How to Analyze Factorial Design?

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Juan Camilo de la Cruz Alzate
回答： Abhimenyu 2024-11-11，8:42
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.

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Abhimenyu
Abhimenyu 2024-11-11，8:42
Ran in:
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!

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