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Experimental designs and Analysis of Variance Factorial designs

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Title: Experimental designs and Analysis of Variance Factorial designs


1
Experimental designsandAnalysis of
VarianceFactorial designs
2
Overview
  • Two-factor designs
  • Model-based approach
  • Randomized-blocks designs
  • Unbalanced designs
  • Three-factor designs
  • General
  • Latin-square designs

3
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

4
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

5
Example Monkeys
y errors factor A drug, factor B hour
6
Example Monkeys
Only significant interaction effect
small (0.01) medium (0.06) large (0.15)
7
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

8
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
9
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

10
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

11
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

12
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

13
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

14
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

15
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

16
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

17
Latin-square design
  • Data Analysis Three-way ANOVA without
    interaction
  • Factor B and C (rows and columns) are typically
    treated as random effects

18
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

19
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

20
Incomplete design
  • A Latin square is an example of an incomplete
    design

21
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!

22
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

23
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

24
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

25
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
26
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)
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