# Why does the estimated sample size decrease as the expected effect gets larger?

Here's a fairly simple conundrum: let's say I'm calculating the sample size needed to determine a correlation of r = 0.7 at 95% power and an alpha (two-tailed) of .05. The null is r = 0 (note I use r throughout to denote a correlation coefficient).

G*Power suggests that I need 20 participants.

Now, if I increase the null to r = 0.5, I need 131. This makes sense as there's a smaller difference between the alternative and null values.

However, if I now change the alternative hypothesis to r = 0.9 (and keep the null at r = 0.5), I need 18 participants to demonstrate the effect. Why would one require less participants to find a stronger effect (and practically speaking, you'd need to have very clean data to find a correlation of 0.9)? Surely you'd need more?

• It's not clear to me whether you are using r to represent a correlation coefficient or a slope. In either case, higher r means it's easier to detect that a relationship is significantly different from the null. If r is a slope, the power analysis isn't telling you how likely you are to conclude that the slope is (at least) 0.9; it's telling you, if the true slope is 0.9, how likely are you to conclude that the slope isn't zero. If r is a correlation coefficient, the analysis is telling you how likely you are to detect an effect in the context of a true relationship that isn't very noisy. – Jacob Socolar Aug 27 '17 at 22:21
• Hi Jacob - I edited my question to be more specific about what I mean by 'r' (although, I thought that r generally represented a correlation and beta represented a slope). That's been really helpful - I will try to earn some more rep to upvote your comment! – PyjamaNinja Aug 28 '17 at 8:53
• The reason I was confused about r is because in the language of null-hypothesis significance testing, the parameters that get tested are the slopes (the betas), not the correlation coefficient. – Jacob Socolar Aug 28 '17 at 13:08