suppress intercept in regression when having more than one categorical variable coded in dummy variables

Question

friends:

according to the following link https://stats.stackexchange.com/a/11068/196391

and what I saw in some papers, we can supress the intercept and consider ALL the dummy variables (which have coded the levels of categorical variable) in the model.

I tried to do so in JMP and it found singularity!

I am wondering which of the following three approaches are correct when we have more than one categorical variable?

1- considering a dummy variable for each level of each categorical variables and include all the dummies in the model except the reference dummies of each of the categorical variables (and keep the intercept in the model)

2- considering ALL the dummy variables of each of the categorical variables and suppress the intercept in the model

3- considering categorical variables in the model as they are (no dummy coding)

Answer 1

I believe the only viable option is the first: include all dummy columns except the reference dummy. The reason is that with a single categorical, the intercept is the mean for the reference group, so you could just include all levels of that categorical and do away with the intercept, if so desired, and just interpret the coefficients in an independent manner. However, with multiple categorical covariates, the intercept is the mean of the groups that is made up of the reference variables for each category. So if you get rid of the intercept and include all levels of your categoricals, it becomes impossible to interpret your results.

As far as option 3, this only makes sense if you are dealing with ordinal categories (e.g. categorical levels 1, 2 and 3, where there is meaning in the fact that 1 is less than 2, for instance if they refer to 1st class, 2nd class and 3rd class, where 1st class is less than 2nd class and so on).