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I have several observations with both categorical and numerical variables relating to the personal details of surveyed individuals (such as age, gender, education and region inside the country they live) and also a categorical variable that I'm particularly interested in, indicating each individual's favourite movie among a list of 5-6 given movies.

I'd like to observe how the personal trait variable affects the choice of the favourite movie, also deciding on the target audience for each movie, relating each movie to a certain group of "typical" people for that movie. Which method of analysis should I use?

My initial idea was to use chi-squared tests for each categorical variable with the choice of movie, but this doesn't necessarily account for combined effects with the numerical variables and can produce erroneous results due to multiple analyses. Another idea was to use multiple regression, substituting numerical codes for categorical variables, but I'm not sure how to deal with categorical variables with more than two levels. Which statistical method should I use for my purpose?

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Since you have 5 or 6 given alternatives without any order or other structure, you could look into multinomial logistic regression. https://en.wikipedia.org/wiki/Multinomial_logistic_regression Some examples (using the nnet package from CRAN, for neural networks) can be found here: http://www.ats.ucla.edu/stat/r/dae/mlogit.htm See also https://stats.stackexchange.com/search?q=multinomial+logistic+regression for a list of questions and answers about multinomial logistic regression on this site.

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