Causal Inference
Causal inference is an experimental design used to isolate causation and then make use of predictive statistics.
Description
To study the impact of a treatment on an outcome, all other variables must be held constant. If a relationship can be observed ceteris paribus, then causation has been identified.
Natural Experiments
A natural experiment or true causal experiment involves random assignment of a treatment such that the treatment and control groups are identical. Methods for causal inference are then applied.
The unit of measurement in social science is almost always people. It isn't possible to form groups of identical people. As a result, causal inference is generally inapplicable. Instead, methods for statistical inference are applied. A tolerance for Type I error (false negatives) is set as a critical value. Experimental data is measured and, if the test statistic is greater than the critical value, the null hypothesis can be rejected.
Quasi-Experiments
A quasi-experiment is distinguished by the assignment of treatment. When the assignment was not completely random, but differences are expected to be controllable, quasi-experimental methods are applied.
Observational Studies
An observational study does not feature any real assignment to treatment. Instead, subjects self-assign a 'treatment' through their own decisions and behavior.
There is no reason to expect treatment and control groups to be identical. There are non-redundant confounders which are related to the treatment and/or the outcome that must be controlled for.