HYPSM on par with Oxbridge with Harvard being "better" than the rest. Don't know why people are constantly trying to bash LSE and comparing it with Harvard and Cambridge. LSE and Imperial pretty much on par with Columbia, Penn...... The only problem they face is that their not multi facilities so some people may not see them legitimately compare to Columbia that offers a top programs in law, business, medical, arts etc.
Other top universities are UCL, St Andrews, Kings, Edinburgh. Not sure how to compare then to US colleges though