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I have a question regarding how to analyze categorical data in this situation. Assume that I have five choices of food for people to choose from and I expect that 50% of my research participants will choose the first choice, while the other 50% will choose the second choice. However, in reality 20% of the participants choose each of the choices (i.e., there is an equal number of people choosing each choice) and the total number of participants is 1000. Also, each participant can choose only one choice.

Obviously, based on the frequency counts the actual data do not fit my expectation. I was wondering what type of inferential statistical test I should perform to investigate whether the data fit my expectation. If I understand correctly, I should not use a Pearson's Chi-square test of goodness of fit because the expected frequencies of three of the choices are zero (the expected frequencies are 500, 500, 0, 0, and 0). For example, the source below says that there must be at least 5 expected frequencies in each group of your categorical variable: https://statistics.laerd.com/spss-tutorials/chi-square-goodness-of-fit-test-in-spss-statistics.php

I would appreciate your suggestions.

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  • $\begingroup$ Hi! There's probably a mistake in your question, if there is equal number of people choosing each choice, then it's 20%, not 25%. Unless people can choose multiple foods, but that's not what I understood from your question. You can edit your question to clarify this. $\endgroup$
    – J-J-J
    Commented Nov 20, 2023 at 6:23
  • $\begingroup$ @J-J-J Sorry for the typo. You are right and I have edited the question. Thank you so much! $\endgroup$
    – silverfox_33
    Commented Nov 20, 2023 at 6:25

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Your expectation is that no one in your population of interest would choose the last three options, and instead would choose equally between the two first options. Obviously, this is not the case, as many people from your sample did choose the last three options, you don't need a test to see that. It's like asking "Do black swans exist at all?", and then observing a bank of black swans. You don't need a statistical test at all to answer that question, in particular when observing just one black swan would be enough.

Now, maybe you should reconsider your null hypothesis: Is this expectation useful to answer the research question you are actually trying to answer? Without using statistical language, what do you want to learn about your population of interest? Does testing for a null hypothesis of $(0.5,0.5, 0, 0, 0)$ really help you to answer that question? Looking back at what you want to learn from your sample would certainly be useful.

Maybe you'll find out that "50% 50% 0% 0% 0%" was not your actual expectation. Or maybe you'll find out that you're not interested in hypothesis testing after all, but in estimating the proportion of people in each category (in which case, you might want to look into confidence intervals for your proportions, rather than hypothesis testing).

Note that even if your actual expectation is something more akin to "anything between 0% and 5% for the last three options", your observed counts are so far removed from that, that a test would certainly be redundant anyway.

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  • $\begingroup$ Thank you so much. I really appreciate your answer. The question was not based on a specific research question but was out of my curiousity as I am learning how to use chi-square tests. If you don't mind, could you elaborate a bit (or suggests a source) on estimating the proportion of people in each category, in which case I should look at C.I. for proportions? $\endgroup$
    – silverfox_33
    Commented Nov 21, 2023 at 7:50
  • $\begingroup$ @silverfox_33 Sure, there is this summary of some possible methods for computing simultaneous confidence intervals for multinomial proportions: blogs.sas.com/content/iml/2017/02/15/… . The Goodman method is a good one. You can find implementations in various software, for instance in the DescTools package in R: rdrr.io/cran/DescTools/man/MultinomCI.html (this page mentions many references btw, if you want to investigate a bit more the theoretical side of it) $\endgroup$
    – J-J-J
    Commented Nov 21, 2023 at 8:14
  • $\begingroup$ Now, if you're interested in confidence intervals, maybe you should ask a separate question about it. By the way, my answer still holds even if you asked it out of curiosity. In your example, conducting a test does not really make sense, if you have an expected value of 0 and observe one occurence of the category. An expectation of 0 is saying "I think that no one in the population has this characteristic"; as it just requires one observation to disprove this hypothesis, a test is essentially useless. $\endgroup$
    – J-J-J
    Commented Nov 21, 2023 at 8:18
  • $\begingroup$ Yes, your answer regarding the lack of necessity to do the test makes a lot of sense. I will check out the links you provided and will definitely start a new thread if I have any questions. Thank you so much! $\endgroup$
    – silverfox_33
    Commented Nov 22, 2023 at 10:47
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    $\begingroup$ I find your answer useful (and also funny), so +1. In particular, pointing out that running a test when the data is manifestly inconsistent with the null hypothesis. CV answers are not consistent on that point, by the way, which is unfortunate. $\endgroup$
    – dipetkov
    Commented Nov 25, 2023 at 11:39

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