Why is VAT a regressive tax?

I admit that I am not very good at sums and that I am easily confused by economic arguments. So I hope that somebody will be able to explain a comment made by Roger Williams on Sunday's Politics Show and by David Milliband on today's Daily Politics - that raising VAT is a regressive tax because it falls most heavily on the poorest in society.

If a poor person spends £20,000 a year and a rich person spends £200,000 I would have thought that the richer person would pay at least 10 times as much in purchase taxes than the poorer person. Because VAT is a purchase tax that is only charged on luxury goods, with essential goods being exempt, I would also have thought that it stands to reason that the richer person would spend a greater percentage of his money on luxuries whilst the poorer person would spend a large proportion of his money on tax exempt essentials. If this is the case then the richer person will pay even more in tax. This doesn't appear regressive or disproportionate to me; it appears to be fair and proportionate. So how do Mr Williams and Mr Milliband come to the opposite conclusion to me?