Full Factorial Designs

R2026b

Full factorial designs measure response variables using every possible treatment (combination of the factor levels). A full factorial design for n factors with N₁, ..., N_n levels requires N₁ × ... × N_n experimental runs—one for each treatment. While advantageous for separating individual effects, full factorial designs can make large demands on data collection. To learn about designs that involve fewer experimental runs, see Fractional Factorial Designs.

Statistics and Machine Learning Toolbox™ offers several ways to work with full factorial designs:

• Create a fullFactorialDOE object by using the fullFactorialDOE function. The fullFactorialDOE function provides these advantages:

  • The fullFactorialDOE function allows you to specify the factor names, which factors are categorical, level values for each factor, and experiment model.

  • In addition to returning the design runs, the fullFactorialDOE function stores your specifications in the fullFactorialDOE object properties.

  After you create a fullFactorialDOE object, you can

  • Fit a linear regression model to the design run responses using the fitlm function.

  • Randomize the run order in the design using the randomizeRunOrder function.

  • Add replicates (duplicates of the original design runs) using the addReplicates function.

  See the examples below and the fullFactorialDOE reference page for more information.

• Use the DOE Explorer app to create a full factorial design table and fit a linear regression model to the design run responses. Perform factor analysis and generate plots and tables to assess the model fit.

Generate Multilevel Full Factorial Design

Suppose a machine shop has three machines labeled 1, 2, and 3 and four operators Op1, Op2, Op3, and Op4. If the same operator always uses the same machine, it is impossible to determine if a machine or an operator is the cause of variation in production. By allowing every operator to use every machine, effects are separated.

Generate a full factorial design by creating a fullFactorialDOE object.

dFF = fullFactorialDOE([1 2 3],["Op1" "Op2" "Op3" "Op4"], ...
    FactorNames=["Machine","Operator"], ...
    CategoricalFactors="all")

dFF = 
  fullFactorialDOE with properties:

                Design: [12×2 table]
      StandardRunOrder: [12×1 double]
    ModelSpecification: "1 + Machine + Operator"
                Levels: {[1 2 3]  ["Op1"    "Op2"    "Op3"    "Op4"]}
    CategoricalFactors: [1 2]
          IsRandomized: 0
         NumReplicates: 0

Display the design table.

dFF.Design

ans = 12×2 table
    Machine    Operator
    _______    ________

       1        "Op1"  
       1        "Op2"  
       1        "Op3"  
       1        "Op4"  
       2        "Op1"  
       2        "Op2"  
       2        "Op3"  
       2        "Op4"  
       3        "Op1"  
       3        "Op2"  
       3        "Op3"  
       3        "Op4"  

Each of the 3×4 = 12 rows of dFF represents one machine/operator combination.

Generate Two-Level Full Factorial Design

Many experiments can be conducted with two-level factors, using two-level designs. Suppose the machine shop in the previous example always keeps the same operator on the same machine, but wants to measure production effects that depend on the composition of the day and night shifts.

Generate a two-level full factorial design by creating a fullFactorialDOE object.

dFF = fullFactorialDOE(4,FactorNames=["Op1" "Op2" "Op3" "Op4"])

dFF = 
  fullFactorialDOE with properties:

                Design: [16×4 table]
      StandardRunOrder: [16×1 double]
    ModelSpecification: "1 + Op1 + Op2 + Op3 + Op4"
                Levels: {[-1 1]  [-1 1]  [-1 1]  [-1 1]}
    CategoricalFactors: []
          IsRandomized: 0
         NumReplicates: 0

Display the design table.

dFF.Design

ans = 16×4 table
    Op1    Op2    Op3    Op4
    ___    ___    ___    ___

    -1     -1     -1     -1 
    -1     -1     -1      1 
    -1     -1      1     -1 
    -1     -1      1      1 
    -1      1     -1     -1 
    -1      1     -1      1 
    -1      1      1     -1 
    -1      1      1      1 
     1     -1     -1     -1 
     1     -1     -1      1 
     1     -1      1     -1 
     1     -1      1      1 
     1      1     -1     -1 
     1      1     -1      1 
     1      1      1     -1 
     1      1      1      1 

Each of the 2^4 = 16 rows of dFF represents one schedule of operators for the day (1) and night (–1) shifts.