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This Cross Validated answer states that it's valid to do a paired t-test to test whether the difference is greater than a certain value (e.g. 5) by changing the specified value of in the null/alternate hypothesis. My question is: how can I calculate the minimum required sample size for this test?

I already know of methods to calculate sample size when testing whether the difference is greater than zero (e.g. this resource) but how does this change when using a different value of in the null/alternate hypotheses? I'm getting confused here between the minimum detectable effect size (e.g. 1.5) and the effect size that we're testing for (5)

Alice
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1 Answers1

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Testing the null hypothesis that the mean difference is greater than 5 is algebraically and conceptually the same as subtracting 5 from all the values in the first treatment, then testing the null that the mean difference is greater than 0.

This means that you can use the same tools for power or sample size, you just need to shift the results afterwards. For example, if for a given power, sample size, etc. you are told that the minimum detectable effect size is 1.5 (for a null mean of 0), then that means your minimum detectable effect size is 6.5 (1.5 + 5) when testing the null that the mean difference equals 5.

If you want to find the sample size that gives you 95% power to detect a true difference of 7 (for a null of 5), then you can just compute the sample size for a difference of 2 (7 - 5 for the null difference of 0).

For both cases you need an estimate of the standard deviation of the differences, but that will not be affected by shifting values by a constant value, so will be the same regardless of the null value.

Greg Snow
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