<?xml version="1.0" encoding="utf-8"?><!DOCTYPE article  PUBLIC '-//OASIS//DTD DocBook XML V4.4//EN'  'http://www.docbook.org/xml/4.4/docbookx.dtd'><article><articleinfo><title>FAQ/pearson</title><revhistory><revision><revnumber>14</revnumber><date>2013-03-08 10:17:59</date><authorinitials>localhost</authorinitials><revremark>converted to 1.6 markup</revremark></revision><revision><revnumber>13</revnumber><date>2010-09-01 15:41:14</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>12</revnumber><date>2010-09-01 15:37:14</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>11</revnumber><date>2010-09-01 15:35:40</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>10</revnumber><date>2010-09-01 15:05:15</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>9</revnumber><date>2010-09-01 15:04:18</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>8</revnumber><date>2009-12-01 12:41:17</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>7</revnumber><date>2009-12-01 12:40:07</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>6</revnumber><date>2009-12-01 12:37:56</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>5</revnumber><date>2009-12-01 12:37:39</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>4</revnumber><date>2009-12-01 12:23:27</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>3</revnumber><date>2009-12-01 12:22:37</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>2</revnumber><date>2009-12-01 12:22:15</date><authorinitials>PeterWatson</authorinitials></revision><revision><revnumber>1</revnumber><date>2009-12-01 12:20:44</date><authorinitials>PeterWatson</authorinitials></revision></revhistory></articleinfo><section><title>The relationship between the Pearson zero-order correlation and a simple regression coefficient</title><para>The Pearson correlation equals the <emphasis>beta</emphasis> 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  <ulink url="https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/pearson/statswiki/StatsCourse2009#">Graduate Statistics Programme October-December 2009</ulink>). 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. </para><para>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. </para><screen><![CDATA[REGRESSION
  /MISSING LISTWISE
  /STATISTICS COEFF OUTS R ANOVA
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN
  /DEPENDENT y
  /METHOD=ENTER x  .]]></screen><screen><![CDATA[CORRELATIONS
  /VARIABLES=y x
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE .]]></screen><para>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.  </para><para>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 <emphasis>regci</emphasis> and <emphasis>olswbtest</emphasis> 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: </para><screen><![CDATA[source('http://www-rcf.usc.edu/~rwilcox/Rallfun-v12')]]></screen><para><emphasis role="underline">Note</emphasis> the 'v12' may change as versions are updated. If there is an error using the above line go to Rand Wilcox's <ulink url="http://www-rcf.usc.edu/~rwilcox/">website</ulink> to see if the version number has changed. </para><para>Details of using these functions are given in the appendix to Ng and Wilcox (2010).  </para><para><emphasis role="underline">References</emphasis> </para><para>Cohen J, &amp; Cohen P (1983). Applied multiple regression/correlation analysis for the behavioral sciences (2nd edition). Hillsdale, NJ: Erlbaum. </para><para>Cohen J, Cohen P, West SG and Aiken LS (2003). Applied multiple regression/correlation analysis for the behavioral sciences. Erlbaum:London. </para><para>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. </para><para>Ng M and Wilcox RR (2010). Comparing the regression slopes of independent groups. <emphasis>British Journal of Mathematical and Statistical Psychology</emphasis> <emphasis role="strong">63</emphasis> 319-340. </para></section></article>