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Imagine experiment in which we show three cars: red, black and violet to responders and ask them which one is 'the coolest'. Question is: is their choice affected by their age?

Now we have two strategies:

  1. Multinomial logit with age as independent and choice of color as dependent variable.
  2. ANOVA (compare ages in red, black and violet groups) essentially linear model with choice of color as independent and age as dependent variable

I know that these two answer very different questions. And I know that first one is better choice. But suppose, I for some reason, used ANOVA and it showed significant differences among groups (assume that ANOVA assumptions were met).

Can I still conclude that age affects choice?

Notes:

  • I know that, technically, ANOVA showed me that distribution of ages is different in different groups
  • I don't want to cut age into intervals and run chi-squared-like test
  • I won't be using it for predictions, I just need yes/no answer if age affects choice
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The outcomes are different, and so the interpretations are different. The ANOVA version tells you if there is a difference between the mean ages for car preference. The second tell you how a one unit change in age influences the likelihood of preference (in terms of an odds ratio).

If the ANOVA provides you with a significant difference, this provides evidence that the mean age for preference between cars is different.

So yes, you can conclude that age and car choice are correlated.

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  • $\begingroup$ "X and Y are correlated" is not the same as "X is affected by Y". Phrase "X is affected by Y" means that Y takes part in process of determining X. Correlation does not tell anything about direction of association. $\endgroup$ – Łukasz Deryło Jul 6 '18 at 13:09
  • $\begingroup$ I'm trying to avoid word "causes" as much as I can not to provoke discussion about causality here... $\endgroup$ – Łukasz Deryło Jul 6 '18 at 13:10
  • $\begingroup$ You can do this with anova. Everything is just in the direction of whatever your baseline car is. So, for example, if blue is your baseline: older people prefer red cars over blue. There is no difference in age preference between blue and yellow cars. Now making a statement about red and yellow cars is trickier, but can be inferred from estimates and confidence intervals. $\endgroup$ – JWH2006 Jul 6 '18 at 13:30
  • $\begingroup$ I'm not sure if this is true. Anova (or post-hoc test, to be precise) tells me that eg. those who choose red are older than those who choose violet. My question is, can I safely convert this conclusion into the one you stated: older people prefer red over violet. $\endgroup$ – Łukasz Deryło Jul 6 '18 at 15:52
  • $\begingroup$ Yes you can, but you need to be very precise with your conclusions. Running the Anova will give you three estimates in your regression: 1. the intercept (your baseline color), 2. beta 1: the difference between color 1 and the baseline color, 3. beta 2: the difference between color 2 and the baseline color. Here you can safely make claims about color 1 or 2 and the baseline color, but making a claim about color 1 and color 2 only is a little more difficult. $\endgroup$ – JWH2006 Jul 6 '18 at 18:11

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