# Presentation: 178692

##  Factorial Designs	

- Michael Thompson

## 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.

## Experimental Units (e.g.)

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

## Types of Treatment Structures

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

## Assignment Rules

- Completely Randomized Design
- Randomized Complete Block Design
- Latin Squares Design

## Measurements (e.g.)

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

## 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.

## 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.

## 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.

## 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

## 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.

## 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

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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 1⁄4 fraction would reduce this to 32 combinations.

## 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.

## Fractional Factorial Design

## 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.

## 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.

## Half-Fraction example

**a  b  c  d         a  b  c  d**
- -  -  -  -         +  -  -  -
- -  -  -  +         +  -  -  +
- -  -  +  -         +  -  +  -
- -  -  +  +         +  -  +  +
- -  +  -  -         +  +  -  -
- -  +  -  +         +  +  -  +
- -  +  +  -         +  +  +  -
- -  +  +  +         +  +  +  +

## Half-Fraction example

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

## Half-Fraction example

**a  b  c  d  e      a  b  c  d  e**
- -  -  -  -  +      +  -  -  -  -
- -  -  -  +  -      +  -  -  +  +
- -  -  +  -  -      +  -  +  -  +
- -  -  +  +  +      +  -  +  +  -
- -  +  -  -  -      +  +  -  -  +
- -  +  -  +  +      +  +  -  +  -
- -  +  +  -  +      +  +  +  -  -
- -  +  +  +  -      +  +  +  +  +

## 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.

## 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.

## 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