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I have 4-point ordinal data on users' opinions about various apps which they use, and I want to run significance tests on the ratings given to each app. None of the apps requires the others to be present, but people may use multiple apps, resulting in some potential overlap in the users of different apps. Are these data sufficiently independent for ANOVA, or would Kruskal-Wallis be more appropriate?

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The answers are clearly not independent if you want to build a model that predicts the ratings of the different apps simultaneously. Consider using mixed models in this case. If you fit separate models per app then your answers are independent.

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