The Multi-factor analysis of variance model is used to evaluate the distribution and prevalence of responses to multi-factor questions on surveys. It involves analyzing each multi-factor question separately, comparing single-factor and multi-factor questions, and creating cross-tabulations of multi-factor questions and their response options to enable more detailed analysis.

The way data is formatted for multi-factor questions is distinct from that of single-choice questions, where each option for the former is considered a unique field. For this platform, binary data is required for multi-factor questions, where 1 denotes selection and 0 represents no selection. If the data does not adhere to the prescribed format, data conversion must be conducted to enable multi-factor analysis.

 In the analysis of multi-factor questions, the response rate pertains to the ratio of options selected in relation to each other, while the penetration rate reflects the general appeal of each option. The disparity between the two is attributed to the respective denominators used in their calculations. For example, if 100 samples are examined, and 200 options are selected across a multiple-choice question, with 70 of those options being a particular selection, the response rate for that option would be 35% (70/200), while its popularity rate would be 70% (70/100).

References：

• Candioti, L. V., De Zan, M. M., Cámara, M. S., & Goicoechea, H. C. (2014). Experimental design and multiple response optimization: Using the desirability function in analytical methods development. Talanta, 124, 123-138.
• Edwards, Y. D., & Allenby, G. M. (2003). Multivariate analysis of multiple response data. Journal of Marketing Research, 40(3), 321-334.
• Thomas, D. R., & Decady, Y. J. (2004). Testing for association using multiple response survey data: Approximate procedures based on the Rao-Scott approach. International Journal of Testing, 4(1), 43-59.