psych (version 2.0.12)

# r.test: Tests of significance for correlations

## Description

Tests the significance of a single correlation, the difference between two independent correlations, the difference between two dependent correlations sharing one variable (Williams's Test), or the difference between two dependent correlations with different variables (Steiger Tests).

## Usage

r.test(n, r12, r34 = NULL, r23 = NULL, r13 = NULL, r14 = NULL, r24 = NULL,
n2 = NULL,pooled=TRUE, twotailed = TRUE)

## Arguments

n

Sample size of first group

r12

Correlation to be tested

r34

Test if this correlation is different from r12, if r23 is specified, but r13 is not, then r34 becomes r13

r23

if ra = r(12) and rb = r(13) then test for differences of dependent correlations given r23

r13

implies ra =r(12) and rb =r(34) test for difference of dependent correlations

r14

implies ra =r(12) and rb =r(34)

r24

ra =r(12) and rb =r(34)

n2

n2 is specified in the case of two independent correlations. n2 defaults to n if if not specified

pooled

use pooled estimates of correlations

twotailed

should a twotailed or one tailed test be used

## Value

test

Label of test done

z

z value for tests 2 or 4

t

t value for tests 1 and 3

p

probability value of z or t

## Details

Depending upon the input, one of four different tests of correlations is done. 1) For a sample size n, find the t value for a single correlation where $$t = \frac{r * \sqrt(n-2)}{\sqrt(1-r^2)}$$ and

$$se = \sqrt{\frac{1-r^2}{n-2}})$$.

2) For sample sizes of n and n2 (n2 = n if not specified) find the z of the difference between the z transformed correlations divided by the standard error of the difference of two z scores: $$z = \frac{z_1 - z_2}{\sqrt{\frac{1}{(n_1 - 3) + (n_2 - 3)}}}$$.

3) For sample size n, and correlations r12, r13 and r23 test for the difference of two dependent correlations (r12 vs r13).

4) For sample size n, test for the difference between two dependent correlations involving different variables.

Consider the correlations from Steiger (1980), Table 1: Because these all from the same subjects, any tests must be of dependent correlations. For dependent correlations, it is necessary to specify at least 3 correlations (e.g., r12, r13, r23)

 Variable M1 F1 V1 M2 F2 V2 M1 1.00 F1 .10 1.00 V1 .40 .50 1.00 M2 .70 .05 .50 1.00 F2 .05 .70 .50 .50 1.00 V2 .45 .50 .80 .50 .60 1.00 Variable M1 F1 V1 M2 F2 V2

For clarity, correlations may be specified by value. If specified by location and if doing the test of dependent correlations, if three correlations are specified, they are assumed to be in the order r12, r13, r23.

Consider the examples from Steiger:

Case A: where Masculinity at time 1 (M1) correlates with Verbal Ability .5 (r12), femininity at time 1 (F1) correlates with Verbal ability r13 =.4, and M1 correlates with F1 (r23= .1). Then, given the correlations: r12 = .4, r13 = .5, and r23 = .1, t = -.89 for n =103, i.e., r.test(n=103, r12=.4, r13=.5,r23=.1)

Case B: Test whether correlation between two variables (e.g., F and V) is the same over time (e.g. F1V1 = F2V2)

r.test(n = 103, r12 = 0.5, r34 = 0.6, r23 = 0.5, r13 = 0.7, r14 = 0.5, r24 = 0.8)

## References

Cohen, J. and Cohen, P. and West, S.G. and Aiken, L.S. (2003) Applied multiple regression/correlation analysis for the behavioral sciences, L.Erlbaum Associates, Mahwah, N.J.

Olkin, I. and Finn, J. D. (1995). Correlations redux. Psychological Bulletin, 118(1):155-164.

Steiger, J.H. (1980), Tests for comparing elements of a correlation matrix, Psychological Bulletin, 87, 245-251.

Williams, E.J. (1959) Regression analysis. Wiley, New York, 1959.

See also corr.test which tests all the elements of a correlation matrix, and cortest.mat to compare two matrices of correlations. r.test extends the tests in paired.r,r.con

## Examples

Run this code
# NOT RUN {
n <- 30
r <- seq(0,.9,.1)
rc <- matrix(r.con(r,n),ncol=2)
test <- r.test(n,r)
r.rc <- data.frame(r=r,z=fisherz(r),lower=rc[,1],upper=rc[,2],t=test$t,p=test$p)
round(r.rc,2)

r.test(50,r)
r.test(30,.4,.6)       #test the difference between two independent correlations
r.test(103,.4,.5,.1)   #Steiger case A of dependent correlations
r.test(n=103, r12=.4, r13=.5,r23=.1)
#for complicated tests, it is probably better to specify correlations by name
r.test(n=103,r12=.5,r34=.6,r13=.7,r23=.5,r14=.5,r24=.8)   #steiger Case B

##By default, the precision of p values is 2 decimals
#Consider three different precisions shown by varying the requested number of digits
r12 =  0.693458895410494
r23 =  0.988475791500198
r13 =  0.695966022434845
print(r.test(n =  5105 ,  r12 =  r12 ,  r23 = r23 , r13 =  r13 )) #probability < 0.1
print(r.test(n =  5105 ,  r12 =  r12, r23 = r23 , r13 =  r13 ),digits=4) #p < 0.1001
print(r.test(n =  5105 , r12 =  r12, r23 = r23 , r13 =  r13 ),digits=8) #p< <0.1000759

#an example of how to compare the elements of two matrices
R1 <- lowerCor(psychTools::bfi[1:200,1:5])  #find one set of Correlations
R2 <- lowerCor(psychTools::bfi[201:400,1:5]) #and now another set sampled
#from the same population
test <- r.test(n=200, r12 = R1, r34 = R2)
round(lowerUpper(R1,R2,diff=TRUE),digits=2)  #show the differences between correlations
#lowerMat(test$p) #show the p values of the difference between the two matrices adjusted <- p.adjust(test$p[upper.tri(test$p)]) both <- test$p