, and Silver's (2003) modification of Dunn and Clark's (1969) z using a backtransformed average Fisher's (1921) Z procedure
The approach to backtransform averaged Fisher's (1921) s was first proposed by Silver and Dunlap (1987) and was applied to the comparison of overlapping correlations by Hittner et al. (2003).
The test is based on Steiger's (1980) approach.
The test statistic
is calculated as
(Hittner et al., 2003, p. 153), where
(Hittner et al., 2003, p. 153),
(Silver and Dunlap, 1987, p. 146, formula 4), and
(Silver and Dunlap, 1987, p. 146).
Zou's (2007) confidence interval
This test calculates the confidence interval of the difference between the two correlation coefficients and
.
If the confidence interval includes zero,
the null hypothesis that the two correlations are equal must be retained.
If the confidence interval does not include zero, the null hypothesis has to be rejected.
A lower and upper bound for the interval (
and
, respectively) is given by
and
(Zou, 2007, p. 409), where
(Zou, 2007, p. 406),
(Zou, 2007, p. 409), and
(Zou, 2007, p. 406).
denotes the desired alpha level of the confidence interval.
Boyer, I. E., Palachek, A. D., & Schucany. W. R. (1983). An empirical study of related correlation coefficients. Journal of Educational Statistics, 8, 75-86. doi:10.2307/1164871
Dunn, O. J. & Clark, V. A. (1969). Correlation coefficients measured on the same individuals. Journal of the American Statistical Association, 64, 366-377. doi:10.2307/2283746
Dunn, O. J. & Clark, V. A. (1971). Comparison of tests of the equality of dependent correlation coefficients. Journal of the American Statistical Association, 66, 904-908. doi:10.2307/2284252
Fisher, R. A. (1921). On the probable error of a coefficient of correlation deduced from a small sample. Metron, 1, 1-32.
Glass, G. V., & Stanley, J. C. (1970). Statistical methods in eduction and psychology. Englewood Cliffs, NJ: Prentice-Hall.
Glass, G. V., & Stanley, J. C. (1984). Statistical methods in eduction and psychology (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall.
Hendrickson, G. F., Stanley J. C., & Hills, J. R. (1970). Olkin's new formula for significance of r13 vs. r23 compared with Hotelling's method. American Educational Research Journal, 7, 189-195. doi:10.2307/1162159
Hendrickson, G. F., & Collins, J. R. (1970). Note correcting the results in "Olkin's new formula for the significance of r13 vs. r23 compared with Hotelling's method". American Educational Research Journal, 7, 639-641. doi:10.2307/1161847
Hittner, J. B., & May, K. (1998). DEPCORR: A SAS program for comparing dependent correlations. Applied Psychological Measurement, 22, 93-94. doi:10.1177/01466216980221010
Hittner, J. B., May, K., & Silver, N. C. (2003). A Monte Carlo evaluation of tests for comparing dependent correlations. The Journal of General Psychology, 130, 149-168. doi:10.1080/00221300309601282
Hotelling, H. (1940). The selection of variates for use in prediction, with some comments on the general problem of nuisance parameters. Annals of Mathematical Statistics, 11, 271-283. doi:10.1214/aoms/1177731867
May, K., & Hittner, J. B., (1997a) - A note on statistics for comparing dependent correlations. Psychological Reports, 80, 475-480. doi:10.2466/pr0.1997.80.2.475
May, K., & Hittner, J. B. (1997b). Tests for comparing dependent correlations revisited: A Monte Carlo study. The Journal of Experimental Education, 65, 257-269. doi:10.1080/00220973.1997.9943458
Meng, X. L., Rosenthal, R., & Rubin, D. B. (1992). Comparing correlated correlation coefficients. Psychological Bulletin, 111, 172-175. doi:10.1037//0033-2909.111.1.172
Neill, J. J., & Dunn, O. J. (1975). Equality of dependent correlation coefficients. Biometrics, 31, 531-543. doi:10.2307/2529435
Olkin, I. (1967). Correlations revisited. In J. C. Stanley (Ed.), Improving experimental design and statistical analysis (pp. 102-128). Chicago, IL: Rand McNally.
