The sample size calculation for heterogeneous variances

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The sample size calculation for heterogeneous
variances  

• Wei-ming Luh
• ???
• National Cheng Kung University,
• ????
• Tainan, Taiwan

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• Selecting an insufficient sample size yields a
  study with inadequate sensitivity, whereas
  selecting an excessive sample size wastes
  resources.

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Sufficient sample size is important.

• Textbooks Mace (1974), Cohen (1988), Kraemer
  Thiemann (1987), and Desu Raghavarao (1990).
• Computer programs Gorman Primavera (1995),
  Lenth (2000), Morse (1999), SAS Institute (1999)
• Thomas (1998) even provided a comprehensive list
  of power-analysis software
  //www.forestry.ubc.ca/conservation/power/

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One-sample t testGiven a, ß

• The minimum sample size needed
• is the standardized effect
  size, which is the difference of the actual and
  hypothesized means in standard deviation, s,
  units

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Guenther (1981)s modification
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Caution!

• Conventional formulas are based on the assumption
  of normality and variance homogeneity.

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Assumption violationHeavy-tailed and asymmetric
distribution
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In the context of long-tailed distributions and
heterogeneous variances

• Trimmed mean method to correct non-normality
• Approximate test to take care of heterogeneity
  (Behrens-Fisher problem )

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1. Trimmed mean
Let be the order
statistics of random sample
Let be the proportion of trimming in each
tail of the distribution So the effective
sample size is
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Yuens method (1974)
which is distributed approximately as the Student
t with degrees of freedom
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Winsorized Variance(replacing)
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Effective sample size
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Original sample size
Always rounding up to the next highest integer.
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Monte Carlo simulation

• 1. We generated data by using SAS RANNOR function
  to create the standard normal observations (Z)
• 2. We used the g-and-h distributions (Hoaglin,
  1985) to transform Z to reflect the target
  distribution shapes

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Five distribution shapes

• The corresponding skew and kurtosis Normal
  (g0, h0) (0, 0)
• heavy-tailed (g0, h.1) (0, 5.5)
• (g0, h.2) (0, 36.22)
• Asymmetrical (g0.5, h0) (1.75, 8.9)
• (g0.5, h.2) (13.16, 42895)

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Sample size table (for d1, one-sided test,
power.80)
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Simulation results (a.05)
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Conclusion

• The heavier the distribution tail, the fewer the
  subjects needed for the trimmed mean method.
• The trimmed sample size can achieve the desired
  statistical power while the conventional sample
  size formulas result in over-sampling.

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2. unequal variances
Yuens method (1974)
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Schouten (1999),
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Monte Carlo simulation
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• Generalized Behrens-Fisher problem
• Schwertman (1987)
• The largest mean difference can be tested as

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More questions

• How much to trim?
• When variances are unknown and possibly unequal,
  the usual unbiased estimate can be used.
• How about confidence interval?
• How about cost constraint?

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More sample size formulas should be developed for
robust statistics.

• Thank you for your listening.

luhwei_at_mail.ncku.edu.tw