# Why Euclidean norm was used in Z-score combination?

The formula for the Z-score combination is,

$$Z_w = \frac{\sum_{i=1}^k w_i Z_i}{\sqrt{\sum_{i=1}^k w_i^2}}$$

Euclidean norm was used as the dividend. Why isn't it absolute-value norm ($$\sum |w|$$)?

Though it was called Stouffer's method, it was invented by Liptak, T. (1958). In the paper, I could not find any discussion on why Euclidean norm was used. It first appeared as formula 2.38 in the paper.

• What are these variables? Commented May 4, 2023 at 11:41
• Stouffer's Z-score method en.wikipedia.org/wiki/… Commented May 4, 2023 at 11:43

Suppose we have exactly two $$z$$ values which are equal and we set the weights equal to one. Then the value of Stouffers method reduces to $$\frac{2z}{\sqrt2}$$ which is larger than $$z$$ and hence leads to a smaller $$p$$-value as we would expect. Now do the same for the $$L_1$$ norm and you get $$\frac{2z}{2}$$ which equals $$z$$ and so having two studies giving the same result does not lead to a smaller $$p$$-value.

• I can understand that. I already made that thought experiment. Can you give any more formal proof? Commented May 4, 2023 at 13:30
• Dear Professor Michael Dewey, Though it was called Stouffer's method, it was invented by Liptak, T. (1958). In the paper, I could not find any discussion on why Euclidean norm was used. It first appeared as formula 2.38 in the paper. Commented May 4, 2023 at 13:37

Got it! I have further looked into the literature, and found that it was first described by F. M. Mosteller, and R. R. Bush in 1954 rather than Liptak, T. (1958). First, we known

$$z = \frac{Δx}{SD}$$

Note it is standardized z that we want to combine rather than unstandardized SD, so the SD can be dropped.

When weights were applied to Δxi, the unit has been changed because of the weights. To convert it back to SD unit, the Euclidean norm was used.

See the original explanation.

F. M. Mosteller, and R. R. Bush, Selected quantitative techniques, In: G. Lindzey (ed.), Handbook of Social Psychology: Vol. 1. Theory and Method, Addison-Wesley, 1954, 289--334.