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Let's say the mean income in New Jersey (group 1) and Florida (group 2) is $80,000 and $55,000 respectively. Using a dataset of relevant predictor variables, I'd like to attempt to explain this gap (i.e. $25,000). My question is what method I'd use to do so. For example, would I take the difference of the means and estimate a regression with it as my DV? Thanks in advance!

Zach Goldberg
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Assuming you are using continuous independent variables to explain the difference, you can use a linear regression of the form:

$$y\sim\beta_1 + \beta_2 x + z_1(\beta_3+\beta_4 x)$$,

where y is your income, $z_1$ is an indicator variable for whether a subject is in group 2, and $x$ is your covariate independent variable. When you fit this model, significance of $\beta_3$ will tell you if there is a difference in the two group's intercept, and significance of $\beta_4$ will tell you if your covariate contribute to the observed difference in group incomes.

Asy
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  • Okay. So there's no need to regress the differences in mean income (i.e. I should just regress the income variable as a whole)? – Zach Goldberg Feb 10 '18 at 01:05
  • Also, what are B3 and B4 in this case? Other IVs? – Zach Goldberg Feb 10 '18 at 01:08
  • A regression is equivalent to conducting a t-test or ANOVA. To see this, suppose you want to see how an IV $x$ affect the income -- you would be checking if the mean income differ for the different levels of $x$ (if it's categorical). In that case the regression would be $$y\sim \beta_1+\beta_2 x$$. – Asy Feb 10 '18 at 06:02
  • If your IV is continuous, you would be conducting an ANCOVA. And that is what my original equation represents. In that case, $\beta_4$ would tell you if the effect of $x$ for your second group is different from that of your first/control group. – Asy Feb 10 '18 at 06:04
  • I suggest using the regression because the coefficient $\beta_2$ in the categorical case, and $\beta_2$, $\beta_4$ in the ANCOVA case gives you a quantitative estimate of your IV's effect. Also see [this post](https://stats.stackexchange.com/a/59048/138234) about the equivalence between regression and t-test/ANOVA – Asy Feb 10 '18 at 06:06