Propensity score matching tries to make a comparison of self-chosen groups look like a randomised one.
It rests on the idea that units with the same chance of treatment can be compared fairly.
Matching on a single score is far easier than matching on many characteristics at once.
Step 1 estimates each unit's probability of treatment from its measured characteristics.
Step 2 pairs each treated unit with an untreated unit whose score is close.
Step 3 checks balance, confirming that the matched groups now have similar characteristics.
Step 4 compares outcomes between the matched groups.
Units with no close match, such as members unlike any non-member, are dropped.
That improves balance but means the estimate applies only to the units that were matched.
Weighting by the score is an alternative to matching that keeps more of the data.
The method assumes that every important confounder was measured and used.
It cannot fix unmeasured confounding, however good the matching looks.
Including variables that the treatment itself changes introduces bias rather than removing it.
In practice, tables comparing the groups before and after matching are the main evidence that the method worked.
The usual misconception is that matching makes data as good as a trial, when it only balances what was measured.