Application of Factor Analysis to a table of count data

Hi,
I am being asked to apply a statistical technique that I do not think is optimal. I have asked in other forums, but did not receive an answer that I thought was conclusive. Please have patience with me.

I am being asked to conduct a factor analysis on a table of data. The table lists statements in the rows, and brands as columns. The cells include counts of people who endorsed the statements for that brand. People who consumed a brand were asked to endorse each statement regarding that brand. Each brand was consumed by a different number of people.

My objective is to understand the underlying reasons consumers purchase these brands, and to maximize the amount of variance predicted by two factors.

Is it correct to conduct a factor analysis on this table, treating the columns (brands) as objects, and the rows (statements) as observations?

Is it more correct to conduct a factor analysis on respondent level data? (Where columns are brands with nested statements, and rows are individual people, and the cells represent a 0 or 1 endorsement of the statement by brand?). Would that even work?


Is it more correct to conduct a correspondence analysis on this data? Is there a reason to prefer correspondence analysis over factor analysis? Should we not do both? Why would I prefer one over the other?


Any direction is much appreciated.