Topic 12

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Topic 12 Further Topics in ANOVA

• Unequal Cell Sizes
• (Chapter 20)

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Overview

• Well start with the Learning Activity.
• More practice in interpreting ANOVA results and
  a baby-step into 3-way ANOVA.
• An illustration of the problems that an
  unbalanced design will cause.
• Well then continue with a discussion of
  unbalanced designs (Chapter 20)

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Collaborative Learning Activity

• Take your time going through this. Ask questions
  as needed!

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Question 1

• Analyze the design elements.

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Design Chart

• Unequal Cell Sizes but there is SOME balance
  achieved
• Single Factor Analyses will be balanced.
• GenderAge 6 observations per cell
• TimeAge 6 observations per cell
• GenderTime Unbalanced

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Question 2

• Analyze AgeTime

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Interaction Plot (ignoring gender)
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Interpretations

• No interaction is evident between age and time
• Seems middle age group gets generally higher
  offers.
• Seems offers during the week are generally higher
  than on the weekend (this effect is not as big as
  the age effect)

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Main Effects Plots
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ANOVA

• Type I vs Type III?

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LSMeans

• 3 (middle aged, weekday) is the highest
• Using Tukey comparisons it is significantly
  higher than all others.
• Slicing will show the same things that we
  guessed from the plots.

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LSMeans (sliced)
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Slicing of LSMeans

• Sums of Squares add to???
• DF add to???
• Effect of slicing is to look at differences for
  one of the two factors at a specific level of the
  other factor.
• Interpretations???

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Question 3

• Analyze AgeGender

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Interaction Plot (ignoring Time)
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Interpretations

• Small interaction is seen might be described as
  follows
• There is still a clear main effect Middle aged
  get higher offers in general
• There seem to be no gender differences for middle
  aged or young.
• For elderly, women may be getting lower offers
  than men.

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LSMeans (sliced comparisons)
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ANOVA / LSMeans

• Only age differences show up in the ANOVA.
• Sliced LSMeans comparisons do pick up gender
  difference within elderly
• Note Type I error rate is uncontrolled. But on
  the other hand sample sizes are also fairly
  small.
• Conclusions?

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Question 4

• Analyze TimeGender

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Interaction Plot (ignoring age)
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Interpretations

• Seems to be a clear interaction For men, there
  is not much difference in the offer between
  weekday/weekend.
• Women should go on the weekdays, where it seems
  they average about 400 more.
• Interestingly, significance is not seen in the
  ANOVA table, but is seen in the sliced LSMeans
  output.
• Remember Type I Error is uncontrolled.

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ANOVA Table

• Why are Type I / Type III SS different here?

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Sliced LSMeans
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Conclusions

• This is an intriguing example, because the ANOVA
  output would lead you to believe there is a small
  time effect, but no gender effect.
• Looking at the interaction plot presents a
  completely different picture (and likely a more
  accurate one). Lets reconsider that, showing
  the sample sizes.

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Interaction Plot (ignoring age)
n 6
n 6
n 12
n 12
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Confounding

• This picture illustrates how the effects of
  gender and time will be confounded.
• Suppose that women do get lower offers than men
  in general. Then because the women received more
  weekend offers (and men more offers on weekdays),
  the average offer on the weekend will by default
  be lower than the weekday.
• Simple example Suppose men get 2 and women get
  1. Then with the sample sizes, the weekday
  average will be 30/18 while the weekend average
  will be only 24/18.

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Questions 5 6

• 3-way ANOVA
• Is Gender Important?

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Modeling

• Removing unimportant terms (starting at the
  interaction level) seems like a reasonable way to
  go.
• Use Type III SS to do this since cell sizes are
  not the same.
• The procedure leads to a model containing only
  Age and Time suggesting that gender is
  unimportant. But we know this may not be
  accurate since gender/time are confounded.

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Confounding

• What exactly does it mean to say that the
  time/gender effects are confounded.
• The biggest thing that it means is that the
  analysis we just did is inappropriate since...
• The time effect may have been seen because more
  women went on the weekend. It may well be a
  gender effect that is disguised as a time effect
  due to the unbalanced design.
• Due to the lack of balance we were forced to
  use Type III SS which (due to collinearity /
  confounding may not tell the whole story).

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Importance of Gender?

• Probably!
• Direct algorithmic analysis suggests both time
  and age are important, while gender is not. But
  due to confounding, that wasnt really
  appropriate.
• The plot for timegender indicates what is
  probably the real story (due to small sample
  sizes it is hard to get significance).
• With a balanced design we would be much better
  off. The effects would not be confounded, and we
  could therefore see an accurate picture.

