Can a power analysis calculate n to detect changes in a population mean over time?

Question

Would it be incorrect to interpret a power analysis as: one possible method of calculating the minimum required sample size to detect a change (%) in a population mean, from one point in time to another (?). As in, what sample size would I need to detect a 30% change in the population mean in a future survey?

Using the Cohen's D effect size formula as an example for my power analysis; if "M" is the sample mean and "σ" is the sample standard deviation (together an estimate of the current population mean), can "µ" be considered the future population mean (the 30% change in the current mean, without sample error)? This is for a "before and after" repeated measures design.

Would it be wrong to then conclude:

# Effect size, Cohen's D: d=(M-μ)/σ
# M=1, σ=0.5, µ=0.7
# 0.6 = (1-0.7)/0.5

> pwr.t.test(d = 0.6, 
+            power = 0.8000, 
+            n = NULL, 
+            type = "paired",
+            alternative = "two.sided",
+            sig.level = 0.05)

     Paired t test power calculation 

              n = 23.79452
              d = 0.6
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

NOTE: n is number of *pairs*

Conclusion: A sample size of 24 would be sufficient to detect a +/-30% change (2-tailed test) from the current population mean in future surveys, with 80% probability and alpha=0.05 .

Is it incorrect to define a power analysis on paired data this way?

I'm guessing this is incorrect wording. The power analysis isn't for detecting changes from one point in time to another, but for testing the accuracy of an estimated population parameter via taking a sample, no?

Answer 1

I think I have things mixed-up here. The effect size I'm using above is for a one-sample test, which does seem like it can be interpreted as a level of how accurately the one-sample mean M and sd estimate the hypothesized population mean µ. As in, if this hypothesized population mean were real, could we detect it (?) (maybe someone else can confirm that wording).

Firstly, what I should be using is the effect size for paired samples:

Where Z is the mean of the differences between paired observations, and s is the standard deviation of the differences. The key part in all this (I think) is that when the "paired" sample has no sampling error (pop. mean estimate), you get the same effect size as the one-sample formula! The interpretation of the paired version, as I understand it, is we want to know what the effect was after a certain treatment. In my case, the treatment is the passage of time, which seems valid.

So, my conclusion could read "...with a sample size of 24, we can detect an effect size equivalent to a +/-30% change in the sample mean, if it exists". Any corrections are welcome!