Interpreting main effects in the presence of interaction effects in regression

Question

I have always been confused about interpreting main effects in the presence of interaction effects.

Put simply,

Y = a sense of belonging to school

X1 = ethnicity (0= white, 1=black) / negative effect

X2 = relationship with teachers (1= very bad ~ 4 very good) / negative effect

X1*X2 = interaction term / positive effect

In this case, Should I say,

Black students have a lower sense of belonging to school than white students do. The students' relationship with teachers is negatively associated with their sense of belonging to school. However, black students have a higher sense of belonging to school when they have good relationships with teachers.

Or,

Black students have a lower sense of belonging to school than white students do. However, black students have a higher sense of belonging to school when they have good relationships with teachers even after controlling the main effect of the relationship with teachers on their sense of belonging to school.

Thanks,

Answer 1

In the presence of an interaction, you should not interpret the main effects. What the estimated effect of $x_2$ is, depends on $x_1$, and vice versa. This is how you specified the model, after all.

In your example, the negative association between $x_2$ and $y$ you've observed is the effect for the reference group: $x_1 = 0$ (white). The fact that the interaction's effect is positive tells you that if $x_1 = 1$ (black) and $x_2$ is high (better relationship with teachers), then $y$ (sense of belonging) increases, compared to the reference group. In fact, if the effect size of the interaction is large enough, it might even be that there is a negative association between $x_2$ and $y$ for white students, while there is a positive effect for $x_2$ and $y$ for black students.

On a different note, from reading your question I assume $y$ and $x_2$ both are self-reported values. In that case it would be slightly more correct to say that (given $x_1 = 0$) students who claim to have a better relationship with their teachers report a weaker sense of belonging to the school. I would also be careful to check whether the groups are otherwise homogeneous before attributing the observed differences to race (e.g. is there a similar ratio of men/women, are the groups similarly aged, etc.).