In a two-factorial design with two levels for factor A and four levels for factor B, which of the following represents a valid multiple regression equation?

In a two-factorial design, which encompasses two independent variables (or factors), especially with multiple levels, the valid multiple regression equation needs to accurately reflect the interaction between these factors.

The correct representation captures both the main effects of each factor and any interaction that may exist between them. When you have two levels for factor A and four levels for factor B, the regression equation must comprise terms that represent these parameters.

The chosen equation, which mentions the longest one that ends with 1 and 4, implies a detailed structure reflecting the combination of levels from both factors. For a two-factorial design, the appropriate regression equation typically includes intercept and coefficients for each effect (the levels of factors A and B), as well as an interaction term that accounts for the combined effect of factors A and B.

In this case, an equation would commonly involve the main effect coefficients relating to each level of factor A and B (which would be part of the longest expression) and would not simply conclude with arbitrary numbers unless they have specific meaning in terms of the levels of A and B. The interaction terms represent a nuanced understanding of how levels of these factors work together and affect the response variable, making it essential in accurately modeling the scenario.

Thus, the reasoning behind