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=SCPXYSCPZZ−SCPZYSCPXZSSXSSZ−SCP2XZ

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=√SSX√SSYrXY

If we substitute all `SCP`

in the numerator accordingly and then simplify we’ll get that the numerator is **proportional** to the quantity

rXY−rZYrXZ

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*