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I have an analysis

$Y = β_0 + β_1X_1 + β_2X_2 + β_3(X_{1} \times X_2) + \epsilon$

, in which:

$Y$ is a continuous outcome variable
$X_1$ is a continuous predictor (it can be thought of as a type of personality trait)
$X_2$ is a binary variable, dummy coded as 0 for participants randomly assigned to experimental condition A and 1 for participants randomly assigned to experimental condition B.

Based on theory we think that being low on the continuous predictor $X_1$ will predict being low on $Y$. We think that being in experimental condition B should have some impact in pushing down scores on $Y$. We also think that participants who are low on $X_1$ and who are exposed to experimental condition B will have a disproportionate reduction in their $Y$ scores, i.e. that there will be an interaction effect. Ultimately it is this interaction effect that I'm most interested in, although we'd also like to interpret the other main effects.

The main thing I'm wondering is whether my method of analysis is appropriate, given the research question I'm interested in. I'm not used to seeing an experiment analyzed in the context of a moderated regression.

I'm also wondering if my method of analysis requires me to think about things beyond what would normally be considered in a moderated regression. For example, one thing I hadn't really considered was the impact of my dummy coding scheme being 0-1, and of my arbitrary decision to code Condition A as 0 and Condition B as 1, rather than vice-versa.

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This is quite standard. See this page for example, clarifying the interpretation of the coefficients in a model with one continuous predictor, one dummy predictor, and their interaction.

The dummy-coding choce will affect the meaning of the coefficients but not the fundamental results of the analysis. If you use treatment contrasts for the analysis (the default for example in R) then $\beta_1$ will represent the slope of $Y$ with respect to $X_1$ at whichever dummy level you coded as 0, and the $\beta_3$ interaction coefficient will represent the difference of slope from $\beta_1$ when the dummy value is 1.

If the interaction term is significant then there really can't be a separate interpretation of main effects; to know the effect of the continuous predictor you then need to know the value of the dummy variable, and vice-versa.

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