# DOE: Full-factorial vs. Fractional-factorials (and notations)

## DOE: Full-factorial vs. Fractional -factorials (and notations)

### Full factorial experiments:

• Examine every possible combination of factors and levels

• Enable us to:

• Determine main effects that the manipulated factors will have on response variables

• Determine effects that factor interactions will have on response variables

• Estimate levels to set factors at for best results

• Provides a mathematical model to predict results

• Provides information about all main effects

• Provides information about all interactions

• Quantifies the Y=f(x) relationship

• Limitations

• Requires more time and resources than fractional factorials

• Sometimes labeled as optimizing designs because they allow you to determine which factor and setting combination will give the best result within the ranges tested . They are conservative, since information about all main effects and variables can be determined.

• Most common are 2-level designs because they provide a lot of information, but require fewer trials than would studying 3 or more levels.

• The general notation for a 2-level full factorial design is:

• 2 is the number of levels for each factor

• k is the number of factors to be investigated

• This is the minimum number of tests required for a full factorial

### Fractional factorial experiments:

• Look at only selected subsets of the possible combinations contained in a full factorial

• Allows you to screen many factors—separate significant from not-significant factors—with smaller investment in research time and costs

• Resources necessary to complete a fractional factorial are manageable (economy of time, money, and personnel)

• Limitations/drawbacks

• Not all interactions will be discovered /known

• These tests are more complicated statistically and require expert input

• General notation to designate a 2-level fractional factorial design is:

• 2 is the number of levels for each factor

• k is the number of factors to be investigated

• 2 -p is the size of the fraction (p = 1 is a 1/2 fraction, p = 2 is a 1/4 fraction, etc.)

• 2 k-p is the number of runs

• R is the resolution, an indicator of what levels of effects and interactions are confounded, meaning you can't separate them in your analysis

### Loss of resolution with fractional factorials

• When using a fractional factorial design, you cannot estimate all of the interactions

• The amount that we are able to estimate is indicated by the resolution of an experiment

• The higher the resolution, the more interactions you can determine

This experiment will test 4 factors at each of 2 levels, in a half-fraction factorial (2 4 would be 16 runs, this experiment is the equivalent of 2 3 = 8 runs).

The resolution of IV means:

• Main effects are confounded with 3-way interactions (1 + 3 = 4). You have to acknowledge that any measured main effects could be influenced by 3-way interactions. Since 3-way interactions are relatively rare, attributing the measured differences to the main effects only is most often a safe assumption.

• 2-way interactions are confounded with each other (2 + 2 = 4). This design would not be a good way to estimate 2-way interactions.

## Interpreting DOE results

Most statistical software packages will give you results for main effects, interactions, and standard deviations.

1. Main effects plots for mean

• Interpretation of slopes is all relative. Lines with steeper slopes (up or down) have a bigger impact on the output means than lines with little or no slope (flat or almost flat lines).

• In this example, the line for shelf placement slopes much more steeply than the others—meaning it has a bigger effect on sales than the other factors. The other lines seem flat or almost flat, so the main effects are less likely to be significant.

2. Main effects plots for standard deviation

• These plots tell you whether variation changes or is the same between factor levels.

• Again, you want to compare slopes in comparison to each other. Here, Design has much more variation one level than at the factors (so you can expect it to have much more variation at one level than at the other level).

3. Pareto chart of the means for main factor effects and higher-order interactions

• You're looking for individual factors (labeled with a single letter) and interactions (labeled with multiple letters ) that have bars that extend beyond the "significance line"

• Here, main factor A and interaction AB have significant effects, meaning placement, and interaction of placement and color have the biggest impact on sales (compare to the "main effects plot for mean," previous page).

4. Pareto chart on the standard deviation of factors and interactions

• Same principle as the Pareto chart on means

• Here, only Factor C (Design) shows a significant change in variation between levels

5. Minitab session window reports

• Shelf Placement and the Shelf Placement* Color interactions are the only significant factors at a 90% confidence internal (if alpha were 0.05 instead of 0.10, only placement would be significant)

Fractional Factorial Fit: Sales versus Shelf Placem, Color, Design, Text

Term

Effect

Coef

SE Coef

T

P

Constant

128.50

0.2500

514.00

0.001

Shelf PI

38.50

19.25

0.2500

77.00

0.008

Color

2.00

1.00

0.2500

4.00

0.156

Design

0.50

0.25

0.2500

1.00

0.500

Text

0.00

0.00

0.2500

0.00

1.000

Shelf PI*Color

3.50

1.75

0.2500

7.00

0.090

Shelf PI*Design

3.00

1.50

0.2500

6.00

0.105

Analysis of Variance for Sales (coded units)

Source

DF

Seq SS

F

P

Main Effects

4

2973.00

2973.00

743.250

1E+03

0.019

2-Way Interactions

2

42.50

42.50

21.250

42.50

0.108

Residual Error

1

0.50

0.50

0.500

Total

7

3016.00

• Design is the only factor that has a significant effect on variation at the 90% confidence level

Fractional Factorial Fit: Std Dev versus Shelf Placement, Color,

Term

Effect

Coef

SE Coef

T

P

Constant

9.0000

0.2500

36.00

0.018

Shelf PI

1.5000

0.7500

0.2500

3.00

0.205

Color

0.0000

0.0000

0.2500

0.00

1.000

Design

6.5000

3.2500

0.2500

13.00

0.049

Text

1.0000

0.5000

0.2500

2.00

0.295

Shelf PI*Color

0.5000

0.2500

0.2500

1.00

0.500

Shelf PI*Design

0.0000

0.0000

0.2500

0.00

1.000

Analysis of Variance for Std (coded units)

Source

DF

Seq SS

F

P

Main Effects

4

91.0000

91.0000

22.7500

45.50

0.111

2-Way Interactions

2

0.5000

0.5000

0.2500

0.50

0.707

Residual Error

1

0.5000

0.5000

0.5000

Total

7

92.0000