Pearson, K., & Filon, L. N. G. (1898). Mathematical contributions to theory of evolution: IV. On the probable errors of frequency constants and on the influence of random selection and correlation. Philosophical Transactions of the Royal Society of London, Series A, 191, 229-311. doi:10.1098/rsta.1898.0007
Silver, N. C , & Dunlap, W. P. (1987). Averaging correlation coefficients: Should Fisher's Z transformation be used? Journal of Applied Psychology, 72, 146-148. doi:10.1037//0021-9010.72.1.146
Silver, N. C., Hittner, J. B., & May, K. (2004). Testing dependent correlations with nonoverlapping variables: A Monte Carlo simulation. Journal of Experimental Education, 73, 53-69. doi:10.3200/JEXE.71.1.53-70
Silver, N. C., Hittner, J. B., & May, K. (2006). A FORTRAN 77 program for comparing dependent correlations. Applied Psychological Measurement, 30, 152-153. doi:10.1177/0146621605277132
Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological Bulletin, 87, 245-251. doi:10.1037//0033-2909.87.2.245
Wilcox, R. R., & Tian, T. (2008). Comparing dependent correlations. The Journal of General Psychology, 135, 105-112. doi:10.3200/GENP.135.1.105-112
Williams, E. J. (1959). The comparison of regression variables. Journal of the Royal Statistical Society, Series B, 21, 396-399. Retrieved from http://www.jstor.org/stable/2983809
Zou, G. Y. (2007). Toward using confidence intervals to compare correlations. Psychological Methods, 12, 399-413. doi:10.1037/1082-989X.12.4.399
cocor, cocor.indep.groups, cocor.dep.groups.nonoverlap, as.htest
# Compare the difference between the correlations (age, intelligence) and # (age, shoe size) measured in the same group (all values are fictional): r.jk <- .2 # Correlation (age, intelligence) r.jh <- .5 # Correlation (age, shoe size) r.kh <- .1 # Correlation (intelligence, shoe size) n <- 315 # Size of the group cocor.dep.groups.overlap(r.jk, r.jh, r.kh, n, var.labels=c("age", "intelligence", "shoe size"))
# Compare the difference between the correlations (age, intelligence) and # (age, shoe size) measured in the same group (all values are fictional): r.jk <- .2 # Correlation (age, intelligence) r.jh <- .5 # Correlation (age, shoe size) r.kh <- .1 # Correlation (intelligence, shoe size) n <- 315 # Size of the group cocor.dep.groups.overlap(r.jk, r.jh, r.kh, n, var.labels=c("age", "intelligence", "shoe size"))
Performs a test of significance for the difference between two correlation coefficients based on independent groups. The function expects correlation coefficients as input.
