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The basis of my experiment are 4 neutral text snippets. For each text snippet, I create an uncertain and a certain version so that I have a total of 8 text snippets (2 versions for each text snippet, 1 certain and 1 uncertain version). The "version" of the text is the IV.

Now, the participants are randomly shown 4 text snippets at the same time (always 2 uncertain and 2 certain text snippets). The participants are asked to indicate if they perceive the text snippet as uncertain or not (DV).

On the one hand, I have a within-subject design because each participant sees uncertain and certain versions of text snippets. On the other hand, I have a between-subject design because a participant only sees either the certain or the uncertain version of a text snippet.

In my analysis, I will first examine the difference between the uncertain and certain versions in general (I sum up the results of the certain and the uncertain versions and compare the results). Then, I will examine the difference between the uncertain and certain versions of each text snippet.

My question is: What is the name of that setting? Is that setting a crossover-design?

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If I understand your description correctly, you have one within-subject factor with two levels (certain/uncertain) and one between-subject factor (based on how you assign the different text snippets to different group, but I'm not sure I understand what exactly you are doing there). So, this would be a repeated-measures design with different groups (based on the assignment of different text snippets) which I would refer to as is a mixed design (https://www.discoveringstatistics.com/statistics-hell-p/porus-comparing-means/mixed-designs/), and you could analyse it, e.g., with a mixed ANOVA (https://statistics.laerd.com/spss-tutorials/mixed-anova-using-spss-statistics.php). I think crossover-design refers to something similar.

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