Sample Size

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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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 views: 17 posted: 5/13/2011 language: English pages: 11