How to Analyze Factorial Design?

I'm been trying to analize a factorial desing 2^3, with 4 center points on matlab. My teacher said i have to analyze it with matlab instead minitab.

I tried to use the function anovan, but i can't find a way to include my especifications for the analysys of variance for my 2^3 desing.

[Scott MacKenzie]

It would be helpful if you posted the data with your question.

Answer 1

[Abhimenyu]

Hi Juan,

I understand that you want to analyze a factorial design of 2^3. A (2^3) factorial design involves three factors, each at two levels, resulting in (2^3 =) 8 experimental runs.

Adding center points to this design helps in detecting any curvature in the response surface, providing insights into potential non-linear relationships between the factors and the response.

Please find the example MATLAB code below to understand the analysis of the factorial design:

• Generate the (2^3) Factorial Design: A design with three factors is defined, each having two levels. This creates a full factorial design. The design matrix uses coded levels (-1, 1) to simplify the interpretation of effects. Coded levels are standard in factorial designs as they center the data, making it easier to identify interactions and main effects.

% Generate the 2^3 factorial design with coded levels
factors = 3; % Number of factors
levels = 2; % Levels for each factor
design = fullfact([levels levels levels]);

% Convert to coded levels (-1, 1)
design(design == 1) = -1;
design(design == 2) = 1;

• Center Points: Center points are added to the design to test for curvature. They are typically placed at the midpoint of the factor levels (coded as 0, 0, 0)

% Add center points (coded as [0, 0, 0])
centerPoints = repmat([0, 0, 0], 4, 1);
design = [design; centerPoints];

• Response Data: Please input your experimental data here. The response variable corresponds to the observed outcomes for each run of the design, including both factorial and center points.

% Generate response data (example data)
% Replace this with your actual response data
response = [10; 12; 14; 16; 18; 20; 22; 24; 15; 15; 15; 15];

• Fit the Linear Model: The design matrix and response data are combined into a table for analysis. The factors are labeled X1, X2, and X3, and the response as Y. The fitlm function fits a linear model including all main effects and interactions (X1*X2*X3).

% Step 3: Fit the linear model
tbl = array2table([design, response], 'VariableNames', {'X1', 'X2', 'X3', 'Y'});
mdl = fitlm(tbl, 'Y ~ X1*X2*X3');

• ANOVA: The analysis of variance (ANOVA) is performed to test the significance of each factor and their interactions. The ANOVA table provides p-values that help determine which effects are statistically significant.

% Step 4: Perform ANOVA
anovaResults = anova(mdl, 'summary');

% Display results
disp(anovaResults);

                       SumSq      DF      MeanSq          F          pValue  
                     _________    __    __________    __________    _________

    Total               178.67    11        16.242                           
    Model                  168     7            24             9     0.025369
    . Linear               168     3            56            21    0.0065485
    . Nonlinear      6.926e-29     4    1.7315e-29    6.4931e-30            1
    Residual            10.667     4        2.6667                           
    . Lack of fit       10.667     1        10.667           Inf            0
    . Pure error             0     3             0                           

disp(mdl);

Linear regression model:
    Y ~ 1 + X1*X2 + X1*X3 + X2*X3 + X1:X2:X3

Estimated Coefficients:
                    Estimate        SE          tStat         pValue  
                   ___________    _______    ___________    __________

    (Intercept)         16.333     0.4714         34.648    4.1402e-06
    X1                       1    0.57735         1.7321        0.1583
    X2                       2    0.57735         3.4641      0.025721
    X3                       4    0.57735         6.9282     0.0022784
    X1:X2           1.3288e-15    0.57735     2.3016e-15             1
    X1:X3           1.3681e-15    0.57735     2.3696e-15             1
    X2:X3           2.1751e-15    0.57735     3.7674e-15             1
    X1:X2:X3       -5.3748e-16    0.57735    -9.3095e-16             1


Number of observations: 12, Error degrees of freedom: 4
Root Mean Squared Error: 1.63
R-squared: 0.94,  Adjusted R-Squared: 0.836
F-statistic vs. constant model: 9, p-value = 0.0254

For more information on the fullfact, fitlm, and anova functions in MATLAB R2024b, please refer to the documentation links provided below:

1. https://www.mathworks.com/help/stats/fullfact.html
2. https://www.mathworks.com/help/stats/fitlm.html
3. https://www.mathworks.com/help/stats/anova.html

I hope this helps!