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

                               A
                 T 
                               ˆ   2

                                   k 2
                              n2
                                                       3
    *Mean Square Pure Error
                    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:

                        A
                                      2
                        ˆ   2

                            k 2
                     n2                                         4
                Sample Size
• This can be rewritten


                       4 2
                         ˆ
                   n  2 k 2
                      A 2




                                5
               Sample Size
How do wefind  ?
               ˆ
• If we have all the data from a replicated
  experiment, we can just use MSPE:



              MSPE
             ˆ
                                              6
               Sample Size
How do wefind  ?
               ˆ
• 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.)

                          k 2    m
                 n*2
     
     ˆ
                   m
                                 E
                                 i1
                                         2
                                         i



                                                     8
                 Sample Size
• Finally, we can use the empirical rule to obtain an
  estimate.




                                                        9
                 Sample Size
• Regardless of the estimate, in order to detect an
  effect of size A, we use as a lower bound:



                          ˆ     2
              n          k 4       2
                      2          A


                                                      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

				
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