I have data for a dependent variable, DV, representing a count of objects, with a range from 0 to 6 (sample size 450), slightly left skewed with a mode at 4:

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Furthermore, I have a single categorical variable with three groups. I wish to perform a test to determine whether to reject the null hypothesis that the mean (or some other location measure) of the number of objects is the same across the three groups. ANOVA was suggested to me, but I believe this would be incorrect. However, could someone please confirm or correct my argument for this position?

The argument posed to me by the other party was that the DV is approximately normal and ANOVA is fairly robust to some deviation from normality. I disagreed with this position for the following reasons:

  1. ANOVA requires a continuous DV at interval or ratio level. At first sight it may seem that count is ratio-level data, in that it has a true zero (you have zero items) and the intervals are equal. However, on Laerd.com, the definition of interval-level data and above includes:

    "...their central characteristic is that they can be measured along a continuum..."

    Going by a dictionary definition of "continuum", it would suggest a space in which you can observe values closer and closer to one another such that in the limit, the difference is almost imperceptible. Or another way of looking at it, two values are infinitely divisible. This is not the case for an integer variable such as a count of objects.

  2. As mentioned earlier, ANOVA requires (roughly) a normally distributed dependent variable with equal variance across the groups but is robust to some degree to nonnormality. However, I don't think an integer variable can validly be modelled with a normal distribution in that it can only hold integer values - even if it has a similar bell shape.

So anyway, because of point (1) I would have to conclude that count data is ordinal-level and a Kruskal-Wallis test would be better suited for this situation. However, it seems wasteful to treat counts as only ordinal, as the integer intervals are equal and there is a zero... so in addition to confirming or rejecting my arguments, any suggestions for other tests are welcome :-)