Idea behind inverse probability weighting

Suppose the full data are

Group:	A			B			C
Response:	1	1	1	2	2	2	3	3	3

The average response is 2. However, we observe:

Group:	A			B			C
Response:	1	?	?	2	2	2	?	3	3

From the observed data, the average response is 13/6, biased.

Notice the probability of response is 1/3 in group A, 1 in group B and 2/3 in group C.

Calculate weighted average, where each observation is weighted by 1/{Probability of response}:

$${\frac{{ 1 \times \frac{3}{1} + (2+2+2)\times 1 + (3+3)\times \frac{3}{2}}}{{\frac{3}{1} + 1+1+1 + \frac{3}{2} + \frac{3}{2}}}}$$ = 2.

IPW has eliminated the bias in this case; more generally it will give estimators the property they 'home in' on the truth as the sample size increases (i.e. they are consistent).