I did not run such a complex model. However, you may try Mplus. You can specify your model for the three way interaction and 3-level analysis (if you have fourth level, it's better control the fourth level using dummy variables.).

I agree with Yae-Yeol Kim,

The key is modeling your interactions appropriately. Interactions become complicated with regression, because effects need to be corrected for crossings with other variables. ANOVA is better with interactions because levels are naturally distinct. 

If you start with these formulas

http://www.ats.ucla.edu/stat/stata/faq/con3way.htm

Then realize when you are estimating effects for a particular variable, you may need to include both of your dummy-coded effects in the equation, so instead of B1X1, you would have (B1aX1a + B1bX1b) as the estimate of the effect. 

I agree that structural equation modeling might be a better fit, depending on the sample size. 

Another option is to group the file (split the file) on one variable and look at the two-way interaction at each level of the 3rd variable. A three-way interaction would be indicated if the two-way interaction of two variables is different at the levels of a third variable. SEM would be ideal, but if you do not have enough participants, this approach might work, and be more clear. You would loose some power by splitting the file.

Such an interaction is fine; I've done similar ones in regular OLS regression (or mixed-effects models, but it's conceptually similar). It just all depends on your predictions. A high-order interaction is no problem if you already had clear predictions what to look for, and therefore know how to decompose (i.e., tease apart) the interaction. Without such predictions, a high-order interaction is a nightmare. But this is not a statistical problem; it's just a problem of how to interpret the patterns in your data.

I can't really say anything more specific unless you ask a more specific question. As it stands right now, I don't know what problem you are facing.

Dan, would appreciate if you can provide reference for the "split the file" method mentioned in your last paragraph (or papers that used that method), and of course thanks Tae-Yeol, Dan, and Stephen for your help.  Really appreciate it!

Sorry for not being clear enough.

The issue here really is the type of data that I am dealing with.  I have

- one dichotomous predictor (measured behavior: yes, no)

- one continuous predictor (measurement scale)

- one 3-level categorical predictor (experimental condition), dummy coded here

Existing research methods/statistics papers that I found only deal with 3-way interactions with all continuous variables or all categorical variable, not the type of variable combination I have (a multi categorical variable with 3 or more levels mixing with a continuous variable).  

While I can multiply out all the terms (e.g., the dummy coded experimental condition x measured behavior), I am not sure if that is an acceptable way to do so.  

From what I found categorical predictor mixing with continuous in mediation or moderation is not that easy (or common) to do, and that might be why there is a separate paper (link below) for that in mediation analysis when Prof Hayes' another paper takes care of continuous mediator.

https://scholar.google.com/citations?view_op=view_citation&hl=en&user=qRrkCbkAAAAJ&sortby=pubdate&citation_for_view=qRrkCbkAAAAJ:BrmTIyaxlBUC

In a nutshell, I hope someone has done something similar (with a mix of 3-level categorical and continuous variable) before so that I can cite it, be it a method paper or analyses done in a typical research paper.  Thanks!

Splitting the data doesn't really need a reference, it's based on the very definition of an interaction. An interaction of X and Y means that the effect of X is different at different levels of Y; therefore, the way to explain it is to look at different levels of Y (i.e., to "split" the data) and to talk about the simple effect of X at each level.

It's still not quite clear to me what kind of problem you're having. Are you unsure how to code the data, or how to interpret the model? In general, though, the issue is no different than it is for an interaction between two factors (or any N number of factors).

Hi Stephen, thanks for your quick feedback.

While what you said might be true, I did find reference that indicates subgroups analysis could be problematic.  It's in the 9.3 section of the following book, which has been widely cited.  

http://www.amazon.com/Introduction-Mediation-Moderation-Conditional-Analysis/dp/1609182308

Thanks

This probably just depends what exactly you mean by "splitting" the data. If you actually throw out all the data from one condition and analyze just the remaining condition then this could be a problem because you are losing the information captured by that. But with proper coding, a model will let you see the simple effect at each level of the factor while also keeping track of the remaining variance. For example, in R syntax, if you have a categorical factor 'Condition' with levels A and B, and continuous variables X and Y, then a model like

lm( Y ~ Condition/X )

will show you the simple effect of X at each level of Condition, without actually removing any of the data.

(for a worked example, load up R and run the following code:

airquality$Month

You don't need to have unique values for different IVs. Consider dummy (treatment) coding for example. IV1 can be coded as {0,1}, and IV2 also coded as {0,1}. Then, one condition has a 0 for IV1 and a 0 for IV2; one condition has a 1 for IV1 and a 0 for IV2; etc. Likewise, for the interaction term, when you multiply these coefficients together, you'll end up with three conditions that have a 0 for the interaction term and one condition that has a 1.

Your prediction sounds pretty straightforward. There are a lot of ways you could plot it. You could plot the interaction of IV1 and IV2 at different levels of IV3. Or you could plot the effect of IV3 (a regression line) at different levels of IV1 and IV2.

You can code with 0/1 (dummy coding) or ..5/-.5 (sum coding), either is fine, depends on how you want to look at the results. See http://www.mypolyuweb.hk/~sjpolit/coding_schemes.html.

For an interaction term you won't necessarily get four different coefficients. The interaction term is just the coefficients for one IV multiplied by the coefficients for the other IV.

If you want those coefficients to be interpretable as main effects, then everything in the interaction needs to be centered (around 0). 1/-1 coding makes X1 and X2 centered, but M also has to be centered so that its mean is zero.

There are many ways to do this: