# The interpretation of Ceteris Paribus or holding all things equal when including multiple 3-way interactions that share interacting variables

I have a regression model that includes multiple 3-way interactions, but where two of the interacting variables occur in all interactions ($X$ and $Z$) and only one changes: let's call these variables of interest, and denote them as $W_1-W_3$

My question is about the interpretation of Ceteris Paribus (or holding other things constant) when analysing one of the variables of interests (more on that further down).

Without including additional continuous or categorical independent variables I may want to control for, the regression model is

$$Y = \beta_0 + \beta_1 X + \beta_2Z + \beta_3 XZ + \underbrace{\beta_4 W_1 + \beta_5 XW_1 + \beta_6 ZW_1 + \beta_7 XZW_1}_\text{First variable of interest} + \underbrace{\beta_8 W_2 + \beta_9 XW_2 + \beta_{10} ZW_2 + \beta_{11} XZW_2}_\text{Second variable of interest} + \underbrace{\beta_{12} W_3 + \beta_{13} XW_3 + \beta_{14} ZW_3 + \beta_{15} XZW_3}_\text{Third variable of interest}$$

Where $Z$ and $W_{1-3}$ are binary variables that can be either be high or low, while $X$ is continuous (time).

If we take one of the variables of interest, $W_1$, we can define the different slopes as follows (same goes for $W_2-W_3$)

$\\Z_{low},&space;W_{low}:&space;\beta_1&space;\\Z_{high},&space;W_{low}:&space;\beta_1&space;+&space;\beta_3&space;\\Z_{low},&space;W_{high}:&space;\beta_1&space;+&space;\beta_5&space;\\Z_{high},&space;W_{high}:&space;\beta_1&space;+&space;\beta_3&space;+&space;\beta_5&space;+&space;\beta_7$

And thereafter compute the mean pairwise difference between any two slopes (here also shown only for $W_1$)

$\\Z_{high}&space;\text{&space;vs&space;}&space;Z_{low}&space;\text{&space;for&space;}&space;W_{low}:&space;\beta_2&space;+&space;\beta_3&space;X&space;\\Z_{high}&space;\text{&space;vs&space;}&space;Z_{low}&space;\text{&space;for&space;}&space;W_{high}:&space;\beta_2&space;+&space;\beta_3&space;+&space;\beta_6&space;+\beta_7&space;X&space;\\Z_{high}&space;\text{&space;for&space;}&space;W_{low}&space;\text{&space;vs&space;}&space;Z_{high}&space;\text{&space;for&space;}&space;W_{high}:&space;\beta_3&space;+\beta_5&space;+&space;\beta_6&space;+&space;\beta_7&space;X&space;\\Z_{low}&space;\text{&space;for&space;}&space;W_{low}&space;\text{&space;vs&space;}&space;Z_{low}&space;\text{&space;for&space;}&space;W_{high}:&space;\beta_3&space;+&space;\beta_5&space;$

What I am wondering about is the interpretation of Ceteris Paribus (or holding other things constant) when analysing one of the variables of interests. Say for example that I am looking at $Z_{high}&space;\text{&space;vs&space;}&space;Z_{low}&space;\text{&space;for&space;}&space;W_1_{high}$. In the case of a continuous independent variables that was not part of the focal interactions, it would be at any value of that variable, or regardless of the levels of a categorical independent variable (also not part of the focal interactions).

But if I analysed the same comparison (i.e. $Z_{high}&space;\text{&space;vs&space;}&space;Z_{low}&space;\text{&space;for&space;}&space;W_1_{high}$), at what level are $W_2&space;$ and $W_3$ at? Are they at their reference categories (i.e. low) or it doesn’t matter?

With treatment/dummy coding, the value of the individual coefficient for a predictor involved in higher-level interactions is the value when all of its interacting predictors are at their reference levels. So the $$\beta_2$$ coefficient for $$Z$$ in your setup is the value when all its interacting binary predictors $$W_1$$, $$W_2$$ and $$W_3$$ are at their reference levels (taken to be 0) and when the continuous predictor $$X$$ has a value of 0.

You can see this for the $$Z_{\text{high}}$$ versus $$Z_{\text{low}}$$ comparison by examining all of the terms involving $$Z$$, as the other terms cancel in that comparison:

$$\beta_2Z + \beta_3 XZ + \beta_6 ZW_1 + \beta_7 XZW_1 + \beta_{10} ZW_2 + \beta_{11} XZW_2 + \beta_{14} ZW_3 + \beta_{15} XZW_3.$$

Your proposed formula for $$Z_{\text{high}}$$ versus $$Z_{\text{low}}$$ at $$W_1 = 0$$, $$\beta_2 + \beta_3X$$, thus only holds for the situation where $$W_2$$ and $$W_3$$ are also at zero. Otherwise, you need to include the further interaction terms involving $$Z$$ and non-zero values of $$W_2$$ and/or $$W_3$$.

• Many thanks for your response! I just want to double check another thing. In the case of the contrast 𝑍_high vs 𝑍_low at e.g. 𝑊_1=1 for a X!=0 (say X=3), the corresponding contrast for 𝑍_high vs 𝑍_low for 𝑊_2 and 𝑊_3 while at 0 for, are also at X=3, right?
– jO.
Feb 28, 2022 at 8:35
• @jO. that's why you should write out the entire formula, particularly when interactions are involved. Write out the entire formula for the regression to start and set up the comparison of interest based on that entire formula. That's what I did to get the displayed formula for the Z-high vs Z-low comparison. (Until I did that I almost provided an incorrect answer myself.) Then plug in all of the covariate values. So yes, if you assume X=3 you must set X=3 in all terms.
– EdM
Feb 28, 2022 at 14:07