cocor.indep.groups( r1.jk, r2.hm, n1, n2, alternative = "two.sided", test = "all", alpha = 0.05, conf.level = 0.95, null.value = 0, data.name = NULL, var.labels = NULL, return.htest = FALSE )
cocor.indep.groups( r1.jk, r2.hm, n1, n2, alternative = "two.sided", test = "all", alpha = 0.05, conf.level = 0.95, null.value = 0, data.name = NULL, var.labels = NULL, return.htest = FALSE )
r1.jk |
A number specifying the correlation between j and k measured in group 1 |
r2.hm |
A number specifying the correlation between h and m measured in group 2 |
n1 |
An integer defining the size of group 1 |
n2 |
An integer defining the size of group 2 |
alternative |
A character string specifying whether the alternative hypothesis is two-sided (" |
test |
A vector of character strings specifying the tests to be used ( |
alpha |
A number defining the alpha level for the hypothesis test. The default value is |
conf.level |
A number defining the level of confidence for the confidence interval (if test |
null.value |
A number defining the hypothesized difference between the two correlations used for testing the null hypothesis. The default value is |
data.name |
A vector of character strings describing the data/groups. The vector may contain one character string to describe both data sets/groups or two character strings to describe each data set/group separately. |
var.labels |
A vector of four character strings specifying the labels for j, k, h, and m (in this order). |
return.htest |
A logical indicating whether the result should be returned as a list containing a list of class 'htest' for each test. The default value is |
Returns an S4 object of class 'cocor.indep.groups' with the following slots:
r1.jk |
Input parameter |
r2.hm |
Input parameter |
n1 |
Input parameter |
n2 |
Input parameter |
alternative |
Input parameter |
alpha |
Input parameter |
conf.level |
Input parameter |
null.value |
Input parameter |
data.name |
Input parameter |
var.labels |
Input parameter |
diff |
Difference between the two correlations, r1.jk and r2.hm, that were compared |
For each test a slot of the same name exists with a list containing the following elements:
statistic |
The value of the test statistic (if test |
distribution |
The distribution of the test statistic (if test |
p.value |
The p-value of the test (if test |
conf.int |
The confidence interval of the difference between the two correlations (if test |
The tests make use of Fisher's -to-
transformation (1921, p. 26):
Fisher's (1925) z
This significance test was first described in Fisher (1925,
pp. 161-168) and its test statistic is calculated as
and
are the two
transformed correlations that are being compared.
and
specify the size of the two groups the correlations are based on.
The equation is also given for example in Peters and van Voorhis (1940, p. 188) and Cohen,
Cohen, West, and Aiken (2003, p. 49, formula 2.8.11).
Zou's (2007) confidence interval
This test calculates the confidence interval of the difference between the two correlation coefficients and
.
If the confidence interval includes zero,
the null hypothesis that the two correlations are equal must be retained.
If the confidence interval does not include zero, the null hypothesis has to be rejected.
A lower and upper bound for the interval (
and
, respectively) is given by
and
(Zou, 2007, p. 409).
A lower and upper bound for the confidence interval of (
and
) and
(
and
) are calculated as
(Zou, 2007, p. 406), where
(Zou, 2007, p. 406).
denotes the desired alpha level of the confidence interval,
whereas
specifies the size of the group the correlation is based on.
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd ed.). Mahwah, NJ: Erlbaum.
Fisher, R. A. (1921). On the probable error of a coefficient of correlation deduced from a small sample. Metron, 1, 1-32.
Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh, Scotland: Oliver and Boyd. Retrieved from http://psychclassics.yorku.ca/
Peters, C. C., & Van Voorhis, W. R. (1940). Statistical procedures and their mathematical bases. New York: McGraw-Hill.
Zou, G. Y. (2007). Toward using confidence intervals to compare correlations. Psychological Methods, 12, 399-413. doi:10.1037/1082-989X.12.4.399
cocor, cocor.dep.groups.overlap, cocor.dep.groups.nonoverlap, as.htest
# Compare the difference between two correlations based # on two independent groups: r1.jk <- .7 # Correlation between age and intelligence measured in group 1 n1 <- 305 # Size of group 1 r2.hm <- .6 # Correlation between age and intelligence measured in group 2 n2 <- 210 # Size of group 2 cocor.indep.groups(r1.jk, r2.hm, n1, n2, data.name=c("group1", "group2"), var.labels=c("age", "intelligence", "age", "intelligence"))
# Compare the difference between two correlations based # on two independent groups: r1.jk <- .7 # Correlation between age and intelligence measured in group 1 n1 <- 305 # Size of group 1 r2.hm <- .6 # Correlation between age and intelligence measured in group 2 n2 <- 210 # Size of group 2 cocor.indep.groups(r1.jk, r2.hm, n1, n2, data.name=c("group1", "group2"), var.labels=c("age", "intelligence", "age", "intelligence"))
Returns input parameters of a cocor result object of class 'cocor.indep.groups', 'cocor.dep.groups.overlap', or 'cocor.dep.groups.nonoverlap' as a list with each slot representing a list element.
