# test.cor: Test for the product-moment correlation coefficient for H0:... In miscor: Miscellaneous Functions for the Correlation Coefficient

## Description

This function tests the product-moment correlation coefficient for H0: ρ = ρ0, so that any value for ρ0 can be specified.

## Usage

 1 2 3 test.cor(x = NULL, y = NULL, r = NULL, n = NULL, rho0 = 0, alternative = c("two.sided", "less", "greater"), reduced = FALSE, conf.level = 0.95, digits = 3, output = TRUE) 

## Arguments

 x a numeric vector. y a numeric vector. r alternative specification, product-moment correlation coefficient. n alternative specification, number of observations. rho0 a number indicating ρ0, the value under the null hypothesis. alternative a character string describing the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". reduced logical: if TRUE, compuatation is based on the reduced formula. conf.level confidence level of the interval. digits integer indicating the number of decimal places to be displayed. output logical: if TRUE, output is shown.

## Details

Computation is based on Fisher's z transformation \textbf{z} = 0.5 /cdot ln(\frac{1 + r}{1 - r}). The difference between the full formula (i.e., reduced = FALSE) and the reduced formula (i.e., reduced = TRUE) is that the full formula includes the term \frac{ρ}{n - 1} in the formula of the exectation E, i.e.,

\emph{E}(\textbf{z}) = 0.5 /cdot ln(\frac{1 + ρ}{1 - ρ}) + \frac{ρ}{n - 1}

whereas the reduced formula does not include this term, i.e.,

\emph{E}(\textbf{z}) = 0.5 /cdot ln(\frac{1 + ρ}{1 - ρ})

It is recommended to always use the full formula, especially in small samples.

## Value

Returns an object of class test.cor with following entries:

 call function call dat data.frame with x and y (if available) spec specification of function arguments res list with results, i.e., t or z (test statistic), df (degree of feedom), pval (significance value), r (correlation coefficient), n (sample size), lower (lower limit of CI), upper (upper limit of CI)

## Author(s)

Takuya Yanagida takuya.yanagida@univie.ac.at,

## References

Cramer, H. (1946). Mathematical methods of statistics. Princeton: Princeton Press.

Rasch, D., Kubinger, K. D., & Yanagida, T. (2011). Statistics in psychology - Using R and SPSS. New York: John Wiley & Sons.

Kubinger, K. D., Rasch, D., & Simeckova, M. (2007). Testing a correlation coefficient's significance: Using H0: 0 < ρ ≤ λ is preferable to H0: ρ = 0. Psychology Science, 49, 74-87.

size.cor, comptest.cor, seqtest.cor
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 #-------------------------------------- # Two-sided test # H0: rho == 0, H1: rho != 0 # r = 0.23, n = 60 test.cor(r = 0.23, n = 120) #-------------------------------------- # Two-sided test # H0: rho == 0.4, H1: rho != 0.4 # r = 0.55, n = 120 test.cor(r = 0.55, n = 120, rho0 = 0.4) #-------------------------------------- # One-sided test # H0: rho <= 0.4, H1: rho > 0.4 # Generate random data dat <- sim.cor(100, rho = 0.4) test.cor(dat$x, dat$y, rho0 = 0.4)