Factorial Designs

1
Factorial Designs

• Michael Thompson

2
Parts of Experimental Design

• Set of experimental units.
• Set of treatments.
• Rules by which treatments are assigned to
  experimental units.
• Measurements made on experimental units following
  application of treatment.

3
Experimental Units (e.g.)

• Patients with heart disease in a drug study.
• Volunteers in a marketing study.
• Corn seeds in an agricultural study.

4
Types of Treatment Structures

• One-Way Treatment Structure
• Two-Way Treatment Structure
• Factorial Arrangement Treatment Structure
• Fractional Factorial Arrangement Treatment
  Structures

5
Assignment Rules

• Completely Randomized Design
• Randomized Complete Block Design
• Latin Squares Design

6
Measurements (e.g.)

• Mortality in a health outcomes study.
• Survey score in marketing study.
• Plant size at time x for agricultural study.

7
Experimental vs. Observational

• In Observational studies the assignment of
  treatments to experimental units is not under the
  control of the researcher.
• Disadvantage of observational studies lies in the
  difficulty in recognizing causal relationships.
• e.g. effects of treatment on very sick patients.

8
Definition of Factorial Design

• An experiment in which the effects of multiple
  factors are investigated simultaneously.
• The treatments consist of all combinations that
  can be formed from the different factors.
• e.g. an experiment with 5 2-level factors would
  result in 32 treatments.

9
Definition of Factorial Design

• The treatments are assigned randomly to the pool
  of experimental units with an equal number of
  units in each treatment.
• The number of experimental units assigned to each
  treatment is referred to as the number of
  replications.

10
2 Factor Model Specification

• Yi B0 B1X1i B2X2i B3X1iX2i ei
• Yi Outcome for ith unit
• B0 Intercept coefficient
• B1 Effect 1 coefficient
• B2 Effect 2 coefficient
• B3 Interaction coefficient
• X1i Level of factor 1 for ith unit
• X2i Level of factor 2 for ith unit
• ei Error term for ith unit

11
Analysis of Factorial Design

• Main Effects effects of each factor independent
  of the remaining factors.
• Interaction Effects 2- to n-way interaction
  effects between all combinations of factors.
• Design provides a lot more information than a
  single factor experiment with potentially not
  much more work.

12
Example

• Experimental units 100 patients with
  depression.
• Set of factors drug therapy (y/n) and
  psychotherapy (y/n)
• Rules - Randomly assign 25 patients to each of
  the possible combinations in (2).
• Measurement Beck Depression Scale

13
Example - Scenario 1

• Both drug and psychotherapy main effects and
  interaction are equal to 0.
• Mean score is 28 with a standard deviation of 9.

14
Example Scenario 1 (cont.)

• No Yes p-value
• Drug 28.9 29.0 0.462
• Psychotherapy 28.9 29.0 0.993
• Interaction - 27.7 0.398
• As expected the null hypothesis that the drug,
  psychotherapy, and interaction effects are equal
  to zero cannot be rejected.

15
Example Scenario 2

• The drug main effect is equal to 7.
• The psychotherapy main effect is equal to 4.
• The interaction effect is equal to 0.
• Base mean score for someone with neither the drug
  nor the psychotherapy effect is 28 with a
  standard deviation of 9.

16
Example Scenario 2 (cont.)

• No Yes p-value
• Drug 28.9 36.0 0.002
• Psychotherapy 28.9 36.0 0.015
• Interaction - 38.7 0.398
• The null hypothesis that the drug and
  psychotherapy effects are equal to zero is
  rejected, but the null hypothesis that the
  interaction is zero is not rejected.

17
Example Scenario 3

• The drug main effect is equal to 7.
• The psychotherapy main effect is equal to 4.
• The interaction effect is equal to 12.
• Base mean score for someone with neither the drug
  nor the psychotherapy effect is 28 with a
  standard deviation of 9.

18
Example Scenario 3 (cont.)

• No Yes p-value
• Drug 28.9 36.0 0.002
• Psychotherapy 28.9 36.0 0.015
• Interaction - 50.7 0.005
• The null hypotheses that the drug, psychotherapy,
  and interactions effects are equal to zero is
  rejected.

19
Fractional Factorial Design

• Only a fraction of all treatments is included in
  the experiment.
• Used with experiments where a large number of
  treatments is investigated.
• For example, a factorial experiment with seven
  2-level factors would require 128 experimental
  units.
• A ¼ fraction would reduce this to 32 combinations.

20
Fractional Factorial Design

• The reduction in data requirements comes with a
  price - some or all interactions cannot be
  modeled. However, in many cases estimating higher
  order interactions is of dubious value.
• Proper fractional designs have the properties of
  being balanced and orthogonal.

21
Fractional Factorial Design
22
The Half Fraction

• Requires half the data that a full factorial
  design needs.
• The main effects and all two way interactions are
  modeled using this approach.

23
Half-Fraction example

• The five factor 2-level case
• The main effects a, b, c, d, and e and two-way
  interaction effects ab, ac, ad, ae, bc, bd, be,
  cd, ce, and de.

24
Half-Fraction example
a b c d a b c d - - - -
- - - - - - - - - -
- - - - - -
- - - - - - -
- - - - -

25
Half-Fraction example

• Use the column product of a - d to calculate e.

26
Half-Fraction example
a b c d e a b c d e - - - -
- - - - - - - - - -
- - - - - - - -
- - - - - - - -
- - - - - -
- - - -

27
Summary - Full Factorial Designs

• Allow the researcher to explore multiple factors
  simultaneously.
• Hypothesis tests can be performed on not only
  main effects, but all possible interactions as
  well.

28
Summary - Fractional Factorial Designs

• Are useful in situations where a factorial design
  is desired, but the number of treatment levels
  required is prohibitively high.
• Cost is the loss of the estimation of some or all
  interaction effects.

29
References

• Cochran, W.G. and Cox, G.M. Experimental Designs,
  2nd ed. John Wiley Sons. 1957.
• Neter, J., Wasserman W., and Kutner, M.H. Applied
  Linear Statistical Models, 3rd ed Irwin 1990
• Box, G., Hunter, W., and Hunter, J. Statistics
  for Experimenters, 1st ed Wiley 1978