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Importance of Gender? (2)

• Differing sample sizes means that
• Estimates for women on weekdays, and men on
  weekends, will have larger standard errors.
• This will reduce our power to detect differences,
  and the effects will overlap to some extent
  because of the unequal sample sizes.
• When we looked at the gendertime interaction,
  the plot suggested there was an important one.
  Further studies should be conducted to determine
  if this is the case.

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Unbalanced Two-Way ANOVA

• Unequal Cell Sizes
• (Chapter 20 skim only)

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Differing Cell Sizes

• Encountered for a variety of reasons including
• Convenience usually if we have an observational
  study, we have very little control over the cell
  sizes.
• Cost Effectiveness sometimes the cost of
  samples is different, and we may use larger
  sample sizes when the cost is less.
• Accidently In experimental studies, you may
  start with a balanced design, but lose that
  balance if some problem occurs.

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Differing Cell Sizes (2)

• What changes?
• Loss of balance brings intercorrelation among
  the predictors.
• Type I and III SS will be different typically
  Type III SS should be used for testing but as we
  have seen even that is not perfect!
• Standard errors for cell means and for multiple
  comparisons will be different (they depend on the
  cell size). For the same reason, confidence
  intervals will have different widths.

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Example

• Examine the effects of gender (A) and anxiety
  level (B) on a toxin level in the bloodstream.
• Three categories of anxiety (Severe, Moderate,
  and Mild).
• We categorize people on this basis after they are
  in the study (it is an observational factor).
• For cost effectiveness, we wouldnt want to throw
  away data just to keep a balanced design.

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Data
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Interaction Plot
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Interpretation

• Effect seems to be greater if anxiety is more
  severe.
• This is an interaction of the enhancement type.
  The effect of anxiety level on toxin levels is
  greater for women than it is for men.
• Remember, we arent saying anything about
  significance here well do that when we look at
  the ANOVA.

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ANOVA Output
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Type I / III SS
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Differences in Type I / III SS

• The more unbalanced the design, the further apart
  these may be.
• There are actually four types of SS
• I Sequential
• II Added Last (Observation)
• III Added Last (Cell)
• IV Added Last (Empty Cells)

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Type I SS

• Sequential Sums of Squares Most appropriate for
  equal cell sizes.
• SS(A), SS(BA), SS(ABA,B)
• Each observation is weighted equally. So the net
  result for an unbalanced design is that some
  treatments will be considered with greater weight
  than others.

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Type II SS

• Variable Added Last SS Generally only used for
  regression because again each observation is
  weighted equally.
• SS(AB,AB), SS(BA,AB), SS(ABA,B)

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Type III SS

• Variable Added Last SS, appropriate for unequal
  cell sizes. Type III SS adjusts for the fact
  that cell sizes are different.
• Each cell is weighted equally, with the result
  that treatments are weighted equally. This means
  that observations in smaller cells will carry
  more weight.
• SS(AB,AB), SS(BA,AB), SS(ABA,B)

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Type IV SS

• Variable Added Last SS and similar to Type III SS
  but further allows for the possibility of empty
  cells.
• It is only necessary to use these if there are
  empty cells (which hopefully there wont be if
  youve designed the experiment well).
• SS(AB,AB), SS(BA,AB), SS(ABA,B)

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General Strategy

• Remember that Type I SS and Type III SS examine
  different null hypotheses.
• Type III SS are preferred when sample sizes are
  not equal, but can be somewhat misleading if
  sample sizes differ greatly.
• Type IV SS are appropriate if there are empty
  cells.
• Can obtain Type IV SS if necessary by using /ss4
  in MODEL statement

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Example (continued)

• The interaction is unimportant, nor is there an
  apparent large effect of gender.
• Now look at comparing different levels of
  anxiety should not change models at this
  point, so just average over gender (LSMeans).

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LSMeans

• Must use LSMeans to adjust all means to the same
  average level of gender.

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Comparisons

• Mild group has significantly lower toxin levels
  than the moderate and severe groups

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Confidence Intervals

• Could get CIs for means and/or differences if
  you wanted them.
• They will be of different widths why?
• It will be harder to detect differences for
  groups with fewer observations.

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Questions?
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Upcoming in Topic 13...

• Random Effects
• (parts of chapters 17 19 that
  were previously skipped)