Sample Size How many replications, n, do I need? What size effects do you want to detect? We will focus on detecting effects of a pre-specified magnitude 1

Sample Size Researchers have difficulty articulating the magnitude of both effects and experimental error Have them specify a range and then use the empirical rule to estimate noise Provide sample sizes for a range of effects 2

*Mean Square Pure Error Sample size We rewrite the t-statistic to replace MSPE* with a general estimator of experimental error (assume the effect of interest is A): *Mean Square Pure Error 3

Sample Size If A is distinguishable from background noise, the absolute value of the t test statistic is larger than the t critical value The t-critical value for 5% risk is approximately 2, so in order to detect an effect of magnitude A, we need: 4

Sample Size This can be rewritten 6

Sample Size If we have all the data from a replicated experiment, we can just use MSPE: 7

Sample Size If we have an unreplicated pilot experiment, or only the effects from a pilot experiment, we can do the following: Let E1,…,Em be the set of negligible effects from a pilot experiment with n* reps (often the experiment is unreplicated so n=1) . 7

Sample Size We can construct an estimate as: (Recall that m is the number of negligible effects.) 8

Sample Size We can use other methods to obtain an estimate Empirical rule sc from centerpoint design Combine MSPE and negligible effects 9

Sample Size Regardless of the estimate, in order to detect an effect of size A, we use as a lower bound: 10

Sample Size U-do-it exercise: Estimate experimental error from the probability plot you obtained from the violin example. Suppose the violinist wanted to detect effects equal to .5 decibels; how many replications are needed? 11