1. 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

2. 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

3. 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

4. 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

5. Sample Size formula
This can be rewritten 6

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

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

8. 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

9. 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

10. 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

11. 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= sum(E(i)^2) Sigmahat= n ge 2.71^2/(1*.5^2)=29.44 round up to 30. 11

12. 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