Stat 401 Lab October 17, 2005

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Stat 401 LabOctober 17, 2005

• James D. Abbey
• Iowa State University

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Contact Information

• James Abbey
• Email jdabbey_at_iastate.edu
• Website www.public.iastate.edu/jdabbey
• Office and Office Hours
• 306 Snedecor Hall
• M and F 1000-1030
• Th 200-300

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Homework

• Schedule for today
• Homework 7
• Quick Review for Homework 6
• Questions Good! (Yes, that is good factorial,
  which is a really big number.)

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Homework 7

• HW7 Concepts
• ANOVA.
• How to check the assumptions
• Graphical and numerical methods
• The overall F-Test
• Confidence Intervals around individual means
• Linear Combinations (and Contrasts)
• Full vs. Reduced Models
• Why contrasts are superior to pair-wise t-tests
• Weighting a Grand Mean

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• Assumptions in ANOVA
• Normality, equal variance and independence.
• Check normality graphically using
• Boxplot
• Does the plot appear symmetric
• Normal Probability Plot
• Do the points follow a straight line
• Check equal variance by
• Graphical Residuals by predicted plot or
  side-by-side boxplots
• Numerical Direct comparison of variance or
  standard deviation
• Rule of thumb Variance ? no variance more than
  4 times larger than any other Std Dev ? no std
  dev more than 2 times larger than any other

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Homework 7

• Examples
• NPP Reasonable ResPred Good

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Homework 7

• Remember the overall F-Test
• F MSB/MSW or FMSModel/MSError
• Ho All treatment means equal
• Ha at least one treatment mean not equal
• Do not double the p-value for a two-sided test.

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Homework 7

• Building 95 CIs around individual means
• Core formula
• Calculate each group mean ( )
• Find your relevant t-quantile (note d.f. n-I)
• Take your MSerror or MSwithin and divide by the
  number of observations in the group of interest.
  Take the square root.
• Apply the remainder of the formula

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Homework 7

• Linear Combinations and Contrasts
• A linear contrast is simply a linear combination
  with the sum of the ci 0.
• Say I have 4 means
• A contrast of mean 1 to mean 2 is (1,-1,0,0)
• A contrast of mean 2 to mean 4 is (0,1,0,-1)
• Is (-1/2 , 1, -1/2 , 0) a contrast?

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Homework 7

• The four means (continued)
• I want to take the difference between the first
  two means and the last two means with respect to
  some other variable of interest (Hint This is
  HW7 question 1 part f sub-part iii)
• So, c (1 , -1 , -1 , 1). Why?

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Homework 7

• Full vs. Reduced Models
• Reduction is based primarily on questions of
  interest.
• However, as an example, say we have 4 means
• Ha
• Ho
• F-test results
• F10 ? p-value lt 0.01 ? Reject Ho ? Must use full
  model
• F1.2 ? p-value gt 0.1 ? Fail to reject Ho ?
  Reduced model just as good as full model

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Homework 7

• Full vs. Reduced Models
• Remember the formula
• Be very careful about your d.f.
• The core question of Full vs. Reduced Models is
  how many means (terms) can be set equal before
  the model no longer accounts for the error
  effectively.

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Homework 7

• Contrasts vs. pair-wise t-tests
• Both contrasts and pair-wise t-tests require
  independence, normality and equal variance.
• Yet, for contrasts, the assumptions are checked
  against all groups in the study.
• Contrasts are superior to pair-wise t-test if
  you, the researcher, want to find information
  about differences or combinations of differences
  regarding more than 2 means.

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Homework 7

• Weighting a grand mean
• Hint This relates to HW7 question 2 part b
• The coefficients take on ratio values of the
  group sample size over the total
• Note that N total sample size (n1n2nk) and
  k the total number of groups.

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Homework 7

• Weighted Grand Mean
• The formula for the grand mean
• The Sp2 is the same as before (the expanded
  version from page 121)
• So, the standard error becomes

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Homework 7 Summary

• Assumptions for ANOVA check both graphically
  and numerically (equal variance)
• Remember that CIs about individual means are
  simply
• A linear contrast is a special case of a linear
  combination where the sum of the coefficients is
  zero

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Homework 7 Summary

• Full vs. Reduced Models
• Reduction is based primarily on questions of
  interest. Ho is the reduced model. Hence,
  failing to reject Ho is our desired outcome!
• Contrasts offer more flexibility than pair-wise
  t-based comparisons

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Homework 6 Summary

• Steps in ANOVA 1) d.f. 2) SS 3) MS 4) F-Stat
  5) p-value
• If you are given an MS value, you can back out
  information without hand calculation. Remember
  the relationships!
• MSE Sp2 (see page 121 of your text)
• Again, note the relationships between SS, MS and
  their d.f.!

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JMP Mini-Lecture

• If you have questions, ask them now.
• JMP mini-lecture is next.

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JMP Mini-Lecture

• Using JMP for ANOVA
• Analyze ? Fit Model
• Saving Residuals for Analyzing Assumptions
• Both topics are covered in the following document
  and video
• Document http//www.public.iastate.edu/7Ejdabbe
  y/Stat401/JMPFiles/ResidDiag.doc
• Video http//www.public.iastate.edu/7Ejdabbey/S
  tat401/JMPFiles/ResidDiag.avi