get.cocor.input(result.object) ## S4 method for signature 'cocor' get.cocor.input(result.object)
get.cocor.input(result.object) ## S4 method for signature 'cocor' get.cocor.input(result.object)
result.object |
A cocor result object of class 'cocor.indep.groups', 'cocor.dep.groups.overlap', or 'cocor.dep.groups.nonoverlap'. |
Returns a list containing all input parameters as list elements:
r1.jk |
A number specifying the correlation between j and k measured in group 1 (only for result objects of class 'cocor.indep.groups'). |
r2.hm |
A number specifying the correlation between h and m measured in group 2 (only for result objects of class 'cocor.indep.groups'). |
n1 |
An integer defining the size of group 1 (only for result objects of class 'cocor.indep.groups'). |
n2 |
An integer defining the size of group 2 (only for result objects of class 'cocor.indep.groups'). |
r.jk |
A number specifying the correlation between j and k (only for result objects of class 'cocor.dep.groups.overlap' and 'cocor.dep.groups.nonoverlap'). |
r.jh |
A number specifying the correlation between j and h (only for result objects of class 'cocor.dep.groups.overlap'). |
r.hm |
A number specifying the correlation between h and m (only for result objects of class 'cocor.dep.groups.nonoverlap'). |
n |
An integer defining the size of the group (only for result objects of class 'cocor.dep.groups.overlap' and 'cocor.dep.groups.nonoverlap'). |
alternative |
A character string specifying whether the alternative hypothesis is two-sided ("two.sided") or one-sided ("greater" or "less", depending on the direction). |
alpha |
A number defining the alpha level for the hypothesis test. |
conf.level |
A number defining the level of confidence for the confidence interval. |
null.value |
A number defining the hypothesized difference between the two correlations used for testing the null hypothesis. |
data.name |
A vector of character strings describing the data/groups. |
var.labels |
A vector of four character strings specifying the labels for j, k, h, and m (in this order). |
get.cocor.results, cocor, cocor.indep.groups, cocor.dep.groups.overlap, cocor.dep.groups.nonoverlap
data("aptitude") cocor.result <- cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) get.cocor.input(cocor.result)
data("aptitude") cocor.result <- cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) get.cocor.input(cocor.result)
Returns result parameters of a cocor result object of class 'cocor.indep.groups', 'cocor.dep.groups.overlap', or 'cocor.dep.groups.nonoverlap' as a list with each slot representing a list element.
get.cocor.results(result.object, test = "all") ## S4 method for signature 'cocor' get.cocor.results(result.object, test = "all")
get.cocor.results(result.object, test = "all") ## S4 method for signature 'cocor' get.cocor.results(result.object, test = "all")
result.object |
A cocor result object of class 'cocor.indep.groups', 'cocor.dep.groups.overlap', or 'cocor.dep.groups.nonoverlap'. |
test |
A vector of character strings specifying the tests to be returned (e.g.,
|
Returns a list containing all result parameters as list elements:
diff |
Difference between the two correlations that were compared. |
statistic |
The value of the test statistic (unless test zou2007 is used). |
distribution |
The distribution of the test statistic (unless test zou2007 is used). |
df |
The degrees of freedom of the distribution of the test statistic (only for result objects of class 'cocor.dep.groups.overlap' if test hotelling1940, hendrickson1970, or williams1959 is used). |
p.value |
The p-value of the test (unless test zou2007 is used). |
conf.int |
The confidence interval of the difference between the two correlations (if test zou2007 is used). |
get.cocor.input, cocor, cocor.indep.groups, cocor.dep.groups.overlap, cocor.dep.groups.nonoverlap
data("aptitude") cocor.result <- cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) get.cocor.results(cocor.result)
data("aptitude") cocor.result <- cocor(~knowledge + intelligence.a | logic + intelligence.a, aptitude$sample1) get.cocor.results(cocor.result)