Has anyone validated, proposed or tested a single-item state-based positive affect or happiness measure? I am particularly interested in something for tracking mood every day.
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Many single item measures of mood can be found in the literature. Those two are based on the idea that affect is bidimensional and that one's current state can be reported using a grid:
You might also want to look at http://studiolab.ide.tudelft.nl/studiolab/pmri/ [Disclaimer: I know the people who developed it]. It is conceptually similar to Russell's AffectGrid but purely non-verbal.
Finally, it seems necessary to mention the Self-Assessment Manikin. Technically, it's not a single-item instrument (the original version has three, many people use only the first two) but it is very popular. Main publication is Bradley, M.M., & Lang, P.J. (1994). Measuring emotion: The self-assessment manikin and the semantic differential. Journal of Behavior Therapy and Experimental Psychiatry, 25 (1), 49-59.
Edited (a long time) after comment: Most authors in this field argue that mood or affect are fundamentally bi-dimensional, if not more, so it makes sense that measures would also be bi-dimensional. The closest you can come to a single Likert-type item would perhaps be the SAM valence scale.
Another issue is that traditional psychometrics is almost entirely focused on dealing with item-specific variance, i.e. how to select items to build a scale, how many items are needed to reach a given level of reliability, how to model latent constructs using different items, how to equate scales or build population norms, how to adapt scales to different populations, etc. There is not much you can do with single items.
Consequently, it's not very complicated to create a new “valence measure” by drawing some smiling faces, altering the response format or slightly changing the wording. If the new measure is not completely idiosyncratic, you will obtain some level of correlation with other valence measures but beyond that what could make it a “valid” or superior measure of mood?