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Does adding depth to a multiple factor ANOVA increase Type I error rate?
I am doing a study on the relationship between different sociodemographic characteristics and the risk of under-age drinking. I have collected data on the sociodemographic characteristics of the student at the time of their first drinking and they fall into the following categories. There are 8 categories for sex, 6 for race, and 4 for socioeconomic status.
Sex: Female, Male, Missing
Race: Black, Other, White
SES: Low, Medium, High
Based on this data, I want to do a multifactorial analysis of variance with age as the continuous independent variable, sex, race and SES as factor levels, and drinking as the dependent variable. Does the fact that I have different levels for sex, race, and SES increase the risk of a Type 1 error? Is there any research that answers this question? If so, could you point me in the right direction to read this research?
A:
One way to think about it is that I might have a single covariate that's measured $p$ different ways, and where one of those measurements may actually be "missing" (observed but negative) and $p-1$ of the measurements may be observed and positive. If you have one missing and $p-1$ positive, the effect of that measurement on your model will be biased upwards in $R^2$. A similar situation holds for factors.
Another way to think about it is that a factor is being measured $p$ times (e.g. "sex" is measured 3 times, "race" is measured twice), and each measurement is typically not orthogonal, which could result in the presence of a factor's correlation with other factors affecting your model.
If you have $p$ different factors measured in your analysis, the number of degrees of freedom increases by $p$, so it's not necessarily true that adding $p$ factors would necessarily increase your type I error.
You have $p$ for the total number of factors you
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