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 Run SIM macro. Show handout SIM has effects of A=1, AB=2 and C=4, with sigma=10; n needs to be BIG to reliably detect them. Sample Size analysis; power analysis. 1

Sample Size Starting the conversation 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

Sample size T statistics 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 Rejection region 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 formula This can be rewritten 6

Experimental error Replicated experiments If we have all the data from a replicated experiment, we can just use MSPE: 7

Experimental Error Pilot studies 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

Exerperimental error Using negligible effects We can construct an estimate as: (Recall that m is the number of negligible effects.) n* is reps in pilot study. k is number of effects. E(i) are negligible effects, m is number of negligible effects. 8

Experimental Error Other approaches We can use other methods to obtain an estimate Empirical rule sc from centerpoint design Combine MSPE and negligible effects 9

Sample size formula Another form Regardless of the estimate, in order to detect an effect of size A, we use as a lower bound: Can substitute A for any effect 10

Sample Size U-do-it 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? Refer to Lecture 14. n*=11, m=11, k=4, E(i)—don’t include AB=-1.3. sum(E(i)^2)-1.84. Sigmahat=2.71. n ge 2.71^2/(1*.5^2)=29.44 round up to 30. 11

Helicopter Example Factors One small paper clip as weight Rotor length (3 in*, 5 in) Rotor width (1.25 in, 2.25 in) Tail length (3 in, 5 in) One small paper clip as weight Left rotor back, right rotor forward *R program prints to 5/6 scale Add paper clips since rotor is too short. Likely that only effect wi.l be main effect for Rotor Length. 11