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The relationship between the Pearson zero-order correlation and a simple regression coefficient

The Pearson correlation equals the beta coefficient in SPSS which is the regression estimate from using one standardized predictor to predict a standardized outcome (see for example Cohen and Cohen (1983), pp 41-44 and the regression talk given as part of the Graduate Statistics Programme October-December 2009). In addition, the t-test in the regression for the unstandardized coefficient, B, is equivalent to testing if a zero-order correlation equals zero. The regression model is assumed to contain the intercept as well as the predictor.

The following syntax for regression and correlations in SPSS give equivalent tests of whether a Pearson correlation between two continuous variables, x and y, is zero.

REGRESSION
  /MISSING LISTWISE
  /STATISTICS COEFF OUTS R ANOVA
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN
  /DEPENDENT y
  /METHOD=ENTER x  .

CORRELATIONS
  /VARIABLES=y x
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE .

Howell (2002) and Cohen and Cohen (1983) also show how two regression coefficients from two independent groups obtained from a simple regression of x on y may be compared using a t-test. Cohen, Cohen, West and Aiken (2003) also show that this t-test may be done using an interaction term.

Ng and Wilcox (2010) suggest two tests of the difference in independent group simple regression coefficients which are more robust (having better accuracy of type I error) to the usual t-test when the groups have different variances and/or are non-normal. R code consisting of two functions called regci and olswbtest is available to fit these two approaches (the Theil-Sen percentile bootstrap and the HC4-based wild bootstrap quasi-t test). These functions may be downloaded by simply typing at a R prompt the one line:

source('http://www-rcf.usc.edu/~rwilcox/Rallfun-v12')

Note the 'v12' may change as versions are updated. If there is an error using the above line go to Rand Wilcox's website to see if the version number has changed.

Details of using these functions are given in the appendix to Ng and Wilcox (2010).

References

Cohen J, & Cohen P (1983). Applied multiple regression/correlation analysis for the behavioral sciences (2nd edition). Hillsdale, NJ: Erlbaum.

Cohen J, Cohen P, West SG and Aiken LS (2003). Applied multiple regression/correlation analysis for the behavioral sciences. Erlbaum:London.

Howell DC (2002). Statistical methoids for psychology. Fifth edition. Wadsworth:London. Details of comparing regression coefficients are also in earlier versions of this book available in the CBSU library.

Ng M and Wilcox RR (2010). Comparing the regression slopes of independent groups. British Journal of Mathematical and Statistical Psychology 63 319-340.