Observational study
In statistics, the goal of an observational study is to draw inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator.  This is in contrast with controlled experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group before the start of the treatment.
The assignment of treatments may be beyond the control of the investigator for a variety of reasons:
- A randomized experiment would violate ethical standards.  Suppose one wanted to investigate the abortionbreast cancer hypothesis, which postulates a causal link between induced abortion and the incidence of breast cancer.  In a hypothetical controlled experiment, one would start with a large subject pool of pregnant women and divide them randomly into a treatment group (receiving induced abortions) and a control group (bearing children), and then conduct regular cancer screenings for women from both groups.  Needless to say, such an experiment would run counter to common ethical principles.  (It would also suffer from various confounds and sources of bias, e.g., it would be impossible to conduct it as a blind experiment.)  The published studies investigating the abortionbreast cancer hypothesis generally start with a group of women who already have received abortions. Membership in this "treated" group is not controlled by the investigator: the group is formed after the "treatment" has been assigned.
- The investigator may simply lack the requisite influence.  Suppose a scientist wants to study the public health effects of a community-wide ban on smoking in public indoor areas.  In a controlled experiment, the investigator would randomly pick a set of communities to be in the treatment group.  However, it is typically up to each community and/or its legislature to enact a smoking ban.  The investigator can be expected to lack the political power to cause precisely those communities in the randomly selected treatment group to pass a smoking ban.  In an observational study, she would typically start with a treatment group consisting of those communities where a smoking ban is already in effect.
- A randomized experiment may be impractical.  Suppose a researcher wants to study the suspected link between a certain medication and a very rare group of symptoms arising as a side effect.  Setting aside any ethical considerations, a randomized experiment would be impractical because of the rarity of the effect.  There may not be a subject pool large enough for the symptoms to be observed in at least one treated subject.  An observational study would typically start with a group of symptomatic subjects and work backwards to find those who were given the medication and later developed the symptoms. Thus a subset of the treated group was determined based on the presence of symptoms, instead of by random assignment.
In all of those cases, if a randomized experiment cannot be carried out, the alternative line of investigation suffers from the problem that the decision which subjects receive the treatment and which subjects do not is not entirely random and thus is a potential source of bias.  A major challenge in conducting observational studies is to draw inferences that are acceptably free from influences by overt biases, as well as to assess the influence of potential hidden biases.
In observational studies, investigators may use propensity score matching (PSM) in order to reduce overt biases.