Transcript No Slide Title

Power
We all know what P(Type I Error) =  is:
 = P (rej. H0 / H0 true)
 = P(Type II Error)
= P (acc. H0 / H0 false)
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Power is : “the ability to detect out
a small deviation from the
hypothesized value.”
“The test which can detect a
smaller deviation is the more
powerful one.”
What is “power” in terms of ,  ?
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H0 true
we accept H0
No Error
1-
we reject H0
Type I
error

H0 false
Type II
error

No Error
1-
 = P (acc. H0 / H0 false)
a correct
1-  = P (rej. H0 / H0 false) = conclusion
1- = “POWER”
(of a test)
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For ANOVA (with one factor) we have,
POWER = f ( , 1, 2, 
 = significance level
1 = df of numerator (of Fcalc)
2 = df of denominator (of Fcalc)
  “Non-Centrality Parameter”
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

 R (• j - ••)2
C
= A MEASURE OF HOW DIFFERENT THE
’s ARE FROM ONE ANOTHER
(includes “sample size” by including R & C)
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1-=
Function of a distribution called “noncentral F dist’n” (which is NOT the
same as the regular F distribution)
However, typically, we don’t know  or the 
values.
So — in practice, you need to “estimate” 
from data and “suppose” a value of  for
purposes of finding POWER
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Example:
One factor ANOVA with  = .05, R = 10, C = 3.
We decide that a working estimate of  = 5
and that we (arbitrarily) choose to calculate
(1-) assuming that the ’s are each 1• apart.
(eg., 17 22 27). Then,
=
=
1
5
1
5
10(52 + 02 + 52)
3
(
500
(3)
1/2
1/2
)
12.9
= 2.58
=
5
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We have
 = 2.58
1 = 2 (3 col)
2 = 27 (30-3)
 = .05
Now go to “Power Tables”: eg., the one for 1
= 2 (they are “indexed” by the value of 1):
This table is for  = .05 and for  = .01, and
for various values of 2 and .
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POWER ( , , 2) at 1 = 2
See next page for chart
Our Answer:
( = 30,  = .05,  = 2.6 )
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POWER  .976
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Figure 3.8 on p.77 (Paul)
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Accept H0 =
All column means are
the same.
All column means are each “<5” apart
with “97.6%” confidence.
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FINDING “REQUIRED”
SAMPLE SIZE
Given C, , 1-, and “” we can find
the minimum sample size required, in
terms of R, the number per column.
 = Max (•j) – min (•j)
= Range of ’s
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Sometimes one estimates  and 
separately; sometimes one estimates the
ratio  and not necessarily the separate , 

values.

A common choice of  is 2 (i.e., the range
of ’s is 2 standard deviations).
Example:

C= 3
 = .05
 = .90
= 2

R=8
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Table 3.3 on p.81 (Paul)
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(1-) Confidence Interval (C.I.)
• For • j :
Y. j  Z1 / 2

R
• For • j - • k :
(Y. j  Y.k )  Z1 / 2

R/2
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If  is unknown,
• Replace  by
ˆ 
MSE
• Replace Z2 by t2 where the df of t is
the df of Error.
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