Experimental designs and Analysis of Variance Factorial designs

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Experimental designsandAnalysis of
VarianceFactorial designs
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Overview

• Two-factor designs
• Model-based approach
• Randomized-blocks designs
• Unbalanced designs
• Three-factor designs
• General
• Latin-square designs

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Two-factor designs

• Two-way factorial design
• a and b treatments (groups), sample of N cases
• Balanced design n cases per group, N n ? a ? b
• Assign cases to levels completely at random
• advantage
• Increase precision (more variance explained) and
  possibility to investigate interactions
• disadvantage
• Size will be large with many factors and
  treatment levels and interactions may be
  difficult to interpret

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Two-factor designs

• Data analysis two-factor ANOVA with interactions
• Test
• H0 no main effect A vs. Ha main effect A
• H0 no main effect B vs. Ha main effect B
• H0 no interaction AB vs. Ha interaction AB
• Model
• with

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Example Monkeys
y errors factor A drug, factor B hour
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Example Monkeys
Only significant interaction effect
small (0.01) medium (0.06) large (0.15)
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Model-based comparisons

• Testing is based on comparing models
• Model under the null hypothesis H0
• Model under the alternative hypothesis Ha
• What is the improvement and is this significant?
• The best alternative model is the model that
  includes all effects
• This model is called the full linear model
• Which model should be used under the null
  hypotheses?
• Depends on which hypothesis is to be tested
• Main effects, interaction effect

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Model-based comparisons

• Null hypothesis models remove effect to test
• Main effect of A
• Main effect of B
• Interaction effect
• Estimate these models and calculate predictions
• Test interaction effect A
• H0 model gives predictions such that the
  (marginal) means determined by levels of A are
  equal
• Ha model gives predictions
• Follow procedure used in single-factor analysis

Example
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Model-based comparisons

• To test the null hypothesis
• Calculate unexplained SS for both models
• unexplained variation (error) in H0
  model
• unexplained variation (error) in Ha
  model
• The SS of the effect equals
• The SS of the error equals

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Example Monkeys

• To test the main effect of A
• unexplained variation (error) in
  H0 model
• unexplained variation (error) in
  Ha model
• SS of A
• SS of the error

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Randomized-blocks design

• Randomized complete blocks design or blocking
• Use characteristic of cases as non-experimental
  factor ? simplest case of two-way design
• Second factor defines blocks error control
• Homogeneous blocks of subjects are formed
  beforehand to reduce within-group variability
• Example of a randomized-blocks design

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Randomized-block design

• Advantages
• Reduction of error variance (SSS/AB smaller)
• Increases comparability of groups by assuring the
  block sizes are equal (making the groups more
  homogeneous)
• Interaction between factors can be detected
• Always include blocking variable, regardless of
  significance
• F test of blocking variable is only approximately
  because it is a non-experimental factor, and
  usually it is a random factor
• Moreover, you can maximize intrablock variance by
  making blocks as different as possible ?
  pointless to test equality of means

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Post-hoc blocking

• Blocking after collection of the data
• Blocking not initially planned in design
• Form blocks post hoc, segregating subjects into
  homogeneous blocks
• Problem often unequal sample sizes
• If blocking variable is continuous, an
  alternative blocking method is ANCOVA

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Latin-square design

• Use characteristic of cases as non-experimental
  factor ? three-way design
• Two extra factors to control sources of (error)
  variation (nuisance variables)
• The square is an arrangement of treatments, i.e.,
  a two dimensional layout of three factors
• rows and columns each represent one nuisance
  factor
• treatment entries inside the matrix

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Latin-square design

• Often there is only 1 observation in each cell of
  the design ? no interaction between the factors
• Each level of the factor (treatment) of interest
  A appears once in each row and column ? effects
  of B and C are spread evenly over various levels
  of A
• Example of a Latin-square design

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Latin-square design

• Factors A, B, and C each p levels
• Sample of N p2 cases
• Assign cases randomly to cells in the design
• Latin squares are often used in repeated measures
  designs
• advantage
• Controlling extra sources of variation
• disadvantage
• Limited number of treatments to test, and the
  number of treatment levels must be the same

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Latin-square design

• Data Analysis Three-way ANOVA without
  interaction
• Factor B and C (rows and columns) are typically
  treated as random effects

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Graeco-Latin square design

• Use characteristic of cases as non-experimental
  factor ? four-way design
• Three extra factors to control sources of (error)
  variation (nuisance variables)
• The square is an arrangement of treatments, i.e.,
  a two dimensional layout of four factors
• rows and columns each represent factors A and B
• factor C and D entries inside the matrix
• There is only 1 observation in each cell of the
  design ? no interaction between the factors

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Graeco-Latin square design

• Each level of the treatment A appears once in
  each row and column ? Latin square for A
• Each level of the factor D appears once in each
  row and column ? Latin square for D
• These designs are examples of incomplete designs
• Example of a Graeco-Latin square design

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Incomplete design

• A Latin square is an example of an incomplete
  design

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Types of experimental designs

• Higher factorial designs Three-factor design
• Latin-square design special case of three-factor
  designs (only main effects, no interactions)
• Generalization of single-factor and two-factor
  designs
• Three-way interaction
• Present when the simple interactions of two
  factors differ with the levels of the third
  factor
• Look at simple interactions two-way interactions
• Interpretation is hard!

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Unbalanced designs

• One-way ANOVA
• Balanced design n cases per group, N n ? a
• Unbalanced design nj cases per group,
• In one-way designs there are no (big) problems
• Only power of the F-test can be smaller
• Factorial ANOVA
• Correlation between the factors will emerge due
  to unequal number of cases per cell

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Unbalanced designs

• Balanced designs equal number of cases per cell
• Interpretation predictors in regression model
  are independent
• Unbalanced designs unequal number of cases per
  cell
• Called non-orthogonal designs because of
  correlation between factors dependence
• Unequal group sizes cause the A, B, and AB
  effects to be partially confounded with each
  other
• In regression this dependence is known as
  multicollinearity, and influences the order in
  which the variables are included in the model
• important for the interpretation of the results

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Unbalanced designs

• Order of the factors (effects) in the model is
  important for testing the significance of the
  effects
• cf. stepwise selection procedures in regression
• correct the effect of a factor for the effects of
  factors already included in the model
• correct the effects (main/interaction) for other
  effects already included in the model

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Unbalanced designs

• For balanced designs correction is not necessary
  the partitioning of total variance is always the
  same, because effects are independent
• For unbalanced designs the partitioning depends
  on order of the variables, because effects are
  not independent

SST SSA SSB SSAB SSS/AB
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Unbalanced designs

• Using the dummy variables, the sums of squares
  can be partitioned in several ways, depending on
  the order of the effects
• Correct each effect for every other effect
  (regression approach in SPSS SS type III)
• Correct interaction effects for all main effects,
  do not correct main effects for interaction
  effects (experimental approach in SPSS SS type
  II)
• Order the effects on theoretical basis, correct
  each effect for the previous one (hierarchical
  approach in SPSS SS type I)