I work in the field of the image processing and I need to compare results of my algorithm with a gold standart results. For this purpose I calculate the Pearson correlation coefficient between the results of my algorithm and gold standart.
Result of my algorithm is 3d image which in turn consists from three 2d slices. So that I calculated correlation between each slice in my image and in the gold standart image. In this way I got the next results:
correlation between MySlice1 and GoldSlice1 is 0.93
correlation between MySlice2 and GoldSlice2 is 0.95
correlation between MySlice2 and GoldSlice3 is 0.96
After that I calculated the correlation between the whole 3d images and I got that correlation equals to 0.9 what is a bit wondering for me. May it be shown that in the general case the total correlation will be less than the least correlation between pair of slices or it's just happened accidently for this particular case and in general it's not a rule?
Pearson correlation coefficient is a non-linear transformation from the data to some value $\in[-1, 1]$. Therefore, one cannot expect a linear combination of the results from three slices to be the result of the correlation coefficient of the $3D$ image.
The key point is the non linear scaling by the standard deviations, which may be an issue in your case.