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I remember reading in a lot of introductory psychology texts that since there can be no random assignment of gender as a factor, the experiment can only be seen as a quasi-experiment. How does this work? Does gender as a factor always make my experiment quasi-experimental?

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If your experiment is preselecting and assigning participants non-randomly, such as using sex as the main factor to partition participant groups and responses, then yes it is quasi-experimental design. If the participant partitioning is random then this is not a quasi-experimental. However many studies may investigate sex differences as part of an exploratory analysis. You may investigate sex differences as a byproduct of a between-, within- or mixed-experimental designs to see if there are any sex differences influencing the data outside of the experimental manipulations. This does not make the design quasi, but is important to investigate because there are often differences between males and females in cognitive task performance e.g. working memory tasks, and neural activity. The examples I have given are easily found in papers on the areas I'm personally investigating at the moment (working memory) which are quasi-experimatal designs because sex was the main grouping factor, but differences can be found in a great many papers which were not looking at sex differences specifically.

Essentially the quasi design basically just means non-random assignment, and is useful in many study types not just for investigating sex differences. To summerise, quasi-design if preselected non-random assignment. Not quasi if used as part of exploratory analysis of a study using random group assignment to the experimental conditions.

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