Example where a simple correlation coefficient has a sign opposite to that of the corresponding partial correlation coefficient

Give some examples where a simple correlation coefficient has a sign opposite to that of the corresponding partial correlation coefficient and comment on it.

It is a question from an examination paper.Please help.

Answer

The sign of partial correlation coefficient is the same as the sign of linear regression coefficient. (In fact, partial r is just one of the ways to standardize regressional b.) So, if we have some variables, for example three, X, Y, Z, and you want to know the sign of rXY.Z – the partial correlation between X and Y – you will be enough to know the sign of bX in regression of Y by X and Z (or bY in regression of X by Y and Z).

If we assume that the three variables are centered (their means were brought to 0), the formula of a linear regression coefficient found in many textbooks could be written as follows:

bX=SCPXYSCPZZSCPZYSCPXZSSXSSZSCP2XZ

where SCP stands for “sum-of-crossproducts” and SS for “sum-of-squares”. The denominator here is always positive, so the sign of bX depends entirely on the numerator. We can expand what is an SCP, for example SCPXY:

SCPXY=SSXSSYrXY

If we substitute all SCP in the numerator accordingly and then simplify we’ll get that the numerator is proportional to the quantity

rXYrZYrXZ

and its sign is the sign of this quantity. So, whatever the sign of zero-order correlation rXY, the sign of partial correlation rXY.Z is determined by the last expression. Below is an example: rXY=.314, rYZ=.589, rXZ=.606, rXY.Z=.067, negative because .314.589.606<0.

       X        Y        Z 

   1.339   -1.097     .014 
    .619    1.022     .792 
   -.722    1.127     .699 
   -.695   -1.081   -2.016 
   1.421     .318    1.068 
   1.467     .002    1.284 
   -.619     .692    -.691 
   -.319    1.228    2.002 
    .478   -1.056   -1.281 
    .490     .704    1.151 
   -.316    1.204     .030 
   -.203     .021    1.176 
    .168    1.732    1.741 
    .763    1.090    1.834 
   2.734    -.227    1.044 
  -1.603    -.447   -2.056 
   -.846    -.024    -.335 
   -.009     .132     .932 
   -.304     .118    -.938 
   -.612   -1.878   -1.655 
  -1.370    -.607    -.499 
   -.921    -.893   -1.136 
   -.534     .312    -.282 
   -.136   -1.189   -1.203 
    .406     .752     .338 
   -.069     .559    -.227 
    .534    -.547     .167 
   -.450     .417    -.512 
   1.364    1.319    1.327 
  -1.019     .190    -.157 
   1.608     .588     .861 
  -1.909    -.871   -1.322 
    .488    -.266     .361 
  -1.492   -1.645   -1.216 
    .533     .006     .791 
   -.341     .890     .939 
   -.862     .873    -.342 
  -2.076   -1.051   -1.160 
    .059    1.314    -.456 
   -.666    -.652   -1.761 
   -.742     .885     .606 
   -.333    -.087   -1.040 
    .789     .684    1.322 
   -.121    1.006     .766 
    .528    -.190     .206 
    .944    1.752    2.055 
   -.368    -.548    -.619 
   -.655     .432    -.141 
   -.663   -1.176   -1.164 
   -.799   -1.607   -1.844 
    .563    -.052    -.011 
   -.959   -1.281     .267 
   1.256     .323     .569 
   -.099     .869    -.693 
    .813   -1.057   -1.393 
   1.443    1.519    1.180 
   1.513    1.662    1.160 
   1.488     .494    -.285 
   -.247     .808     .324 
   -.903     .086    -.912 
    .750   -1.304     .717 
  -1.665    -.847   -1.045 
  -1.945    -.480    -.439 
    .105     .804    1.303 
   -.524    1.251    1.201 
   -.277   -1.400    -.391 
   -.936   -1.406    -.215 
   2.029     .318    1.128 
  -1.214    1.002   -1.313 
   -.180     .205    -.845 
   -.364    1.176    -.428 
   1.087    1.167    1.743 
   -.736    -.779   -1.038 
   -.386    1.176     .167 
    .022     .120    1.399 
    .749   -1.324    1.507 
   -.262    -.438   -1.634 
  -1.199    -.206    -.439 
   -.339   -1.687   -1.082 
  -1.529   -1.969   -1.179 
  -1.028    -.806   -1.331 
  -1.080   -1.855   -1.958 
    .072    -.523     .044 
   -.096     .481    -.214 
    .220    -.221     .931 
   1.217    -.801     .412 
  -1.542     .398    -.735 
  -1.238    1.301    -.361 
    .320     .806     .951 
   -.039    -.198    -.526 
    .588    -.001     .860 
   -.682   -1.109    -.607 
    .767    -.381     .255 
   -.783     .338     .475 
    .120    1.227     .345 
   -.207    -.607     .130 
   1.450    1.145     .721 
   -.903     .127     .646 
   1.567    1.106     .477 
    .382    -.942     .404

Attribution
Source : Link , Question Author : Argha , Answer Author : ttnphns

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