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I'm trying to compare mean interest levels of 3 products. The same sample was asked about each of the three products. How can I test for equal means among the three products?

Other information: Interest level was quantified using Likert scale. Sample size is 200.

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  • $\begingroup$ Ordinal regression. $\endgroup$ – user2974951 Aug 6 '19 at 7:53
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In order to compare multiple means within the same subjects you can use 'paired' analyses. In case of two means, you can choose to use a paired t-test. In case of 3 or more means, you could use a paired version of an ANOVA: repeated measures ANOVA with a within-subject factor (product).

However, it is important to note that these kinds of analyses assume a normally distributed, continuous outcome. Your outcome has an ordinal distribution, so it could be that another analytical method would be a better fit. One option is to use a non-parametric test for paired data that does not assume normality, e.g., Wilcoxon signed-rank test. Another option is to use a form of ordinal regression that takes into account the nested structure of the data. Here, you could for instance consider Generalized Estimating Equations.

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