Fit Linear Model Using "fitlm(" With 3-Way and 4-Way Interactions

I want to create a linear regression model with 2, 3, and 4-way interactions. However, currently, my code will only return a 2-way interaction. How do I get the 3-way and 4-way interactions?

Here's my current code and the results:

y =[45; 71; 48; 65; 68; 60; 80; 65; 43; 100; 45; 104; 75; 86; 70; 96];
g1 = {'−'; '+'; '−'; '+'; '−'; '+'; '−'; '+'; '−'; '+'; '−'; '+'; '−'; '+'; '−'; '+'};
g2 = {'−'; '−'; '+'; '+'; '−'; '−'; '+'; '+'; '−'; '−'; '+'; '+'; '−'; '−'; '+'; '+'};
g3 = {'−'; '−'; '−'; '−'; '+'; '+'; '+'; '+'; '−'; '−'; '−'; '−'; '+'; '+'; '+'; '+'};
g4 = {'−'; '−'; '−'; '−'; '−'; '−'; '−'; '−'; '+'; '+'; '+'; '+'; '+'; '+'; '+'; '+'};
 
tbl = table(y,g1,g2,g3,g4);
lm = fitlm(tbl, 'interactions')




lm = 


Linear regression model:
    y ~ 1 + g1*g2 + g1*g3 + g1*g4 + g2*g3 + g2*g4 + g3*g4

Estimated Coefficients:
                   Estimate      SE        tStat        pValue  
                   ________    ______    _________    __________

    (Intercept)     45.188     4.1922       10.779    0.00011917
    g1_+                23     5.0559       4.5491     0.0061175
    g2_+                 1     5.0559      0.19779         0.851
    g3_+             26.75     5.0559       5.2908     0.0032162
    g4_+              -0.5     5.0559    -0.098894       0.92507
    g1_+:g2_+         0.25     5.0559     0.049447       0.96248
    g1_+:g3_+       -36.25     5.0559      -7.1698    0.00082085
    g1_+:g4_+        33.25     5.0559       6.5764       0.00122
    g2_+:g3_+         4.75     5.0559      0.93949       0.39061
    g2_+:g4_+        -0.75     5.0559     -0.14834       0.88787
    g3_+:g4_+        -2.25     5.0559     -0.44502       0.67491


Number of observations: 16, Error degrees of freedom: 5
Root Mean Squared Error: 5.06
R-squared: 0.978,  Adjusted R-Squared 0.933
F-statistic vs. constant model: 21.9, p-value = 0.00163

Answer 1

[Samayochita]

Hi Sarah,

The issue is in the model specification string you are passing to “fitlm”. The model specification is specified as “interactions” which only includes up to two-way interactions.

To include 3-way and 4-way interactions, you need to manually specify the formula in the form “y ~ terms”, where the terms are in Wilkinson Notation.

Here is how you can include all interactions up to 4-way:

lm = fitlm(tbl, 'y ~ g1*g2*g3*g4');

For more information, please refer to the following documentation links:

https://www.mathworks.com/help/stats/fitlm.html

https://www.mathworks.com/help/stats/fitlm.html#bt0ck7o_sep_shared-DefinitionWilkinsonNotation

I hope this answer was helpful!