# Chi-Square test - respondents may be assigned to multiple categories?

I am referring to this example: http://www.sthda.com/english/wiki/chi-square-test-of-independence-in-r

The data used are frequencies for household tasks:

Wife Alternating Husband Jointly
Laundry 156 14 2 4
Main_meal 124 20 5 4
Dinner 77 11 7 13
Breakfast 82 36 15 7
Tidying 53 11 1 57
Dishes 32 24 4 53
Shopping 33 23 9 55
Official 12 46 23 15
Driving 10 51 75 3
Finances 13 13 21 66
Insurance 8 1 53 77
Repairs 0 3 160 2
Holidays 0 1 6 153

My questions:

1. Normally, in a contingency table for chi square, the categories (rows) should be independent. In this data set, I am not sure if the surveyed households were assigned to more rows. In such a case the chi square would be not appropriate or?

2. Considering each household was just asked about one activity (row), would it then be ok to use the chi square test like in the example?

To give some context, the table comes from p.381 of "Nonsymmetric Correspondence Analysis: A Tool for Analysing Contingency Tables With a Dependence Structure" by Kroonenberg & Lombardo (1999) https://doi.org/10.1207/S15327906MBR3403_4 (in fact they adapted data collected by other researchers in the 1970s in Germany, their article explains all of this in detail). The article is also freely available here for download, the part about the household tasks study is pp.379-384.

223 young, childless married couples were asked to answer who performs primarily a given task in the household, for each task. So the same household is theoretically counted in each row, which indeed would be problematic for conducting a chi-square test on this dataset (see Is it okay to run a chi square if each participant is contributing multiple counts?). But anyway, as Peter Flom says in his answer, it would be redundant to conduct a chi-square test in the first place, given that there are very obvious differences between rows.

You may have noticed that the sum of each row does not add up to 223, and is different between each row. It's due to responses being excluded from the table when the husband and the wife disagreed on who performs the task. In their article, Kroonenberg and Lombardo discuss why a "disagreement" column was not included in the table to take into account such cases.

The core of their article also discusses a possible method to analyze this kind of data using a variant of correspondence analysis (nonsymmetric correspondence analysis), but using a regression as suggested by Peter Flom in his answer may be a good option depending on the question you want to answer ultimately. Note that in their analysis, Kroonenberg and Lombardo treated "tasks" as predictor variable, and "who performed the task" as the response variable.

As a side note, it looks like that the author of the webpage you link to in your question is unaware of the original source of the data and its study design, as they don't mention it. It probably explains why they thought a chi-square test was suitable for this table.

• thank you very much! In case we assume that each household was asked just about one activity, would a chi-square test, in the way the author did on his webpage, be appropriate and legitimate? Oct 28, 2023 at 8:17
• @Gustav Good question. Leaving aside the issue of wife/husband possible disagreement, and assuming the survey was conducted on 1744 different couples who were each asked about a single activity, the test would be valid but pretty useless in this case, given that the difference between rows is very obvious without conducting any test. In other words, what would a chi-square test tell you that you don't already know when looking at the table? Oct 28, 2023 at 9:07
• I completely agree that a test with these frequencies is not needed. Just wondering if a study design asking different persons for each row would be a valid base for a chi-square (did not see a chi-square in this way). I find the analysis done on the website quite helpful (with the balloonplot, mosaic plot, and the residuals) - and also the text descriptions on how to interpret them. Want to use these for my interpretation of chi-square tests - was just a bit afraid it is incorrect as the basis contingency table was "strange". Oct 28, 2023 at 10:04
• @Gustav The underlying issue here is probably "Is this study design adequate to answer the research question?", which leads to asking "What is the research question we want to answer?". For instance, if we have questions about the division of tasks in couples, then you should ask each couple about every task in the household - otherwise you simply have missing data about their answers to the other tasks, which means you could miss some important things or phenomenons. (Plus it would be a waste of resources: why bother contacting 1700 people and asking them a single question?). Oct 28, 2023 at 10:43
• @Gustav: In other words, interesting research questions we may have relative to this dataset probably cannot be answered using a chi-square test (which is the point Peter Flom made in his answer), no matter if couples were asked a single or multiples questions. Oct 28, 2023 at 10:49

I don't think a chi-square test answers a useful question here.

First, the data pass the IOTT. That's the interocular trauma test (it hits you between the eyes). When one row is 156-14-2-4 and another row is 0-3-160-2 then you don't need a test of association.

Second, if you want a test, I would do some kind of regression. There is clearly a dependent and an independent variable. Tasks in household might be assigned by sex, but sex can't be assigned by task. This would also let you add covariates, which is where the interesting results might be found. E.g. Do younger couples assign tasks differently than older ones? Do couples with kids differ from those without? How about by who is working outside the home (man, woman, both, neither)? Or whether they have any outside help with these tasks?

Third, another possibility is to look (somehow) at the nature of tasks and how they are assigned. One could try to find some measure of "tradition" in these tasks, for instance.

• Considering this as a regression also allows for the possibility of modeling a multi-label outcome.
– Dave
Oct 25, 2023 at 12:21