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I'd like to do an ANOVA on the following problem:

The only dependent variable is the number of children a person has. The two independent variables are the person's age and the person's income. Of course there might be a non-negligible interaction between age and income. We want to neglect the fact that neither age nor income (I presume) are actually normally distributed as I learned that ANOVA is quite robust to violation of that condition (or is it?).

Let's consider persons aged 20 to 70 with incomes from 0 to 250k/year.

I am wondering how the sizes of the factor levels can change the results of the analysis, i.e. can I expect analyses that consider 10 age groups (20 to 25, 25 to 30,..., 60 to 70) to yield "better/worse" results (be more/less statistically significant) than analyses that use only 5 age groups (20 to 30, 30 to 40,...)?

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    $\begingroup$ ANOVA makes no assumptions about the distributions of the explanatory variables (the factors). In fact, you don't have an ANOVA problem, you have a regression that you're trying to shoehorn into the ANOVA framework by binning your variables. Why not solve it using appropriate regression techniques? $\endgroup$ – whuber Jul 27 '16 at 22:29
  • $\begingroup$ @whuber Why is this not an ANOVA problem? $\endgroup$ – AlphaOmega Jul 28 '16 at 10:31
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    $\begingroup$ If you have the actual values for age in years and income in currency units then you should fit a model using them as variables not factors. If you do indeed only have them categorised then that is an ANOVA problem. From your question it is hard to tell which you have. $\endgroup$ – mdewey Jul 28 '16 at 11:55
  • $\begingroup$ Well I do have the actual numbers, so I could use a regression model. But I might eas well just categorise them (young, middle, old; low income, medium income, high income) and use an ANOVA... or not? $\endgroup$ – AlphaOmega Jul 29 '16 at 9:02
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Partially answered in comments:

ANOVA makes no assumptions about the distributions of the explanatory variables (the factors). In fact, you don't have an ANOVA problem, you have a regression that you're trying to shoehorn into the ANOVA framework by binning your variables. Why not solve it using appropriate regression techniques?

– whuber

If you have the actual values for age in years and income in currency units then you should fit a model using them as variables not factors. If you do indeed only have them categorised then that is an ANOVA problem. From your question it is hard to tell which you have.

– mdewey

In neither case is there an assumption of normality (or any other distribution) on the explanatory variables.

Well I do have the actual numbers, so I could use a regression model. But I might as well just categorise them (young, middle, old; low income, medium income, high income) and use an ANOVA... or not?

– AlphaOmega

You could, but should not. Binning (categorizing) is missing information, see What is the justification for unsupervised discretization of continuous variables?. You should rather use a regression model with the variables (age in years, income) represented via splines. See for instance Advanced regression modeling examples.

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