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My experiment tests how frequently people in two different countries use different types of computers. Assume, that there are only three types of computers (desktop, laptop and tablet) . I am interested to see the difference of usage between types (within each country) as well as between two countries (for each type). I conducted the survey in which participants ranked how frequently they use each type (1 - low frequency, 10 - high frequency). I got the following avrage results (error bars are SEM):

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For desktop there is almost no difference between countries, but there is quite a large one-side difference for laptops and desktops. But given that there are only three types of computers, how it can be that higher frequency for two types in country 1 is not compensated by higher values for other type for country 2? Am I right with my concern or there is a flow in my logic?

To resolve the issue, for each participant I calculated relative frequency by dividing the frequency value of each type by sum of frequencies. So, now the previous problem was solved (see the graph below). But: a) the applied normalization altered the relationship, so now the biggest difference between countries is for desktops; and b) how I can run now 2-way ANOVA for these data given that the values are not independent (in each country they summed up to 100).

What is the correct way to tackle my problem?

enter image description here

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With respect to your first question, it could be that people in country 2, overall, use computers (of any type) less frequently than those in country 1, or that "frequently" has a slightly different meaning to people in country 2 than country 1, or any of several other reasons that mean your data, as collected, is valid. Given that there are several straightforward and actually quite likely reasons for the overall reported frequency of computer usage to be different between the two countries, there is no need to normalize the data.

With respect to your second question, I'd probably set up my model to have a country factor, a computer type factor, and an interaction term. The country factor would address the issue brought up by your first question, but remember that if "country" is significant it doesn't mean there's a significant difference in actual computer usage! The detailed analysis that you are interested in would depend on whether the interaction term was significant or not.

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  • $\begingroup$ Thank you for the answer! The question is whether I am allowed to used ANOVA in the second case given that types in each country are not independent (i.e., type3 = 1 - type1 - type2) $\endgroup$ – student Sep 9 '18 at 6:55
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    $\begingroup$ Well that's true of all ANOVA with categorical variables; ANOVA on, say, frequency of drinking by men / women has a "type" where the percentage of men = 1 - the percentage of women, etc. $\endgroup$ – jbowman Sep 9 '18 at 15:28
  • $\begingroup$ Thanks. But I opened a separate thread because it's not clear whether I can or cannot run this test stats.stackexchange.com/questions/366346/… $\endgroup$ – student Sep 11 '18 at 8:09
  • $\begingroup$ That thread asks about a completely different problem. In that thread, you are trying to analyze percentages that sum to 100; in this thread, you are trying to analyze frequencies. As I pointed out in my answer above, there is no need to normalize those frequencies to sum to 100%; by doing so, you gain nothing and lose the ability to analyze what you really want. $\endgroup$ – jbowman Sep 11 '18 at 14:00

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