Quasi-experiments are studies which lack the control of a true experiment because one or more of its requirements cannot be met, ex: deliberate use of an independent variable or random assignment of subjects to different groups.

Studies of the effects of drugs on pregnant women are based on data about women who have already been pregnant and either taken or not taken drugs. Researcher can’t control assignment of subjects or choices presented to them, but can measure differences between the two populations and obtain significant findings. Such studies provide a basis for investigations that would otherwise be impossible.

Source: Gale Encyclopedia of Psychology
A quasi-experiment is one that applies an experimental interpretation to results that do not meet all the requirements of a true experiment. It means when the situation is such that the experimenter has some control over the manipulation of independent variables but fails to arrange for the other basic requirement of a true experiment, which is, creating equivalent groups.

The various types of Quasi-experimental designs are listed below:

Time-series design

When a control group or comparison group cannot be included in an experiment because of the situation in
which the experiments is being conducted, but the experimenter wants to have a design which may exercise a better control over the extraneous variables time series design is used.

Equivalent time –samples design

It is an extension of the time-series design with the repeated introduction of the treatment or the experimental variable. Like the time series design, a single group is used and the group is exposed to repeated treatments in some systematic way.

Non-equivalent control group design

Except for random assignment of subjects to the experimental and the control conditions, which occur in case of the pretest-posttest control group design, this is identical to it.

Counterbalanced design

In counterbalanced design experimental control is achieved by randomly applying experimental treatments. Such designs are called crossover designs, switch-over designs and rotation experiments. A counterbalancing design in which four treatments have been randomly given to four groups on four different occasions.

Separate-Sample Pretest-Posttest Design

It is suited to situations where experimenter cannot assign treatments to all subjects at a time. Hence, he is forced to select a sample and administer the treatment. Then, again another sample is taken and the same treatment is repeated.

Patched-up design

The experimenter starts with an inadequate design and then, adds some features so that recurrent factors producing invalidity may be maximally controlled. The patched-up design, shown below is a combination of two different pre-experimental designs, neither of which is adequate in itself, but which become adequate when combined.

Longitudinal design

In longitudinal design the researcher usually measures a group of subjects in order to observe the effect of passage of time. In fact, such designs are confounded by extraneous events that occur during the study and they may not generalize over time.

Cross-section design

Cross-sectional design is a between subjects quasi-experiment in which the researcher observes the subjects at different ages or at different points in temporal sequence. Thus, the researcher may select a cross-section of ages, testing the vocabulary of a group of 5 yr olds, another group of 6 yr olds and so on.

Cohort design

In a cohort design the researcher conducts a longitudinal study of several groups, each from a different generation.

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There are a number of different techniques of quasi-experimental research some of which have been discussed above. Depending on the particular situation and factors like data availability, a researcher may choose the technique that appears to be best.

Sources:
Tests, Measurements and Research Methods in Behavioural Sciences, AK Singh